j. differential geometry 57 (2001) 67-171

PRESCRIBING SCALAR CURVATURE ON S^{N}, PART 1: APRIORI ESTIMATES

CHIUN-CHUAN CHEN & CHANG-SHOU LIN

Abstract

In this paper, we describe great details of the bubbling behavior for a sequence of solutions w_{i} of

Lw_{i} + R_{i}wi^{\frac{n+2}{n−2}} = 0 on S^{n},

where L is the conformal Laplacian operator of (S^{n}, g_{0}) and R_{i} = n(n−2)+ t_{i}R̂, R̂ ∈ C^{1}(S^{n}). As t_{i} ↓ 0, we prove among other things the location of blowup points, the spherical Harnack inequality near each blowup point and the asymptotic formulas for the interaction of different blowup points. This is the first step toward computing the topological degree for the nonlinear PDE.

1. Introduction

This is the first of a series of papers to study the problem of prescribing scalar curvature on S^{n}, the n-dimensional sphere with n ≥ 3. Let g_{0} be the metric on S^{n} induced from the flat metric of ℝ^{n+1}, and R be a given C^{1} positive function on S^{n}. We are interested in the question whether there exists a metric g conformal to g_{0} such that R is the scalar curvature of g. Set g = c_{n}w^{\frac{4}{n−2}} g_{0} for a suitable positive constant c_{n}. Then the question above is equivalent to finding a smooth positive solution of

(1.1)  Lw + Rw^{\frac{n+2}{n−2}} = 0 on S^{n},

Received October 9, 2000.

where L = Δ_{g_{0}} − \frac{n(n − 2)}{4} is the conformal Laplacian operator of (S^{n}, g_{0}). In general, the same question can be studied in any Riemannian manifold. For a compact Riemannian manifold and a constant R, this problem is called the Yamabe problem, which was solved in early 80s through the works by Trudinger [22], Aubin [1] and Schoen [19]. For a historic account, we refer the readers to Lee and Parker [14] and references therein. For the last three decades, Equation (1.1) has been continuing to be one of major subjects in nonlinear elliptic PDEs. For recent developments, see [1], [2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [21] and the references therein.

In [5], Chang-Gursky-Yang considered Equation (1.1) when n = 3 and R is a positive Morse function on S^{3}. Under some nondegenerate conditions on the critical points of R, Chang-Gursky-Yang were able to obtain the apriori bound for positive solutions of Equation (1.1). Furthermore, they computed the Leray-Schauder degree d for Equation (1.1) by the following formula

(1.2)  d = − (1 + ∑_{p∈Γ^{−}}(−1)^{ind(p)}) ,

where Γ^{−} = {p ∈ S^{3} | p is a critical point of R satisfying Δ_{g_{0}}R(p) < 0} and ind(p) is the Morse index of the Hessian of R at p. When the righthand side of (1.2) is assumed to be nonzero, the existence of positive solutions to Equation (1.1) was previously obtained by Bahri-Coron [3] and Schoen-Zhang [21]. However, the degree-counting formula (1.2) provides us more information about Equation (1.1). Particularly, it tells us when the concentration phenomenon for solutions of (1.1) could occur. Li [16] proved the apriori bound for Equation (1.1) on S^{4} and derived the formula for the Leray-Schauder degree by adding the effect of the interaction of multiple blow-up points. In this series of papers, we will generalize the results of [5] and [16] on S^{3} and S^{4} to higher dimensions.

prescribing scalar curvature on S^{n} 69

As in our previous works [8], [9], it is more convenient for us to study (1.1) in ℝ^{n}. Without loss of generality, we may assume that the north pole of S^{n} is not a critical point of R. By using the stereographic projection π from S^{n} to ℝ^{n}, we set u(x) = 2^{\frac{n−2}{2}} (1 + |x|^{2})^{\frac{2−n}{2}} w(π^{−1}(x)) for x ∈ ℝ^{n}. Then u(x) satisfies

(1.3)  \begin{cases}Δu(x) + K(x)u^{\frac{n+2}{n−2}} = 0 & in ℝ^{n}, \\ u(x) = O(|x|^{2−n}) & at ∞, \end{cases}

where K(x) = R(π^{−1}(x)) for x ∈ ℝ^{n}.

When K(x) is a constant, solutions of (1.1) can be classified completely. See [13] and [4]. For nonconstant R(x), it is well-known that existence of solutions depends on K in a very subtle way. So, throughout the paper and [10], we always assume 0 < a ≤ K(x) ≤ b and K(x) has a finite set of critical points {q_{1}, …, q_{N}}. Near each q_{j}, by Taylor’s expansion, K(x) can be written as

K(x) = K(q_{j}) + Q_{j}(x − q_{j}) + R_{j}(x),

where Q_{j}(x) is a C^{1} homogeneous function of degree β_{j} > 1, i.e., Q_{j}(λx) = λ^{β_{j}}Q_{j}(x) for λ > 0 and R_{j} satisfies

lim_{x→q_{j}} |x − q_{j}|^{−β_{j}}R_{j}(x) = lim_{x→q_{j}} |x − q_{j}|^{1−β_{j}}| ▽ R_{j}|(x) = 0.

Here, β_{j} is not necessarily an integer. Of course, if K(x) ∈ C^{∞}, then β_{j} must be an integer.

(K0) | ▽ Q_{j}(x)| ≥ c_{1}|x|^{β_{j}−1} for some c_{1} > 0.

Let U_{1}(x) = (1 + |x|^{2})^{− \frac{n−2}{2}} .

(K1) At each critical point q_{j}, according to β_{j}, K satisfies one of the following conditions (i), (ii) and (iii):

(i) If β_{j} < n, Q_{j} satisfies

(1.4)  (\begin{matrix}∫_{ℝ^{n}} ▽Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx \\ ∫_{ℝ^{n}} Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx \end{matrix}) ≠ (\begin{matrix}0 \\ 0 \end{matrix}) ,

for any ξ ∈ ℝ^{n}.

(ii) If β_{j} = n, then

(1.5)  ∫_{S^{n−1}} Q_{j}(x)dσ ≠ 0

provided that there exists a vector ξ ∈ ℝ^{n} satisfying

(1.6)  ∫_{ℝ^{n}} ▽Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (y)dy = 0.

(iii) If β_{j} > n,

(1.7)  ∫_{ℝ^{n}}⟨x − q_{j}, ▽K⟩|x − q_{j}|^{−2n}dx ≠ 0.

We note that all integrals in (1.4)–(1.7) are L^{1}(ℝ^{n}). In [5], [9] and [16], we knew that only part of critical points of K might be blowup points for certain solutions. Denote by Γ^{−} those critical points of K. More precisely:

Definition 1.1. Assume that K satisfies (K0). We say q_{j} ∈ Γ^{−} if and only if K satisfies one of the following conditions (i), (ii) and (iii) at q_{j} according to β_{j}:

(i) If β_{j} < n, there exists ξ ∈ ℝ^{n} such that

(1.8)  \{ \begin{matrix}∫_{ℝ^{n}} ▽Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx = 0 and \\ ∫_{ℝ^{n}} Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx < 0. \end{matrix}

(ii) If β_{j} = n, there exists ξ ∈ ℝ^{n} satisfying

(1.9)  \{ \begin{matrix}∫ ▽Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx = 0 and \\ ∫_{S^{n−1}} Q_{j}(x)dσ < 0. \end{matrix}

(iii) If β_{j} > n,

(1.10)  ∫_{ℝ^{n}}⟨x − q_{j}, ▽K⟩|x − q_{j}|^{−2n}dx < 0.

Clearly, the notion q_{j} ∈ Γ^{−} and conditions (K0)–(K1) are invariant under the conformal transformations.We list several examples of Q to explain conditions (K0) and (K1).

Example 1.2.

1. Q(y) = ∑_{j=1}^{n} a_{j}y_{j}^{2}. Clearly a_{j} ≠ 0 for all j iff (K0) holds. It is easy to see that ξ = 0 is the only vector satisfying ∫_{ℝ^{n}} ▽Q(y +

ξ)U1^{\frac{2n}{n−2}} (y)dy = 0 and ∫_{ℝ^{n}} Q(y)U1^{\frac{2n}{n−2}} (y)dy = c_{n} ∑_{j=1}^{n} a_{j} for some

positive constant c_{n}. Thus, (K0) and (K1) hold for a Morse function R on S^{n} satisfying ΔR(q) ≠ 0 for any critical point q of R. And q ∈ Γ^{−} iff ΔR(q) < 0.

2. Q(y) = ∑_{j=1}^{n} a_{j}y_{j}^{3}, a_{j} ≠ 0, for j = 1, 2, …, n. Clearly, no ξ ∈ ℝ^{n}

satisfies ∫_{ℝ^{n}} ▽Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy = 0.

3. Q(y) = y_{1}^{3} − λy_{1} ∑_{j=2}^{n} y_{j}^{2}. For λ > \frac{3}{n−2}, Q(y) satisfies (K0) and (1.4). In fact, there are exactly two solutions ξ = ±ξ_{0} of

∫_{ℝ^{n}} ▽Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy = 0, where ξ_{0} = (ξ_{0,1}, 0, …, 0) for some

ξ_{0,1} > 0. Direct computations show

∫ Q(y + ξ_{0})U1^{\frac{2n}{n−2}} (y)dy = − ∫ Q(y − ξ_{0})U1^{\frac{2n}{n−2}} (y)dy < 0.

The main purpose of our work is to show that homogeneous functions Q_{j}(x) for q_{j} ∈ Γ^{−} completely determine the structure of solutions of (1.1). Conditions (K0) and (K1) are already enough for our purpose. However, in order to make our presentation transparent here, each Q_{j} at q_{j} ∈ Γ^{−} is assumed to satisfy

(K2) For each q_{j} ∈ Γ^{−} with β_{j} < n, assume that

(1.11) ∫_{ℝ^{n}} Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx < 0 whenever

∫_{ℝ^{n}} ▽Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx = 0.

To state our main theorem, we introduce the notion Λ^{−}. Assume (K0) and (K1). Let Λ^{−} be a collection of subsets of Γ^{−} such that a subset A of Γ^{−} is an element in Λ^{−} if and only if A satisfies the following conditions.

1. The number of the elements in A ≥ 2.

2. For any two elements q_{j} ≠ q_{k} in A, the exponents β_{j} and β_{k}

satisfies

\frac{1}{β_{j}^{∗}} + \frac{1}{β_{k}^{∗}} > \frac{2}{n − 2},

where

(1.12)  β_{j}^{∗} = min(β_{j}, n).

Now we can state a special case of the Main Theorem we are going to prove in this paper and the subsequent one [10].

Theorem 1.1. Assume that K satisfies (K0) and (K1) such that β_{j}

of Q_{j} at each critical point q_{j} in Γ^{−} satisfies β_{j} > \frac{n − 2}{2}. In addition, we assume

(1.13)  \frac{1}{β_{j}^{∗}} + \frac{1}{β_{k}^{∗}} ≠ \frac{2}{n − 2}

for q_{j} ≠ q_{k} ∈ Γ^{−}. Then there exists a constant c > 0 such that for any solution w of (1.1), we have

(1.14)  c^{−1} ≤ w(y) ≤ c for y ∈ S^{n}.

Let d denote the Leray-Schauder degree for the nonlinear map w + L^{−1}(Rw^{\frac{n+2}{n−2}} ) on C^{2,α}(S^{n}) with 0 < α < 1. Moreover, if (K2) holds additionally, then d satisfies

(1.15) d = − [1 + ∑_{j∈Γ^{−}}(−1)^{n+1} deg F_{j} + ∑_{A∈Λ^{−}} ∏_{k∈A}((−1)^{n+1} deg F_{k})] ,

where deg F_{j} denotes the standard topological degree of the mapping F_{j}(x) = ▽Q_{j}(x) from S^{n−1} to ℝ^{n}\{0}, and Γ^{−} and Λ^{−} are defined as above.

We remark that the assumption β_{j} > \frac{n−2}{2} in Theorem 1.1 is an also necessary condition for the existence of apriori bounds for solutions of Equation (1.1). In [11], we constructed blowing up solutions of (1.1) for some K satisfying (K0) and (K1) with β_{j} < \frac{n−2}{2}. To establish the apriori bound (1.14), the first step is to understand the details of blowing-up behavior of a sequence of solutions w_{i} near each blow-up point. In [8], [9] for a sequence of local solutions u_{i} of

(1.16)  Δu_{i} + K_{i}(x)ui^{\frac{n+2}{n−2}} = 0 in B_{2} = \{x | |x| < 2\}

where 0 is assumed the only blowup point, we have completely classified types of concentrations of u_{i} according to the flatness β of Q at the blowup point 0. In particular, if \frac{n−2}{2} < β < n then

(1.17)  u_{i}(x) ∼ M_{i}^{−γ}

in any compact set of B_{1}\{0}, where

γ = \begin{cases}\frac{2β}{n−2} − 1 & if β < n − 2 \\ 1 & if β ≥ n − 2, \end{cases}

and M_{i} is the maximum of u_{i} in B_{1}. Hereafter, the notation a_{i} ∼ b_{i} for two sequences of positive numbers denotes that the ratio a_{i}/b_{i} is bounded above and below by two positive constants independent of i. Thus, u_{i}(x) ↓ 0 in C_{loc}^{2}(B_{1}\{0}). The result (1.17) is important when global solutions u_{i} of Equation (1.3) are considered, because those local maxima must satisfy certain rules according to (1.17). Together with the Pohozaev identity, we must have \frac{1}{β_{j}^{∗}} + \frac{1}{β_{k}^{∗}} = \frac{2}{n−2} for some blowup points q_{j} and q_{k}. The apriori bound (1.14) then follows from this. We will give a complete proof of this result in Section 10 of the paper. When n − 2 < β_{j} < n for any critical point q_{j}, the apriori bound was obtained previously in [15].

The degree counting formula (1.15) is more difficult to prove. Usually, there are two ways to establish the Leray-Schauder degree. One is to approach the nonlinear term in Equation (1.1) by subcritical exponents. Another one is to deform the curvature function R, e.g., replace R in Equation (1.1) by R_{t} = 1 + t(R − 1) for 0 ≤ t ≤ 1. For the latter case, if one can show for any ε > 0, solutions of (1.1) with R replaced by R_{t} are uniformly bounded for ε ≤ t ≤ 1, then the Leray-Schauder degree is the same for each t ≠ 0. Thus, for our purpose, it suffices to compute the Leray-Schauder degree for small t > 0. In the situation when t is small enough, the degree theory developed by Chang-Yang [6] can be applied very well. But, Chang-Yang was only able to prove the degree counting formulas (1.2) for the class of Morse functions. More seriously, as we will see, the degree formula in [6] did not count all possible solutions. Roughly speaking, their results only covered the case when solutions of (1.1) possess at most one blow-up point as t tends to zero. Later in this paper, we will prove that under assumptions (K0) and (K1), if a sequence of solutions w_{i} of (1.1) with R_{t_{i}} as the scalar curvature blows up as t_{i} → 0, then the number of blow-up points must be greater than one. Therefore, solutions obtained in [6] only consist of bounded solutions as t → 0. We also remark that if the degree β_{j} for each q_{j} ∈ Γ^{−} is no less than n−2, then any sequence of solutions of (1.1) with R replaced by R_{t_{i}} remains uniformly bounded as t_{i} → 0. In this case, Λ^{−} is an empty set and the degree-counting formula (1.15) reduces to d = −[1 + ∑_{j∈Γ^{−}}(−1)^{n+1} deg F_{j}]. When R is a Morse function on S^{3}, this is the degree counting (1.2).

In this paper, we consider a sequence of solutions u_{i} of (1.3) with curvature functions K_{i} set by

(1.18)  K_{i}(x) = n(n − 2) + t_{i}K̂(x),

where we assume t_{i} → 0. Here, K̂ is a C^{1} function satisfying the nondegenerate conditions (K0)–(K1). Solutions u_{i} are always assumed to blow up at some points of ℝ^{n}. The main purpose of this article is to study blowup behavior of u_{i} near a blowup point and to study the effect due to the interaction between different blowup points. This is the first step for computing the degree-counting formula. Based on these, we will construct all possible blowup solutions of (1.1) as t_{i} ↓ 0 in [10] and then we are able to compute the “local degree” for each blowup solution. In [10], we will give a complete proof of the degree formula. From the analytic point of view, the main difference between this paper and [9] are: First, we consider the degenerate case lim_{i→∞} K_{i} = constant here, which can not be covered by the results for nondegenerate lim_{i→∞} K_{i} in [9]. Second, we allow the number β_{j} defined in (K0) to be greater than or equal to n in this paper, while we assume 1 < β_{j} ≤ n − 2 in [9]. Third, we also consider the interaction between different blow-up points here, while we mainly study local behavior near a blow-up point in [9].

The first interesting question concerning a sequence of blowup solutions is to find the location of blowing up points. A general result states that if K_{i} converges to K in C^{1}, then any blowup point must be a critical point (see [21], [16], [8]). Obviously, this result could not be of any help for our present situation because the limit function of K_{i} is identically a constant. Nevertheless, by using more delicate estimates than the nondegenerate case, we are still able to prove the following.

Theorem 1.2. Suppose K̂ satisfies (K0) and u_{i} is a sequence of solutions of (1.3) with K = K_{i} given in (1.18). Then ▽K̂(q) = 0 for any blowup point q of u_{i}.

Throughout the paper, we let {q_{1}, …, q_{m}} be the set of blowup points for {u_{i}}, and β_{j} be the degree of Q_{j} of K̂ at q_{j}. To analyze the blowup behavior of u_{i} more accurately, the important step is to show the isolatedness of blowup points, that is, to prove the spherical Harnack inequality (1.19):

(1.19)  max_{|x−q_{j}|=r} u_{i}(x) ≤ c min_{|x−q_{j}|=r} u_{i}(x) for 0 ≤ r ≤ r_{0}.

For nondegenerate case, the spherical Harnack inequality (1.19) was proved even for local solutions. See [8], [9] of the reference. For the degenerate case, we do not know whether the spherical Harnack inequality holds or not for local solutions. In Section 4, we study the situation when it fails. Due to the analysis there and the effect of interactions of different blowup points, nevertheless, the spherical Harnack inequaltiy is proved for global solutions.

Theorem 1.3. Suppose that K̂ satisfies (K0) and (K1). Assume

β_{j} ≥ \frac{2(n − 2)}{n} for each q_{j} ∈ Γ^{−}. Then any blowup point is isolated. Furthermore, if β_{j} < n + 1 at a blowup point q_{j}, then u_{i} satisfies

(1.20)  u_{i}(x) ≤ c |x − q_{j}|^{− \frac{n−2}{2}}

for |x − q_{j}| ≤ δ_{0} with some positive constants δ_{0} and c.

By the theory of elliptic equations and the scaling property of Equation (1.3), inequality (1.20) implies (1.19). Hence, we also call (1.20) the spherical Harnack inequality. We note that in Theorem 1.3, (K1) is required only for those q_{j} where β_{j} < n − 2.

For each blowup point q_{j}, we let M_{i,j} and q_{i,j} denote the local maximum and a local maximum point of u_{i} near q_{j}, that is,

(1.21)  M_{i,j} = u_{i}(q_{i,j}) = max_{|x−q_{j}|≤δ_{0}} u_{i}(x),

where δ_{0} is a small positive number such that the distance of q_{j} and q_{k} are greater than 2δ_{0}. The following theorem is concerned with the asymptotic relations of M_{i,j} for different blowup points. Let l denote the nonnegative positive integer such that q_{1}, …, q_{l} are simple blowup points and q_{l+1}, …, q_{m} are not simple blowup points. For the notion of simple blowup points, we refer the reader to [8], [9] or Section 2 of this paper.

Theorem 1.4. Assume that K̂ satisfies (K0) and (K1) and assume β of Q > \frac{n−2}{2} at any q ∈ Γ^{−}. Let \{q_{j}\}_{j=1}^{m} be the set of blowup points for u_{i}, and M_{i,j}, q_{i,j} and l be defined as above. Then m ≥ 2, l ≥ 1 and β_{1} = … = β_{l} > β_{j} for l + 1 ≤ j ≤ m. Furthemore, the following conclusions hold:

(i) We have q_{j} ∈ Γ^{−} for 1 ≤ j ≤ m and there exists a constant c > 0 such that

(1.22) |q_{i,j} − q_{j}| ≤ c \begin{cases}M_{i,j}^{− \frac{2}{n−2}} & if β_{j} < n + 1, \\ M_{i,j}^{− \frac{2}{n−2}} (log M_{i,j})^{\frac{1}{n}} & if β_{j} = n + 1, \\ M_{i,j}^{− \frac{2}{n−2} \frac{n}{β_{j} −1}} & if β_{j} > n + 1. \end{cases}

Moreover, the limit vector ξ = lim_{i→+∞} Mi,j^{\frac{2}{n−2}} (q_{i,j} − q_{j}) satisfies (1.8) if β_{j} < n, and satisfies (1.6) if n ≤ β_{j} < n + 1

(ii) Assume that l = 1. We index q_{j} according to the ordering of β_{j} : β_{1} > β_{2} = … = β_{l_{1}} > β_{l_{1}+1} ≥ … ≥ β_{m} for some positive integer l_{1}. Then

(1.23)  \frac{1}{β_{1}^{∗}} + \frac{1}{β_{2}} > \frac{2}{n − 2},

M_{i,j} satisfies (1.24)

\begin{matrix}t_{i}M_{i,1}^{− \frac{2β^{∗}_{1}}{n−2}} & if β_{1} ≠ n \\ t_{i}M_{i,1}^{− \frac{2n}{n−2}} log M_{i,1} & if β_{1} = n \end{matrix}\} = (1 + o(1)) ∑_{j=2}^{l_{1}} η_{1,j}M_{i,j}^{−1}M_{i,1}^{−1},

and

(1.25) t_{i}Mi,j^{\frac{−2β_{j}}{n−2}} = (1 + o(1))η_{j,1}M_{i,j}^{−1}M_{i,1}^{−1} for 2 ≤ j ≤ m,

where

(1.26)  η_{j,k} = \frac{n(n − 2)|S^{n−1}||q_{j} − q_{k}|^{−n+2}}{|b_{j}|} ,

and

(1.27) b_{j} = \begin{cases}β_{j} ∫_{ℝ^{n}} Q_{j}(x + ξ)U1^{\frac{2n}{n−2}} (x)dx \\ with ξ = lim_{i→+∞} M_{i,j}(q_{i,j} − q_{j}) & if β_{j} < n \\ n ∫_{S^{n−1}} Q_{j}(x)dσ & if β_{j} = n \\ ∫_{ℝ^{n}}⟨x − q_{j}, ▽K̂⟩|x − q_{j}|^{−2n}dx & if β_{j} > n \end{cases}

(iii) Assume l ≥ 2. Then β_{1} = … = β_{l} < n − 2 and M_{i,j} satisfies

(1.28) t_{i}M_{i,j}^{− \frac{2β_{1}}{n−2}} = (1 + o(1)) ∑_{k=1,k≠j}^{l} η_{j,k}M_{i,j}^{−1}M_{i,k}^{−1} for 1 ≤ j ≤ l,

and

(1.29) t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} = (1+o(1)) ∑_{k=1}^{l} η_{j,k}M_{i,k}^{−1}M_{i,j}^{−1} for l+1 ≤ j ≤ m.

Theorem 1.4 gives us rather complete information about blowup solutions, that is, the local maxima of blowup solutions must satisfy the necessary conditions (1.24) and (1.25), or (1.28) and (1.29). Conversely, in [10] we will construct such blowup solutions satisfying these relations and compute the contribution of these solutions to the Leray-Schauder degree of Equation (1.1). We note that the third term of the right hand side of (1.15) corresponds to the effect of multiple blowup points.

The paper is organized as follows: In Section 2–Section 9, we consider the degenerate case for Equation (1.3), that is, K_{i}(x) = n(n−2)+ t_{i}K̂(x) with t_{i} ↓ 0. In Section 2, main results for local solutions are stated and their proofs are given in the subsequent sections. There are two main issues in Section 2. The first one is the quantity L_{i}, which is associated with each “good” local maximum point of solutions. The quantity L_{i} is introduced in Sections 2 and will play an important role because it decides how large of the range where u_{i} behaves “simply”. We will give its proof in Sections 3 and this is the major step where the method of moving planes is applied. Another important issue in Section 2 is the spherical Harnack inequality (1.20). We will see that when the flatness β ≥ \frac{n−2}{2}, the spherical Harnack inequality always holds. See Theorem 2.4. The case β < \frac{n−2}{2} is the difficult one for our analysis, even when the Harnack inequality holds. In the general principle, we can obtain the local bubbling informations through the Pohozaev identities. However, we have to compute each term in the identity very accurately and the Harnack inequality itself is not enough for us to achieve this goal. We need a sharper estimate for the error term of the solution and the approximation bubbles. This is a very delicate analysis because in general the solutions might lose the energy more than one bubble. In Section 5, we show that a method of ODE surprisingly gives us fine estimates when the spherical Harnack inequality is validated. Together with suitably chosen comparision functions, we complete the proof of our desired estimate in Sections 5. See Theorem 2.7. This is one of two difficult jobs in the paper. These estimates for the error term are required in the proof of Lemma 7.1 in Section 7. Lemma 7.1 exactly tells us how, through the Pohozaev identities, the local informations can be put together to obtain more global one. Section 4 will deal with the situation when the spherical Harnack inequaltiy (2.19) fails. Here, we employ a technique of Schoen to localize blowup points. Combined with the method of moving planes developed in Section 3, this provides a clear picture for the case when the Harnack inequality does not hold. Based on the analysis in Section 4 and Lemma 7.1, Theorem 1.3 and Theorem 1.4 are proved in Section 8 and Section 9, respectively. We will prove Theorem 1.2 in Section 6 as a direct consequence of results in Section 2. Finally, we will prove the apriori bound of Theorem 1.1 in Section 10.

Acknowledgement

Part of this work has been done while both authors visited the National Center for Theoretical Sciences of NSC in Taiwan. We want to thank NCTS for the warm hospitality and the stimulating environment.

2. Estimates for local solutions

For the convenience of the reader, we briefly review some of previous results from [8] and [9], which would be useful later. Let u_{i} be a solution of

(2.1)  Δu_{i} + K_{i}(x)ui^{\frac{n+2}{n−2}} = 0 in Ω,

where Ω is an open set in ℝ^{n}. Let x_{0} be a blowup point. Following Schoen’s idea, a blowup point x_{0} is called simple if there exists a constant c > 0 and a sequence of local maximum points x_{i} of u_{i} such that

(2.2)  x_{0} = lim_{i→+∞} x_{i},

and

(2.3)  u_{i}(x_{i} + x) ≤ c U_{λ_{i}}(x) for |x| ≤ r_{0},

where r_{0} > 0 is independent of i, λ_{i} = u_{i}(x_{i})^{− \frac{2}{n−2}} tends to zero as i → +∞ and

(2.4)  U_{λ}(x) = (\frac{λ}{λ^{2} + |x|^{2}} )^{\frac{n−2}{2}} for x ∈ ℝ^{n}.

For any λ > 0, by elementary calculation, U_{λ}(x) satisfies

ΔU_{λ} + n(n − 2)Uλ^{\frac{n+2}{n−2}} (x) = 0 in ℝ^{n}.

We note that the definition of a simple blowup point is different from the original one given by Schoen. However, it is not difficult to prove that these two definitions are equivalent.

Instead of (2.3), the inequality

(2.5)  u_{i}(x_{i} + x) ≤ c u_{i}(x_{i})^{−1}|x|^{−n+2}

is often used when x_{0} is a simple blowup point. Also, by (2.4), we have

(2.6)  U_{λ}(x) ≤ (2|x|)^{− \frac{n−2}{2}} for x ≠ 0,

which implies that if x_{0} is a simple blowup point, then

(2.7)  u_{i}(x_{i} + x) ≤ c |x|^{− \frac{n−2}{2}} for |x| ≤ r_{0}.

A blowup point x_{0} is called isolated if (2.7) holds for some c and r_{0} > 0. It is easy to see a simple blowup point must be isolated. The inequality (2.7) is important because it implies that the Harnack inequality holds for each sphere with center x_{i}, i.e., there exists a positive constant c > 0 such that

(2.8)  max_{|x−x_{i}|=r} u_{i}(x) ≤ c min_{|x−x_{i}|=r} u_{i}(x)

for 0 ≤ r ≤ r_{0}.

Suppose that x_{0} is a blowup point of u_{i}. Theorem 1.3 in [8] states that x_{0} is a simple blowup point if K_{i}(x) → K(x) in C^{1} and K_{i} satisfies for some constant c either (i) | ▽ K_{i}(x)| ≤ c if n = 3 or (ii)

(2.9)  | ▽^{j} K_{i}(x)| ≤ c| ▽ K_{i}(x)|^{\frac{β−j}{β−1}}

if n ≥ 4 in a neighborhood of x_{0} for 1 ≤ j ≤ β = n − 2. Also see [15] for the same conclusion when global solutions are considered. We make some remarks here. First, if K_{j} = n(n − 2) + t_{i}K̂ with K̂ satisfying (2.9), then (2.9) holds for K_{i} also with the same constant c. Thus Theorem 1.3 in [8] can apply to our case. Second, if K̂ is smooth and | ▽ K̂(x_{0})| ≥ c > 0, then obviously condition (2.9) holds for K_{i} also. Actually, from the first step of the proof of Theorem 1.3 in [8], the smoothness assumption of K̂ can be removed if x_{0} is not a critical point of K̂. Even when x_{0} is a critical point, it is not necessary to assume that K̂ is smooth. In this case, condition (2.9) can be replaced by

c_{1}|x − x_{0}|^{β−1} ≤ | ▽ K̂(x)| ≤ c_{2}|x − x_{0}|^{β−1} in a neighborhood of x_{0} for some constants c_{2} > c_{1} and β > 1.

(2.10)

Thus, Theorem 1.3 of [8] can be restated as follows:

Theorem A. Let u_{i} be a solutions of (2.1) with K_{i} = n(n−2)+t_{i}K̂ and x_{0} ∈ Ω be a blowup point of u_{i}. Assume that either x_{0} is not a critical point of K̂ or x_{0} is a critical point of K̂ and K̂ satisfies (2.10) for some β ≥ n − 2. Then x_{0} is a simple blowup point.

Obviously, if x_{0} is a simple blow-up point, then there are no blowup points in a small neighborhood of x_{0}. If we further assume that K̂ has a discrete set of critical points in Ω, then by Theorem A, u_{i} has a discrete set of blowup points at most. Hence, throughout Section 2 to Section 5, we always assume that u_{i} is a solution of

(2.11)  \begin{cases}Δu_{i} + K_{i}(x)ui^{\frac{n+2}{n−2}} (x) = 0 on B_{2}\\{0\}, \\ u_{i}(x) is uniformly bounded in any compact \\ set of B_{2}\\{0\}, \end{cases}

where B_{2} = {x : |x| < 2}, and K_{i}(x) = n(n − 2) + t_{i}K̂ where K̂ satisfies (2.10) with x_{0} = 0 for x ∈ B_{2} and some β ≥ 1. Here, solutions u_{i} is assumed to blow up at 0. Let M̂_{i} denote the maximum of u_{i} and x_{i} be a maximum point of u_{i}, i.e.,

(2.12)  M̂_{i} = u_{i}(x_{i}) = max_{|x|≤2} u_{i}(x) → +∞

as i → +∞. Clearly x_{i} → 0. If β = 1 or β ≥ n − 2, by Theorem A, (2.3) holds for some constant c > 0. When 1 < β < n − 2, the situation is more complicated as shown in [9].

A solution u_{i} may have local maximum points beside x_{i}. Let z_{i} be any local maximum point of u_{i} with u_{i}(z_{i}) → +∞. Then by assumption (2.11), lim_{i→∞} z_{i} = 0. Let v_{i}(y) be the scaled function defined by

(2.13)  v_{i}(y) = M_{i}^{−1}u_{i}(z_{i} + M_{i}^{− \frac{2}{n−2}} y) with M_{i} = u_{i}(z_{i}).

Obviously, v_{i}(y) is well-defined for |y| ≤ Mi^{\frac{2}{n−2}} when i is large. In the paper, we will always reduce the arguments to the situation when

v_{i}(y) is uniformly bounded in any compact set of ℝ^{n}, that is, for any ε > 0, there exists a sequence of R_{i} → +∞ such that (2.14)

|v_{i}(y) − U_{1}(y)| ≤ εU_{1}(y) for |y| ≤ R_{i}.

In this case, by passing to a subsequence, v_{i}(y) converges to U_{1}(y) in C_{loc}^{2}(ℝ^{n}), where U_{1}(y) is given in (2.4) with λ = 1.

For such “good” local maximum point z_{i}, we set

(2.15) L_{i}(z_{i}) = min [(t_{i}^{−1}u_{i}(z_{i})^{\frac{2}{n−2}} |z_{i}|^{1−β})^{\frac{1}{n−2}} , (t_{i}^{−1}u_{i}(z_{i})^{\frac{2β̂}{n−2}} )^{\frac{1}{n−2}} ],

where β̂ = β if β < n and β̂ is any positive number in (n − 1, n) if β ≥ n. One of the main themes for local solutions is to know if the

scaled vector Mi^{\frac{2}{n−2}} z_{i} is bounded. This is closely related to the quantity L_{i}(z_{i}). To see this, let us assume β < n for simplicity. In this case, if

lim_{i→+∞} u_{i}(z_{i})|z_{i}|^{\frac{n−2}{2}} = +∞, then

u_{i}(z_{i})^{\frac{2}{n−2}} |z_{i}|^{1−β} = (u_{i}(z_{i})^{\frac{2}{n−2}} |z_{i}|)^{1−β}u_{i}(z_{i})^{\frac{2β}{n−2}} = o(1)u_{i}(z_{i})^{\frac{2β}{n−2}}

and

L_{i}(z_{i}) = (t_{i}^{−1}u_{i}(z_{i})^{\frac{2}{n−2}} |z_{i}|^{1−β})^{\frac{1}{n−2}} .

On the other hand, if

lim_{i→+∞} u_{i}(z_{i})|z_{i}|^{\frac{n−2}{2}} < +∞,

then it is easy to see

L_{i}(z_{i}) ∼ (t_{i}^{−1}u_{i}(z_{i})^{\frac{2β}{n−2}} )^{\frac{1}{n−2}} .

The quantity L_{i}(z_{i}) plays an important role for us to understand the bubbling profile of u_{i}. Our first result concerns with L_{i}(x_{i}) and the simple blowup at 0. We recall x_{i} is a maximum point of u_{i} and M̂_{i} = u_{i}(x_{i}) is the maximum of u_{i}. See (2.12).

Theorem 2.1. Suppose u_{i} is a solution of (2.11) and K̂ satisfies (2.10) for some β ≥ 1. Assume (1.4) in addition if β < n − 2. Then after passing to a subsequence, 0 is a simple blow-up point if and only if there exists a constant c > 0 independent of i such that

M̂i^{\frac{2}{n−2}} ≤ c L_{i}(x_{i})

for all i.

An interesting case is when the ratio M̂_{i}^{− \frac{2}{n−2}} L(x_{i}) tends to +∞ as i → +∞. If u_{i} is a global solution of (1.3), by applying the method of moving planes, we can prove that 0 is the only simple blowup point. See (6.8).

On the other hand, when the ratio M̂_{i}^{− \frac{2}{n−2}} L(x_{i}) is bounded, we have the following result.

Theorem 2.2. Let u_{i} and K̂ satisfy the assumptions of Theorem 2.1 and let x_{i}, M̂_{i} and L_{i}(x_{i}) be defined in (2.12) and (2.15), respectively. Suppose that there is c > 0 such that

L_{i}(x_{i}) ≤ c M̂i^{\frac{2}{n−2}} ,

then M̂_{i}|x_{i}|^{\frac{n−2}{2}} is bounded and β < n − 2. Furthermore, if assume in addition that K̂ satisfies (K0) with Q being the homogeneous function

and lim_{i→+∞} ξ_{i} = ξ exists with ξ_{i} = M̂i^{\frac{2}{n−2}} x_{i}, then ξ satisfies

(2.16)  ∫_{ℝ^{n}} ▽Q(x + ξ)U1^{\frac{2n}{n−2}} (x)dx = 0.

The following consequence of Theorem 2.2 is important when we come to determine the position of blowup points for global solutions of (1.3).

Corollary 2.3. Let u_{i} and K_{i} satisfy the assumptions of Theorem 2.1. Assume that either ▽K̂(0) ≠ 0 or ▽K̂(0) = 0 with β ≥ n − 2,

then lim_{i→+∞} L_{i}(x_{i})M̂_{i}^{− \frac{2}{n−2}} = +∞.

Both proofs of Theorem 2.1 and 2.2 are given in Section 3, where the application of the reflection method are discussed. By Theorem A, the flatness β of K̂ at 0 determines the bubbling behavior of u_{i}. Conventionally, u_{i} is said to lose the energy of one bubble at 0 if u_{i} converges to 0 in C_{loc}^{1}(B_{2} \ {0}) and

(2.17)  lim_{i→+∞} ∫_{|x|≤1} ui^{\frac{2n}{n−2}} (x)dx = (\frac{S_{n}}{n(n − 2)})^{\frac{n}{2}} ,

where S_{n} is the Sobolev best constant. Clearly, if u_{i} blows up at 0 simply, then u_{i} lost one bubble.

Theorem 2.4. Assume that K̂ satisfies (K0) and (K1) at 0 with \frac{n−2}{2} ≤ β, and u_{i} is a solution satisfying (2.11). Then u_{i} loses the energy of only one bubble at 0. Suppose in addition that \overline{lim}_{i→+∞}L_{i}(x_{i})M̂_{i}^{− \frac{2}{n−2}} <

+∞. Then there exists a constant c > 0 such that

(2.18)  u_{i}(x) ≤ c |x|^{\frac{2−n}{2}} for |x| ≤ 1.

Set ξ_{i} = M̂i^{\frac{2}{n−2}} x_{i}. Then after passing to a subsequence, the limit ξ = lim_{i→+∞} ξ_{i} satisfies (1.8).

When β < \frac{n−2}{2}, it is possible that (2.18) does not hold and it is also possible that u_{i} loses energy of more than one bubble even (2.18) holds. We first consider the case when inequality (2.18) does not hold. There are two alternatives in this case.

Theorem 2.5. Assume that K̂ satisfies (K0) and (K1) at 0, and u_{i} is a solution of (2.11). Suppose

(2.19)  lim_{i→+∞} sup(u_{i}(x)|x|^{\frac{n−2}{2}} ) = +∞.

Then one of the followings holds:

(i) The origin is a simple blowup point and consequently, an isolated blowup point. More precisely, we have

(2.20)  \{ \begin{matrix}u_{i}(x_{i} + x) ≤ c U_{λ_{i}}(x) for |x| ≤ 1, and \\ lim_{i→+∞} M̂_{i}|x_{i}|^{\frac{n−2}{2}} = +∞, \end{matrix}

where λ_{i} = M̂_{i}^{− \frac{2}{n−2}} .

(ii) The origin is not a simple blowup point and is not an isolated

blowup point. In this case, we have β < \frac{n − 2}{2} and there exists a local maximum point z_{i} of u_{i} satisfying (2.21)

u_{i}(z_{i})|z_{i}|^{\frac{n−2}{2}} → ∞ and L_{i}(z_{i})u_{i}(z_{i})^{− \frac{n−2}{2}} → ∞ as i → +∞

such that for any δ > 0, u_{i}(x) is a simple blowup with center z_{i} for x ∉ B(0, δ|z_{i}|), i.e.,

(2.22)  u_{i}(x) ≤ c U_{λ_{i}}(x − z_{i})

for |x| ≥ δ|z_{i}|, where λ_{i} = u_{i}(z_{i})^{− \frac{2}{n−2}} . Also, for x ∉ B(z_{i}, δ|z_{i}|), we have

(2.23)  u_{i}(x)|x|^{\frac{n−2}{2}} ≤ c

with c = c(δ) independent of i. Moreover, u_{i}(z_{i}) = o(1)M̂_{i}, where o(1) tends to 0 as i → +∞ and M̂_{i} = max_{|x|≤2} u_{i}(x).

Remark 2.6. Two consequences follow from Theorem 2.5. First, since (2.22) implies

(2.24)  min_{|x|=1} u_{i}(x) ∼ u_{i}(z_{i})^{−1},

the spherical Harnack inequality (2.18) holds if u_{i}(x) ≥ c > 0 on B_{2} for some c independent of i. Second, by (2.21),

lim_{i→+∞} L_{i}(z_{i})u_{i}(z_{i})^{− \frac{n−2}{2}} = +∞.

We will see later that this implies if u_{i} is a sequence of global solutions, then the number of the type of blowup points described in (ii) of Theorem 2.5 is at most one. See (6.8). By using this fact, we then are able to apply Lemma 7.1 to get rid of the blowup point of the type of behavior in case (ii) of Theorem 2.5. This is indeed Theorem 1.3.

When u_{i} converges to zero in C_{loc}^{1}(B_{2}\{0}), we say u_{i} loses energy of more than one bubble near 0 if

(2.25)  lim_{i→+∞} ∫_{|x|≤1} ui^{\frac{2n}{n−2}} (x)dx > (\frac{S_{n}}{n(n − 2)})^{\frac{n}{2}} .

In this case, we have β < \frac{n−2}{2} by Theorem A and Theorem 2.4. It is easy to see the blowup described in (ii) of Theorem 2.5 belongs to this case. Actually, when β < \frac{n−2}{2}, it is possible for u_{i} to lose infinite energy. See [11] for the existence for such solutions.

To estimate u_{i} more accurately when it satisfies (2.18) and loses energy of more than one bubble, let

(2.26)  u_{i}(r) = \frac{1}{|∂B_{r}|} ∫_{|x|=r} u_{i}dσ

be the spherical average of u_{i}, and

(2.27)  w_{i}(s) = u_{i}(r)r^{\frac{n−2}{2}} with r = e^{s}.

Obviously, w_{i}(s) is well-defined for s ≤ 0. Since 0 is a blowup point, w_{i} has at least one maximum point. Let s_{i} ≤ 0 be the local maximum point of w_{i}, which is nearest to zero. Set

(2.28)  M_{i} = e^{−( \frac{n−2}{2} )s_{i}},

(2.29)  L_{i} = (t_{i}^{−1}Mi^{\frac{2β}{n−2}} )^{\frac{1}{n−2}} ,

(2.30)  R_{i} = L_{i}^{γ}, γ = \frac{1}{1 − \frac{2β}{n−2}} , and

(2.31)  ũ_{i} = M_{i}^{−1}u_{i} (Mi^{\frac{−2}{n−2}} x) .

Then we have the following estimates:

Theorem 2.7. Suppose that K̂ satisfies (K0) and (K1) at 0 with

1 < β < \frac{n − 2}{2}, and u_{i} is a solution of (2.11) which converges uniformly

to zero in any compact set of B_{2}\{0} and satisfies (2.18) and (2.25). Define w_{i}, s_{i}, M_{i}, L_{i}, R_{i} and ũ_{i} as above. Then lim_{i→+∞} M_{i} = +∞ and

there are c > 0, a_{i} → 1, z_{i} ∈ ℝ^{n} and λ_{i} > 0 such that the following hold:

(i) lim_{i→∞} λ_{i} = λ and lim_{i→+∞} z_{i} = z, where λ and z satisfy

(2.32)  1 = λ^{2} + |z|^{2}.

Set ξ = \sqrt{λ}z. Then ξ satisfies (1.8).

(ii) ũ_{i} satisfies

(2.33)  |ũ_{i}(x)| ≤ c |x|^{− \frac{n−2}{2}} for |x| ≤ R_{i}^{−2}, and

\begin{matrix}|ũ_{i}(x) − a_{i}U_{λ_{i}}(x − z_{i})| \\ ≤ c (L_{i}^{−n+2} + R_{i}^{−n+2}|x|^{−n+2} + max_{|y|=M \frac{2}{n−2} ; i} |ũ_{i}(y) − a_{i}U_{λ_{i}}(y − z_{i})|), \end{matrix} (2.34)

for R_{i}^{−2} ≤ |x| ≤ Mi^{\frac{2}{n−2}} .

Remark 2.8. If L_{i}M_{i}^{− \frac{2}{n−2}} ≤ c for some constant c > 0, then from the proof of Theorem 2.7, we will see that ũ_{i}(y) ≤ c_{1}L_{i}^{−n+2} for some

constant c_{1} when |y| = Mi^{\frac{2}{n−2}} . Thus, the third term in the right hand

side of (2.34) can be absorbed by L_{i}^{−n+2} when L_{i}M_{i}^{− \frac{2}{n−2}} ≤ c.

To extend the notion of simple blowup to cover the case when u_{i} loses energy of more than one bubble, we modify (2.3) as follows. Let B_{r}(y) denote {x : |x − y| < r}.

Definition 2.9. Assume 0 is a blowup point. The blowup point 0 is called simple-like if there exist c > 0, r_{0} > 0, a sequence of numbers {λ_{i}}, a sequence of points {z_{i}} and a sequence of balls {B_{r_{i}}(y_{i})} such that lim_{i→∞} λ_{i} = 0, lim_{i→∞} z_{i} = lim_{i→∞} y_{i} = 0, lim_{i→∞} r_{i}λ_{i}^{−1} = 0, and

u_{i}(x + z_{i}) ≤ c U_{λ_{i}}(x) on B_{r_{0}}(0) \ B_{r_{i}}(y_{i}).

According to the definition, it is not difficult to see that there are exactly three types of simple-like blowup point: simple blowup, the blowup described in (ii) of Theorem 2.5, and the blowup in Theorem 2.7

when L_{i} ≥ cMi^{\frac{2}{n−2}} for some constant c > 0. On the other hand, if 0 is non-simple-like, then by Theorem 2.5, inequality (2.18) holds and 0 must be isolated.

Remark 2.10. When the assumption (K1) is concerned in the theorems of this section, (K1) is required only when β < n − 2.

3. Applications of the method of moving planes

In this section, we will collect some well-known results and prove some lemmas which will be used in the proofs of the theorems in Section 2. In the proofs, we often assume there is a sequence of local maximum points z_{i} of u_{i} such that the scaled function v_{i} in (2.13) satisfies (2.14). By applying the method of moving planes, we can improve the result of (2.14). When K_{i} satisfies if the nondegenerate conditions (K0) and (K1) with 1 < β ≤ n − 2, we proved that u_{i}(z_{i} + x) could

be bounded by c U_{λ_{i}}(x) with λ_{i} = u_{i}(z_{i})^{− \frac{2}{n−2}} for |x| ≤ L_{i}M_{i}^{− \frac{2}{n−2}} . See Lemma 3.1 in [9]. Actually the proof there can apply to the degenerate case. In the following, we give a brief sketch of the proof for the convenience of readers. In fact, Lemma 3.1 below deals with the case more general than the one considered in [9], namely, u_{i} is allowed to have very large values, compared with u_{i}(z_{i}), in some small region. Let d(B, 0) denote the distance from the origin to a ball B.

Lemma 3.1. Suppose that u_{i} is a solution of (2.11), z_{i} is a local maximum point of u_{i} and v_{i} is given as in (2.13). Let B be a closed ball in ℝ^{n} with d(B, 0) > 0 and ε be a positive (small) number. Suppose that there is a sequence of R_{i} → +∞ as i → +∞ such that

|v_{i}(y) − U_{1}(y)| ≤ εU_{1}(y)

for |y| ≤ R_{i} and y ∉ B. Then there exists δ = δ(ε, d(B, 0)) > 0 such that

(3.1)  min_{|y|≤r} v_{i}(y) ≤ (1 + 2ϵ)U_{1}(r)

for 0 ≤ r ≤ L_{i}^{∗}(δ), where L_{i}^{∗}(δ) = min(δL_{i}(z_{i}), Mi^{\frac{2}{n−2}} ).

Proof. When B is an empty set and 1 ≤ β ≤ n−2, this is Lemma 3.1 in [9]. Thus, we only sketch the proof below. For the details, we refer the interested readers to [9].

Let e_{1} = (1, 0, ⋯, 0) and τ = d(B, 0). We may assume the center of B is r_{0}e_{1} for some r_{0} > τ. Let

\begin{matrix}F (x) = \frac{τ^{2}x}{|x|^{2}} + τe_{1}, \\ v_{i}(x) = (\frac{τ}{|x|})^{n−2}v_{i}(\frac{τ^{2}x}{|x|^{2}} + τe_{1}), \\ U_{1}(x) = (\frac{τ}{|x|})^{n−2}U_{1}(\frac{τ^{2}x}{|x|^{2}} + τe_{1}). \end{matrix} (3.2)

By a straighforward calculation, we have

U_{1}(x) = (\frac{λ}{λ^{2} + |x − x_{0}|^{2}} )^{\frac{n−2}{2}} ,

where λ = \frac{τ^{2}}{τ^{2} + 1} and x_{0} = −\frac{τ^{3}e_{1}}{τ^{2} + 1}. Also we have F^{−1}(B) = \{x :

x = F^{−1}(y), y ∈ B} ⊂ {(x_{1}, x_{2}, ⋯, x_{n}) : x_{1} > 0}, d(F (B), 0) > 0 and v_{i} satisfies

△v_{i} + K_{i}(x)v_{i}^{\frac{n+2}{n−2}} = 0

for x ∉ F^{−1}(B), where K_{i}(x) = K_{i}(z_{i} + M_{i}^{− \frac{2}{n−2}} F (x)).

Now assume that the conclusion of Lemma 3.1 does not hold. Then by passing to a subsequence, there is a sequence of positive number r_{i} such that r_{i} ≤ L_{i}^{∗}(δ) and

(3.3)  min_{|y|≤r_{i}} v_{i}(y) ≥ (1 + 2ϵ)U_{1}(r_{i}),

where δ = δ(ε) will be chosen later. By the assumptions, it is easy to see r_{i} ≥ R_{i} → +∞ as i → +∞. Since by (3.2), v_{i}(x) uniformly converges to U_{1}(x) in C_{loc}^{2}(ℝ^{n}\{0}), v_{i} has a local maximum at some point q_{i} near x_{0}. Now we are going to apply the method of moving planes to obtain a contradiction.

For any λ < 0, let Σ_{λ} = {x | x_{1} > λ}, T_{λ} = {x | x_{1} = λ} and x^{λ} denote the reflection point of x with respect to T_{λ}. We also let Σ′_{λ} = Σ_{λ} ∩ {x | |x| ≥ τ^{2}(r_{i} − τ)^{−1}}. In the following, we will choose

a number λ_{0} satisfying −|x_{0}| < λ_{0} < −\frac{|x_{0}|}{2} and show that for λ ≤ λ_{0}, there exists i_{0} = i_{0}(λ_{0}) such that

(3.4)  v_{i}(x^{λ}) ≤ v_{i}(x)

for x ∈ Σ′_{λ}, λ ≤ λ_{0} and i ≥ i_{0}. This yields a contradiction to the fact that v_{i} has a local maximum near x_{0}. Note that the local maximum point q_{i} tends to x_{0} as i → ∞.

Let w_{λ}(x) = v_{i}(x) − v_{i}(x^{λ}). Then w_{λ} satisfies

(3.5)  △w_{λ} + b_{λ}(x)w_{λ}(x) = Q_{λ}(x)  in Σ′_{λ},

where

\{ \begin{matrix}b_{λ}(x) = K_{i}(x)\frac{(v_{i}(x)^{\frac{n+2}{n−2}} − (v_{i}(x^{λ})^{\frac{n+2}{n−2}} )}{v_{i}(x) − v_{i}(x^{λ})} \\ Q_{λ}(x) = (K_{i}(x^{λ}) − K_{i}(x)) v_{i}(x^{λ})^{\frac{n+2}{n−2}} . \end{matrix}

By (3.2) and (3.3), we have for |x| = τ^{2}(r_{i} − τ)^{−1},

(3.6)  v_{i}(x) ≥ (\frac{r_{i} − τ}{τ} )^{n−2} min_{|y|≤r_{i}} v_{i} ≥ (1 + ε)U_{1}(0)

for i large. On the other hand, v_{i}(x^{−|x_{0}|}) converges to U_{1}(0^{−|x_{0}|}) = U_{1}(0) uniformly for |x| = τ^{2}r_{i}^{−1}, where x^{−|x_{0}|} and 0^{−|x_{0}|} are the reflection points of x and 0 with respect to the hyperplane T_{−|x_{0}|}. Hence

there exists −|x_{0}| < λ_{0} < \frac{−|x_{0}|}{2} such that

v_{i}(x^{λ}) ≤ (1 + \frac{ε}{2})U_{1}(0)

for |x| = τ^{2}(r_{i} −τ)^{−1}, λ ≤ λ_{0} and large i. Together with (3.6), it implies for |x| = τ^{2}(r_{i} − τ)^{−1},

w_{λ}(x) ≥ \frac{ε}{2}U_{1}(0)

for λ ≤ λ_{0} and large i. In the following, we fix this λ_{0}. Then there is a small c_{0} such that

(3.7)  w_{λ}(x) ≥ \frac{ε}{2}U(0) ≥ c_{0}r_{i}^{−n+2}G^{λ}(x, 0)

holds for |x| = τ^{2}(r_{i} − τ)^{−1}, λ ≤ λ_{0} and large i, where G^{λ}(x, y) is

G^{λ}(x, y) = c_{n}(\frac{1}{|y − x|^{n−2}} − \frac{1}{|y^{λ} − x|^{n−2}} ),

the Green function of −△ on Σ_{λ} = {x : x_{1} > λ}.

If λ_{1} < 0 and |λ_{1}| is large, then we have

(3.8)  w_{λ}(x) ≥ \frac{c_{0}}{2} r_{i}^{−n+2}G^{λ}(x, 0)

for λ ≤ λ_{1}, x ∈ Σ′_{λ} and large i. For the details, see [9].

For λ > λ_{1}, let Q_{λ}^{+} = max(0, Q_{λ}), L_{i} = L_{i}(z_{i}) and

(3.9)  h_{λ}(x) = aL_{i}^{−n+2}G^{λ}(x, 0) − ∫_{Σ′_{λ}} G^{λ}(x, η)Q_{λ}^{+}(η) dη,

where a is a positive number to be chosen later. Obviously, h_{λ} satisfies

△h_{λ} = Q_{λ}^{+} ≥ Q_{λ} in Σ′_{λ}.

For λ ≤ λ_{0} and η ∈ Σ_{λ}, since |η^{λ}| ≥ |η| and |η^{λ}| ≥ |λ_{0}| ≥ \frac{|x_{0}|}{2} > 0, one has by (3.2)

|v_{i}(η^{λ})| ≤ c_{1}(1 + |η^{λ}|)^{−(n−2)}.

Here, we use F^{−1}(B) ⊂ Σ_{λ} also. For η ∈ Σ′_{λ}, we have

|η| ≥ τ^{2}(r_{i} − τ)^{−1} ≥ \frac{τ^{2}}{2} L_{i}^{∗}(δ) ≥ \frac{τ^{2}}{2} M_{i}^{− \frac{2}{n−2}} .

To estimate the integral term in (3.9), we note

Q_{λ}^{+}(η) ≤ c_{2}(1+|η^{λ}|)^{−(n+2)}|K_{i}(z_{i}+M_{i}^{− \frac{2}{n−2}} F (η^{λ}))−K_{i}(z_{i}+M_{i}^{− \frac{2}{n−2}} F (η))|.

By (2.10), when η ∈ Σ′_{λ},

\begin{matrix}|K_{i}(z_{i} + M_{i}^{− \frac{2}{n−2}} F (η)) − K_{i}(z_{i})| \\ ≤ ct_{i}M_{i}^{− \frac{2}{n−2}} |F (η)| \{|z_{i}|^{β−1} + M_{i}^{− \frac{2(β̂−1)}{n−2}} |F (η)|^{β̂−1}\} \\ ≤ c_{3}t_{i}M_{i}^{− \frac{2}{n−2}} (1 + |η|^{−1}) \{|z_{i}|^{β−1} + M_{i}^{− \frac{2(β̂−1)}{n−2}} (1 + |η|^{1−β̂})\} \\ ≤ c_{4}L_{i}^{2−n}(1 + |η|^{−β̂}), \end{matrix} (3.10)

where |η| ≥ \frac{τ^{2}}{2} M_{i}^{− \frac{2}{n−2}} is used and β̂ is the number in (2.15). Thus, we have

(3.11)  Q_{λ}^{+}(η) ≤ c_{5}L_{i}^{−n+2}(1 + |η|^{−β̂})(1 + |η^{λ}|)^{−(n+2)}.

By (3.11), following the computation in the proof of Lemma 3.1 in [9], we obtain

(3.12)  ∫_{Σ′_{λ}} G^{λ}(x, η)Q_{λ}^{+}(η)dη ≤ c_{6}L_{i}^{−n+2}G^{λ}(x, 0)

for x ∈ Σ′_{λ}, where c_{6} is a constant depending on the constants in (2.10), τ and n only.

Set a = 2c_{6} in (3.9). Then

(3.13) 0 < \frac{a}{2}[L(z_{i})]^{−n+2}G^{λ}(x, 0) ≤ h_{λ}(x) ≤ a[L(z_{i})]^{−n+2}G^{λ}(x, 0).

Recall that r_{i} ≤ δL_{i}(z_{i}). Choose δ to be sufficiently small such that c_{0}δ^{−n+2} ≥ 2a. Then by (3.7) and (3.8), for i large,

w_{λ}(x) > h_{λ}(x)

holds for x ∈ Σ′_{λ} if λ = λ_{1}, and holds for |x| = τ^{2}(r_{i} − τ)^{−1} and λ ≤ λ_{0}. It follows that h_{λ} satisfies the assumptions of Lemma 2.1 in [9] with λ_{1} ≤ λ ≤ λ_{0} when i is large. Applying Lemma 2.1 in [9], w_{λ}(x) > h_{λ}(x) > 0 for x ∈ Σ′_{λ} and λ ≤ λ_{0}. Hence, (3.4) is proved, and then the proof of Lemma 3.1 is finished. q.e.d.

Note that if u_{i} is a global solution defined in the whole space ℝ^{n}, then we can choose

L_{i}^{∗}(δ) = min(L_{i}^{∗}(δ), λMi^{\frac{2}{n−2}} )

for any λ > 0. Inequality (3.1) is very useful when the Harnack inequality holds for v_{i} on each sphere |y| = r. Actually, under some extra condition on u_{i}, we can derive the spherical Harnack inequality from (3.1) itself by using the Green representation formula. We will explain this in Lemma 3.4, which tells us how to derive the Harnack inequality. Before that, we have to state two well-known lemmas. For their proofs, see [9].

Lemma 3.2. Suppose ϕ(x) satisfies

△ϕ(x) + n(n + 2)U1^{\frac{4}{n−2}} ϕ(x) = 0 in ℝ^{n}

with ϕ(x) → 0 as |y| → ∞. Then ϕ(x) can be written as

ϕ(x) = c_{0}ψ_{0}(x) + ∑_{j=1}^{n} c_{j}ψ_{j}(x)

for some c_{j} ∈ ℝ, j = 0, 1, …, n, where ψ_{j}(x) = \frac{∂U_{1}}{∂x_{j}} for 1 ≤ j ≤ n and

ψ_{0}(x) = \frac{n − 2}{2}U_{1} + x · ∇U_{1}.

Lemma 3.3. Suppose that u is a positive smooth solution of

△u + K(x)u^{\frac{n+2}{n−2}} = 0 in B_{r},

where |K(x)| ≤ b. Then there exists a small ϵ_{o} > 0, depending on b and n only, such that if ||u||_{L^{\frac{2n}{n−2}}} ≤ ϵ_{o}, then the Harnack inequality

u(x) ≤ c u(y)

holds for |x|, |y| ≤ r/4, where c > 0 depends on b and n only.

In Lemma 3.4, we consider a more general setting, which is needed later. Assume that 0 < a ≤ K(x) ≤ b, u is a solution of

(3.14)  △u + K(x)u^{\frac{n+2}{n−2}} = 0, u > 0 for |x| ≤ l_{0},

and U is the solution of

(3.15)  \{ \begin{matrix}△U + K_{0}U^{\frac{n+2}{n−2}} = 0, U > 0 in ℝ^{n}, \\ U(0) = max_{ℝ^{n}}U = 1, \end{matrix}

where K_{0} is a positive constant. Let B_{r} = {x : |x| < r}.

Lemma 3.4. Let u, U and l_{0} be as above. Suppose 0 < σ < 1,

R ≤ \frac{l_{0}}{8} , and E ⊆ B_{R/2} such that

(3.16)  |u(x) − U(x)| ≤ σU(x)

for x ∈ B_{R}\E,

(3.17)  ∫_{|x|≤R} |K(x) − K_{0}|U^{\frac{n+2}{n−2}} dx ≤ σ,

(3.18)  ∫_{E} U^{\frac{n+2}{n−2}} dx < σ, and

(3.19)  min_{|x|=l} u(x) ≤ (1 + σ)U(r)

for some l ∈ [R, \frac{l_{0}}{4} ]. Then there is a constant c_{1} depending on n and b only such that

(3.20)  ∫_{R≤|x|≤l} u^{\frac{n+2}{n−2}} dx ≤ c_{1}(R^{−2} + σ + (\frac{l}{l_{0}} )^{n−2}),

Furthermore, if

(3.21)  u(x) ≤ c_{2} (R^{−2} + σ + (\frac{l}{l_{0}} )^{n−2})^{−1}, and

(3.22)  min_{|x|=r} u(x) ≤ c_{3} U(r)

for R ≤ r ≤ l where c_{2} = c_{2}(n, a, b) is a small positive constant and c_{3} > 0, then

(3.23)  u(x) ≤ c_{4}U(x)

for |x| ≤ \frac{l}{2} and x ∉ E, where c_{4} depends on c_{2} and c_{3}.

Proof. For r > 0, let B_{r} = {x : |x| < r}. Let G(x, η) be the Green function of the Laplacian operator −△ on the ball B_{l_{0}} with zero boundary value. Let x_{0} be a point satisfy |x_{0}| = l and u(x_{0}) = min_{|x|≤l} u(x). By the Green identity and (3.19),

(3.24)  (1 + σ)U(x_{0}) ≥ u(x_{0}) ≥ ∫_{B_{l_{0}}} G(x_{0}, η)K(η)u^{\frac{n+2}{n−2}} (η)dη,

and

\begin{matrix}U(x_{0}) = & ∫_{B_{l_{0}}} G(x_{0}, η)K_{0}U^{\frac{n+2}{n−2}} dη + U(l_{0}) \\ ≤ & ∫_{B_{l_{0}}} G(x_{0}, η)K_{0}U^{\frac{n+2}{n−2}} dη + U(l_{0}) \end{matrix} (3.25)

Hence there is c_{n} depending on n only such that

\begin{matrix}a c_{n} ∫_{\frac{R}{2} ≤η≤ \frac{l_{0}}{2}} (l + |η|)^{−n+2}u^{\frac{n+2}{n−2}} dη \\ ≤ u(x_{0}) − ∫_{B_{\frac{R}{2}} \E} G(x_{0}, η)K(η)u^{\frac{n+2}{n−2}} dη \\ ≤ (1 + σ)U(x_{0}) − ∫_{B_{\frac{R}{2}} \E} G(x_{0}, η)K(η)u^{\frac{n+2}{n−2}} dη. \end{matrix} (3.26)

By the assumptions (3.16) and (3.17), there is c_{4} depending on n and b only such that

∫_{B_{\frac{R}{2}} \E} G(x_{0}, η)K(η)u^{\frac{n+2}{n−2}} dη
≥ ∫_{B_{\frac{R}{2}} \E} G(x_{0}, η)\{K_{0}U^{\frac{n+2}{n−2}} + K(η)(u^{\frac{n+2}{n−2}} − U^{\frac{n+2}{n−2}} )
− |K(η) − K_{0}|U^{\frac{n+2}{n−2}} \} dη
≥ ∫_{B_{\frac{R}{2}}} G(x_{0}, η)K_{0}U^{\frac{n+2}{n−2}} dη − c_{4}l^{−n+2}σ.

Together with (3.25), it leads to

a c_{n} ∫_{\frac{R}{2} ≤η≤ \frac{l_{0}}{2}} (1 + l^{−1}|x|)^{−n+2}u^{\frac{n+2}{n−2}} dη

≤ l^{n−2} [∫_{B_{l_{0}}\B_{\frac{R}{2}}} G(x_{0}, η)K_{0}U^{\frac{n+2}{n−2}} dη + c_{4}l^{−n+2}σ]

+ l^{n−2} [σU(x_{0}) + U(l_{0})]

≤ c_{5} (σ + R^{−2} + (\frac{l}{l_{0}} )^{n−2}),

where c_{5} depends on n and b only. Obviously the inequality (3.20) follows immediately.

Let ϵ_{0} be the number in Lemma 3.3 and c_{2} be a small number such that

c_{2}c_{5}(c_{n}a)^{−1} < ϵ_{0}.

If u(x) ≤ c_{2}(R^{−2} + σ + (\frac{l}{l_{0}} )^{n−2})^{−1} for \frac{R}{2} ≤ |x| ≤ l, then

∫_{\frac{R}{2} ≤|η|≤l} u^{\frac{2n}{n−2}} dη < ∫_{\frac{R}{2} ≤|η|≤l} u^{\frac{n+2}{n−2}} dη ( max_{\frac{R}{2} ≤|η|≤l} u) < ϵ_{0}.

By Lemma 3.3, the Harnack inequality holds for u on {x : |x| = r} with

R ≤ r ≤ \frac{l}{2}. The inequality (3.23) then follows from it and (3.22) for

R ≤ r ≤ \frac{l}{2}. Together with (3.16), (3.23) holds for all |x| ≤ \frac{l}{2} and x ∉ E. q.e.d.

Let z_{i} be a local maximum point and v_{i} be the scaled solution in (2.13) such that (2.14) holds and U_{i}(y) be the solution of (3.15) with K_{0} = K_{i}(z_{i}). In the next step, we are going to estimate the difference between v_{i} and U_{i}(y). By (2.14), for any ε > 0, we have a sequence of R_{i} → +∞ such that

|v_{i}(y) − U_{i}(y)| ≤ εU_{i}(y) for |y| ≤ R_{i}.

By Lemma 3.1, there exists δ_{0} = δ_{0}(ε) > 0 such that

(3.27)  min_{|y|=r} v_{i}(y) ≤ (1 + 2ε)U_{i}(r)

for 0 ≤ r ≤ L_{i}^{∗}(δ_{0}). Then Lemma 3.4 yields the following important result.

Lemma 3.5. Let v_{i} and U_{i} be described as above. Suppose that there is a sequence of positive number l_{i} ≤ L_{i}^{∗}(δ_{0}) such that

(3.28)  v_{i}(y) ≤ c_{1} for |y| ≤ l_{i}.

Then there exists a small d > 0 such that

(3.29)  v_{i}(y) ≤ c_{2}U_{i}(y), and
(3.30)  |v_{i}(y) − U_{i}(y)| ≤ c_{2}r_{i}^{−n+2}

for |y| ≤ r_{i} = dl_{i} where d is a constant depending on n only. Further-

more, let Q̃_{i}(y) = K_{i}(z_{i}) − K_{i}(z_{i} + M_{i}^{− \frac{2}{n−2}} y). Then for r ≤ r_{i},

(3.31)  |∫_{|y|≤r} Q̃(y)Ui^{\frac{n+2}{n−2}} (y)ψ_{0}(y)dy | ≤ c_{1}r^{−n+2},

and

(3.32)  |∫_{|y|≤r} Q̃(y)Ui^{\frac{n+2}{n−2}} (y)ψ_{j}(y)dy | ≤ c_{1}r^{−n+1}

for 1 ≤ j ≤ n, where ψ_{j}(x) are given in Lemma 3.2.

Proof. Without loss of generality, we might assume R_{i} << l_{i}. Otherwise, (3.29)–(3.30) hold automatically. By Lemma 3.1, (3.27) holds for 0 ≤ r ≤ l_{i}. Since K_{i} = n(n − 2) + t_{i}K̂, we have

∫_{|x|≤R_{i}} |K̃_{i}(x) − K_{i}(z_{i})|U^{\frac{n+2}{n−2}} (x)dx ≤ c t_{i} ≤ ε,

for t_{i} small, where K̃_{i}(x) = K_{i}(z_{i} + M_{i}^{− \frac{2}{n−2}} x). Thus, v_{i} satisfies assumptions (3.16) ∼ (3.19) with an empty set E, R = R_{i}, l = dl_{i} and

l_{0} = Mi^{\frac{2}{n−2}} . Let d be small such that

c_{1}(R_{i}^{−2} + ε + d^{n−2}) < c_{2},

where c_{2} is the constant in (3.21). Then by (3.28), we have

v_{i}(y) ≤ c_{2}(R_{i}^{−2} + ε + d^{n−2})^{−1} for |y| ≤ l_{i}.

Then (3.29) follows immediately from Lemma 3.4. The inequality (3.30) can be proved by the same argument as in Lemma 3.3 of [9]. Hence, we omit the proof here.

96 chiun-chuan chen & chang-shou lin

To Prove (3.31) and (3.32), we let w_{i} = v_{i}(y) − U_{i}(y). Then w_{i} satisfies

(3.33)  Δw_{i} + b̃_{i}(y)w_{i}(y) = Q̃_{i}(y)Ui^{\frac{n+2}{n−2}} (y),

where

(3.34)  \begin{cases}b̃_{i}(y) = K̃_{i}(y) (\frac{vi^{\frac{n+2}{n−2}} − Ui^{\frac{n+2}{n−2}}}{v_{i} − U_{i}} ) , \\ K̃_{i}(y) = K_{i}(z_{i} + M_{i}^{− \frac{2}{n−2}} y), and \\ Q̃_{i}(y) = K_{i}(z_{i}) − K̃_{i}(y). \end{cases}

Multiplying (3.33) by ψ_{j}, one has

\begin{matrix}∫_{|y|≤r} w_{i}(Δψ_{j} + b̃_{i}ψ_{j})dy + ∫_{|y|=r} (ψ_{j} \frac{∂w_{i}}{∂ν} − w_{i} \frac{∂ψ_{j}}{∂ν} ) dσ \\ = ∫_{|y|≤r} Q̃_{i}Ui^{\frac{n+2}{n−2}} ψ_{j}dy \end{matrix} (3.35)

for 0 ≤ j ≤ n. Let r_{i} = dl_{i}. By (3.30), we have for |y| ≤ r_{i},

(3.36)  |v_{i}(y) − U_{i}(y)| ≤ c_{2}r_{i}^{2−n}.

To estimate the first term of (3.35), we recall

Δψ_{j} + \frac{n + 2}{n − 2}K_{i}(z_{i})Ui^{\frac{4}{n−2}} ψ_{j} = 0,

and then

\begin{matrix}w_{i}(Δψ_{j} + b̃_{i}ψ_{j}) = & (K̃_{i}(y) − K_{i}(z_{i}))(vi^{\frac{n+2}{n−2}} − Ui^{\frac{n+2}{n−2}} )ψ_{j} \\ + K_{i}(z_{i})(vi^{\frac{n+2}{n−2}} − Ui^{\frac{n+2}{n−2}} − \frac{n + 2}{n − 2}Ui^{\frac{4}{n−2}} w_{i})ψ_{j}. \end{matrix}

Hence for j = 0, we have as in (3.10)

\begin{matrix}|w_{i}(Δψ_{0} + b̃_{i}ψ_{0})| \\ ≤ c\{r_{i}^{2−n}L_{i}(z_{i})^{−n+2}(1 + |y|)^{β̂−n−2} + r_{i}^{2(2−n)}(1 + |y|)^{−4}\} \\ ≤ 2c r_{i}^{2(2−n)}(1 + |y|)^{−2}, \end{matrix} (3.37)

where |ψ_{0}(y)| ≤ c(1 + |y|)^{2−n} and β̂ < n are used. Similarly, by

|ψ_{j}(y)| ≤ c(1 + |y|)^{1−n} for 1 ≤ j ≤ n,

we have

(3.38)  |w_{i}(Δψ_{j} + b̃_{i}ψ_{j})| ≤ c r_{i}^{2(2−n)}(1 + |y|)^{−3}.

By applying (3.37) and (3.38), we have

| ∫_{B_{r}} w_{i}(Δψ_{j} + b̃_{i}ψ_{j})dy | = O(r^{2−n})

for j = 0, and

| ∫_{B_{r}} w_{i}(Δψ_{j} + b̃_{i}ψ_{j})dy | = O(r^{1−n})

for 1 ≤ j ≤ n. When |y| = r, we have

| ▽ v_{i}(y)| ≤ c|y|^{−1}v_{i}(y) = O(|y|^{1−n})

by the gradient estimate. Therefore, the boundary term of (3.35) is bounded by O(r^{2−n}) for j = 0 and is bounded by O(r^{1−n}) for 1 ≤ j ≤ n. Both (3.31) and (3.32) then follow from (3.35). q.e.d.

Proof of Theorem 2.2. We prove Theorem 2.2 by contradiction. Suppose lim_{i→+∞} M̂_{i} |x_{i}|^{\frac{n−2}{2}} = +∞. If β ≥ n − 2, by the definition (2.15)

and the assumption that L_{i}(x_{i})M̂_{i}^{− \frac{2}{n−2}} is bounded, we have

(3.39)  L_{i}(x_{i}) = (t_{i}^{−1}M̂i^{\frac{2}{n−2}} |x_{i}|^{1−β})^{\frac{1}{n−2}} .

If 1 ≤ β < n − 2, then

\begin{aligned}t_{i}^{−1}M̂i^{\frac{2}{n−2}} |x_{i}|^{1−β} & = t_{i}^{−1}M̂i^{\frac{2β}{n−2}} (M̂i^{\frac{2}{n−2}} |x_{i}|)^{1−β} \\ & ≤ t_{i}^{−1}M̂i^{\frac{2β}{n−2}} , \end{aligned}

which implies (3.39) also.

Let v_{i}(y) be defined as (2.13) with z_{i} = x_{i}. Obviously, v_{i}(y) ≤ 1 for

|y| ≤ M̂i^{\frac{2}{n−2}} . By Lemma 3.5, there exists a δ_{2} > 0 such that (3.29)– (3.32) hold with dl_{i} replaced by δ_{2}L_{i}(x_{i}). Recall the quantity Q̃_{i} in Lemma 3.5. We may assume lim_{i→+∞} \frac{▽K̂(x_{i})}{|▽K̂(x_{i})|} = e_{1} = (1, 0, …, 0). Then

\begin{aligned}−Q̃_{i} & = K_{i}(x_{i} + M̂i^{\frac{−2}{n−2}} y) − K_{i}(x_{i}) \\ & = t_{i}M̂i^{\frac{−2}{n−2}} (▽K̂(x_{i}), y) + c(δ, i)t_{i}M_{i}^{− \frac{2}{n−2}} | ▽ K̂(x_{i})||y| \\ & = t_{i}M̂i^{\frac{−2}{n−2}} | ▽ K̂(x_{i})|y_{1} + c(δ, i)t_{i}M_{i}^{− \frac{2}{n−2}} | ▽ K̂(x_{i})||y| \end{aligned} (3.40)

for |y| ≤ δM̂i^{\frac{2}{n−2}} |x_{i}|, where c(δ, i) could be arbitrarily small if i is large and δ is small. Therefore, we can choose δ small enough so that

(3.41)  \begin{aligned}∫_{|y|≤r_{i}}(−Q̃_{i})Ui^{\frac{n+2}{n−2}} (y)ψ_{1}(y)dy & ≥ c t_{i}M̂_{i}^{− \frac{2}{n−2}} |x_{i}|^{β−1} \\ & = c (L_{i}(x_{i}))^{2−n} \end{aligned}

for some c > 0 where r_{i} = δM̂i^{\frac{2}{n−2}} |x_{i}|. For the simplicity of notations, we let l_{i} = δ_{2}L_{i}(x_{i}). If r_{i} ≥ l_{i}, then by (3.41), we have

(3.42)  ∫_{|y|≤l_{i}}(−Q̃_{i})Ui^{\frac{n+2}{n−2}} (y)|ψ_{1}(y)|dy ≥ c_{1}(L_{i}(x_{i}))^{2−n}.

If l_{i} ≥ r_{i}, as in (3.10), we have

∫_{r_{i}≤|y|≤l_{i}} |Q̃_{i}|Ui^{\frac{n+2}{n−2}} (y)ψ_{1}(y)dy

≤ c ∫_{r_{i}≤|y|≤l_{i}}(L_{i}(x_{i})^{−n+2}|y|^{−2n} + t_{i}M̂_{i}^{− \frac{2β̂}{n−2}} |y|^{−n−1})dy
= o(1)L_{i}(x_{i})^{2−n}.

Together with (3.41), it implies that (3.42) holds also in the case of l_{i} ≥ r_{i}.

On the other hand, by (3.32), we have

|∫_{|y|≤l_{i}} Q̃_{i}Ui^{\frac{n+2}{n−2}} ψ_{1}dy| ≤ c_{1}L_{i}(x_{i})^{−n+1}.

This contradicts (3.42). Hence we conclude M̂i^{\frac{2}{n−2}} |x_{i}| is bounded.

Suppose β ≥ n − 2. Since M̂i^{\frac{2}{n−2}} |x_{i}| is bounded, we have

\begin{aligned}M̂i^{\frac{2}{n−2}} |x_{i}|^{1−β} & = (M̂i^{\frac{2}{n−2}} |x_{i}|)^{1−β} M̂i^{\frac{2β}{n−2}} \\ & ≥ c_{1}M̂_{i}^{2}. \end{aligned}

Hence,

lim_{i→+∞} L_{i}^{n−2}(x_{i})M̂_{i}^{−2} ≥ c_{1} lim_{i→+∞} t_{i}^{−1} = +∞,

which yields a contradiction to our assumptions. Thus, β < n − 2 must hold.

To prove (2.16), we let w_{i}(y) = l_{i}^{n−2}(v_{i}(y) − U_{i}(y)) where l_{i} = δ_{2}L_{i}(x_{i}). Then w_{i} satisfies (3.33) with Q̃_{i}(y) replaced l_{i}^{n−2}Q̃_{i} in the right hand side. By (K0),

(3.43)  \begin{aligned}Q̃_{i}(y) & = K_{i}(x_{i}) − K_{i}(x_{i} + M̂i^{\frac{2}{n−2}} y) \\ & = −t_{i} [Q(x_{i} + M̂_{i}^{− \frac{2}{n−2}} y) + R(x_{i} + M̂_{i}^{− \frac{2}{n−2}} y)] \\ & + (K_{i}(x_{i}) − K_{i}(0)) \\ & = −t_{i}M̂_{i}^{− \frac{2β}{n−2}} [Q(ξ_{i} + y) + o(1)(|y|^{β} + 1)] \\ & + (K_{i}(x_{i}) − K_{i}(0)), \end{aligned}

where ξ_{i} = M̂i^{\frac{2}{n−2}} x_{i}. By (3.30) of Lemma 3.5, w_{i}(y) is uniformly bounded in ℝ^{n}. After passing to a subsequence, we may assume that

w_{i}(y) converges to w(y) in C_{loc}^{2}(ℝ^{n}). Since β < n − 2 and M̂i^{\frac{2}{n−2}} |x_{i}| is

bounded, we have L_{i}^{−n+2} ∼ t_{i}M̂_{i}^{− \frac{2β}{n−2}} . We may assume

c = lim_{i→∞} t_{i}l_{i}^{n−2}M̂i^{\frac{−2β}{n−2}} > 0

exists. Multiplying both sides of (3.33) by ψ_{j} = \frac{∂U_{i}}{∂y_{j}}, we have by integration by parts,

\begin{aligned}∫_{B_{l_{i}}} l_{i}^{n−2}Q̃_{i}(y)Ui^{\frac{n+2}{n−2}} ψ_{j}(y)dy = & ∫_{B_{l_{i}}} w_{i}(Δψ_{j} + b̃_{i}(y)ψ_{j})dy \\ & + ∫_{∂B_{l_{i}}} (ψ_{j} \frac{∂w_{i}}{∂ν} − w_{i} \frac{∂ψ_{j}}{∂ν} )dσ. \end{aligned}

By (3.30), the boundary term = O(l_{i}^{−1}) → 0 as i → +∞, and

\begin{aligned}|Δψ_{j} + b̃_{i}(y)ψ_{j}| & ≤ |b̃_{i}(y)ψ_{j}(y)| + (n + 2)n|U1^{\frac{4}{n−2}} (y)ψ_{j}(y)| \\ & ≤ c(1 + |y|)^{−(n+2)}. \end{aligned}

Thus, by Lebseque’s convergence theorem, the right hand side converges to

∫_{ℝ^{n}} w(Δψ_{j} + n(n + 2)U1^{\frac{4}{n−2}} ψ_{j})dy = 0.

Together with (3.43), it implies

0 = lim_{i→+∞} ∫_{B_{l_{i}}} l_{i}^{n−2}Q̃_{i}(y)Ui^{\frac{n+2}{n−2}} ψ_{j}(y)dy

= c lim_{i→+∞} ∫_{B_{l_{i}}} Q(ξ_{i} + y)U1^{\frac{n+2}{n−2}} (y)\frac{∂U_{1}(y)}{∂y_{j}}dy

= \frac{(n − 2)c}{2n} ∫_{ℝ^{n}} Q(ξ + y)\frac{∂}{∂y_{j}} U1^{\frac{2n}{n−2}} (y)dy

= \frac{−(n − 2)c}{2n} ∫_{ℝ^{n}} \frac{∂}{∂y_{j}} Q(ξ + y)U1^{\frac{2n}{n−2}} (y)dy,

where U_{1} is defined in (1.4). Here, we have used the fact that ψ_{j}(y) is odd in y_{j}, and

∫_{B_{l_{i}}} (K_{i}(x_{i}) − K_{i}(0))ψ_{j}(y)Ui^{\frac{n+2}{n−2}} (y)dy = 0.

The proof of Theorem 2.2 is complete. q.e.d.

Proof of Theorem 2.1. Note that in Section 8, (2.16) is also proved when β < n + 1. This holds only for global solutions. See Lemma 8.1. Let x_{i} and M̂_{i} be the maximum point and the maximum of u_{i} defined in (2.12). We first prove the “if” part. Assume there is a constant c > 0 such that

(3.44)  L_{i}(x_{i}) ≥ cM̂i^{\frac{2}{n−2}} .

Let v_{i}(y) be the scaled solution defined in (2.13) with z_{i} = x_{i}. Obviously,

v_{i}(y) ≤ 1 for |y| ≤ M̂i^{\frac{2}{n−2}} . By Lemma 3.1, Lemma 3.5 and (3.44), there exists a small positive number δ > 0 such that v_{i}(y) ≤ c U_{1}(y) for

|y| ≤ δM̂i^{\frac{2}{n−2}} and for some c > 0. Therefore, 0 is a simple blow-up point.

To prove the “only if” part, we assume

(3.45)  lim_{i→+∞} L_{i}(x_{i})M̂_{i}^{− \frac{2}{n−2}} = 0.

Suppose that 0 is a simple blowup point. Then there exists positive constants c and δ_{0} < 1 such that

(3.46)  v_{i}(y) ≤ c U_{1}(y)

for |y| ≤ δ_{0}M̂i^{\frac{2}{n−2}} . Following the notations of Lemma 3.5, we let w_{i}(y) = v_{i}(y) − U_{i}(y) and ψ_{0}(y) = \frac{n−2}{2}U_{i}(y) + y · ▽U_{i}(y). By the gradient estimate, we have by (3.46), | ▽ v_{i}(y)| = O(|y|^{−n+1}) for |y| ≥ 1. Thus,

(3.47)  ∫_{|x|=r̂_{i}}(ψ_{0} \frac{∂w_{i}}{∂ν} − w_{i} \frac{∂ψ_{0}}{∂ν} )dσ = O(r̂_{i}^{−n+2}) = O(M̂_{i}^{−2}),

where r̂_{i} = δ_{0}M̂i^{\frac{2}{n−2}} . To estimate the first term of (3.35), we have by Lemma 3.5

\begin{aligned}∫_{Br̂_{i}} w_{i}(Δψ_{0} + b̃_{i}ψ_{0})dy & = ∫_{B_{r_{i}}} w_{i}(Δψ_{0} + b̃_{i}ψ_{0})dy \\ & + ∫_{Br̂_{i}\B_{r_{i}}} w_{i}(Δψ_{0} + b̃_{i}ψ_{0})dy, \end{aligned}

where r_{i} = δ_{0}L_{i}(x_{i}). By Theorem 2.2, we have 1 ≤ β < n − 2. Similar to (3.37), we have by the fact β < n − 2 that

|w_{i}(Δψ_{0} + b̃_{i}ψ_{0})| ≤ c r_{i}^{2(n−2)}(1 + |y|)^{−4}

for 1 ≤ r ≤ r_{i}. Hence

|∫_{B_{r_{i}}} w_{i}(Δψ_{0} + b̃_{i}ψ_{0})dy| = O(r_{i}^{−n+1}).

We note that Lemma 3.5 is crucial in the estimate above. By applying |v_{i}(y)| + |U_{i}(y)| ≤ c|y|^{−n+2} and |ψ_{0}(y)| ≤ c|y|^{−n+2} for r_{i} ≤ |y| ≤ r̂_{i},

|∫_{Br̂_{i}\B_{r_{i}}} w_{i}(Δψ_{0} + b̃_{i}ψ_{0})dy| = O(r_{i}^{−n+1}).

Together with these two estimates, we have

(3.48)  |∫_{Br̂_{i}} w_{i}(Δψ_{0} + b̃_{i}ψ_{0})dy| = O(r_{i}^{−n+1}).

By Theorem 2.2, ξ_{i} = M̂i^{\frac{2}{n−2}} x_{i} is bounded. We may assume ξ = lim_{i→+∞} ξ_{i}. Then ξ satisfies

(3.49)  ∫_{ℝ^{n}} ▽Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy = 0.

Also, the right hand side of (3.35) converges to

\begin{matrix}lim_{i→+∞} t_{i}^{−1}M̂i^{\frac{2β}{n−2}} (∫_{Br̂_{i}} Q̃_{i}Ui^{\frac{n+2}{n−2}} ψ_{0}(y)dy) \\ = − ∫_{ℝ^{n}} Q(y + ξ)U1^{\frac{n+2}{n−2}} (y)ψ_{0}(y)dy. \end{matrix} (3.50)

Recall that ψ_{0}(y) = \frac{n−2}{2}U_{1}(y) + y ▽ U_{1}(y). From integration by parts, (3.49) and y · ▽Q(y) = βQ(y), we have

(3.51)  \begin{matrix}− ∫_{ℝ^{n}} Q(y + ξ)U1^{\frac{n+2}{n−2}} (y)ψ_{0}(y)dy \\ = \frac{n − 2}{2n} ∫_{ℝ^{n}} y · ▽Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy \\ = \frac{β(n − 2)}{2n} ∫_{ℝ^{n}} Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy ≠ 0. \end{matrix}

The last term does not vanish due to (K1).

Recall L_{i}(x_{i})^{n−2} ∼ t_{i}^{−1}M̂i^{\frac{2β}{n−2}} . Putting (3.50), (3.51) and (3.48) together these estimates, we have

\begin{aligned}L_{i}(x_{i})^{2−n} & ≤ c |∫_{Br̂_{i}} Q̃_{i}Ui^{\frac{n+2}{n−2}} ψ_{0}(y)dy | \\ & ≤ c (L_{i}(x_{i})^{1−n} + M̂_{i}^{−2}), \end{aligned}

which yields a contradiction to (3.45). Therefore, the proof of Theorem 2.1 is complete. q.e.d.

4. The method of localizing blow-up points

In this section, we will employ the method of localization of blow-up points to prove Theorem 2.4 and Theorem 2.5. This technique was due to R. Schoen. In the previous work [9], we have used this method to prove the isolatedness of blow-up points. For other applications of this method, see [17], [18]. We begin with the following lemma.

Lemma 4.1. Let δ, σ and ε be small positive numbers and R > 1. Then there exist positive constants R = R(δ, σ) and C_{0} = C_{0}(δ, σ, R, ε) independent of i such that the following statements hold:

(i) If u_{i}(y_{0})|y_{0}|^{\frac{n−2}{2}} ≥ C_{0}, then there exists a local maximum point z ∈ B(y_{0}, 2δ|y_{0}|) of u_{i} such that

(4.1)  u_{i}(y_{0}) ≤ u_{i}(z)

and the rescaled function

v_{i}(y) = u_{i}(z)^{−1}u_{i}(u_{i}(z)^{− \frac{2}{n−2}} y + z)

satisfies

(4.2) \begin{cases}the origin 0 is the only local maximum point of v_{i} \\ in B(0, 4R), and |v_{i} − U_{1}|_{C^{2}(B(0,4R))} ≤ σ(4R)^{2−n}. \end{cases}

(ii) Let {z_{j}^{i}}_{j=1}^{s_{i}} denote all local maximum points of u_{i} in the ball B_{1} which satisfy u_{i}(z_{j}^{i})|z_{j}^{i}|^{\frac{n−2}{2}} ≥ C_{0} and (4.2) with z = z_{j}^{i}. Assume u_{i}(z_{1}^{i}) ≥ u_{i}(z_{2}^{i}) ⋯ ≥ u_{i}(z_{s_{i}}^{i}). Then

(a) u_{i}(y) ≤ 2C_{0}|y|^{− \frac{n−2}{2}} for y ∉ Ω_{i} where Ω_{i} = ∪_{j}B(z_{j}^{i}, 2δ|z_{j}^{i}|). Furthermore,

|z_{j}^{i} − z_{k}^{i}| ≥ 4Ru_{i}(z_{j}^{i})^{− \frac{2}{n−2}}

for j ≠ k.

(b) u_{i}(x) ≤ 2u_{i}(z_{j}^{i}) holds for x ∈ B(z_{j}^{i}, 2δ|z_{j}^{i}|) and

|z_{j}^{i}| ≤ ε|z_{k}^{i}| for j < k ≤ s_{i}. (4.3)

Lemma 4.1 can be proved by the blow-up method of Schoen and the method of moving planes, Lemma 3.1. See Lemma 4.1–Lemma 4.4 in [9]. In fact, we can prove more in Lemma 4.2 below. In the following, z_{j}^{i} is indexed by the ordering u_{i}(z_{1}^{i}) ≥ … ≥ u_{i}(z_{s_{i}}^{i}).

Lemma 4.2. Let {z_{j}^{i}}_{j=1}^{s_{i}} be the local maximum points in Lemma 4.1 and δ > 0 be a small number. Then we have the following statements if the positive constant C_{0} in Lemma 4.1 is large enough.

(i) The inequality

L_{i}(z_{j}^{i}) ≥ (δu_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|)^{\frac{n−1}{n−2}}

holds for 1 ≤ j ≤ s_{i}.

(ii) Let

L_{i}^{∗}(z_{j}^{i}) = min(L_{i}(z_{j}^{i}), u_{i}(z_{j}^{i})^{\frac{2}{n−2}} )

and

D_{j}^{i} = \{y : |y − z_{j}^{i}| ≤ cL_{i}^{∗}(z_{j}^{i})u_{i}(z_{j}^{i})^{− \frac{2}{n−2}} \}

with c small. Then

z_{k}^{i} ∉ D_{j}^{i}

when k > j.

Proof. We follow notations in Section 3. Let v_{i} be defined in (2.13) with z_{i} = z_{j}^{i} and U_{i} be the solution to (3.15) with K_{0} = K_{i}(z_{i}).

We may assume C_{0} is very large. If 1 < β < n, by (2.15) and u_{i}(z_{j}^{i})|z_{j}^{i}|^{\frac{n−2}{2}} ≥ C_{0}, we have

u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|^{1−β} = (u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|)^{1−β}u_{i}(z_{j}^{i})^{\frac{2β}{n−2}} < u_{i}(z_{j}^{i})^{\frac{2β}{n−2}}

and

(4.4)  L_{i}(z_{j}^{i}) = (t_{i}^{−1}u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|^{1−β})^{\frac{1}{n−2}} .

If β ≥ n and L_{i}(z_{j}^{i}) ≠ (t_{i}^{−1}u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|^{1−β})^{\frac{1}{n−2}} , then by (2.15),

L_{i}(z_{j}^{i}) = (t_{i}^{−1}u_{i}(z_{j}^{i})^{\frac{2β̂}{n−2}} )^{\frac{1}{n−2}} ≥ (u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|)^{\frac{n−1}{n−2}}

for large i since β̂ > n − 1, that is, (i) holds in this case. Hence in order to prove (i), we may assume L_{i}(z_{j}^{i}) = (t_{i}^{−1}u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|^{1−β})^{\frac{1}{n−2}} and 1 < β < n.

Let δ be small enough, M_{i} = u_{i}(z_{j}^{i}), L_{i} = L_{i}(z_{j}^{i}) and

r_{i} = δ min(L_{i}, Mi^{\frac{2}{n−2}} |z_{j}^{i}|).

. Then by (b) of part (ii) in Lemma 4.1,

v_{i}(y) ≤ 2 for |y| ≤ r_{i}.

By Lemma 3.1, Lemma 3.4 and Lemma 3.5, we have

v_{i}(x) ≤ c U_{i}(x)

and

|v_{i}(x) − U_{i}(x)| ≤ c r_{i}^{−2+n}

for |x| ≤ r_{i}, where c is a constant independent of δ and i. For the sake of simplicity, δ always denotes a small positive number, but could change from line to line. Assume ∇K̂(z_{j}^{i}) is in the direction e_{1} = (1, 0, ⋯, 0).

Let ψ_{1} = \frac{∂U_{i}}{∂y_{1}} . By (3.32) of Lemma 3.5,

|∫_{|x|≤r_{i}} Q̃_{i}Ui^{\frac{n+2}{n−2}} ψ_{1} dx| ≤ c_{1}r_{i}^{−n+1}. (4.5)

By (3.40), we have

−Q̃_{i}(y) = t_{i}M_{i}^{− \frac{2}{n−2}} | ▽ K̂(z_{j}^{i})|(y_{1} + o(1)|y|)

for |y| ≤ r_{i}, where o(1) could be arbitrarily small if δ is small. Since ψ_{1}(y)y_{1} ≥ 0, we have

(4.6)  |∫_{B_{r_{i}}} Q̃_{i}Ui^{\frac{n+2}{n−2}} ψ_{1}dy| ≥ c_{2}t_{i}M_{i}^{− \frac{2}{n−2}} |z_{j}^{i}|^{β−1}

for some c_{2} > 0. Since we assume L_{i}(z_{j}^{i}) = (t_{i}^{−1}u_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|^{1−β})^{\frac{1}{n−2}} , it follows from (4.5) and (4.6) that

(4.7)  L_{i}^{−n+2} ≤ c_{3}r_{i}^{−n+1}.

Since r_{i} ≤ L_{i}, C_{0} is large and L_{i} → +∞ as i → +∞, we conclude r_{i}/L_{i} is small from (4.7). Thus r_{i} = δu_{i}(z_{j}^{i})^{\frac{2}{n−2}} |z_{j}^{i}|. Since both c_{1} and c_{2} are independent of δ and i, part (i) of Lemma 4.2 follows from (4.7) if δ is chosen to be small enough.

We prove (ii) by contradiction. Assume that after passing to a sequence, there exists j_{i} < k_{i} such that z_{k_{i}}^{i} ∈ D_{j_{i}}^{i} and both u_{i}(z_{j_{i}}^{i})|z_{j_{i}}^{i}|^{\frac{n−2}{2}} and u_{i}(z_{k_{i}}^{i})|z_{k_{i}}^{i}|^{\frac{n−2}{2}} tend to +∞. For simplicity of notations, we let z_{i} = z_{j_{i}}^{i} and w_{i} = z_{k_{i}}^{i}. Recall that u_{i} satisfies

(4.8)  u_{i}(w_{i}) ≤ u_{i}(z_{i}).

Let M_{i} = u_{i}(z_{i}) and v_{i}(y) be the solution in (2.13) scaled with respect

to the local maximum point z_{i}. Since Mi^{\frac{2}{n−2}} |z_{i}| → +∞ and (4.2) holds, we have for any σ > 0, by Lemma 3.1

(4.9)  min_{|y|≤r} v_{i}(y) ≤ (1 + 2σ)U_{1}(r)

if i is large and 0 ≤ r ≤ 4d_{0}L_{i}^{∗}(z_{i}) with some d_{0} = d_{0}(σ) > 0. Let

l_{i} = d_{0}L_{i}^{∗}(z_{i}). Applying Lemma 3.5 with an empty set E, l_{0} = Mi^{\frac{2}{n−2}} and l = l_{i}, there is a constant c_{1} independent of σ and i

(4.10)  ∫_{R≤|y|≤l_{i}} vi^{\frac{n+2}{n−2}} (y)dy ≤ c_{1}σ,

provided that d_{0} < σ^{\frac{1}{2}} and R ≥ σ^{− \frac{1}{2}} .

Set

B_{i} = \{x | |x − w_{i}| ≤ u_{i}(w_{i})^{− \frac{2}{n−2}} \}

and

B̂_{i} = \{y | M_{i}^{− \frac{2}{n−2}} (y + z_{i}) ∈ B_{i}\}.

By (ii) of Lemma 4.1 and (4.8),

4R ≤ u_{i}(w_{i})^{\frac{2}{n−2}} |z_{i} − w_{i}| ≤ Mi^{\frac{2}{n−2}} |z_{i} − w_{i}| ≤ cL_{i}^{∗}(z_{i})

because w_{i} ∈ D_{i}. By (ii) of Lemma 4.1, we have |z_{i}| = o(1)|w_{i}| and

\begin{matrix}Mi^{\frac{2}{n−2}} u_{i}(w_{i})^{− \frac{2}{n−2}} << Mi^{\frac{2}{n−2}} |w_{i}| \\ = (1 + o(1))Mi^{\frac{2}{n−2}} |z_{i} − w_{i}| ≤ cL_{i}^{∗}(z_{i}). \end{matrix}

Thus, B_{i} ⊆ 2D_{i}.

Since u_{i}(x) ≤ u_{i}(w_{i}) ≤ u_{i}(z_{i}) for x ∈ B_{i}, we have v_{i}(y) ≤ 1 for y ∈ B̂_{i}. Since by Lemma 4.1, 0 is the unique local maximum of v_{i}(y) for |y| ≤ 4R, we have B̂_{i} ⊆ {y | R ≤ |y| ≤ l_{i}} if the constant c in D_{j}^{i} is small. Again by (i) of Lemma 4.1, we have for some constant c_{2} > 0,

\begin{aligned}0 < c_{2} & ≤ ∫_{B_{i}} ui^{\frac{2n}{n−2}} (x)dx = ∫B̂_{i} vi^{\frac{2n}{n−2}} dy ≤ ∫B̂_{i} vi^{\frac{n+2}{n−2}} (y)dy \\ & ≤ ∫_{R≤|y|≤l_{i}} vi^{\frac{n+2}{n−2}} (y)dy ≤ c_{1}σ, \end{aligned}

which yields a contradiction if σ is small enough. Therefore, (ii) is proved. q.e.d.

Proof of Theorem 2.4. Let L_{i} = L_{i}(x_{i}) and M̂_{i} = u_{i}(x_{i}). Suppose

that L_{i}M̂_{i}^{− \frac{2}{n−2}} → +∞, then by Theorem 2.1, 0 is a simple blowup point and u_{i} loses the energy of one bubble at 0. Therefore, we suppose that

lim_{i→+∞} L_{i}M̂_{i}^{− \frac{2}{n−2}} < +∞.

By Theorem 2.2, M̂i^{\frac{2}{n−2}} |x_{i}|^{\frac{n−2}{2}} is bounded, β < n − 2 and ξ =

lim_{i→+∞} M̂i^{\frac{2}{n−2}} x_{i} satisfies (2.16). From the definition (2.15) of L_{i}, we have

L_{i}(x_{i}) ∼ (t_{i}^{−1}M̂i^{\frac{2β}{n−2}} )^{\frac{1}{n−2}} . Applying Lemma 3.1 and Lemma 3.5, u_{i} satisfies

(4.11)  c_{1}M̂_{i}^{−1}|x|^{2−n} ≤ u_{i}(x) ≤ c_{2}M̂_{i}^{−1}|x|^{2−n}

for

M̂_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ δ(t_{i}^{−1}M̂_{i}^{\frac{2β}{n−2} −2})^{\frac{1}{n−2}}

with a small δ > 0. Let r_{i} = δ(t_{i}^{−1}M̂_{i}^{\frac{2β}{n−2} −2})^{\frac{1}{n−2}} . Then, we have

(4.12)  min_{|x|=r_{i}} u_{i}(x) ∼ t_{i}M̂_{i}^{1− \frac{2β}{n−2}} .

Now suppose

lim_{i→+∞} \begin{matrix}sup \\ B & 2 \end{matrix}(u_{i}(x)|x|^{\frac{n−2}{2}} ) = +∞.

Let z_{i} = z_{1}^{i}, where z_{1}^{i} is the local maximum point in Lemma 4.2. Let

M_{i} = u_{i}(z_{i}). Since Mi^{\frac{2}{n−2}} |z_{i}| ≥ C_{0} is very large, we have

(4.13)  L_{i}(z_{i}) ≤ (t_{i}^{−1}Mi^{\frac{2β}{n−2}} )^{\frac{1}{n−2}} ,

and by (i) of Lemma 4.2,

(4.14)  Mi^{\frac{2}{n−2}} |z_{i}| << L_{i}(z_{i}).

Since u_{i}(x) is a positive superharmonic function, there exists a small constant c > 0 such that

(4.15)  u_{i}(z_{i} + x) ≥ c M_{i}^{−1}(|x|^{2−n} − (3/2)^{2−n})

for M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ \frac{3}{2}. In particular, we have

(4.16)  min_{|x−z_{i}|≤min(r̂_{i},1)} u_{i}(x) ≥ cM_{i}^{−1}r̂_{i}^{2−n} ≥ ct_{i}M_{i}^{1− \frac{2β}{n−2}} ,

where r̂_{i} = (t_{i}^{−1}M_{i}^{\frac{2β}{n−2} −2})^{\frac{1}{n−2}} . Since u_{i}(x) has the only maximum point x_{i} in the region {x | |x| ≤ r_{i}}, we have by (4.14)

r_{i} ≤ |z_{i}| << r̂_{i},

namely, the ball B_{r_{i}}(0) is contained inside of the ball B(z_{i}, r̂_{i}). Hence, if r̂_{i} is bounded, by (4.12), (4.16) and the maximum principle, we have

\begin{aligned}t_{i}M̂_{i}^{1− \frac{2β}{n−2}} ∼ min_{|x|=r_{i}} u_{i} & ≥ min_{|x−z_{i}|≤r̂_{i}} u_{i} \\ & ≥ ct_{i}M_{i}^{1− \frac{2β}{n−2}} . \end{aligned}

First we consider the case when β > \frac{n−2}{2}. Since β > \frac{n−2}{2} and M̂_{i} is the maximum of u_{i}, it implies M̂_{i} ∼ M_{i}. Hence, the function v_{i}(y) rescaled with respect to the center z_{i} satisfies

v_{i}(y) ≤ c

for some constant c > 0 and |y| ≤ Mi^{\frac{2}{n−2}} . Thus, v_{i}(y) ∼ U_{1}(y) for |y| ≤ δL_{i}(z_{i}) by Lemma 3.4. Particularly, we have

M̂_{i} ∼ M_{i}U_{1}(|z_{i}|Mi^{\frac{2}{n−2}} ) = M_{i}(M_{i}|z_{i}|^{\frac{n−2}{2}} )^{−2} = o(1)M_{i},

which obviously yields a contradiction.

For the case β = \frac{n−2}{2}, we have by (4.14), (4.15) and the maximum principle,

\begin{aligned}t_{i} & = t_{i}M̂_{i}^{1− \frac{2β}{n−2}} ∼ min_{|x|=r_{i}} u_{i} \\ & ≥ min_{|x−z_{i}|≤2|z_{i}|} u_{i} ≥ cM_{i}^{−1}|z_{i}|^{2−n}, \end{aligned}

which implies

r̂_{i} = (t_{i}^{−1}M_{i}^{−1})^{\frac{1}{n−2}} ≤ c|z_{i}|.

But by (4.14), |z_{i}| << r̂_{i} for large i. Thus, we obtain a contradiction and then (2.18) is proved.

Once that (2.18) is established, (1.8) follows from Lemma 5.2 of Section 5. Also, from (2.18), the energy outside the region, where u_{i} is not simple, tends to zero. Therefore (2.17) is obtained, and then Theorem 2.4 is proved. q.e.d.

Proof of Theorem 2.5. Suppose that u_{i} satisfies

lim_{i→+∞} \begin{matrix}sup \\ B & 2 \end{matrix}(u_{i}(x)|x|^{\frac{n−2}{2}} ) = +∞.

Assume that 0 is not a simple blowup point. Then β < n − 2 by Corollary 2.3. Let δ, R, C_{0} and the local maximum points {z_{j}^{i}}_{j=1}^{s_{i}} of u_{i} satisfy the assumptions of Lemma 4.1 and Lemma 4.2. We will prove s_{i} = 1 for i large.

Let z_{i} = z_{1}^{i}, L_{i} = L_{i}(z_{i}), M_{i} = u_{i}(z_{i}) and v_{i}(y) be the scaled function defined in (2.13). We claim

(4.17)  lim_{i→+∞} L_{i}M_{i}^{− \frac{2}{n−2}} = +∞.

We prove (4.17) by contradiction. Suppose

lim_{i→+∞} L_{i}M_{i}^{− \frac{2}{n−2}} < +∞.

Then for any small number σ > 0, by Lemma 3.1 and Lemma 3.5, there is a small positive number d_{0} = d_{0}(σ) such that

(4.18)  min_{|y|=r} v_{i}(y) ≤ (1 + σ)U_{1}(r)

for 0 ≤ r ≤ d_{0}L_{i}, and

(4.19) ∫_{R≤|x|≤d_{0}L_{i}} vi^{\frac{n+2}{n−2}} (y)dy ≤ c_{1}(σ + R^{−2} + (\frac{d_{0}L_{i}}{L_{i}})^{n−2}) ≡ c_{1}σ̃,

where R is very large and c_{1} is a positive constant independent of σ and

i. Note that L_{i}^{∗}(d_{0}) = min(d_{0}L_{i}, Mi^{\frac{2}{n−2}} ) = d_{0}L_{i} due to the assumption

L_{i} ≤ c Mi^{\frac{2}{n−2}} .

Let Ω_{i} be the set in Lemma 4.1. Let σ be a small positive number, which will be chosen later. For |x| ≥ δ|z_{i}| and z ∉ Ω_{i}, we have by Lemma 4.1,

u_{i}(x)|x|^{\frac{n−2}{2}} ≤ 2C_{0} ≤ 2M_{i}|z_{i}|^{\frac{n−2}{2}}

for i large, which implies that

u_{i}(x) ≤ c M_{i}

for some c = c(δ) > 0. If x ∈ Ω_{i}, then for some j,

u_{i}(x) ≤ 2u_{i}(z_{j}^{i}) ≤ 2u_{i}(z_{i}).

Hence, there is c_{1} = c_{1}(δ) > 0 such that

(4.20)  u_{i}(x) ≤ c_{1}M_{i}

for |x| ≥ δ|z_{i}|.

If σ and d_{0} are small and R is large, then by (4.19) and (4.20), Lemma 3.5 can be applied to obtain the Harnack inequality for v_{i}(y) on each sphere |y| = r ≤ d_{0}L_{i} if the annulus \{y | \frac{r}{2} ≤ |y| ≤ 2r\} does not

intersect with the set \{y | |y + Mi^{\frac{2}{n−2}} z_{i}| ≤ δMi^{\frac{2}{n−2}} |z_{i}|\}. In particular,

(4.21)  v_{i}(y) ≤ c U_{i}(y),

holds for 2Mi^{\frac{2}{n−2}} |z_{i}| ≤ |y| ≤ d_{0}L_{i}, where c is a constant independent of i and δ. Let

(4.22)  r_{i} = d_{0}L_{i}M_{i}^{− \frac{2}{n−2}} .

Going back to the function u_{i}, (4.21) implies

(4.23)  u_{i}(z_{i} + x) + |x|| ▽ u(z_{i} + x)| ≤ cM_{i}^{−1}|x|^{2−n}

for 2δ|z_{i}| ≤ |x| ≤ r_{i}.

Let e_{i} = | ▽ K̂(z_{i})|^{−1} ▽ K̂(z_{i}) and e = lim_{i→+∞} e_{i}. Applying the

Pohozaev identity,

\begin{matrix}\frac{n − 2}{2n} ∫_{B(z_{i},r_{i})}⟨e, ▽K_{i}⟩ui^{\frac{2n}{n−2}} dx \\ = ∫_{∂B(z_{i},r_{i})} [⟨e, ▽u_{i}⟩\frac{∂u_{i}}{∂ν} − ⟨e, ν⟩\frac{| ▽ u_{i}|^{2}}{2} \\ +\frac{n − 2}{2n}⟨e, ν⟩K_{i}ui^{\frac{2n}{n−2}} ] dσ. \end{matrix} (4.24)

By (4.23), the right hand side of (4.24) is dominated by r_{i}^{−n+1}M_{i}^{−2}. To find a lower bound, we decompose B(z_{i}, r_{i}) into four parts: A_{1} = {x |

|x − z_{i}| ≤ M_{i}^{− \frac{2}{n−2}} R_{0}\}, A_{2} = \{x | |x| ≤ 3δ|z_{i}|\}, A_{3} = \{x | 3δ|z_{i}| ≤ |x| ≤

2|z_{i}|, |z_{i} − x| ≥ M_{i}^{− \frac{2}{n−2}} R_{0}\} and A_{4} = \{x | 2|z_{i}| ≤ |x| ≤ r_{i}\}, where R_{0} is a positive number.

For x ∈ A_{2}, we have by Lemma 4.1

u_{i}(x) ≤ 2C_{0}|x|^{− \frac{n−2}{2}} .

Then

(4.25)  ∫_{A_{2}} | ▽ K_{i}|ui^{\frac{2n}{n−2}} (x)dx ≤ c_{2}(δ|z_{i}|)^{β−1}t_{i}.

For x ∈ A_{3}, we have

∫_{A_{3}} | ▽ K_{i}|ui^{\frac{2n}{n−2}} (x)dx ≤ c t_{i}|z_{i}|^{β−1} ∫_{A_{3}} ui^{\frac{2n}{n−2}} (x)dx.

By (4.19) and v_{i}(y) ≤ c_{1}(δ), we have

\begin{aligned}∫_{A_{3}} | ▽ K_{i}|ui^{\frac{2n}{n−2}} dx & ≤ c t_{i}|z_{i}|^{β−1} ∫_{R_{0}≤|y|≤d_{0}L_{i}} vi^{\frac{2n}{n−2}} (y)dy \\ & ≤ c t_{i}|z_{i}|^{β−1}(c_{2}(δ)σ̃ + R_{0}^{−n}), \end{aligned} (4.26)

where the estimate,

∫_{R_{0}≤|y|≤R} vi^{\frac{2n}{n−2}} (y)dy ≤ c ∫_{R_{0}≤|y|≤R} |y|^{−2n}dy ≤ cR_{0}^{−n}

is used.

For x ∈ A_{4}, we apply (4.21)

u_{i}(x) ≤ cM_{i}^{−1}|x|^{2−n}.

Hence

\begin{matrix}∫_{A_{4}} | ▽ K_{i}|ui^{\frac{2n}{n−2}} (x)dx \\ ≤ c t_{i}M_{i}^{− \frac{2n}{n−2}} ∫_{A_{4}} |x|^{−2n+β−1}dx \\ ≤ c t_{i}M_{i}^{− \frac{2n}{n−2}} |z_{i}|^{−(n+1)+β} \\ = c (t_{i}|z_{i}|^{β−1})(M_{i}|z_{i}|^{\frac{n−2}{2}} )^{\frac{−2n}{n−2}} . \end{matrix} (4.27)

For x ∈ A_{1}, we have a positive c_{0} > 0 such that

(4.28)  ∫_{A_{1}}⟨e, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx ≥ c_{0}(t_{i}|z_{i}|^{β−1}).

If we chose σ, d_{0} to be small and R_{0} to be large, then by (4.25) ∼ (4.28), the left hand side of (4.24) has

(4.29)  ∫_{B(z_{i},r_{i})}⟨e, ▽K_{i}⟩ui^{\frac{2n}{n−2}} dx ≥ (c_{0}/2)t_{i}|z_{i}|^{β−1}

when i is large. Combining the estimates of both sides of (4.24), one has

t_{i}|z_{i}|^{β−1} ≤ c r_{i}^{−n+1}M_{i}^{−2} = c_{1}L_{i}^{−n+1}Mi^{\frac{2}{n−2}} ,

namely,

L_{i}(z_{i})^{−n+2} ≤ c_{1}L_{i}(z_{i})^{−n+1},

which obviously yields a contradiction. Hence (4.17) is proved.

If s_{i} > 1, then by (4.17), L_{i}^{∗}(z_{1}^{i})u_{i}(z_{1}^{i})^{− \frac{2}{n−2}} ≥ 1 with L_{i}^{∗}(z_{1}^{i}) = min(L_{i}(z_{i}), u_{i}(z_{i})^{\frac{2}{n−2}} ) defined in Lemma 4.2. Since z_{2}^{i} ∉ D_{1}^{i}, we have |z_{2}^{i}| ≥ c_{2} for some c_{2} > 0. On the other hand, by (2.11) and the Harnack inequality, we have u_{i} converges to 0 uniformly on any compact subset of B_{1} \ {0}. Thus,

u_{i}(z_{2}^{i}) ≤ max_{|x|=c_{2}} u_{i}(x) → 0 as i → +∞,

which yields a contradiction again. Therefore, s_{i} = 1. We note that x_{i} ≠ z_{i} because 0 is not a simple blowup point. The other conclusions of Theorem 2.5 follow from (4.17) and the lemmas in Section 3. Hence, the proof of Theorem 2.5 completely finished. q.e.d.

5. An ODE approach

In this sectin, we consider a sequence of solution u_{i} of (2.11) such that

sup_{|x|≤1}(u_{i}(x)|x|^{\frac{n−2}{2}} ) ≤ c_{1} and u_{i}(x) converges

to 0 in C_{loc}^{2}(B_{1}\{0}).

(5.1)

From (5.1) and the theory of elliptic equations, it is easy to see

max_{|x|=r} u_{i}(x) ≤ c min_{|x|=r} u_{i}(x)

for 0 ≤ r ≤ \frac{1}{2} and some c > 0 depending on c_{1} only. Let u_{i}(r), w_{i}(s), s_{i}, M_{i} and L_{i} are defined in (2.26) ∼ (2.31), respectively. By (5.1), w_{i}(s) ≤ c_{1} for s ≤ 0. Throughout this section, we set

(5.2)  R_{i} = L_{i}^{γ} and γ = \frac{1}{1 − \frac{2β}{n−2}}

By a straightforward computation, w_{i} satisfies

(5.3)  w′′_{i} − (\frac{n − 2}{2})^{2}w_{i} + K_{i}(s)wi^{\frac{n+2}{n−2}} = 0 for s ≤ 0,

where

K_{i}(s) = |∂B_{e^{s}}(0)|^{−1}w_{i}^{− \frac{n+2}{n−2}} (s) ∫_{|x|=e^{s}} K_{i}(x) (u(x)|x|^{\frac{n−2}{2}} )^{\frac{n+2}{n−2}} dσ

and B_{e^{s}}(0) is the ball with radius e^{s} and center 0. Since we assume K_{i} is bounded between two positive constants, by (5.1), there are â and b̂ such that K_{i}(s) satisfies

(5.4)  0 < â ≤ K_{i}(s) ≤ b̂.

From (5.3) and (5.4), there is a constant c_{2} > 0 such that if s is a local maximum point of w_{i}, then

(5.5)  w_{i}(s) ≥ c_{2} > 0.

In particular, we have w_{i}(s_{i}) ≥ c_{2} > 0. Since u_{i}(x) converges to zero in C_{loc}^{2}(B_{1}\{0}), s_{i} → −∞ as i → +∞. Thus, we have by (5.5),

(5.6)  lim_{i→+∞} M_{i} = +∞.

We can obtain some basic estimates for w_{i} as in the following. For the proof, see [9]. Let w_{i} be denoted by w.

Lemma 5.1. There is a small number ϵ_{0} > 0 and large M such that the following statements hold:

(i) Suppose that w(s) is nonincreasing in (s_{o}, s_{1}) with w(s_{o}) ≤ ϵ_{0}. Then there exists a constant c depending on â and b̂ only such that

(5.7)  s_{1} − s_{o} ≤ \frac{2}{n − 2} log \frac{w(s_{o})}{w(s_{1})} + c

holds. Futhermore, if s_{1} is a local minimum point of w, then

(5.8)  s_{1} − s_{o} ≥ \frac{2}{n − 2} log \frac{w(s_{o})}{w(s_{1})}.

(ii) Suppose that w(s) is nondecreasing in (s_{1}, s_{2}) with w(s_{2}) ≤ ϵ_{0}. Then there exists a constant c depending on â and b̂ only such that

(5.9)  s_{2} − s_{1} ≤ \frac{2}{n − 2} log \frac{w(s_{2})}{w(s_{1})} + c

holds. Futhermore, if s_{1} is a local minimum point of w, then

(5.10)  s_{2} − s_{1} ≥ \frac{2}{n − 2} log \frac{w(s_{2})}{w(s_{1})}.

Proof Theorem 2.7. The proof of Theorem 2.7 is very long. So, we devide it into two steps. The first step is to estimate u_{i} via Lemma 5.1, and the second step can refine the estimate further by using comparison functions. First, we want to prove

Step 1. There is a constant c such that

(5.11)  u_{i}(x) ≤ c (t_{i}M_{i}^{−1})^{γ}|x|^{−n+2}

for R_{i}^{−2}M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ R_{i}^{−1}M_{i}^{− \frac{2}{n−2}} , and γ = (1 − \frac{2β}{n−2})^{−1},

(5.12)  u_{i}(x) ≤ c M_{i}

for R_{i}^{−1}M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ M_{i}^{− \frac{2}{n−2}} ,

(5.13)  u_{i}(x) ≤ cM_{i}^{−1}|x|^{−n+2} for M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ 1

if L_{i}M_{i}^{− \frac{2}{n−2}} ≥ c_{1} > 0, and

(5.14) u_{i}(x) \{ \begin{matrix}≤ c M_{i}^{−1}|x|^{−n+2} for M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ L_{i}M_{i}^{− \frac{2}{n−2}} \\ ≤ c M_{i}^{−1}L_{i}^{−n+2} for L_{i}M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ 1, \end{matrix}

provided that lim_{i→+∞} L_{i}Mi^{\frac{−2}{n−2}} = 0.

Recall w_{i}(s) = u_{i}(r)r^{\frac{n−2}{2}} with s = log r ≤ 0. Let ŝ_{i} be a local

maximum point of w_{i}. By (5.5), w_{i}(ŝ_{i}) ≥ c > 0. Set û_{i}(x) = r̂i^{\frac{n−2}{2}} u_{i}(r̂_{i}x) with r̂_{i} = eŝ_{i}. Then û_{i}(x) ≤ c|x|^{\frac{2−n}{2}} for 0 ≤ |x| ≤ r̂_{i}^{−1}. By passing to a subsequence, û_{i}(x) converges to Û(x) in C_{loc}^{2}(ℝ^{n}\{0}). In Lemma 5.2 (below), we will show that Û(x) = [λ̂(λ̂^{2} + |x − q̂|^{2})^{−1}]^{\frac{n−2}{2}} for some λ̂ > 0 and q̂ ∈ ℝ^{n}. A direct computations show that \overline{Û(r)}r^{\frac{n−2}{2}} has a

unique critical point at r = \sqrt{λ̂^{2} + |q|^{2}}, which is also nondegenerate. From here, we deduce that for each large i, w_{i}(s) has a sequence of local maximum point s_{j,i} and local minimum point \underline{s}_{j,i} for j = 1, 2, …, N(i). Such that the following holds:

s_{j,i} < \underline{s}_{j,i} < s_{j+1,i} with s_{N(i)+1,i} = s_{i}, w(s) is decreasing

for s ∈ (s_{j,i}, \underline{s}_{j,i}) and w(s) is increasing for s ∈ (\underline{s}_{j,i}, s_{j+1,i})

(5.15)

for 1 ≤ j ≤ N(i). Furthermore, w(\underline{s}_{j,i}) → 0 as i → +∞ for j = 1, 2, …, N(i), and,

s_{j+1,i} − \underline{s}_{j,i} and \underline{s}_{j,i} − s_{j,i} → +∞ as i → +∞
for any j = 1, 2, …, N(i). Consequently, M_{j,i}/M_{j+1,i} → 0
as i → +∞ for y ∈ {1, 2, …, N(i)}.

(5.16)

Note that N(i) ≥ 1 due to the assumption that u_{i} loses the energy of more than one bubble. For j = 1, 2, …, N(i), we set û_{i}(x) =

rj,i^{\frac{n−2}{2}} u_{i}(r_{j,i}x) with r_{j,i} = e^{s_{j,i}} and Û to be the limit of û_{i} in C_{loc}^{2}(ℝ^{n}\\{0\}).

Then we have

Lemma 5.2. Let Û be described as above. Then

(5.17)  Û(x) = (\frac{λ̂}{λ̂^{2} + |x − q̂|^{2}} )^{\frac{n−2}{2}} ,

where

(5.18)  1 = λ̂^{2} + |q̂|^{2}.

Furthermore, if set ξ_{0} = \sqrt{λ̂}q̂, then ξ_{0} satisfies

(5.19)  ∫_{ℝ^{n}} ▽Q(ξ_{0} + y)U1^{\frac{2n}{n−2}} (y)dy = 0 and

(5.20)  ∫_{ℝ^{n}} Q(ξ_{0} + y)U1^{\frac{2n}{n−2}} (y)dy < 0.

The proof of Lemma 5.2 will be given at the end of this section. Now, we go back to the proof of Step 1. By the remark above, we denote ŝ_{i} and \underline{s}_{i} to be the local maximum point s_{N(i),i} and local minimum point \underline{s}_{N(i),i}, respectively. Since w_{i}(\underline{s}_{i}) → 0 as i → +∞, there are ŝ_{i} < a_{i} < \underline{s}_{i} < b_{i} < s_{i} such that w_{i}(a_{i}) = w_{i}(b_{i}) = ϵ_{0}, where ϵ_{0} is the small positive number in Lemma 5.1. By a simple scaling argument,

(5.21)  s_{i} − b_{i} ≤ c_{3} = c_{3} (ϵ_{0})

for some constant c_{3} independent of i. By Lemma 5.1,

\begin{aligned}\frac{2}{n − 2} log \frac{ϵ_{0}}{w_{i}(\underline{s}_{i})} & ≤ \underline{s}_{i} − a_{i}, b_{i} − \underline{s}_{i} \\ & ≤ \frac{2}{n − 2} log \frac{ϵ_{0}}{w_{i}(\underline{s}_{i})} + c. \end{aligned} (5.22)

To obtain some estimate for \underline{s}_{i} − a_{i} and b_{i} − \underline{s}_{i}, we need to find upper and lower bounds for w_{i}(\underline{s}_{i}). First, we show that

(5.23)  ( min_{|x|=r_{i}} u_{i})^{−1} max_{|x|=r_{i}} u_{i} → 1 and r_{i} = e^{\underline{s}^{i}}.

uniformly as i → ∞. To see it, let x̂_{i} be any sequence of points with |x̂_{i}| = r_{i}. Let h_{i}(η) = u_{i}(x̂_{i})^{−1}u_{i}(r_{i}η). Since w_{i}(\underline{s}_{i}) → 0 as i → +∞, after passing to a subsequence, h_{i}(η) converges to h(η) in C_{loc}^{2}(ℝ^{n}\{0}) and satisfies

(5.24)  Δh(η) = 0 in ℝ^{n}\{0}.

Let h(r) be the spherical average of h. Since \underline{s}_{i} is a local minimum point of w_{i}, we have

(5.25)  \frac{d}{dr} (h(r)r^{\frac{n−2}{2}} ) = 0 at r = 1.

By the Liouville Theorem, (5.25) implies

(5.26)  \{ \begin{matrix}h(η) = a|η|^{2−n} + b, \\ a = b > 0 \end{matrix}

Clearly, from it we obtain (5.23) and

(5.27)  | ▽ u_{i}|(x) = −u′_{i}(r_{i})(1 + o(1))

for |x| = r_{i} as i → +∞. By (5.23) and (5.27), the Pohozaev identity implies

\begin{matrix}P (r_{i}, u_{i}) = & |S^{n−1}|\{\frac{1}{2}w′_{i}^{2}(\underline{s}_{i}) − \frac{1}{2}(\frac{n − 2}{2})^{2}w_{i}^{2}(\underline{s}_{i})\} \\ + o(1)(w′_{i}^{2}(\underline{s}_{i}) + w_{i}^{2}(\underline{s}_{i})), \end{matrix} (5.28)

where

(5.29)  P (r; u_{i}) = \frac{n − 2}{2n} ∫_{|x|≤r}(x · ▽K_{i})ui^{\frac{2n}{n−2}} (x)dx.

Hence,

\begin{matrix}(1 + o(1))w_{i}^{2}(\underline{s}_{i}) = & − c_{n}P_{n}(r_{i}, u_{i}) \\ ≤ & c_{n}\{ ∫_{e^{a_{i}}≤|x|≤r_{i}} |x|| ▽ K(x)|ui^{\frac{2n}{n−2}} dx \\ + ∫_{|x|≤e^{a_{i}}} |x|| ▽ K(x)|ui^{\frac{2n}{n−2}} dx\} \\ ≡ & I_{1} + I_{2}. \end{matrix} (5.30)

Since | ▽ K_{i}(x)| ≤ c|x|^{β−1}, by (5.1),

|I_{2}| ≤ c t_{i} exp(βa_{i}).

By Lemma 5.1, we have for a_{i} ≤ s ≤ \underline{s}_{i}

c w_{i}(\underline{s}_{i}) exp[\frac{n − 2}{2}(\underline{s}_{i} − s)] ≤ w_{i}(s) ≤ w_{i}(\underline{s}_{i}) exp[\frac{n − 2}{2}(\underline{s}_{i} − s)].

118 chiun-chuan chen & chang-shou lin

Therefore

|I_{1}| ≤ c t_{i}wi^{\frac{2n}{n−2}} (\underline{s}_{i}) exp(n\underline{s}_{i}) ∫_{a_{i}}^{\underline{s}_{i}} exp[(−n + β)s] ds

≤ c t_{i}wi^{\frac{2n}{n−2}} (\underline{s}_{i}) exp(n\underline{s}_{i}) exp[(−n + β)a_{i}].

By Lemma 5.1 again,

\begin{matrix}w_{i}(a_{i}) exp (\frac{(n − 2)(a_{i} − \underline{s}_{i})}{2}) ≤ & w_{i}(\underline{s}_{i}) \\ ≤ & c w_{i}(a_{i}) exp (\frac{(n − 2)(a_{i} − \underline{s}_{i})}{2}) . \end{matrix}

These estimates imply

(5.31)  |I_{1}| ≤ c t_{i}ϵ0^{\frac{2n}{n−2}} exp(βa_{i}).

Hence, we obtain

(5.32)  w_{i}(\underline{s}_{i}) ≤ c ti^{\frac{1}{2}} exp (\frac{βa_{i}}{2} ) .

Together with (5.22), it implies

(5.33)  \underline{s}_{i} − a_{i} ≥ \frac{1}{n − 2}(− log t_{i} − βa_{i}) − c(ε_{0}).

To obtain a lower bound for w_{i}(\underline{s}_{i}), we recall ŝ_{i} < a_{i} < s_{i} to be the next local maximum point of w_{i}. Set

(5.34) û_{i}(y) = M̂_{i}^{−1}u_{i}(M̂_{i}^{− \frac{2}{n−2}} y),

where M̂_{i} = exp(−\frac{n−2}{2} ŝ_{i}). By Lemma 5.2, by passing to a subsequence, û_{i}(y) converges to Û(y) in C_{loc}^{2}(ℝ^{n}\{0}) with

Û(y) = (\frac{λ̂}{λ̂ + |y − q̂|^{2}} )^{\frac{n−2}{2}} .

prescribing scalar curvature on S^{n} 119

Let r̂_{i} = δ exp ŝ_{i} for a small δ > 0. By (5.28) and (5.31),

\begin{matrix}w_{i}^{2}(\underline{s}_{i}) ≥ & |∫_{r̂_{i}≤|x|≤e^{a_{i}}} ⟨x, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx| \\ − c \{t_{i} ∫_{|x|≤r̂_{i}} |x|^{β}ui^{\frac{2n}{n−2}} (x)dx + t_{i} ∫_{e^{a_{i}}≤|x|≤r_{i}} |x|^{β}ui^{\frac{2n}{n−2}} (x)dx\} \\ ≥ & c_{n}t_{i} \{|∫_{r̂_{i}≤|x|≤e^{a_{i}}} ⟨x, ▽K̂⟩ûi^{\frac{2n}{n−2}} dx| − r̂_{i}^{β} − ε0^{\frac{2n}{n−2}} exp(βa_{i})\} . \end{matrix} (5.35)

Since w_{i}(a_{i}) = ϵ_{0}, by the scaling property of Û(y), we have

exp(a_{i} − ŝ_{i}) ∼ ε_{0}^{− \frac{2}{n−2}} >> 1.

By the scaling (5.34),

\begin{matrix}|∫_{r̂_{i}≤|x|≤e^{a_{i}}} ⟨x, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx| \\ = βt_{i}M̂_{i}^{− \frac{2β}{n−2}} |∫_{δ≤|y|≤exp(a_{i}−ŝ_{i})} Q(y)û_{i}(y)^{\frac{2n}{n−2}} dy| \\ = βt_{i}M̂_{i}^{− \frac{2β}{n−2}} (∫_{ℝ^{n}} −Q(y)Û(y)dy) (1 + o(1)), \end{matrix} (5.36)

where o(1) is small provided that both δ and ϵ_{0} be small. Thus, by (5.20), (5.35) yields

(5.37)  w_{i}(\underline{s}_{i}) ≥ c_{1} t_{i}^{1/2} exp(βŝ_{i}/2) ≥ c_{2}(ϵ_{0})t_{i}^{1/2} exp(βa_{i}/2)

for some c_{2}(ϵ_{0}) > 0.

By (5.22),(5.32) and (5.37),

(5.38) \begin{cases}2(\underline{s}_{i} − a_{i}) ≤ b_{i} − a_{i} \\ |\underline{s}_{i} − (1 − \frac{β}{n−2})a_{i} + \frac{1}{n−2} log t_{i}| ≤ c(ϵ_{0}) \\ |b_{i} − (1 − \frac{2β}{n−2})a_{i} + \frac{2}{n−2} log t_{i}| ≤ c(ϵ_{0}). \end{cases}

for some constant c(ϵ_{0}) > 0. Hence we have

\begin{matrix}a_{i} ≤ & (1 − \frac{2β}{n − 2})^{−1}[s_{i} + \frac{2}{n − 2} log t_{i}] + c \\ ≤ & log[R_{i}^{−2}M_{i}^{− \frac{2}{n−2}} ] + c, \end{matrix} (5.39)

and

\begin{matrix}\underline{s}_{i} ≤ & \frac{1}{2}(b_{i} − a_{i}) + a_{i} + c \\ ≤ & \frac{(1 − \frac{β}{n−2})}{(1 − \frac{2β}{n−2})}s_{i} + \frac{1}{(n − 2)(1 − \frac{2β}{n−2})} log t_{i} + c \\ ≤ & log[R_{i}^{−1}M_{i}^{− \frac{2}{n−2}} ] + c, \end{matrix} (5.40)

where R_{i} is defined in (5.2). These estimates together with Lemma 5.1 and (5.22) imply

w_{i}(s) ≤ w_{i}(\underline{s}_{i}) exp[\frac{n − 2}{2}(\underline{s}_{i} − s)]

≤ c_{1} ti^{\frac{1}{2}} exp(−\frac{n − 2}{2}s) exp[\frac{n − 2}{2}\underline{s}_{i} + \frac{1}{2}βa_{i}]

≤ c(ϵ_{0}) exp(−\frac{n − 2}{2}s) exp \{ \frac{\frac{n−2}{2}b_{i} + log t_{i}}{1 − \frac{2β}{n−2}} \}

(5.41)

for a_{i} ≤ s ≤ \underline{s}_{i}, and

\begin{matrix}w_{i}(\underline{s}_{i}) exp[\frac{n − 2}{2}(s − \underline{s}_{i})] ≤ & w_{i}(s) \\ ≤ & w_{i}(\underline{s}_{i}) exp[\frac{n − 2}{2}(s − \underline{s}_{i})] \end{matrix} (5.42)

for \underline{s}_{i} ≤ s ≤ b_{i}. Using (5.41), it follows

\begin{matrix}u_{i}(x) ≤ & c(ϵ_{0}) exp \{ \frac{\frac{n−2}{2}s_{i} + log t_{i}}{1 − \frac{2β}{n−2}} \} |x|^{−n+2} \\ = & c(ϵ_{0})(t_{i}M_{i}^{−1})^{\frac{1}{1− \frac{2β}{n−2}}} |x|^{−n+2} \end{matrix} (5.43)

for exp(a_{i}) ≤ |x| ≤ exp(\underline{s}_{i}), and by Lemma 5.1,

(5.44) cw_{i}(\underline{s}_{i}) exp[−\frac{n − 2}{2}\underline{s}_{i}] ≤ u_{i}(x) ≤ c_{1}(ϵ_{0}) w_{i}(\underline{s}_{i}) exp[−\frac{n − 2}{2}\underline{s}_{i}]

for exp(\underline{s}_{i}) ≤ |x| ≤ exp(s_{i}) and some c_{1}(ϵ_{0}). Since u_{i}(x) ∼ exp(−\frac{n−2}{2}s_{i}) for |x| = exp(s_{i}), (5.44) leads to

(5.45)  u_{i}(x) ∼ exp(−\frac{n − 2}{2}s_{i}) ∼ M_{i}

for exp(\underline{s}_{i}) ≤ |x| ≤ M_{i}^{− \frac{2}{n−2}} . Now (5.39), (5.40), (5.44) and (5.45) imply

(5.46)  u_{i}(x) ≤ c (t_{i}M_{i}^{−1})^{\frac{1}{1− \frac{2β}{n−2}}} |x|^{−n+2}

for R_{i}^{−2}M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ R_{i}^{−1}M_{i}^{− \frac{2}{n−2}} , and

(5.47)  u_{i}(x) ≤ cM_{i}

for R_{i}^{−1}M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ M_{i}^{− \frac{2}{n−2}} ∼ e^{s_{i}}.

Finally, we want to estimate u_{i}(x) for |x| ≥ Mi^{\frac{−2}{n−2}} . Set s_{i}^{∗} to be a local minimum point of w_{i}(s) in (s_{i}, 0) if there is one. Otherwise s_{i}^{∗} = 0. we claim

s_{i}^{∗} → 0 if and only if L_{i}M_{i}^{− \frac{2}{n−2}} → 0 and i → +∞.

Moreover, if s_{i}^{∗} → 0, then e^{s_{i}^{∗}} ∼ L_{i}Mi^{\frac{−2}{n−2}} .

(5.48)

First suppose L_{i}Mi^{\frac{−2}{n−2}} → 0 and s_{i}^{∗} ≥ c > 0. Set

ũ_{i}(y) = M_{i}^{−1}u_{i}(M_{i}^{− \frac{2}{n−2}} y).

By Lemma 5.1,

(5.49)  ũ_{i}(y) ≤ c |y|^{2−n} for 1 ≤ |y| ≤ Mi^{\frac{2}{n−2}} ,

because s_{i}^{∗} ≥ c > 0. The scaled ũ_{i}(y) converges to

U(y) = [λ(λ^{2} + |y − q|^{2})]^{\frac{2−n}{2}}

for λ > 0 and q ∈ ℝ^{n}. Then by Remark 5.3 (below), we have

(5.50)  ∫_{ℝ^{n}} ▽Q(y)U^{\frac{2n}{n−2}} (y)dy = 0.

Note ũ_{i} satisfies Δũ_{i} + K̃_{i}(y)ui^{\frac{n+2}{n−2}} = 0 and

K̃_{i}(y) = K_{i}(M_{i}^{− \frac{2}{n−2}} y).

Clearly,

y · ▽K̃_{i}(y) = t_{i}M_{i}^{− \frac{2β}{n−2}} [Q(y) + O(|y|^{β−1})]

and L_{i}^{n−2} = t_{i}^{−1}Mi^{\frac{2β}{n−2}} . Thus, the Pohozave identity yields

\begin{matrix}\frac{(n − 2)β}{2n} ∫_{ℝ^{n}} Q(y)U^{\frac{2n}{n−2}} (y)dy \\ = lim_{i→+∞} \frac{n − 2}{2n}L_{i}^{n−2} ∫_{|y|≤M \frac{2}{n−2} ; i} ⟨y, ▽K̃_{i}⟩ui^{\frac{2n}{n−2}} (y)dy \\ = lim_{i→+∞} L_{i}^{n−2} ∫_{|y|=M \frac{2}{n−2} ; i} (\frac{n − 2}{2}ũ_{i} \frac{∂ũ_{i}}{∂r} + |\frac{∂ũ_{i}}{∂r} |^{2} r \\ − \frac{1}{2}| ▽ ũ_{i}|^{2}r + \frac{n − 2}{2n}K̃_{i}(y)ui^{\frac{2n}{n−2}} r)dσ → 0, \end{matrix} (5.51)

because the boundary term = O(M_{i}^{−2}). By (5.50) and (K2),

∫ Q(y)U^{\frac{2}{n−2}} (y)dy ≠ 0.

Thus, (5.51) yields a contradiction.

Conversely, we assume s_{i}^{∗} → 0. Then as the second inequality in (5.38), we have

(5.52)  s_{i}^{∗} = (1 − \frac{β}{n − 2})s_{i} − \frac{1}{n − 2} log t_{i} + O(1),

which yields

(5.53)  e^{s_{i}^{∗}} ∼ L_{i}M_{i}^{− \frac{2}{n−2}} ,

and it implies L_{i}M_{i}^{− \frac{2}{n−2}} → 0 as i → +∞. Hence, (5.48) is proved. Clearly, (5.13) and (5.14) follows from Lemma 5.1 and (5.48). Therefore, we have proved Step 1.

Step 2. Recall that ũ_{i}(x) = M_{i}^{−1}u_{i}(M_{i}^{− \frac{2}{n−2}} x). After passing to a subsequence, ũ_{i}(x) converges to U(x − q) in C_{loc}^{2}(ℝ^{n}\{0}) with

(5.54)  U(x) = (\frac{λ}{λ^{2} + |x|^{2}} )^{\frac{n−2}{2}} .

Now we can estimate the difference of ũ_{i} and U(x − q) more precisely if we rescale U(x − q) and translate the position of its maximum point suitably.

If q ≠ 0, then there is a local maximum point q_{i} of ũ_{i} with lim_{i→∞} q_{i} = q. For suitable a_{i} → 1 and λ_{i} → 1, we let the function U_{i}(x) =

a_{i}λ_{i}^{− \frac{n−2}{2}} U(λ_{i}^{−1}(x − q_{i})) > 0 satisfy

(5.55)  \{ \begin{matrix}△U_{i} + K̃_{i}(0)Ui^{\frac{n+2}{n−2}} = 0 in ℝ^{n}, \\ U_{i}(q_{i}) = max_{ℝ^{n}}U_{i} = ũ_{i}(q_{i}) \end{matrix}

with K̃_{i}(x) = K_{i}(M_{i}^{− \frac{2}{n−2}} x), and

(5.56)  ∇(ũ_{i}(q_{i}) − U_{i}(q_{i})) = 0.

Note that λ_{i} and a_{i} are uniquely determined because they satisfy

(5.57)  \{ \begin{matrix}a_{i}λ_{i}^{− \frac{n−2}{2}} U(0) = ũ_{i}(q_{i}) and \\ K̃_{i}(0) = a_{i}^{− \frac{4}{n−2}} n(n − 2). \end{matrix}

If q = 0, let δ_{o} > 0 be a small number which is independent of i and will be chosen later. Then there is q_{i} = q_{i}(δ_{0}) such that

lim_{i→∞} q_{i} = 0, and

(5.58)  ∫_{|x−q_{i}|=δ_{o}}(x − q_{i})ũ_{i} ds = 0,

since ũ_{i}(x) converges to U(x) in C_{loc}^{2}(ℝ^{n} \ {0}). For suitable a_{i} → 1

and λ_{i} → λ, we may let the function U_{i} = a_{i}λ_{i}^{− \frac{n−2}{2}} U_{1}(λ_{i}^{−1}(x − q_{i})) > 0 satisfy

(5.59)  \{ \begin{matrix}△U_{i} + K̃_{i}(0)Ui^{\frac{n+2}{n−2}} = 0 in ℝ^{n}, \\ ∫_{|x−q_{i}|=δ_{o}} U_{i} ds = ∫_{|x−q_{i}|=δ_{o}} ũ_{i} ds, \\ ∫_{|x−q_{i}|=δ_{o}}(x − q_{i})U_{i} ds = 0. \end{matrix}

Set U_{i} as above, let g_{i}(x) = ũ_{i}(x) − U_{i}(x). Then g_{i} satisfies

△g_{i} + b(x)g_{i} = Q̃(x)Ui^{\frac{n+2}{n−2}} ,

where

\begin{matrix}b(x) = & K̃_{i}(x)\frac{ui^{\frac{n+2}{n−2}} − Ui^{\frac{n+2}{n−2}}}{ũ_{i} − U_{i}} , \\ Q̃(x) = & K̃_{i}(0) − K̃_{i}(x). \end{matrix}

Let f_{i}(x) be defined as follows.

f_{i}(x) = |x|^{− \frac{n−2}{2}} for 0 ≤ |x| ≤ R_{i}^{−2},

f_{i}(x) = \{L_{i}^{−n+2} + R_{i}^{−n+2}|x|^{−n+2} + max_{|y|=M \frac{2}{n−2} ; i} |ũ_{i}(y) − U_{i}(y)|\}

for R_{i}^{−2} ≤ |x| ≤ Mi^{\frac{2}{n−2}} ,

and

N_{i} = max_{|x|≤M \frac{2}{n−2} ; i} f_{i}^{−1}(x)|g_{i}(x)|.

Let x_{i} be a point satisfy |x_{i}| ≤ Mi^{\frac{2}{n−2}} and satisfy N_{i} = f_{i}^{−1}(x_{i})|g_{i}(x_{i})|. To prove part (ii), it suffices to show sup_{i≥1} N_{i} < ∞.

Assume that N_{i} is unbounded. Without loss of generality, we may

assume lim_{i→∞} N_{i} = +∞. Let r_{i} = min(L_{i}, Mi^{\frac{2}{n−2}} ). By (5.1), (5.11), (5.12), (5.13) and (5.14), we can see that ũ_{i} satisfies

(5.60)  ũ_{i}(x) ≤ c |x|^{− \frac{n−2}{2}} for |x| ≤ Mi^{\frac{2}{n−2}} ,

\begin{aligned}ũ_{i}(x) & ≤ c(t_{i}M_{i}^{−1})^{γ}M_{i}|x|^{−n+2} \\ & = c R_{i}^{−n+2}|x|^{−n+2} for R_{i}^{−2} ≤ |x| ≤ R_{i}^{−1}, \end{aligned} (5.61)

(5.62)  ũ_{i}(x) ≤ c U(x), for R_{i}^{−1} ≤ |x| ≤ r_{i}, and

(5.63)

If L_{i}M_{i}^{− \frac{2}{n−2}} is bounded, then ũ_{i}(x) ≤ c L_{i}^{−n+2} for r_{i} ≤ |x| ≤ Mi^{\frac{2}{n−2}} .

We note that if L_{i}M_{i}^{− \frac{2}{n−2}} is unbounded, then w_{i}(s) has no local min-

imum for s_{i} ≤ s ≤ 0. Thus, r_{i} = Mi^{\frac{2}{n−2}} and by (5.13), we have for |x| = r_{i},

(5.64)  ũ_{i}(x) ∼ M_{i}^{−2} >> L_{i}^{−n+2},

i.e., (5.63) does not hold in this case.

Since N_{i} is unbounded, we have by (5.60), (5.61) and (5.63),

(5.65)  r_{i} ≥ |x_{i}| ≥ R_{i}^{−1}.

By Green’s identity, we have for r_{i} ≥ |x| ≥ R_{i}^{−1}

\begin{aligned}g_{i}(x) = & ∫_{|η|≤r_{i}} G(x, η)(b(η)g_{i} − Q̃(η)Ui^{\frac{n+2}{n−2}} ) dη \\ & − ∫_{|η|=r_{i}} \frac{∂G(x, η)}{∂ν}g_{i}(η) ds \end{aligned} (5.66)

and

\begin{aligned}∇g_{i}(x) = & ∫_{|η|≤r_{i}} ∇_{x}G(x, η)(b(η)g_{i} − Q̃(η)Ui^{\frac{n+2}{n−2}} ) dη \\ & − ∫_{|η|=r_{i}} \frac{∂∇_{x}G(x, η)}{∂ν}g_{i}(η) ds, \end{aligned} (5.67)

where G(x, η) is the Green function of −△ on {x : |x| ≤ r_{i}}. Since we assume \frac{n − 2}{2} ≥ β > 1, it implies n ≥ 4. By the inequality g_{i} ≤ N_{i}f_{i} and G(x, η) ≤ c_{n}|x − η|^{2−n}, we have the following estimates for R_{i}^{−1} ≤ |x| ≤ r_{i}. Their proofs are elementary and are omitted here. By (5.60), we have

|b(η)g_{i}(η)| ≤ c |η|^{− \frac{n+2}{2}} for |η| ≤ R_{i}^{−2}.

Hence

(5.68)  ∫_{|η|≤R_{i}^{−1}} G(x, η)b(η)g_{i} dη = O(R_{i}^{−n+2}|x|^{−n+2}).

By (5.62) and (5.63), we have

ũ_{i}(x) ≤ c U(x) for R_{i}^{−1} ≤ |x| ≤ r_{i},

which implies

|b(η)| ≤ c(1 + |η|)^{−4} for R_{i}^{−1} ≤ |η| ≤ r_{i}.

Hence

\begin{matrix}∫_{R_{i}^{−1}≤|η|≤r_{i}} G(x, η)b(η)g_{i} dη \\ = O[ ∫_{R_{i}^{−1}≤|η|≤r_{i}} G(x, η)Ui^{\frac{4}{n−2}} N_{i}f_{i} dη] \\ = N_{i} O[R_{i}^{−n+2} \begin{cases}|x|^{−n+4}(1 + |x|)^{−2} & n > 4 \\ log(2 + |x|^{−1})(1 + |x|)^{−2} & n = 4 \end{cases} \\ + (1 + |x|)^{−2}(L_{i}^{−n+2} + max_{|η|=M \frac{2}{n−2} ; i} |g_{i}(η)|)] \\ = O(1)(1 + |x|)^{−2}N_{i}f_{i}(x). \end{matrix} (5.69)

Note that for (5.69), we have used R_{i}^{−n+2}|x|^{−n+2} = o(1)L_{i}^{−n+2} for |x| ≥ 1,

\begin{matrix}∫_{|η|≤r_{i}} G(x, η)Q̃(η)Ui^{\frac{n+2}{n−2}} dη \\ = O [\frac{1}{(1 + |x|)^{n−2}} + \frac{1}{(1 + |x|)^{n−β}} ] L_{i}^{−n+2} \\ = o(1)N_{i}f_{i}(x), \end{matrix} (5.70)

where

|Q̃(η)| = |K_{i}(0) − K_{i}(M_{i}^{− \frac{2}{n−2}} η)|
≤ c t_{i}M_{i}^{− \frac{2β}{n−2}} |η|^{β}
= c L_{i}^{−n+2}|η|^{β}.
(5.71)  ∫_{|η|=r_{i}} \frac{∂G(x, η)}{∂ν}g_{i}(η) ds = O[ max_{|η|=r_{i}} |g_{i}(η)|].

From (5.62) and (5.63), there is ĉ > 0 such that

max_{|η|=r_{i}} |g_{i}(η)| ≤ ĉ min_{|x|≤M \frac{2}{n−2} ; i} f_{i}(x).

Putting these estimates together, we obtain

\begin{matrix}g_{i}(x) = & O[(1 + |x|)^{−2}N_{i}f_{i}(x) + max_{|η|=r_{i}} |g_{i}(η)|] \\ = & O[(1 + |x|)^{−2} + o(1)]N_{i}f_{i}(x) \end{matrix} (5.72)

for r_{i} ≥ |x| ≥ R_{i}^{−1}. Similarly, we have the following estimates for derivatives:

(5.73)  ∫_{|η|≤R_{i}^{−1}} ∇_{x}G(x, η)b(η)g_{i} dη = O(R_{i}^{−n+2}|x|^{−n+1}),

\begin{matrix}∫_{R_{i}^{−1}≤|η|≤r_{i}} ∇_{x}G(x, η)b(η)g_{i} dη \\ = O[ ∫_{R_{i}^{−1}≤|η|≤r_{i}} ∇_{x}G(x, η)Ui^{\frac{4}{n−2}} N_{i}f_{i} dη] \\ = N_{i} O[R_{i}^{−n+2}|x|^{−n+3}(1 + |x|)^{−2} \\ + (1 + |x|)^{−3}(L_{i}^{−n+2} + max_{|η|=M \frac{2}{n−2} ; i} |g_{i}(η)|)] \\ = O(1)(1 + |x|)^{−2}N_{i}f_{i}(x), \end{matrix} (5.74)

\begin{matrix}∫_{|η|≤r_{i}} ∇_{x}G(x, η)Q̃(η)Ui^{\frac{n+2}{n−2}} dη \\ = O(1)[log(2 + |x|)(1 + |x|)^{−n+1} \\ + (1 + |x|)^{−n−1+β}]L_{i}^{−n+2}, \end{matrix} (5.75)

(5.76)  ∫_{|η|=r_{i}} \frac{∂∇_{x}G(x, η)}{∂ν}g_{i}(η) ds = O[r_{i}^{−1} max_{|η|=r_{i}} |g_{i}(η)|]

for r_{i} ≥ |x| ≥ R_{i}^{−1}. It follows from these estimates

\begin{matrix}∇g_{i}(x) = & O[R_{i}^{−n+2}|x|^{−n+1} + (1 + |x|)^{−2}N_{i}f_{i}(x) \\ + r_{i}^{−1} max_{|η|=r_{i}} |g_{i}(η)|] \\ = & O[R_{i}^{−n+2}|x|^{−n+1} + ((1 + |x|)^{−2} + o(1))N_{i}f_{i}(x)] \end{matrix} (5.77)

for r_{i} ≥ |x| ≥ R_{i}^{−1}.

Let x = x_{i} in (5.72). We obtain

N_{i}f_{i}(x_{i}) = |g_{i}(x_{i})| ≤ c[(1 + |x_{i}|)^{−2} + o(1)]N_{i}f_{i}(x_{i})

for some c independent of i. Hence x_{i} must be bounded and

|x_{i}| ≤ c_{1}

for some c_{1} independent of i.

Since R_{i}^{−1} ≤ |x_{i}| ≤ c_{1}, we have

f_{i}(x_{i}) = \{L_{i}^{−n+2} + R_{i}^{−n+2}|x_{i}|^{−n+2} + max_{|y|=M \frac{2}{n−2} ; i} |ũ_{i}(y) − U_{i}(y)|\} .

Note that L_{i} << R_{i}. For any r > 0, if |x| ≥ r, then

(5.78)  \frac{f_{i}(x)}{f_{i}(x_{i})} ≤ 2 + (\frac{L_{i}}{R_{i}} )^{n−2} r^{−n+2}.

By (5.72) and (5.78), |g_{i}(x_{i})|^{−1}g_{i}(x) satisfies for |x| ≥ r > 0,

\begin{aligned}\frac{|g_{i}(x)|}{|g_{i}(x_{i})|} & ≤ c (1 + |x|)^{−2} \frac{f_{i}(x)}{f_{i}(x_{i})} \\ & ≤ c [(1 + |x|)^{−2} + (\frac{L_{i}}{R_{i}} )^{n−2} r^{−n+2}] . \end{aligned}

After passing to a subsequence, the sequence g_{i}(x_{i})^{−1}g_{i}(x) converges in C_{loc}^{2}(ℝ^{n} \ {0}) to a function ϕ which satisfies

(5.79)  \{ \begin{matrix}△ϕ + n(n + 2)U^{\frac{4}{n−2}} ϕ = 0 in ℝ^{n} \ \{0\}, \\ |ϕ| ≤ c (1 + |x|)^{−2}, \end{matrix}

where U is given in (5.54). Since ϕ(x) is bounded, by the regularity of elliptic equations, ϕ satisfies (5.79) in ℝ^{n}. Now we show that ϕ ≢ 0. Since x_{i} is bounded, without loss of generality, we may assume x_{i} → x_{0}. If x_{0} ≠ 0, then ϕ(x_{0}) = 1. Obviously, ϕ(x) ≢ 0 in ℝ^{n}. Now we assume x_{0} = 0. Let δ_{1} be a small positive number. For y_{i} = δ_{1}|x_{i}|^{−1}x_{i}, we have by (5.77) and the fact |x_{i}| > R_{i}^{−1} that

|g_{i}(y_{i}) − g_{i}(x_{i})| ≤ ∫_{|x_{i}|}^{|y_{i}|} |∇g_{i}(s|x_{i}|^{−1}x_{i})|ds

≤ c(R_{i}^{−n+2}|x_{i}|^{−n+2} + δ_{1}N_{i}f_{i}(x_{i}))

≤ \frac{1}{2}N_{i}f_{i}(x_{i}) ≤ \frac{1}{2}|g_{i}(x_{i})|

if N_{i} is large and δ_{1} is small. This implies

|g_{i}(x_{i})^{−1}g(y_{i})| ≥ \frac{1}{2}

for large i and consequently,

min_{|x|=δ_{1}} |ϕ(x)| ≥ \frac{1}{2}.

We conclude that ϕ ≢ 0.

By Lemma 3.2,

ϕ = ∑ γ_{j}ψ_{j}

with ψ_{0} = \frac{n−2}{2}U + (x − q) · ∇U(x − q) and ψ_{j} = \frac{∂U}{∂x_{j}} , 1 ≤ j ≤ n. By (5.56) and (5.59), we have either q ≠ 0, ϕ(q) = 0 and ∇ϕ(q) = 0 or q = 0, ∫_{|x|=δ_{o}} ϕ ds = 0 and ∫_{|x|=δ_{0}} x_{j}ϕds = 0, 1 ≤ j ≤ n, which implies γ_{j} = 0 for 0 ≤ j ≤ n. We obtain a contradiction. Hence N_{i} must be bounded. The proof of Theorem 2.7 is complete. q.e.d.

Proof of Lemma 5.2. We follow the notations in the proof of Theorem 2.7. Recall that û_{i}(y) converges to Û(y) in C_{loc}^{2}(ℝ^{n}\{0}), where Û satisfies (5.16). By the Pohozaev identity

(5.80)  \frac{n − 2}{2n} ∫_{|x|≤1}⟨x, ∇K̂_{i}⟩ûi^{\frac{2n}{n−2}} dx = P (1, û_{i}),

where

K̂_{i}(x) = K_{i}(M̂_{i}^{− \frac{2}{n−2}} x), M̂_{i} = e^{− \frac{n−2}{2} ŝ_{i}},

and

\begin{matrix}P (r, û_{i}) = ∫_{|x|=r} (\frac{n − 2}{2}û_{i} \frac{∂û_{i}}{∂ν} − \frac{1}{2}r|∇û_{i}|^{2} \\ +r |\frac{∂û_{i}}{∂ν} |^{2} + \frac{n − 2}{2n}rK̂_{i}ûi^{\frac{2n}{n−2}} ) dσ. \end{matrix}

Since û_{i}(x) ≤ c|x|^{− \frac{n−2}{2}} , the left hand side of (5.80) tends to 0 as i → ∞, which implies

P (1, U) = lim_{i→∞} P (1, ũ_{i}) = 0.

Since P (r, u) ≡ constant < 0 for any singular solution u of (5.16), Û is smooth at 0. Hence

Û(y) = (\frac{λ̂}{λ̂ + |y − q̂|^{2}} )^{\frac{n−2}{2}} .

130 chiun-chuan chen & chang-shou lin

Since

\frac{d}{dr} û_{i}(r)r^{\frac{n−2}{2}} |_{r=1} = 0,

we have

\frac{d}{dr} \overline{Û}(r)r^{\frac{n−2}{2}} |_{r=1} = 0.

By a straightforward computation, we have

\begin{aligned}\frac{d}{dr} \overline{Û}(r)r^{\frac{n−2}{2}} & = \frac{d}{dr} \frac{1}{|S^{n−1}|} ∫_{S^{n−1}} \frac{(rλ̂)^{\frac{n−2}{2}} dσ}{(λ̂^{2} + |ry − q̂|^{2})^{\frac{n−2}{2}}} \\ & = \frac{(n − 2)λ̂^{\frac{n−2}{2}} r^{\frac{n−4}{2}}}{2|S^{n−1}|} ∫_{S^{n−1}} \frac{(λ̂^{2} + |q̂|^{2} − r^{2})dσ}{(λ̂^{2} + |ry − q̂|^{2})^{\frac{n}{2}}} . \end{aligned}

Thus, r_{0} = \sqrt{λ̂^{2} + |q̂|^{2}} is the only critical point of \overline{Û}(r)r^{\frac{n−2}{2}} and

\frac{d^{2}}{dr^{2}} (\overline{Û}(r)r^{\frac{n−2}{2}} ) |_{r_{0}}< 0.

(5.18) follows readily.

We want to prove that Û(y) satisfies

(5.81)  ∫_{ℝ^{n}} ▽Q(y)Û(y)^{\frac{2n}{n−2}} dy = 0,

and

(5.82)  ∫_{ℝ^{n}} Q(y)Û(y)^{\frac{2n}{n−2}} dy ≤ 0.

By a simple scaling argument, we have (5.22), i.e.,

a_{i} − ŝ_{i} ≤ c (ε_{0}).

Hence, by (5.33),

(5.83)  \underline{s}_{i} − ŝ_{i} > \underline{s}_{i} − a_{i} ≥ \frac{1}{n − 2}(− log t_{i} − βŝ_{i}) − c.

Recall that

M̂_{i} = exp (−\frac{n − 2}{2}ŝ_{i}) and L̂_{i} = (t_{i}^{−1}M̂i^{\frac{2β}{n−2}} )^{\frac{1}{n−2}} .

prescribing scalar curvature on S^{n} 131

By (5.83),

r_{i} ≡ e^{\underline{s}^{i}} M̂i^{\frac{2}{n−2}} ≥ c (t_{i}^{−1}M̂i^{\frac{2β}{n−2}} )^{\frac{1}{n−2}} = cL̂_{i}^{n−2}.

Applying Lemma 5.1, we have

(5.84) û_{i}(y) ≤ c |y|^{2−n}

for 1 ≤ |y| ≤ e^{\underline{s}^{i}} M̂i^{\frac{2}{n−2}} = r_{i}. Since û_{i} satisfies

Δû_{i} + K̂_{i}ûi^{\frac{n+2}{n−2}} = 0 for |y| ≤ M̂i^{\frac{2}{n−2}} ,

where

K̂_{i}(y) = K_{i}(M̂_{i}^{− \frac{2}{n−2}} y).

Let e_{j}, 1 ≤ j ≤ n, be the standard orthorgonal base for ℝ^{n}. Applying Pohozaev’s identities, we have

\begin{matrix}\frac{n − 2}{2n} ∫_{B(0,r_{i})}⟨e_{j}, ▽K̂_{i}⟩ûi^{\frac{2n}{n−2}} (y)dy \\ = ∫_{∂B(0,r_{i})}⟨e_{j}, ▽û_{i}⟩\frac{∂û_{i}}{∂ν} − ⟨e_{j}, ν⟩\frac{| ▽ û_{i}|^{2}}{2} \\ + \frac{n − 2}{2n}⟨e_{j}, ν⟩K̂_{i}ûi^{\frac{2n}{n−2}} dσ \\ = O(r_{i}^{−n+1}), \end{matrix} (5.85)

by (5.85) and the gradient estimate. From (5.28), we have

\begin{matrix}\frac{n − 2}{2n} ∫_{B(0,r_{i})}⟨y, ▽K̂_{i}⟩ûi^{\frac{2n}{n−2}} (y)dy \\ = −\frac{|S^{n−1}|}{2}w_{i}^{2}(\underline{s}_{i})(1 + o(1)). \end{matrix} (5.86)

Since t_{i}M̂_{i}^{− \frac{2β}{n−2}} = L̂_{i}^{−n+2} and

▽K̂_{i}(y) = t_{i}M̂_{i}^{− \frac{2β}{n−2}} (▽Q(y) + o(1)|y|^{β−1}) for |y| ≤ M̂i^{\frac{2}{n−2}} ,

(5.84) and (5.85) yield

\begin{aligned}lim_{i→+∞} & |L̂_{i}^{n−2} ∫_{B(0,r_{i})} t_{i}M̂_{i}^{− \frac{2β}{n−2}} ((\frac{∂Q(y)}{∂y_{j}}) + o(1)|y|^{β−1}) ûj^{\frac{2n}{n−2}} (y)dy| \\ & = |∫_{ℝ^{n}} \frac{∂Q}{∂y_{j}} Û^{\frac{2n}{n−2}} (y)dy| ≤ c L̂_{i}^{−1} → 0 \end{aligned}

as i → +∞, which is (5.81).

To prove (5.82), we note

(y, ▽K̂_{i}(y)) = t_{i}M̂_{i}^{− \frac{2β}{n−2}} (βQ(y) + o(1)|y|^{β}).

Thus, (5.86) yields

β ∫_{ℝ^{n}} Q(y)Û^{\frac{2n}{n−2}} (y)dy = lim_{i→+∞} (L̂_{i}^{n−2} ∫_{B(0,r_{i})}(y, ▽K̂_{i})ûi^{\frac{2n}{n−2}} (y)dy)

= −\frac{n(n − 2)|S^{n−1}|}{4} lim_{i→+∞}(L̂_{i}^{n−2}w_{i}^{2}(\underline{s}_{i}))

≤ 0,

which is (5.82). The proof of Lemma 5.2 is complete. q.e.d.

Remark 5.3. The proof of (5.19) holds also for ũ_{i} of (5.49), when L_{i}M_{i}^{− \frac{2}{n−2}} → 0. Because the left hand side of (5.85) = \frac{n−2}{2n}t_{i}M_{i}^{− \frac{2β}{n−2}} ×

(∫_{δ≤|y|≤M \frac{2}{n−2} ; i} ⟨e_{j}, ▽Q(y)⟩U^{\frac{2n}{n−2}} (y)dy + O(1) ∫_{|y|≤δ} |y|^{β−1−n}dy) ,

(5.85) yields

|∫_{ℝ^{n}} ▽Q(y)U^{\frac{2n}{n−2}} (y)dy| ≤ c L_{i}^{n−2}(M_{i}^{− \frac{2}{n−2}} )^{(n−1)} → 0

as n → +∞, which is (5.50).

Remark 5.4. If L_{i}M_{i}^{− \frac{2}{n−2}} ≥ c > 0 for some constant c > 0, then (5.13) yields u_{i}(x) ≤ c M^{−1}|x|^{2−n} for M_{i}^{− \frac{2}{n−2}} ≤ |x| ≤ 1. By passing to a subsequence, M_{i}u_{i}(x) converges to a positive harmonic function h(x) in C_{loc}^{2}(B_{2}\{0}). We claim

If lim_{i→+∞} L_{i}M_{i}^{− \frac{2}{n−2}} = +∞, then h(x) = \frac{a}{|x|^{n−2}} + O(|x|) near 0

for some a > 0.

(5.87)

Let h(x) = \frac{a}{|x|^{n−2}} + b + O(|x|) for a > 0 and b ∈ ℝ. By applying the Pohozaev identity, we have

\begin{matrix}\frac{n − 2}{2n} ∫_{|x|≤1}⟨x, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx \\ = ∫_{|x|=1} (\frac{n − 2}{2} \frac{∂u_{i}}{∂r} u + r |\frac{∂u_{i}}{∂r} |^{2} \\ −\frac{1}{2}| ▽ u_{i}|^{2}r − \frac{n − 2}{2n}K_{i}(x)ui^{\frac{2n}{n−2}} ) dσ \end{matrix} (5.88)

By scaling, it is easy to see the left hand side of (5.88)

= \frac{(n − 2)β}{2n} L_{i}^{2−n} (∫_{ℝ^{n}} Q(y)U^{\frac{2n}{n−2}} (y)dy + o(1)) ,

and the right hand side = −c_{n}abM_{i}^{−2}(1 + o(1)). Since

∫_{ℝ^{n}} ▽Q(y)U^{\frac{2n}{n−2}} (y)dy = 0

by (5.50), (K1) yields ∫_{ℝ^{n}} Q(y)U^{\frac{2n}{n−2}} (y)dy ≠ 0. Hence, if

lim_{i→+∞} L_{i}^{n−2}M_{i}^{−2} = +∞,

then ab = 0, i.e., b = 0. Thus, the claim (5.87) is proved.

6. Preliminary results of global solutions

From now on, u_{i}(x) is considered to be a solution of (1.3) defined in the whole ℝ^{n}. Theorem 1.2 implies that after passing to a subsequence, {u_{i}} blows up only at finite points. We will prove this later and for the proof of Theorem 1.2, we assume first that {q̂_{j}}_{j=1}^{m} is the set of blowup points for {u_{i}} with m ≥ 1, and u_{i} → 0 on any compact subset of ℝ^{n}\{q̂_{1}, …, q̂_{m}}. Let l ≤ m be the nonnegative integer such that q̂_{1}, …, q̂_{l} are simple-like blowup points and q̂_{l+1}, …, q̂_{m} are non-simple-like blowup points. For the definition of simple-like blowup points, see the end of Section 2. If there are no simple-like blowup points, we let l = 0.

For each blowup point q̂_{j}, we define the local maximum M_{i,j} and the local maximum point in the following ways. Let δ_{0} be a small positive number such that the distance d(q̂_{j}, q̂_{k}) from q̂_{j} to q̂_{k} is greater than 2δ_{0}. If u_{i} loses energy of one bubble near q̂_{j}, that is, if (2.17) holds, then M_{i,j} and q̂_{i,j} are defined by

(6.1)  M_{i,j} = u_{i}(q̂_{i,j}) = max_{|q̂_{j}−x|≤δ_{0}} u_{i}(x).

Let L_{i,j} = L_{i}(q̂_{i,j}) be the number defined in (2.15). If u_{i} loses energy of more than one bubble at q̂_{j}, then there are two cases. The first one is described in (ii) of Theorem 2.5. In this case, q̂_{i,j} denotes the local maximum point z_{i} in the statement of (ii) of Theorem 2.5, and M_{i,j} = u_{i}(q̂_{i,j}). Note that in this case, q̂_{j} is a simple-like blowup point,

(6.2) lim_{i→+∞} L_{i,j}M_{i,j}^{− \frac{2}{n−2}} = +∞, and lim_{i→+∞} (|q̂_{i,j} − q̂_{j}|Mi,j^{\frac{2}{n−2}} ) = +∞

by Theorem 2.5. The second case is described in Theorem 2.7. In this case, M_{i,j} and L_{i,j} are defined as in (2.28) and (2.29), and q̂_{i,j} is defined

to be q̂_{j} + M_{i,j}^{− \frac{2}{n−2}} z_{i}, where z_{i} is in the statement of Theorem 2.7.

By Theorem 2.1, Theorem 2.5, and Theorem 2.7 and the remark after Definition 2.9, q̂_{j} is a simple-like blowup point if and only if

(6.3)  L_{i,j}M_{i,j}^{− \frac{2}{n−2}} ≥ c > 0.

Also, for j ≤ l, we have

(6.4)  min_{|x−q̂_{j}|≤δ_{0}} u_{i}(x) ∼ M_{i,j}^{−1}.

For l + 1 ≤ j ≤ m, we have then

(6.5)  min_{|x−q̂_{j}|≤δ_{0}} u_{i}(x) ∼ L_{i,j}^{2−n}M_{i,j}

and

(6.6)  u_{i}(x) ≤ c|x − q̂_{j}|^{− \frac{n−2}{2}}

for |x − q̂_{j}| ≤ δ_{0} since they are non-simple-like blowup points.

One important situation is that for some j,

(6.7)  lim_{i→+∞} L_{i,j}M_{i,j}^{− \frac{2}{n−2}} = +∞

occurs. We claim

(6.8) If (6.7) holds for some j, then q̂_{j} is the only simple-like blowup point, that is, l = 1.

Proof of (6.8). If u_{i}(x) satisfies the assumption of Theorem 2.7 at q_{j}, then m_{i} ∼ M_{i,j}^{−1} by (6.4). Set h(x) to be the limit of m_{i}^{−1}u_{i}(x). Since

lim_{i→+∞} L_{i,j}M_{i,j}^{− \frac{2}{n−2}} = +∞, (5.87) yields that h(x) = \frac{a}{|x−q_{j}|^{n−2}} for some a > 0. By Lemma 6.1 (below), q_{j} is the only simple-like blowup point. So, we might assume either q_{j} is a simple blowup point or q_{j} is the one described by (ii) of Theorem 2.5. We note that for both cases, by letting an empty set E, R = R_{i}, l = δL_{i,j} and l_{0} = +∞, Lemma 3.5 yields

∫_{R_{i}≤|x|≤L^{∗}+_{i,j}(δ)} vi^{\frac{n+2}{n−2}} (y)dy ≤ c_{1}(R_{i}^{−2} + ε),

where v_{i}(y) = M_{i,j}^{−1}u_{i}(q_{i,j} +M_{i,j}^{− \frac{2}{n−2}} ), R_{i} is given in (2.14), and L_{i,j}^{∗}(δ) =

min(δL_{i,j}, λMi,j^{\frac{2}{n−2}} ) for some fixed δ > 0. Now let x_{0} be another simple-like blowup point, i.e., either x_{0} is a simple blowup point or the one in case (ii) of Theorem 2.5. Say x_{0} = q_{1} ≠ q_{j}. In any case, there is a small neighborhood ω of q_{0} such that min_{ω} u_{i}(x) ∼ M_{i,1}^{−1}. Clearly, min_{ω} u_{i}(x) ∼ min_{|x−q_{j}|≤1} u_{i}(x). Hence M_{i,j} ∼ M_{i,1}. Let

ω_{i}^{∗} = \{y | q_{i,j} + M_{i,j}^{− \frac{2}{n−2}} y ∈ ω\}. Then, v_{i}(y) ≤ c for y ∈ ω^{∗}. Since

lim_{i→+∞} L_{i,j}M_{i,j}^{− \frac{2}{n−2}} = +∞, L_{i,j} >> |q_{j} − q_{1}|Mi,j^{\frac{2}{n−2}} for large i. There-

fore, by choosing λ ≥ 2|q_{j} − q_{1}|, we have L_{i,j}^{∗}(δ) = λMi,j^{\frac{2}{n−2}} , and

0 < c_{2} ≤ ∫_{ω} ui^{\frac{2n}{n−2}} (x)dx

= ∫_{ω_{i}^{∗}} vi^{\frac{2n}{n−2}} dy ≤ c ∫_{ω_{i}^{∗}} vi^{\frac{n+2}{n−2}} (y)dy

≤ c ∫_{R_{i}≤|x|≤L_{i,j}^{∗}(δ)} vi^{\frac{n+2}{n−2}} (y)dy

≤ c c_{1}(R_{i}^{−2} + ε).

Clearly, this yields a contradiction. Then (6.8) is proved.

One important consequence of (6.8) is that if l = 1 and j ≥ 2 or if l ≥ 2 and j ≥ 1, the inequality (6.6) always holds near q̂_{j}. From it, we

have L_{i,j}^{n−2} ∼ t_{i}^{−1}Mi,j^{\frac{2β_{j}}{n−2}} which follows definition of L_{i,j}. To show (6.6) holds in these cases, it suffices for us to consider the case l ≥ 2. By (6.8), lim_{i→∞} L_{i,j}M_{i,j}^{− \frac{2}{n−2}} < ∞ for all j. If (6.6) does not hold near q̂_{j}, then Theorem 2.5 and (2.21) imply that q̂_{j} is a simply blowup point.

However, Theorem 2.2 implies |q̂_{i,j} − q̂_{j}|Mi,j^{\frac{2}{n−2}} ≤ c. Together with the fact that q̂_{j} is a simply blowup point, (6.6) holds at q̂_{j}. Then it yields a contradiction again. Hence we prove the claim.

Now, we prove Theorem 1.2.

Proof of Theorem 1.2. Recall that {q_{1}, ⋯, q_{N}} are the critical points of K̂. Let q be a blowup point of {u_{i}}. We want to prove ▽K̂(q) = 0. We may assume q ≠ ∞. Now suppose that q is not a critical point of K̂. Then by Corollary 2.3 and (6.8), we conclude that after passing to a subsequence, q is the only simple-like blowup point. Therefore, ▽K̂(q̂) = 0 for any other blowup point q̂ ≠ q, and it implies there are at most finite blowup points {q̂_{1}, ⋯, q̂_{m}} which are contained in {q_{1}, ⋯, q_{N}} ∪ {q}. Also by the Harnack inequality, u_{i} → 0 uniformly on any compact subset of ℝ^{n} \ {q̂_{1}, ⋯, q̂_{m}}.

Let M_{i,j} and q̂_{i,j} be defined as above. We may assume q̂_{1} = q. Then

(6.9)  u_{i}(x) ≤ c M_{i,1}^{−1}|x − q̂_{1}|^{2−n}

for x ∉ ⋃_{j≥2}^{m} B(q̂_{j}, δ_{0}), and by (6.6),

(6.10)  u_{i}(x) ≤ c |x − q̂_{j}|^{− \frac{n−2}{2}}

holds for |x − q̂_{j}| ≤ δ_{0} and j ≥ 2. Let e_{1} = (1, 0, ⋯, 0) and Ω_{i} =

ℝ^{n}\ ⋃_{j=1}^{m} B(q̂_{j}, δ_{0}). We may assume e_{1} = \frac{▽K̂(q̂_{1})}{| ▽ K̂(q̂_{1})|}. By the Pohozaev

identity,

\begin{matrix}∫_{B(q̂_{1},δ_{0})} \frac{∂K_{i}(x)}{∂x_{1}}ui^{\frac{2n}{n−2}} (x)dx = & − ∫_{ℝ^{n}\B(q̂_{1},δ_{0})} \frac{∂K_{i}}{∂x_{1}} ui^{\frac{2n}{n−2}} dx \\ ≤ & ∑_{j=2}^{m} ∫_{B(q̂_{j},δ_{0})} | ▽ K_{i}|ui^{\frac{2n}{n−2}} (x)dx \\ + ∫_{Ω_{i}} | ▽ K_{i}|ui^{\frac{2n}{n−2}} dx \\ ≤ & c t_{i} \{∑_{j=2}^{m} δ_{0}^{β_{j}−1} + M_{i,1}^{− \frac{2n}{n−2}} \} , \end{matrix} (6.11)

where inequalities (6.9) and (6.10) are used.

On the other hand, since q̂_{1} is a blowup point,

(6.12)  ∫_{B(q̂_{1},δ_{0})} ui^{\frac{2n}{n−2}} (x)dx ≥ c_{n} > 0

for some constant c_{n} > 0. Since β_{j} > 1 for j ≥ 2, (6.11) implies

c_{n} t_{i} ≤ ∫_{B(q̂_{1},δ_{0})} \frac{∂K_{i}}{∂x_{1}} ui^{\frac{2n}{n−2}} (x)dx ≤ c t_{i} \{∑_{j=2}^{m} δ_{0}^{β_{j}−1} + M_{i,1}^{− \frac{2n}{n−2}} \} ,

which obviously yields a contradiction when δ_{0} is small. The proof is finished q.e.d.

From now on, by passing to a subsequence, we may assume the blowup points are {q_{1}, ⋯, q_{m}} ⊂ {q_{1}, ⋯, q_{N}} and u_{i} → 0 uniformly on any compact subset of ℝ^{n} \ {q_{1}, ⋯, q_{m}}. Let l ≤ m be the nonnegative integer such that q_{1}, …, q_{l} are simple-like blowup points and q_{l+1}, …, q_{m} are non-simple-like blowup points. Set

(6.13)  m_{i} = inf_{ℝ^{n}}(u_{i}(x)(1 + |x|)^{n−2}).

Since u_{i}(x) → 0 for x ∉ {q_{1}, …, q_{m}}, m_{i} → 0 as i → +∞. Let

h_{i}(x) = m_{i}^{−1}u_{i}(x) for x ∈ ℝ^{n}.

Then h_{i}(x) is bounded in C_{loc}^{2}(ℝ^{n}\{q_{1}, …, q_{m}}). After passing to a subsequence, h_{i}(x) converges to h(x) in C_{loc}^{2}(ℝ^{n}\{q_{1}, …, q_{m}}). Since m_{i} → 0, h(x) satisfies

\begin{cases}Δh(x) = 0 in ℝ^{n}\\{q_{1}, …, q_{m}\}, \\ h(x) > 0. \end{cases}

By the Liouville Theorem, we have

(6.14)  h(x) = ∑_{j=1}^{m} \frac{μ_{j}}{|x − q_{j}|},

where μ_{j} ≥ 0 and ∑_{j=1}^{m} μ_{j} ≠ 0.

Lemma 6.1. μ_{j} > 0 if and only if q_{j} is a simple-like blowup point.

Proof. Let q_{j} be a simple-like blowup point. By (6.4),

m_{i} ∼ M_{i,j}^{−1}.

Thus,

\begin{matrix}\frac{1}{m_{i}} ∫_{|x−q_{j}|≤δ_{0}} K_{i}(x)ui^{\frac{n+2}{n−2}} (x)dx \\ ≥ c_{1} M_{i,j} ∫_{|x−q_{j}|≤δ_{0}} ui^{\frac{n+2}{n−2}} dx ≥ c_{2} > 0. \end{matrix}

It implies μ_{j} > 0.

Conversely, if q_{j} is not a simple-like blowup point, then by (6.6) and (6.5),

(6.15)

u_{i}(x) ≤ \begin{cases}|x − q_{j}|^{− \frac{n−2}{2}} & for |x| ≤ M_{i}^{− \frac{2}{n−2}} \\ M_{i}^{−1}|x − q_{j}|^{−n+2} & for M_{i}^{− \frac{2}{n−2}} ≤ |x − q_{j}| ≤ L_{i}M_{i}^{− \frac{2}{n−2}} \\ L_{i}^{2−n}M_{i} & for L_{i}M_{i}^{− \frac{2}{n−2}} ≤ |x − q_{j}| ≤ δ_{0}, \end{cases}

where for the simplicity of notations, M_{i} and L_{i} denote M_{i,j} and L_{i,j}, respectively. Hence

(6.16)  m_{i} ∼ L_{i}^{2−n}M_{i}.

Applying (6.16), a straightforward computation shows

\frac{1}{m_{i}} ∫_{|x−q_{j}|≤δ_{0}} K_{i}(x)ui^{\frac{n+2}{n−2}} (x)dx ≤ \frac{c}{m_{i}} \{M_{i}^{−1} + mi^{\frac{n+2}{n−2}} \} → 0.

Here we have used m_{i}M_{i} ∼ L_{i}^{2−n}M_{i}^{2} → +∞ as i → +∞ by (6.2). Therefore, μ_{j} = 0. q.e.d.

From Lemma 6.1, we immediately have l ≥ 1. The next lemma tell us that there are some constraints for a collection of critical points to be a set of blowup points.

Lemma 6.2.

(i) If l ≥ 2, then we have β_{j} > \frac{n−2}{2} for all j, or β_{j} = \frac{n−2}{2} for all j, or β_{j} < \frac{n−2}{2} for all j. Moreover, β_{1} = β_{2} = … = β_{l} always holds, β_{1} > β_{j} for j ≥ l + 1 if β_{j} > \frac{n−2}{2} for all j, and β_{1} < β_{j} for j ≥ l + 1 if β_{j} < \frac{n−2}{2} for all j.

(ii) If l = 1, then we have β_{j} > \frac{n−2}{2} for 2 ≤ j ≤ m, or β_{j} < \frac{n−2}{2} for 2 ≤ j ≤ m, or β_{j} = \frac{n−2}{2} for 2 ≤ j ≤ m. Furthermore, if

β_{1} ≤ \frac{n−2}{2}, then β_{j} < \frac{n−2}{2} for 2 ≤ j ≤ m. If β_{j} > \frac{n − 2}{2} for

2 ≤ j ≤ m, then β_{1} > β_{j} for j ≥ 2.

Proof. We prove (i) first. Since l ≥ 2, by (6.8), the inequality (6.6)

holds near any q_{j} and L_{i,j}^{2−n} ∼ t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} . By (6.8) again and the fact

q_{1}, …, q_{l} are simple-like blowup points, we also have L_{i,j} ∼ Mi,j^{\frac{2}{n−2}} for 1 ≤ j ≤ l. Thus, M_{i,j} satisfies

(6.17)  L_{i,j}^{2−n} ∼ t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} for 1 ≤ j ≤ m,

(6.18)  m_{i} ∼ L_{i,j}^{2−n}M_{i,j} for 1 ≤ j ≤ m,

and by Lemma 6.1

(6.19)  M_{i,j} = o(1)M_{i,k} for 1 ≤ j ≤ l and k ≥ l + 1.

By (6.17) and (6.18), for j ≠ k,

(6.20)  M_{i,j}^{1− \frac{2β_{j}}{n−2}} ∼ M_{i,k}^{1− \frac{2β_{k}}{n−2}} ,

which implies that there are only three possibilities: β_{j} > \frac{n−2}{2} for all j, or β_{j} = \frac{n−2}{2} all j, or β_{j} < \frac{n−2}{2} for all j. Since M_{i,j} ∼ M_{i,k} if 1 ≤ j, k ≤ l, by (6.20), we have β_{j} = β_{k}. Again by (6.20) and (6.19), we obtain the inequalities: β_{1} > β_{j} for j ≥ l + 1 if β_{1} > \frac{n−2}{2}, or β_{1} < β_{j} for j ≥ l + 1 if β_{1} < \frac{n−2}{2}.

To prove (ii), we note that by (6.15), (6.17) and (6.18) holds for 2 ≤ j ≤ m. Thus, (6.20) holds for j ≠ k ≥ 2, and then we have β_{j} > \frac{n−2}{2} for all j ≥ 2, or β_{j} = \frac{n−2}{2} for all j ≥ 2, or β_{j} < \frac{n−2}{2} for all j ≥ 2.

By (6.19) and

m_{i} ∼ M_{i,1}^{−1} >> L_{i,1}^{2−n}M_{i,1} ≥ t_{i}M_{i,1}^{1− \frac{2β_{1}}{n−2}} ,

we have for j ≥ 2,

M_{i,j}^{1− \frac{2β_{j}}{n−2}} >> M_{i,1}^{1− \frac{2β_{1}}{n−2}} .

Hence, if β_{1} ≤ \frac{n−2}{2}, we have β_{j} < \frac{n−2}{2} for all j ≥ 2. If β_{j} ≥ \frac{n−2}{2} for j ≥ 2, then β_{1} > \frac{n−2}{2} and for j ≥ 2

(\frac{2β_{j}}{n − 2} − 1) log M_{i,1} << (\frac{2β_{j}}{n − 2} − 1) log M_{i,j}

<< (\frac{2β_{1}}{n − 2} − 1) log M_{i,1},

which implies β_{1} > β_{j}. q.e.d.

7. Estimates for the Pohozaev identity

As in Section 6, let q_{1}, …, q_{l} denote all the simple-like blowup points, and let q_{l+1}, …, q_{m} denote the non-simple-like blowup points. Also, let M_{i,j}, q_{i,j} and L_{i,j} be defined as in Section 6. Recall m_{i}^{−1} ∼ M_{i,1}. Hereafter, h(x) denotes the limit of M_{i,1}u_{i}(x). By Lemma 6.1,

h(x) = ∑_{j=1}^{l} \frac{μ_{j}}{|x − q_{j}|^{n−2}} ,

where μ_{j} > 0. For 1 ≤ j ≤ l, the regular part of h at q_{j} is denoted by

h_{j}(x) = ∑_{k=1,k≠j}^{l} \frac{μ_{k}}{|x − q_{k}|^{n−2}} .

The Pohozaev identity plays an important role when we come to study the interaction of different blowup points. Therefore, we have to compute the terms appearing in the Pohozaev identity very precisely. For example, we consider the case when q_{j} is not a simple-like blowup point. Then h(x) of Section 6 is smooth at q_{j}. By a direct computation, the Pohozaev identity leads to

\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ_{0}}⟨x − q_{j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} dx

= ∫_{|x−q_{j}|=δ_{0}} (\frac{n − 2}{2}u_{i} \frac{∂u_{i}}{∂ν} − \frac{1}{2}δ_{0}| ▽ u_{i}|^{2}

+δ_{0}|\frac{∂u_{i}}{∂ν} |^{2} + \frac{n − 2}{2n}δ_{0}K_{i}ui^{\frac{2n}{n−2}} ) dσ

= o(1)M_{i,1}^{−2},

because h(x) = lim_{i→+∞} M_{i,1}u_{i} is smooth at q_{j}. However, it does not

show any information about M_{i,j}. The following lemma improves the estimate.

Lemma 7.1. Suppose β_{j} ≥ \frac{2(n−2)}{n} for all j. Then the following hold:

(1) For m ≥ j ≥ l + 1, we have

\begin{matrix}\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ_{0}} ▽K_{i}(x)ui^{\frac{2n}{n−2}} (x)dx \\ = −(1 + o(1) + c_{1}(δ_{0}))(n − 2)|S^{n−1}| ▽ h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1} \\ + o (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + O (δ_{0}^{n−1}t_{i}Mi,1^{\frac{−2n}{n−2}} ) , and \end{matrix} (7.1)

\begin{matrix}\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ_{0}}⟨x − q_{j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx \\ = −(1 + o(1) + c_{2}(δ_{0}))\frac{(n − 2)^{2}}{2} |S^{n−1}|h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1} \\ + o (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + O (δ_{0}^{n−1}t_{i}M_{i,1}^{− \frac{2n}{n−2}} ) , \end{matrix} (7.2)

where o(1) → 0 as i → +∞, c_{1}(δ) and c_{2}(δ) → 0 as δ → 0.

(2) If l ≥ 2, then 1 ≤ j ≤ l,

\begin{matrix}\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ_{0}} ▽K_{i}(x)ui^{\frac{2n}{n−2}} dx \\ = −(1 + o(1) + c_{1}(δ))(n − 2)|S^{n−1}| ▽ h_{j}(q_{j})M_{i,1}^{−1}M_{i,j}^{−1} \\ + o (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + O (δ_{0}^{n−1}t_{i}M_{i,1}^{− \frac{2n}{n−2}} ) , \end{matrix} (7.3)

and

\begin{matrix}\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ_{0}}⟨x − q_{j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} dx \\ = −(1 + o(1) + c_{2}(δ))\frac{(n − 2)^{2}}{2} |S^{n−1}|h_{j}(q_{j})M_{i,1}^{−1}M_{i,j}^{−1} \\ + o (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + O (δ_{0}^{n−1}t_{i}M_{i,1}^{− \frac{2n}{n−2}} ) . \end{matrix} (7.4)

Proof of Lemma 7.1. For each q_{j} considered here, u_{i}(x) satisfies

(7.5)  \{ \begin{matrix}u_{i}(x) ≤ c |x − q_{j}|^{− \frac{n−2}{2}} for |x − q_{j}| ≤ δ_{0} \\ L_{i,j}^{2−n} ∼ m_{i}M_{i,j}^{−1} ∼ t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} , and \\ β_{j} < n − 2, \end{matrix}

due to (6.8) and Corollary 1.3, where m_{i} is the minimum of u_{i} in (6.13). We separate our argument into two cases which require different estimates. Case (I) is when u_{i} loses energy of one bubble only and Case (II) is when u_{i} loses energy of more than one bubble.

For Case (I), let

(7.6)  ũ_{i}(x) = M_{i,j}^{−1}u_{i}(q_{i,j} + M_{i,j}^{− \frac{2}{n−2}} x).

Then by Lemma 3.5 and (7.5), we have

(7.7)  |ũ_{i}(x) − U_{i}(x)| ≤ c L_{i,j}^{−n+2} for |x| ≤ δ_{0}Mi,j^{\frac{2}{n−2}} .

(Note that in this case, q_{i,j} is the local maximum given in (2.12)), where U_{i} is the solution of

(7.8)  ΔU_{i} + K_{i}(q_{i,j})Ui^{\frac{n+2}{n−2}} = 0 in ℝ^{n}

with U_{i}(0) = max_{ℝ^{n}}U_{1}(x) = 1.

For Case (II), we can apply Theorem 2.7 to estimate the difference

between ũ_{i} and a_{i}U_{λ_{i}}. In this case, β_{j} < \frac{n − 2}{2} always.

In the following, let

L_{i,j}^{∗} = min(L_{i,j}, δ_{0}Mi,j^{\frac{2}{n−2}} ) and l_{i} = δ_{0}Mi,j^{\frac{2}{n−2}}

for the simplicity of notations. Let U_{i} denote the solution of (7.8) for Case (I) and denote a_{i}U_{λ_{i}} for Case (II). Set g_{i}(x) = ũ_{i}(x)−U_{i}(x). Then g_{i} satisfies

(7.9)  △g_{i} + pK_{i}(q_{i,j})U_{i}^{p−1}g_{i} = (K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p} + H_{1},

where p = \frac{n + 2}{n − 2}, K̃_{i}(x) = K_{i}(q_{i,j} + M_{i,j}^{− \frac{2}{n−2}} x), and

(7.10)  H_{1}(x) = K_{i}(q_{i,j})[U_{i}^{p} − u_{i}^{p} + pU_{i}^{p−1}g_{i}].

To estimate the term H_{1}, we consider Case (I) first. By Lemma 3.5, we have

(7.11)  |H_{1}(x)| ≤ c_{1}U_{i}^{p−2}|g_{i}|^{2} ≤ c_{2}U_{i}^{p−2} (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )^{2}

when |x| ≤ L_{i,j}^{∗}, and

(7.12)  |H_{1}(x)| ≤ c_{1} (m_{i}M_{i,j}^{−1})^{p}

when L_{i,j}^{∗} ≤ |x| ≤ δ_{0}Mi,j^{\frac{2}{n−2}} .

For Case (II), we apply Theorem 2.7 to obtain

|H_{1}(x)| ≤ c|x|^{− \frac{n+2}{2}} for |x| ≤ R_{i}^{−2}

|H_{1}(x)| ≤ cR_{i}^{−n−2}|x|^{−n−2} for R_{i}^{−2} ≤ |x| ≤ R_{i}^{−1}

|H_{1}(x)| ≤ cU_{i}^{p−2}|g_{i}|^{2} ≤ c_{2}U_{i}^{p−2}(R_{i}^{−2n+4}|x|^{−2n+4} + L_{i,j}^{−2n+4})

for R_{i}^{−1} ≤ |x| ≤ L_{i,j}^{∗}

|H_{1}(x)| ≤ c L_{i,j}^{−(n−2)p} for L_{i,j}^{∗} ≤ |x| ≤ δ_{0}Mi,j^{\frac{2}{n−2}} ,

(7.13)

where R_{i} = L_{i,j}^{γ} and γ = (1 − \frac{2β_{j}}{n − 2})^{−1}.

Let

∂_{λ}U_{i} = −\frac{n − 2}{2}U_{i}(x) − x · ▽U_{i}

and

∂_{λ}ũ_{i}(x) = −\frac{n − 2}{2}ũ_{i}(x) − x · ▽ũ_{i}(x).

144 chiun-chuan chen & chang-shou lin

Multiplying (7.9) by ▽U_{i}, we have

\begin{matrix}∫_{|x|≤l_{i}} ∇U_{i}(△g_{i} + pK_{i}(q_{i,j})Ui^{\frac{4}{n−2}} g_{i}) dx \\ = ∫_{|x|≤l_{i}}(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}∇ũ_{i} dx \\ + ∫_{|x|≤l_{i}}(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}(∇U_{i} − ∇ũ_{i}) dx \\ + ∫_{|x|≤l_{i}} H_{1}(x)∇U_{i} dx \\ ≡ I + II + III. \end{matrix} (7.14)

Multiplying (7.9) by ∂_{λ}U_{i}, we have

\begin{matrix}∫_{|x|≤l_{i}} ∂_{λ}U_{i}(△g_{i} + pK_{i}(q_{i,j})Ui^{\frac{4}{n−2}} g_{i}) dx \\ = ∫_{|x|≤l_{i}}(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}∂_{λ}ũ_{i} dx \\ + ∫_{|x|≤l_{i}}(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}(∂_{λ}U_{i} − ∂_{λ}ũ_{i}) dx \\ + ∫_{|x|≤l_{i}} H_{1}(x)∂_{λ}U_{i} dx \\ ≡ I^{a} + II^{a} + III^{a}. \end{matrix} (7.15)

Let y = M_{i,j}^{− \frac{2}{n−2}} x. By integration by parts,

\begin{aligned}I = & \frac{1}{p + 1} ∫_{|x|≤l_{i}} ∇_{x}K̃_{i}u_{i}^{p+1} dx \\ & + O( ∫_{|x|=l_{i}} |K_{i}(q_{i,j}) − K̃_{i}(x)|u_{i}^{p+1} ds) \\ = & \frac{1}{p + 1}M_{i,j}^{− \frac{2}{n−2}} ∫_{|y|≤δ_{0}} ∇_{y}K_{i}u_{i}^{p+1} dy \\ & + O(δ_{0}^{n−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} (m_{i})^{\frac{2n}{n−2}} ), \end{aligned} (7.16)

By scaling, we have

\begin{aligned}I^{a} = & \frac{1}{p + 1} ∫_{|x|≤l_{i}}⟨x, ∇_{x}K̃_{i}⟩u_{i}^{p+1} dx \\ & + O( ∫_{|x|=l_{i}} l_{i}|K_{i}(q_{i,j}) − K̃_{i}(x)|u_{i}^{p+1} ds) \\ = & \frac{1}{p + 1} ∫_{|y|≤δ_{0}}⟨y, ∇_{y}K_{i}⟩u_{i}^{p+1} dy \\ & + O(δ_{0}^{n−1}t_{i}(m_{i})^{\frac{2n}{n−2}} ). \end{aligned} (7.17)

To estimate the terms II, III, II^{a} and III^{a}, we consider Case (I) first. By (7.7) and integration by parts,

\begin{matrix}|II| ≤ & c ∫_{|x|≤l_{i}}\{|(∇_{x}K̃_{i})u_{i}^{p}| + |(K_{i}(q_{i,j}) − K̃_{i}(x))∇_{x}u_{i}^{p}|\}|(ũ_{i} − U_{i})| dy \\ + c ∫_{|x|=l_{i}} |(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}(ũ_{i} − U_{i})| ds \\ ≤ & c ∫_{|x|≤L_{i,j}^{∗}} \frac{L_{i,j}^{−2n+4}}{(1 + |x|)^{n−β_{j}+3}} dx + O(δ_{0}^{n−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} mi^{\frac{2n}{n−2}} ) \\ ≤ & O [L_{i,j}^{−2n+4} \{ \begin{matrix}1, & β_{j} < 3 \\ log L_{i,j}, & β_{j} = 3 \\ L_{i,j}^{β_{j}−3}, & β_{j} > 3 \end{matrix}\} + δ_{0}^{n−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} mi^{\frac{2n}{n−2}} ] \\ = & o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ) + O(δ_{0}^{n−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} (m_{i})^{\frac{2n}{n−2}} ) \end{matrix} (7.18)

as i → ∞. Here we have used the fact M_{i,j}|q_{j} − q_{i,j}|^{\frac{n−2}{2}} is bounded and the following estimates:

(7.19)  \{ \begin{matrix}ũ_{i}(x) ∼ m_{i}M_{i,j}^{−1} for |x| ≥ L_{i,j}^{∗}, and \\ |K_{i}(q_{i,j}) − K̃_{i}(x)| ≤ c_{1} t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} (1 + |x|^{β_{j}}) \\ | ▽ K̃_{i}(x)| ≤ c_{1} t_{i}M_{i}^{− \frac{2(β_{j} −1)}{n−2}} (1 + |x|^{β_{j}−1}). \end{matrix}

Similarly, we have

\begin{matrix}|II^{a}| ≤ & c ∫_{|x|≤l_{i}} |⟨x, ∇_{x}[(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}]⟩(ũ_{i} − U_{i})| dy \\ + c ∫_{|x|=l_{i}} |(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}(u_{i} − U_{i})| ds \\ ≤ & c ∫_{|x|≤L_{i,j}^{∗}} \frac{L_{i,j}^{−2n+4}}{(1 + |x|)^{n−β_{j}+2}} dx + O(δ_{0}^{n−1}t_{i}mi^{\frac{2n}{n−2}} ) \\ ≤ & O [L_{i,j}^{−2n+4} \{ \begin{matrix}1, & β_{j} < 2 \\ log L_{i,j}, & β_{j} = 2 \\ L_{i,j}^{β_{j}−2}, & β_{j} > 2 \end{matrix}\}] + O(δ_{0}^{n−1}t_{i}mi^{\frac{2n}{n−2}} ) \\ = & o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ) + O(δ_{0}^{n−1}t_{i}(m_{i})^{\frac{2n}{n−2}} ) \end{matrix} (7.20)

as i → ∞. Here we have used the fact that by (7.5) and m_{i} → 0, which implies

(7.21)  L_{i,j}^{−1} ∼ o(M_{i,j}^{− \frac{1}{n−2}} ).

For the terms III and III^{a}, we have by (7.11) and (7.12),

\begin{matrix}|III| ≤ & c ∫_{|x|≤L_{i,j}^{∗}} (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )^{2} \frac{1}{(1 + |x|)^{5}} dx + O(δ_{0}^{n−1}t_{i}M_{i}^{− \frac{2}{n−2}} mi^{\frac{2n}{n−2}} ) \\ ≤ & O [L_{i,j}^{−2n+4} \{ \begin{matrix}1, & n < 5 \\ log L_{i,j}, & n = 5 \\ L_{i,j}^{n−5}, & n > 5 \end{matrix}\} + δ_{0}^{n−1}t_{i}M_{i}^{− \frac{2}{n−2}} mi^{\frac{2n}{n−2}} ] \\ = & o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ) + O(δ_{0}^{n−1}t_{i}M_{i}^{− \frac{2}{n−2}} mi^{\frac{2n}{n−2}} ), \end{matrix} (7.22)

and

\begin{matrix}|III^{a}| ≤ & c ∫_{|x|≤L_{i,j}^{∗}} (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )^{2} \frac{1}{(1 + |x|)^{4}} dx + O(δ_{0}^{n−1}t_{i}mi^{\frac{2n}{n−2}} ) \\ ≤ & O [L_{i,j}^{−2n+4} \{ \begin{matrix}1, & n < 4 \\ log L_{i,j}, & n = 4 \\ L_{i,j}^{n−4}, & n > 4 \end{matrix}\} + δ_{0}^{n−1}t_{i}mi^{\frac{2n}{n−2}} ] \\ = & o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + O(δ_{0}^{n−1}t_{i}mi^{\frac{2n}{n−2}} ) \end{matrix} (7.23)

prescribing scalar curvature on S^{n} 147

as i → ∞. Thus for Case (I), from (7.16), (7.18) and (7.22), we obtain

\begin{matrix}∫_{|x|≤l_{i}} ∇U_{i}(△g_{i} + pK_{i}(q_{i,j})Ui^{\frac{4}{n−2}} g_{i}) dx \\ = \frac{1}{p + 1}M_{i,j}^{− \frac{2}{n−2}} ∫_{|y|≤δ_{1}} ∇_{y}K_{i}u_{i}^{p+1} dy + o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ) \\ + O(δ_{0}^{n−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} (m_{i})^{\frac{2n}{n−2}} ) \end{matrix} (7.24)

as i → ∞. From (7.17), (7.20) and (7.23), we have

\begin{matrix}∫_{|x|≤l_{i}} ∂_{λ}U_{i}(△g_{i} + pK_{i}(q_{i,j})Ui^{\frac{4}{n−2}} g_{i}) dx \\ = \frac{1}{p + 1} ∫_{|y|≤δ_{0}}⟨y, ∇_{y}K_{i}⟩u_{i}^{p+1} dy + o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) \\ + O(δ_{0}^{n−1}t_{i}(m_{i})^{\frac{2n}{n−2}} ) \end{matrix} (7.25)

as i → ∞.

For Case (II), we have 1 < β_{j} < \frac{n − 2}{2} and n > 4. By using (ii) of Theorem 2.7, we decompose II and II^{a} into three terms respectively.

\begin{matrix}II = & ∫_{|x|≤R_{i}^{−1}} + ∫_{R_{i}^{−1}≤|x|≤L_{i,j}^{∗}} + ∫_{L_{i,j}^{∗}≤|x|≤l_{i}} \\ ≡ & II_{1} + II_{2} + II_{3} \end{matrix}

and

\begin{matrix}II^{a} = & ∫_{|x|≤R_{i}^{−1}} + ∫_{R_{i}^{−1}≤|x|≤L_{i,j}^{∗}} + ∫_{L_{i,j}^{∗}≤|x|≤l_{i}} \\ ≡ & II_{1}^{a} + II_{2}^{a} + II_{3}^{a} \end{matrix}

From integration by parts, (7.19), the fact M_{i,j}|q_{j} − q_{i,j}|^{\frac{n−2}{2}} is bounded and Theorem 2.7,

II_{1} = − \frac{1}{p + 1} ∫_{|x|≤R_{i}^{−1}} ∇_{x}K̃_{i}u_{i}^{p+1} dx

+ O[∫_{|x|=R_{i}^{−1}} t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−β_{j}} ds]

+ ∫_{|x|≤R_{i}^{−1}}(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}∇U_{i} dx

= − \frac{1}{p + 1}(∫_{|x|≤R_{i}^{−2}} + ∫_{R_{i}^{−2}≤|x|≤R_{i}^{−1}}) + O(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−n+1−β_{j}})

+ O (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} (1 + R_{i}^{−β_{j}}) ∫_{|x|≤R_{i}^{−1}} |x|^{− \frac{n−2}{2} p}M_{i,j}^{− \frac{2}{n−2}} dx)

=O (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−2β_{j}+2} + t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−n+1−β_{j}} + t_{i}M_{i,j}^{− \frac{2β_{j} +2}{n−2}} R_{i}^{− \frac{n−2}{2}} ) .

II_{1}^{a} =\frac{−1}{p + 1} ∫_{|x|≤R_{i}^{−1}}⟨x, ∇_{x}K̃_{i}⟩u_{i}^{p+1} dx

+ O[ ∫_{|x|=R_{i}^{−1}} R_{i}^{−1}t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−β_{j}} ds]

+ ∫_{|x|≤R_{i}^{−1}}(K_{i}(q_{i,j}) − K̃_{i}(x))u_{i}^{p}∂_{λ}U_{i} dx

=\frac{−1}{p + 1}(∫_{|x|≤R_{i}^{−2}} + ∫_{R_{i}^{−2}≤|x|≤R_{i}^{−1}}) + O(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−n−β_{j}})

+ O (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} (1 + R_{i}^{−β_{j}}) ∫_{|x|≤R_{i}^{−1}} |x|^{− \frac{n−2}{2} p} dx)

=O (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−2β_{j}} + t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{−n−β_{j}} + t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} R_{i}^{− \frac{n−2}{2}} ) .

Recall that R_{i}^{−1} = L_{i,j}^{−γ} = o (Mi,j^{\frac{− \frac{1}{n−2}}{1− \frac{2β_{j}}{n−2}}} ). Thus,

\begin{matrix}II_{1} = o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ) \\ II_{1}^{a} = o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) \end{matrix}

as i → ∞ if β_{j} ≥ \frac{2(n − 2)}{n}. Here is the place we need β_{j} ≥ \frac{2(n − 2)}{n}.

prescribing scalar curvature on S^{n} 149

From integration by parts and Theorem 2.7,

\begin{aligned}|II_{2}| ≤ & c t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ∫_{R_{i}^{−1}≤|x|≤L_{i,j}^{∗}} \frac{|x|^{β_{j}−1}}{(1 + |x|)^{n+2}} \\ & · (R_{i}^{−n+2}|x|^{−n+2} + L_{i,j}^{−n+2}) dx \\ ≤ & c(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )^{2} \{ \begin{matrix}1, & β_{j} < 3 \\ log L_{i,j}, & β_{j} = 3 \\ L_{i,j}^{β_{j}−3}, & β_{j} > 3 \end{matrix}\} \\ = & o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ), and \end{aligned} (7.26)

\begin{aligned}|II_{2}^{a}| ≤ & c t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ∫_{R_{i}^{−1}≤|x|≤L_{i,j}^{∗}} \frac{|x|^{β_{j}}}{(1 + |x|)^{n+2}} \\ & · (R_{i}^{−n+2}|x|^{−n+2} + L_{i,j}^{−n+2}) dx \\ ≤ & c(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )^{2} \{ \begin{matrix}1, & β_{j} < 2 \\ log L_{i,j}, & β_{j} = 2 \\ L_{i,j}^{β_{j}−2}, & β_{j} > 2 \end{matrix}\} \\ = & o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ). \end{aligned} (7.27)

For II_{3} and II_{3}^{a}, we have

\begin{matrix}|II_{3}| ≤ & c t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ∫_{L_{i,j}^{∗}≤|x|≤l_{i}} |x|^{β_{j}−1}(L_{i,j}^{−n+2})^{p+1} dx \\ = & O[δ_{0}^{n+β_{j}−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} (M_{i,j}L_{i,j}^{−n+2})^{p+1}] \\ = & O[δ_{0}^{n−1}t_{i}M_{i,j}^{− \frac{2}{n−2}} (m_{i})^{p+1}], and, \end{matrix} (7.28)

\begin{matrix}|II_{3}^{a}| ≤ & c t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ∫_{L_{i,j}≤|x|≤l_{i}}(L_{i,j}^{−n+2})^{p+1} dx \\ = & O[δ_{0}^{n+β_{j}}t_{i}(M_{i,j}L_{i,j}^{−n+2})^{p+1}] \\ = & O[δ_{0}^{n−1}t_{i}(m_{i})^{p+1}]. \end{matrix} (7.29)

To estimate III, note that n ≥ 4 and then

|III| ≤c [∫_{|x|≤R_{i}^{−2}} |x|^{− \frac{n+2}{2}} dx + ∫_{R_{i}^{−2}≤|x|≤R_{i}^{−1}} R_{i}^{−n−2}|x|^{−n−2} dx

+ ∫_{R_{i}^{−1}≤|x|≤L_{i,j}^{∗}} \frac{1}{(1 + |x|)^{5}} (R_{i}^{−2n+4}|x|^{−2n+4} + L_{i,j}^{−2n+4}) dx

+ ∫_{L_{i,j}^{∗}≤|x|≤l_{i}} \frac{L_{i,j}^{−n−2}}{(1 + |x|)^{n−1}} dx]

≤ c[R_{i}^{−n+2} + R_{i}^{−n−2}R_{i}^{4}

+ (R_{i}^{−2n+4} \{ \begin{matrix}R_{i}^{n−4}, & n > 4 \\ log R_{i}, & n = 4 \end{matrix} + L_{i,j}^{−2n+4} \{ \begin{matrix}1, & n < 5 \\ log L_{i,j}, & n = 5 \\ L_{i,j}^{n−5}, & n > 5 \end{matrix} )

+ L_{i,j}^{−n−2}Mi,j^{\frac{2}{n−2}} ],

and

|III^{a}| ≤ c [∫_{|x|≤R_{i}^{−2}} |x|^{− \frac{n+2}{2}} dx + ∫_{R_{i}^{−2}≤|x|≤R_{i}^{−1}} R_{i}^{−n−2}|x|^{−n−2} dx

+ ∫_{R_{i}^{−1}≤|x|≤L_{i,j}^{∗}} \frac{1}{(1 + |x|)^{4}} (R_{i}^{−2n+4}|x|^{−2n+4} + L_{i,j}^{−2n+4}) dx

+ ∫_{L_{k,i}^{∗}≤|x|≤l_{i}} \frac{L_{i,j}^{−n−2}}{(1 + |x|)^{n−2}} dx]

≤ c[R_{i}^{−n+2} + R_{i}^{−n−2}R_{i}^{4}

+ (R_{i}^{−2n+4} \{ \begin{matrix}R_{i}^{n−4}, & n > 4 \\ log R_{i}, & n = 4 \end{matrix} + L_{i,j}^{−2n+4} \{ \begin{matrix}log L_{i,j}, & n = 4 \\ L_{i,j}^{n−4}, & n > 4 \end{matrix} )

+ L_{i,j}^{−n−2}Mi,j^{\frac{2}{n−2}} ].

Since R_{i} = L_{i,j}L_{i,j}^{\frac{2β_{j}}{n−2} /(1− \frac{2β_{j}}{n−2} )} ≥ L_{i,j}Li,j^{\frac{2β_{j}}{n−2}} , L_{i,j}^{−n+2} ∼ t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} and L_{i,j}^{−1} =

o(M_{i,j}^{− \frac{1}{n−2}} ), we have

R_{i}^{−n+2} ≤ L_{i,j}^{−n+2}L_{i,j}^{−2β_{j}} = o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2β_{j}}{n−2}} ), and

R_{i}^{−2n+4} max(R_{i}^{n−4}, log R_{i}) ≤ R_{i}^{−n+2} = o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ),

L_{i,j}^{−2n+4} \{ \begin{matrix}1, & n < 5 \\ log L_{i,j}, & n = 5 \\ L_{i,j}^{n−5}, & n > 5 \end{matrix} ≤ L_{i,j}^{−n+2} \{ \begin{matrix}L_{i,j}^{−n+2}, & n < 5 \\ L_{i,j}^{−3} log L_{i,j}, & n = 5 \\ L_{i,j}^{−3}, & n > 5, \end{matrix}

L_{i,j}^{−2n+4} \{ \begin{matrix}log L_{i,j}, & n = 4 \\ L_{i,j}^{n−4}, & n > 4 \end{matrix} ≤ L_{i,j}^{−n+2} \{ \begin{matrix}L_{i,j}^{−2} log L_{i,j}, & n = 4 \\ L_{i,j}^{−2}, & n > 4, \end{matrix}

L_{i,j}^{−n−2}Mi,j^{\frac{2}{n−2}} ≤ L_{i,j}^{−n+2}o(1)M_{i,j}^{− \frac{4}{n−2}} Mi,j^{\frac{2}{n−2}} .

Putting these estimates together, we have by L_{i,j}^{−n+2} ∼ t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} and

L_{i,j}^{−1} = o(M_{i,j}^{− \frac{1}{n−2}} ),

(7.30)  |III| = o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2} − \frac{2}{n−2}} ), and |III^{a}| = o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )

From (7.26) ∼ (7.30), we obtain (7.24) and (7.25) for Case (II) also.

By Lemma 6.1, after passing to a subsequence of {u_{i}}, M_{i,1}u_{i} converges to h = ∑_{j=1}^{l} \frac{μ_{j}}{|x−q_{j}|^{n−2}} . From integration by parts and the facts

△(∇U_{i}) + pK_{i}(q_{i,j})U_{i}^{p−1}∇U_{i} = 0,
△(∂_{λ}U_{i}) + pK_{i}(q_{i,j})U_{i}^{p−1}∂_{λ}U_{i} = 0,

the left hand sides of (7.24) and (7.25) are equal to

(7.31)  ∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} (\frac{∂g_{i}}{∂r} ∇U_{i} − g_{i} \frac{∂∇U_{i}}{∂r} ) dσ

and

(7.32)  ∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} (\frac{∂g_{i}}{∂r} ∂_{λ}U_{i} − g_{i} \frac{∂(∂_{λ}U_{i})}{∂r}) dσ,

respectively.

For 1 ≤ j ≤ l, since q_{j} is a simple-like blowing-up point, we have

(7.33)  m_{i} ∼ M_{i,j}^{−1}.

When j ≥ l + 1, L_{i,j}M_{i,j}^{− \frac{2}{n−2}} → 0 as i → ∞. Thus,

(7.34)  M_{i,1}^{−1} >> M_{i,j}^{−1}.

152 chiun-chuan chen & chang-shou lin

Now assume j ≥ l + 1. On \{x : |x| = δ_{0}Mi,j^{\frac{2}{n−2}} \},

g_{i} = ũ_{i} + O(M_{i,j}^{−2}) and ∇g_{i} = ∇ũ_{i} + O(M_{i,j}^{−2− \frac{2}{n−2}} ).

By (7.34), we have on \{x : |x| = δ_{0}Mi,j^{\frac{2}{n−2}} \},

\begin{aligned}g_{i}(x) & = (1 + o(1))M_{1,i}^{−1}M_{i,j}^{−1}h(q_{j} + M_{i,j}^{− \frac{2}{n−2}} x), \\ ∇_{x}g_{i}(x) & = (1 + o(1))M_{i,1}^{−1}M_{i,j}^{−1− \frac{2}{n−2}} ∇_{y}h(q_{j} + M_{i,j}^{− \frac{2}{n−2}} x), \end{aligned}

where y = q_{j} + M_{i,j}^{− \frac{2}{n−2}} x. We have

∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} \frac{∂g_{i}}{∂r} ∇U_{i}

= −(1 + o(1) + c_{1}(δ_{0}))\frac{n − 2}{n}|S^{n−1}|∇_{y}h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1− \frac{2}{n−2}} ,

− ∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} g_{i} \frac{∂∇U_{i}}{∂r} = −(1 + o(1) + c_{2}(δ_{0}))

× \frac{(n − 1)(n − 2)}{n}|S^{n−1}|∇_{y}h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1− \frac{2}{n−2}} ,

∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} \frac{∂g_{i}}{∂r} ∂_{λ}U_{i}

= −(o(1) + c_{3}(δ_{0}))\frac{n − 2}{n}|S^{n−1}|∇_{y}h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1},

− ∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} g_{i} \frac{∂∂_{λ}U_{i}}{∂r}

= −(1 + o(1) + c_{4}(δ_{0}))\frac{(n − 2)^{2}}{2} |S^{n−1}|∇_{y}h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1},

where c_{j}(δ_{0}) → 0 as δ_{0} → 0. Hence as i → ∞

\begin{matrix}− ∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} (\frac{∂g_{i}}{∂r} ∇U_{i} − g_{i} \frac{∂∇U_{i}}{∂r} ) dσ \\ = −(1 + o(1) + c(δ_{0}))(n − 2)|S^{n−1}|∇_{y}h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1− \frac{2}{n−2}} , \end{matrix} (7.35)

and

\begin{matrix}∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} (\frac{∂g_{i}}{∂r} ∂_{λ}U_{i,λ} − g_{i} \frac{∂∂_{λ}U_{i,λ}}{∂r} ) dσ \\ = −(1 + o(1) + c̃(δ_{0}))\frac{(n − 2)^{2}}{2} |S^{n−1}|h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1}, \end{matrix} (7.36)

where c(δ_{0}), c̃(δ_{0}) → 0 as δ_{0} → 0. Now from (7.24), (7.25), (7.35) and (7.36), we obtain (7.1) and

\frac{n − 2}{2n} ∫_{|x−q_{i,j}|≤δ_{0}}⟨x − q_{i,j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx

= −(1 + o(1) + c_{2}(δ_{0}))\frac{(n − 2)^{2}}{2} |S^{n−1}|h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1}

+ o (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + O (δ_{0}^{n−1}t_{i}M_{i,1}^{− \frac{2n}{n−2}} ) .

By (7.1) and the fact M_{i,j}|q_{j} − q_{i,j}|^{\frac{n−2}{2}} is bounded,

∫_{|x−q_{j}|≤δ_{0}}⟨x − q_{j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx

= ∫_{|x−q_{j}|≤δ_{0}}⟨x − q_{i,j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx

+ ∫_{|x−q_{j}|≤δ_{0}}⟨q_{i,j} − q_{j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx

= ∫_{|x−q_{i,j}|≤δ_{0}}⟨x − q_{i,j}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx

+ o(1)M_{i,1}^{−1}M_{i,j}^{−1} + o (t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) + o (δ_{0}^{n−1}t_{i}M_{i,1}^{− \frac{2n}{n−2}} ) .

We obtain (7.2).

When l ≥ 2 and 1 ≤ j ≤ l, after passing to a subsequence of {u_{i}}, we

have ∞ > lim L_{i,j}M_{i,j}^{− \frac{2}{n−2}} = c > 0. Therefore on \{x : |x| = δ_{0}Mi,j^{\frac{2}{n−2}} \},

we have

g_{i}(x) ∼ U_{i}(x),

g_{i}(x) = (1 + o(1)) [M_{i,1}^{−1}M_{i,j}^{−1}h(q_{j} + M_{i,j}^{− \frac{2}{n−2}} x) − \frac{1}{|x|^{n−2}} ] ,

∇_{x}g_{i}(x) = (1 + o(1)) [M_{1,i}^{−1}M_{i,j}^{−1}∇_{y}h(q_{j} + M_{i,j}^{− \frac{2}{n−2}} x) + \frac{(n − 2)x}{|x|^{n}}] ,

From these estimates, we obtain

\begin{matrix}∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} (\frac{∂g_{i}}{∂r} ∇U_{i} − g_{i} \frac{∂∇U_{i}}{∂r} ) dσ \\ = −(1 + o(1) + c(δ_{0}))(n − 2)|S^{n−1}|∇_{y}h_{j}(q_{j})M_{i,1}^{−1}M_{i,j}^{−1− \frac{2}{n−2}} , \end{matrix} (7.37)

and

\begin{matrix}∫_{|x|=δ_{0}M \frac{2}{n−2} ; i,j} (\frac{∂g_{i}}{∂r} ∂_{λ}U_{i,λ} − g_{i} \frac{∂∂_{λ}U_{i,λ}}{∂r} ) dσ \\ = −(1 + o(1) + c̃(δ_{0}))\frac{(n − 2)^{2}}{2} |S^{n−1}|h_{j}(q_{j})M_{i,1}^{−1}M_{i,j}^{−1}, \end{matrix} (7.39)

where h_{j} = h − \frac{μ_{j}}{|x−q_{j}|^{n−2}} , c(δ_{0}), c̃(δ_{0}) → 0 as δ_{0} → 0. Putting these estimates into (7.24) and (7.25), we obtain (7.3) and (7.4). q.e.d.

8. Isolated blowing up

Proof of Theorem 1.3. Suppose that there exists a blowup point q which is not isolated. Then by Theorem 2.1, Corollary 2.3, Theorem 2.4, Theorem 2.5 and (6.8), q is the only simple-like blowup point. Thus

l = 1, q = q_{1} and β_{1} < \frac{n − 2}{2}. By (ii) of Theorem 2.5,

(8.1)  u_{i}(x) ≤ c|x − q_{1}|^{− \frac{n−2}{2}}

for x ∈ B_{i} = {x | |x − q_{1}| ≤ δ|q_{1} − q_{i,1}|}, where c is independent of δ if

δ ≤ \frac{1}{2}, and

(8.2)  u_{i}(x) ≤ c_{1}U_{λ_{i}}(x − q_{i,1})

for x ∉ B_{i}, where λ_{i} = u_{i}(q_{i,1})^{− \frac{2}{n − 2}} and c_{1} = c_{1}(δ).

In particular, we have

(8.3)  m_{i} ∼ M_{i,1}^{−1} = u_{i}(q_{i,1})^{−1}.

Now, let {q_{j}}_{j=2}^{m} be the other blowup points, and Ω_{i} = ⋃_{j=1}^{m} B(q_{j}, δ_{0}).

Then, (8.3) implies

u_{i}(x) ≤ cM_{i,1}^{−1}(1 + |x|)^{2−n}

prescribing scalar curvature on S^{n} 155

for x ∉ Ω_{i}. By the Pohozaev identity,

(8.4)  ∫_{ℝ^{n}}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx = 0, and
(8.5)  ∫_{ℝ^{n}}⟨e_{i}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx = 0,

where e_{i} = \frac{▽K̂(q_{i,1})}{| ▽ K̂(q_{i,1})|}. By (8.1) and (8.2), we have

\begin{matrix}|∫_{|x−q_{1}|≤δ_{0}}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ ≤ c t_{i} \{∫_{B_{i}} |x − q_{1}|^{β_{1}−n}dx \\ + ∫_{B(q_{1},δ_{0})\B_{i}} |x − q_{1}|^{β_{1}}Uλ_{i}^{\frac{2n}{n−2}} (x − q_{i,1})dx\} \\ ≤ c t_{i}|q_{i,1} − q_{1}|^{β_{1}}, \end{matrix} (8.6)

where lim_{i→+∞} (u(q_{i,1})|q_{i,1} − q_{1}|^{\frac{n−2}{2}} ) = +∞ is used. As in (4.29), we can obtain the lower bound

(8.7)  ∫_{B(q_{1},δ_{0})}⟨e_{i}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx ≥ c_{2} t_{i}|q_{i,1} − q_{1}|^{β_{1}−1},

provided that δ is small enought.

On the other hand, we have

(8.8)  |∫_{ℝ^{n}\Ω_{i}}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| = O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} ,
(8.9)  |∫_{ℝ^{n}\Ω_{i}}⟨e_{i}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| = O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} ,

and by (7.2) of Lemma 7.1,

\begin{matrix}− ∑_{j=2}^{m} ∫_{B(q_{j},δ_{0})}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx \\ = ∑_{j=2}^{m} [− ∫_{B(q_{j},δ_{0})}⟨q_{j} − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx \\ − ∫_{B(q_{j},δ_{0})}⟨x − q_{j}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx] \\ = (1 + ϵ)(n − 2)|S^{n−1}| ∑_{j=2}^{m}(⟨q_{j} − q_{1}, ▽h(q_{j})⟩ \\ + \frac{n − 2}{2}h(q_{j})) · M_{i,1}^{−1}M_{i,j}^{−1} + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} , \\ = −(1 + ϵ)\frac{(n − 2)^{2}}{2} |S^{n−1}| ∑_{j=2}^{m} h(q_{j})M_{i,1}^{−1}M_{i,j}^{−1} + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} \end{matrix} (8.10)

with some small ϵ. By (7.3) and (8.10),

\begin{aligned}∑_{j=2}^{m} & |∫_{B(q_{j},δ_{0})}⟨e_{i}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ & ≤ c ∑_{j=2}^{m} M_{i,1}^{−1}M_{i,j}^{−1} + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} \\ & ≤ c_{1} ∑_{j=2}^{m} ∫_{B(q_{j},δ_{0})}⟨x − q_{1}, ▽K_{i}(x)⟩ui^{\frac{2n}{n−2}} (x)dx \\ & + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} . \end{aligned} (8.11)

Note that h(x) = \frac{μ_{1}}{|x − q_{1}|^{n−2}} . Hence, we can use the following identity in (8.10)

(8.12)  ⟨q_{j} − q_{1}, ▽h(q_{j})⟩ = −(n − 2)h(q_{j}).

By (8.5) ∼ (8.11), we have

\begin{matrix}c_{3} t_{i}|q_{i,1} − q_{1}|^{β_{1}−1} \\ ≤ ∫_{B(q_{1},δ_{0})}⟨e_{i}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx \\ ≤ c ∑_{j=2}^{m} M_{i,1}^{−1}M_{i,j}^{−1} + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} \\ ≤ c_{1} |∫_{ℝ^{n}\B(q_{1},δ_{0})}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} \\ = c_{1} |∫_{B(q_{1},δ_{0})}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} \\ ≤ c_{2} \{t_{i}|q_{i,1} − q_{1}|^{β_{1}} + t_{i}M_{i,1}^{− \frac{2n}{n−2}} \} . \end{matrix} (8.13)

Therefore,

(8.14)  |q_{i,1} − q_{1}|^{β_{1}−1} ≤ c M_{i,1}^{− \frac{2n}{n−2}} .

Recall that β_{1} < \frac{n − 2}{2}. Then (8.14) yields a contradiction to the as-

sumption that lim_{i→+∞}(|q_{i,1} − q_{1}|Mi,1^{\frac{2}{n−2}} ) = +∞. We have proved that every blowup point must be isolated.

To prove the second part, let us assume that q_{j} is a blowup point with β_{j} < n + 1 and lim_{i→+∞} sup_{B(q_{j},δ_{0})}(u_{i}(x)|x − q_{j}|^{\frac{n−2}{2}} ) = +∞. Since (ii) of Theorem 2.5 is excluded, q_{j} must be a simple blowup point. Thus, u_{i} lose the energy of only one bubble at q_{j} and then, q_{i,j} is the local maximum point defined by (6.1). By the assumptions, we have

(8.15)  lim_{i→+∞} (|q_{i,j} − q_{j}|Mi,j^{\frac{2}{n−2}} ) = +∞ and
(8.16)  u_{i}(x) ≤ c U_{λ_{i}}(x − q_{i,j}) for |x − q_{j}| ≤ δ_{0},

where λ_{i} = M_{i,j}^{− \frac{2}{n−2}} . Applying Theorem 2.2, (8.14) implies

lim_{i→+∞} L_{i,j}M_{i,j}^{− \frac{2}{n−2}} = +∞.

Hence q_{j} is the only simple-like blowup point. By repeating the same argument as above, we can reach the same conclusion as (8.14), that is,

|q_{i,j} − q_{j}|^{β_{j}−1} ≤ c M_{i,j}^{− \frac{2n}{n−2}}

for some constant c > 0. Since β_{j} < n + 1, the inequality yields a contradiction to (8.15). Hence (1.20) is proved. q.e.d.

Set q_{i,j} to be the local maximum point of u_{i} defined by (1.21) and

ξ_{i} = Mi,j^{\frac{2}{n−2}} (q_{i,j} − q_{j}). Let ξ be any limit of ξ_{i}. Then we claim:

Lemma 8.1. ξ satisfies

(8.17)  ∫_{ℝ^{n}} ▽Q_{j}(y + ξ)U1^{\frac{2n}{n−2}} (y)dy = 0.

Proof. If L_{i}(q_{i,j})M_{i,j}^{− \frac{2}{n−2}} is bounded where M_{i,j} = u_{i}(q_{i,j}), then (8.17) is proved by Theorem 2.2. So, we may assume

lim_{i→+∞} L_{i}(q_{i,j})M_{i,j}^{− \frac{2}{n−2}} = +∞.

Thus, q_{j} is the only simple-like blowup points. Hence Lemma 7.1 can be applied to all blowup point q_{k}, k ≠ j. For the simplicity, we assume j = 1. By using (7.2) of Lemma 7.1, (8.4), (8.5), (8.10) and (8.11), we have the same conclusion as (8.13), i.e.,

\begin{matrix}|∫_{B(q_{1},δ_{0})} ▽K_{i}(x)ui^{\frac{2n}{n−2}} (x)dx| \\ ≤ c_{1} |∫_{B(q_{1},δ_{0})}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ + O(1)t_{i}M_{i,1}^{− \frac{2n}{n−2}} \\ ≤ c_{2} t_{i} \{∫_{B(q_{1},δ_{0})} |x − q_{1}|^{β_{1}}ui^{\frac{2n}{n−2}} (x)dx + Mi,1^{\frac{−2n}{n−2}} \} \\ ≤ c_{3} t_{i} \begin{cases}M_{i,1}^{− \frac{2n}{n−2}} log(M_{i,1}) if β_{1} = n \\ M_{i,1}^{− \frac{2β^{∗}_{1}}{n−2}} if β_{1} ≠ n, \end{cases} \end{matrix} (8.18)

where β_{1}^{∗} = min(β_{1}, n). On the other hand, by the scaling and (1.20), we have

\begin{matrix}∫_{B(q_{1},δ_{0})} ▽K_{i}(x)ui^{\frac{2n}{n−2}} (x)dx \\ = (∫_{ℝ^{n}} ▽Q_{1}(y + ξ)U1^{\frac{2n}{n−2}} (y)dy + o(1)) t_{i}M_{i,1}^{− \frac{2(β_{1}−1)}{n−2}} . \end{matrix} (8.19)

Since β_{1} − 1 < n, (8.17) follows from (8.18) and (8.19) readily. q.e.d.

9. Asymptotic behaviors of M_{i,j}

Proof of Theorem 1.4. We first prove (1.22). By (1.20) in Theorem 1.3, we only need to consider the case β_{j} ≥ n+1. Suppose β_{j} ≥ n+1 and (1.22) does not hold. Then we have |q_{i,j} − q_{j}|^{\frac{n−2}{2}} M_{i,j} → ∞ as

i → ∞. By Theorem 2.2 and (6.7), j = l = 1. Let e_{i} = \frac{∇K_{i}(q_{i,1})}{|∇K_{i}(q_{i,1})|}. By

(7.1) and (7.2),

∫_{|x−q_{1}|≤δ_{0}}⟨x − q_{1}, ∇K_{i}⟩ui^{\frac{2n}{n−2}} dx

≥ c_{1} ∑_{k=2}^{m}\{M_{i,1}^{−1}M_{i,k}^{−1} + o(t_{i}Mi,k^{\frac{−2β_{1}}{n−2}} )\}

+ O(t_{i}Mi,1^{\frac{−2n}{n−2}} ),

for some c_{1} > 0 and

\begin{matrix}∫_{|x−q_{1}|≤δ_{0}}⟨e_{i}, ∇K_{i}⟩ui^{\frac{2n}{n−2}} dx \\ = O(∑_{j=2}^{s}\{M_{i,1}^{−1}M_{i,j}^{−1} + o(t_{i}Mi,k^{\frac{−2β_{1}}{n−2}} )\}) + O(t_{i}Mi,1^{\frac{−2n}{n−2}} ) \end{matrix}

as i → ∞. On the other hand,

\begin{matrix}∫_{|x−q_{1}|≤δ_{0}}⟨x − q_{1}, ∇K_{i}⟩ui^{\frac{2n}{n−2}} dx \\ ≤ c t_{i}|q_{i,1} − q_{1}|^{β_{1}} + c t_{i}M_{1,i}^{− \frac{2n}{n−2}} \end{matrix}

and

∫_{|x−q_{1}|≤δ_{0}}⟨e_{i}, ∇K_{i}⟩ui^{\frac{2n}{n−2}} dx

≥ ct_{i}|q_{i,1} − q_{1}|^{β_{1}−1} − c_{1} \begin{cases}t_{i}M_{i,1}^{− \frac{2n}{n−2}} (− log |q_{i,1} − q_{1}|) & β_{1} = n + 1 \\ t_{i}M_{i,1}^{− \frac{2n}{n−2}} & β_{1} > n + 1 \end{cases}

≥ ct_{i}|q_{i,1} − q_{1}|^{β_{1}−1} − c_{1} \begin{cases}t_{i}M_{i,1}^{− \frac{2n}{n−2}} (log M_{i,1}) & β_{1} = n + 1 \\ t_{i}M_{i,1}^{− \frac{2n}{n−2}} & β_{1} > n + 1 \end{cases}

for some c > 0 and c_{1} > 0. Putting the estimates above together, we obtain

|q_{i,1} − q_{1}|^{β_{1}−1} ≤ c|q_{i,1} − q_{1}|^{β_{1}} + c \begin{cases}t_{i}M_{i,1}^{− \frac{2(β_{1}−1)}{n−2}} (log M_{1,i}) & β_{1} = n + 1 \\ t_{i}M_{1,i}^{− \frac{2n}{n−2}} & β_{1} > n + 1. \end{cases}

Since |q_{i,1} − q_{1}| → 0 as i → ∞, we conclude

|q_{i,1} − q_{1}| = O(M_{i,1}^{− \frac{2}{n−2}} (log M_{i,1})^{\frac{1}{n}}) for β_{1} = n + 1
|q_{i,1} − q_{1}| = O(M_{i,1}^{− \frac{2}{n−2} \frac{n}{β_{1}−1}} )  for β_{1} > n + 1.

From these, we conclude that (1.22) holds.

Now we prove m ≥ 2. Suppose m = 1. Then q = q_{1} is the only blowup point and it must be simple. If β_{1} < n, then by Theorem 1.3,

(9.1)  |q_{i,1} − q_{1}| ≤ c M_{i,1}^{− \frac{2}{n−2}} .

By the Pohozaev identity,

\begin{matrix}|∫_{|x−q_{1}|≤δ_{0}}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ = |∫_{|x−q_{1}|>δ_{0}}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx| \\ ≤ c_{1}t_{i}M_{i,1}^{− \frac{2n}{n−2}} . \end{matrix} (9.2)

By scaling and (9.1), it is not difficult to see that the left hand side of (9.2) is

|∫_{|x−q_{1}|≤δ_{0}}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx|

= t_{i}M_{i,1}^{− \frac{2β_{1}}{n−2}} |∫_{ℝ^{n}} Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy|

≥ c_{1} t_{i}M_{i,1}^{− \frac{2β_{1}}{n−2}} .

for some c_{2} > 0, where ξ = lim_{i→+∞} Mi,1^{\frac{2}{n−2}} (q_{i,1} − q_{1}). Thus, it yields a

contradiction to β_{1} < n.

If β_{1} = n, the left hand side of (9.2) is greater than c_{1}M_{i,1}^{− \frac{2β_{1}}{n−2}} log M_{i,1} for some c_{1} > 0, which also yields a contradiction. Now we assume β_{1} > n. The Pohozaev identity gives

(9.3)  ∫_{ℝ^{n}}⟨x − q_{1}, ▽K̂(x)⟩ui^{\frac{2n}{n−2}} (x)dx = 0.

Since M_{i,1}u_{i}(x) → \frac{μ_{1}}{|x − q_{1}|^{n−2}} for some μ_{1} > 0 and

M_{i,1}u_{i}(x) ≤ \frac{c}{|x − q_{i,1}|^{n−2}}

for some constant c > 0, by multipling both sides of (9.3) by Mi,1^{\frac{2n}{n−2}} and using (1.22), we obtain

∫_{ℝ^{n}}⟨x − q_{1}, ▽K̂(x)⟩|x − q_{1}|^{−2n}dx = 0,

a contradiction to our assumptions. Hence m ≥ 2 is proved. Let

{q_{1}, …, q_{l}, q_{l+1}, …, q_{m}} be indexed by the ordering β_{1} = … = β_{l} > β_{l+1} = … = β_{l_{1}} > β_{l_{1}+1} ≥ … ≥ β_{m} as in Lemma 6.2. To find the asymptotic behavior of M_{i,j}, we consider the case l = 1 first. Let h_{i}(x) = M_{i,1}u_{i}(x). Then h_{i}(x) converges to μ_{1}|x − q_{1}|^{2−n} for some μ_{1} > 0 by Lemma 6.1. To compute μ_{1}, we use

\begin{aligned}μ_{1}(n − 2)|S^{n−1}| & = lim_{i→+∞} (− ∫_{|x−q_{1}|=δ_{0}} \frac{∂h_{i}}{∂ν} dσ) \\ & = lim_{i→+∞} M_{i,1} ∫_{|x−q_{1}|≤δ_{0}} K_{i}(x)ui^{\frac{n+2}{n−2}} (x)dx \\ & = n(n − 2) ∫_{ℝ^{n}} U1^{\frac{n+2}{n−2}} (y)dy = (n − 2)|S^{n−1}|. \end{aligned} (9.4)

From (9.4), μ_{1} = 1, that is,

(9.5)  h(x) = \frac{1}{|x − q_{1}|^{n−2}} .

In (7.2), since after passing to a subsequence, the left hand side is of

order O(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ), we may drop the term o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} ) in the right hand side. Let Ω = ∪_{j=1}^{m}B(q_{j}, δ_{0}). When β_{1} < n, together with (7.1) and (7.2), the Pohozave identity implies

\begin{matrix}\frac{n − 2}{2n} ∫_{B(q_{1},δ_{0})}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx \\ = − ∑_{j=2}^{m} \frac{n − 2}{2n} (∫_{B(q_{j},δ_{0})}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} dx \\ + ∫_{ℝ^{n}\Ω}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} dx) \\ = −(1 + o(1) + c_{1}(d_{0}))\frac{(n − 2)^{2}}{2} |S^{n−1}| \\ ∑_{j=2}^{m} & (|q_{1} − q_{j}|^{2−n}M_{i,1}^{−1}M_{i,j}^{−1} + o(t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} )) \\ + O(t_{i}M_{i,1}^{− \frac{2n}{n−2}} ), \end{matrix} (9.6)

where (9.5) is used. On the other hand, the left hand side of (9.6) is equal to

β_{1}(\frac{n − 2}{2n})t_{i}M_{i,1}^{− \frac{2β_{1}}{n−2}} (∫_{ℝ^{n}} Q_{1}(y + ξ)U1^{\frac{2n}{n−2}} (y)dy) (1 + o(1)).

Thus, we have

(9.7)  t_{i}M_{i,1}^{− \frac{2β_{1}}{n−2}} = ∑_{j=2}^{l_{1}} η_{1,j}M_{i,1}^{−1}M_{i,j}^{−1}(1 + o(1)),

where

(9.8)  η_{1,j} = \frac{n(n − 2)|S^{n−1}||q_{1} − q_{j}|^{−n+2}}{β_{1}| ∫ Q(y + ξ)U1^{\frac{2n}{n−2}} (y)dy|} .

prescribing scalar curvature on S^{n} 163

When β_{1} = n, we have

t_{i}M_{i,1}^{− \frac{2n}{n−2}} log M_{i,1} = ∑_{j=2}^{l_{1}} η_{1,j}M_{i,1}^{−1}M_{i,j}^{−1},

where

(9.9)  η_{1,j} = \frac{(n − 2)|S^{n−1}||q_{1} − q_{j}|^{−n+2}}{|∫_{S^{n−1}} Q(y)dσ|}

by noting that the left hand side of (9.6) will give

n(\frac{n − 2}{2n})t_{i}M_{i,1}^{− \frac{2n}{n−2}} (∫_{S^{n−1}} Q(y)dσ) log M_{i,1}(1 + o(1)).

When β_{1} > n, we have

\begin{matrix}lim_{i→+∞} & (Mi,1^{\frac{2n}{n−2}} ∫_{ℝ^{n}\ ⋃_{j=2}^{m} B(q_{j},δ)}⟨x − q_{1}, ▽K̂⟩ui^{\frac{2n}{n−2}} dx) \\ = ∫_{ℝ^{n}\ ⋃_{j=2}^{m} B(q_{j},δ)}⟨x − q_{1}, ▽K̂⟩|x − q_{1}|^{−2n}dx \end{matrix}

for any δ > 0. By letting δ → 0, we have

t_{i}M_{i,1}^{− \frac{2n}{n−2}} = (1 + o(1)) ∑_{j=1}^{l_{1}} η_{1,j}M_{i,1}^{−1}M_{i,j}^{−1},

where

(9.10)  η_{1,j} = \frac{n(n − 2)|S^{n−1}||q_{1} − q_{j}|^{−n+2}}{|∫_{ℝ^{n}}⟨x − q_{1}, ▽K̂⟩|x − q_{1}|^{−2n}dx|}.

Thus, (1.24) is proved.

To prove (1.25), we note β_{j} < n − 2 for j ≥ 2. By (1.24), tM_{1,i}^{− \frac{2n}{n−2}} = O(M_{i,1}^{−1}M_{i,j}^{−1}). Hence if we let d tend to 0 suitably, (7.2) implies

β_{j} \frac{n − 2}{2n}t_{i}M_{i,j}^{− \frac{2β_{j}}{n−2}} |∫_{ℝ^{n}} Q_{j}(y + ξ)U1^{\frac{2n}{n−2}} (y)dy|

= (1 + o(1))\frac{n − 2}{2n} ∫_{B(q_{j},δ)}⟨x − q_{1}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} dx

= (1 + o(1))\frac{(n − 2)^{2}}{2} |S^{n−1}||q_{1} − q_{j}|^{−n+2}M_{i,1}^{−1}M_{i,j}^{−1},

which is (1.25).

To prove (1.28), it is enough to prove (1.28) for j = 1. As the proof of (9.5), we have

(9.11)  h(x) = ∑_{k=1}^{l} \frac{μ_{k}}{|x − q_{k}|^{n−2}}

and

(9.12)  μ_{1} = 1.

Since M_{i,k}u_{i}(x) also converges to h̃(x) where

h̃(x) = \frac{1}{|x − q_{k}|^{n−2}} + ∑_{j≠k}^{l} \frac{μ̃_{j}}{|x − q_{j}|^{n−2}} ,

we have

(9.13)  μ_{k} = lim_{i→+∞} \frac{M_{i,1}}{M_{i,k}} .

Since l ≥ 2, we recall that Theorem 2.2 and Lemma 6.2 imply β_{j} < n−2 for all j. By (7.4), we have for j = 1

β_{1} \frac{n − 2}{2n}t_{i}M_{i,1}^{− \frac{2β_{1}}{n−2}} (∫_{ℝ^{n}} Q_{1}(y + ξ)U1^{\frac{2n}{n−2}} (y)dy) (1 + o(1))

= −\frac{(n − 2)^{2}}{2} |S^{n−1}|h_{1}(q_{1})M_{i,1}^{−1}M_{i,1}^{−1}

= −\frac{(n − 2)^{2}}{2} |S^{n−1}| (∑_{j=2}^{l} \frac{μ_{j}}{|q_{j} − q_{1}|^{n−2}} M_{i,1}^{−1}) M_{i,1}^{−1}

= −\frac{n − 2}{2}|S^{n−1}| ∑_{j=2}^{l} \frac{1}{|q_{j} − q_{1}|^{n−2}} M_{i,1}^{−1}M_{i,j}^{−1},

where the last equality comes from (9.13). Clearly, (1.28) follows immediately. Identity (1.29) also follows from (7.2) and (9.13) immediately. Thus, the proof of Theorem 1.4 is complete. q.e.d.

10. Apriori estimates

In this final section, we are going to prove the apriori bound of Theorem 1.1. Here, we consider a sequence of blowing up solutions of Equation (1.3) with K = K_{i} more general than the one in previous sections. We assume that K_{i} converges to a function, say K, in C^{1}, and for simplicity, assume K_{i} has the same set of critical points {q_{1}, q_{2}, …, q_{N}}. Let Q_{i,j}(y) be the homogeneous function in (K0) for K_{i} at q_{j}. Assume that K satisfies (K0) ∼ (K1) and Q_{i,j}(y) → Q_{j}(y) in C^{1}. Let β_{i,j} be the degree of Q_{i,j} and

(10.1)  β_{j} = lim_{i→+∞} β_{i,j} > \frac{n − 2}{2}

for all j such that q_{j} ∈ Γ^{−}, where Γ^{−} is defined in Section 1.

By results of [8], [9], it is known that any blowup point is isolated. Without loss of generality, the point +∞ is assumed not to be a blowup point. Let {q_{1}, …, q_{m}} be the set of blowup points such that q_{1}, …, q_{l} are all simple blowup points and q_{l+1}, …, q_{m} are non-simple blowup points. Following the same proof of Lemma 6.1 and part (i) of Theorem 1.4, we have l ≥ 1 and m ≥ 2. Another important result in [8], [9] is that q_{j} is simple if and only if β_{j} ≥ n − 2. This result follows from Theorem 1.3 of [8], [9] when β_{j} ≠ n − 2. For the case β_{j} = n − 2, it follows from the following lemma similar to Lemma 7.1.

Lemma 10.1. For 2 ≤ j ≤ m if l = 1 and 1 ≤ j ≤ m if l ≥ 2, we have

\begin{matrix}\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ} ▽K_{i}(x)ui^{\frac{2n}{n−2}} (x)dx \\ = −(1 + o(1) + c_{1}(δ)) \\ · (n − 2)|S^{n−1}| (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{2}} \\ ▽ h̃_{j}(q_{j})M̂_{i,1}^{−1}M̂_{i,j}^{−1− \frac{2}{n−2}} \\ + o(M̂_{i,j}^{− \frac{2β_{i,j}}{n−2}} ) + O(δ^{n−1}M̂_{i,1}^{− \frac{2n}{n−2}} ), \end{matrix} (10.2)

\begin{matrix}\frac{n − 2}{2n} ∫_{|x−q_{j}|≤δ}⟨x − q_{j}, ▽K_{i}⟩ui^{\frac{2n}{n−2}} (x)dx \\ = −(1 + o(1) + c_{2}(δ)) \\ · \frac{(n − 2)^{2}}{2} |S^{n−1}| (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{2}} h̃_{j}(q_{j})M̂_{i,1}^{−1}M̂_{i,j}^{−1} \\ + o(M̂_{i,j}^{− \frac{2β_{i,j}}{n−2}} ) + O(δ^{n−1}M̂_{i,1}^{− \frac{2n}{n−2}} ), \end{matrix} (10.3)

where

h(x) = lim_{i→+∞} M̂_{i,1}u_{i}(x) = ∑_{j=1}^{l} \frac{μ_{j}}{|x − q_{j}|^{n−2}} ,

h̃_{j}(x) = h(x) if j ≥ l + 1, and

h̃_{j}(x) = h(x) − \frac{μ_{j}}{|x − q_{j}|^{n−2}}

if 1 ≤ j ≤ l and l ≥ 2.

Here, M̂_{i,j} and q_{i,j} are the local maximum and a local maximum point of u_{i} near q_{j} satisfying

(10.4)  M̂_{i,j} = u_{i}(q_{i,j}) = max_{|x−q_{j}|≤δ_{0}} u_{i}(x)

We can prove Lemma 10.1 by the same argument as in Lemma 7.1, but

the proof is simpler because β_{j} > \frac{n − 2}{2} for all j. The position q_{i,j} also satisfies (1.22) for some constant c > 0. When β_{j} ≤ n − 2, it was proved in [9]. When β_{j} > n − 2, it is a consequence of Lemma 10.1, as shown in the previous sections.

Another important consequence of Lemma 10.1 is the asymptotic behavior of M̂_{i,j} which is similar to Theorem 1.4.

Theorem 10.2. Assume that K satisfies (K0) and (K1) and β_{j} > \frac{n−2}{2} for all q_{j} ∈ Γ^{−}. Let q_{1}, …, q_{l} are simple blowup points and q_{l+1}, …, q_{m} are not simple blowup points. Set

M_{i,j} = (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{4}} M̂_{i,j}

where M̂_{i,j} is the local maximum in (10.4). Then m ≥ 2, l ≥ 1, β_{1} = … = β_{l} > β_{j} for j ≥ l + 1, and the following hold:

(i) If l = 1 and q_{j} is indexed by the ordering β_{1} > β_{2} = … = β_{l_{1}} >

prescribing scalar curvature on S^{n} 167

β_{l_{1}+1} ≥ … ≥ β_{m}, then

\begin{matrix}\begin{matrix}|b_{1}| (\frac{n(n−2)}{K(q_{1})})^{\frac{n}{2}} M_{i,1}^{− \frac{2β^{∗}_{1}}{n−2}} & if β_{1} ≠ n \\ (\frac{n(n−2)}{K(q_{1})})^{\frac{n}{2}} M_{i,1}^{− \frac{2β_{1}}{n−2}} log M_{i,1} & if β_{1} = n \end{matrix} & \} \\ = (1 + o(1))n(n − 2)|S^{n−1}| ∑_{j=2}^{l_{1}} (\frac{n(n − 2)}{K(q_{1})})^{\frac{n−2}{4}} \\ · (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{4}} |q_{1} − q_{j}|^{−n+2}M_{i,1}^{−1}M_{i,j}^{−1} \end{matrix} (10.6)

and

\begin{matrix}|b_{j}| (\frac{n(n − 2)}{K(q_{j})})^{\frac{n}{2}} M_{i,j}^{− \frac{2β_{j}}{n−2}} \\ = (1 + o(1))n(n − 2)|S^{n−1}| (\frac{n(n − 2)}{K(q_{1})})^{\frac{n−2}{4}} \\ · (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{4}} M_{i,1}^{−1}M_{i,j}^{−1} \end{matrix} (10.7)

for 2 ≤ j ≤ m. (ii) If l ≥ 2, then β_{1} = … = β_{l} = n − 2,

\begin{matrix}|b_{j}| (\frac{n(n − 2)}{K(q_{j})})^{\frac{n}{2}} M_{i,j}^{−2} \\ = (1 + o(1))n(n − 2)|S^{n−1}| ∑_{k=1,k≠j}^{l} (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{4}} \\ · (\frac{n(n − 2)}{K(q_{k})})^{\frac{n−2}{4}} |q_{j} − q_{k}|^{−n+2}M_{i,j}^{−1}M_{i,k}^{−1} \end{matrix} (10.8)

for 1 ≤ j ≤ l and,

\begin{matrix}|b_{j}| (\frac{n(n − 2)}{K(q_{j})})^{\frac{n}{2}} M_{i,j}^{− \frac{2β_{j}}{n−2}} \\ = n(n − 2)|S^{n−1}|(1 + o(1)) ∑_{k=1}^{l} (\frac{n(n − 2)}{K(q_{j})})^{\frac{n−2}{4}} \\ · \frac{n(n − 2)}{K(q_{k})}|q_{j} − q_{k}|^{−n+2}M_{i,j}^{−1}M_{i,k}^{−1} \end{matrix} (10.9)

for l + 1 ≤ j ≤ m where b_{j} is given in (1.27).

Now we are in the position to prove the apriori bound of Theorem 1.1. In fact, we are going to prove the result for more general situations. Let A = {q_{k_{1}}, …, q_{k_{m}}} be a subset of Γ^{−} where β_{k_{1}} ≥ β_{k_{2}} ≥ … ≥ β_{k_{m}}. A is called admissible if m ≥ 2 and one of the following conditions holds:

(i) n ≠ β_{k_{1}} > β_{k_{2}} and

(10.10)  \frac{1}{β_{k_{1}}^{∗}} + \frac{1}{β_{k_{2}}^{∗}} = \frac{2}{n − 2},

where β_{j}^{∗} = min(β_{j}, n).

(ii) There exists an integer l ≥ 2 such that

(10.11) n − 2 = β_{k_{1}} = β_{k_{2}} = … = β_{k_{l}} > β_{k_{l+1}} ≥ … ≥ β_{m}.

For an admissible set A of case (i), for simplicity, assume it is {q_{1}, …, q_{m}} with β_{1} > β_{2} = … = β_{l_{1}} > β_{l_{1}+1} ≥ … ≥ β_{m},, we define η = η(A) by

\begin{aligned}η(A) & = (n(n − 2)|S^{n−1}|)^{\frac{2β^{∗}_{1}}{n−2}} (\frac{n(n − 2)}{K(q_{1})})^{\frac{β^{∗}_{1}−n}{2}} \\ & ∑_{j=2}^{l_{1}} (\frac{n(n − 2)}{K(q_{j})})^{−(1+ \frac{(n+2)β^{∗}_{1}}{2(n−2)} )} |b_{j}|^{1− \frac{2β^{∗}_{1}}{n−2}} |q_{1} − q_{j}|^{2−n}. \end{aligned} (10.12)

For A = {q_{1}, …, q_{l}, …, q_{m}} of case (ii), we associate with a l×l matrix η_{ij}(A):

(10.13) η_{jk}(A) = \{ \begin{matrix}|b_{j}| (\frac{n(n−2)}{K(q_{j})})^{\frac{n}{2}} & if j = k \\ −n(n − 2)|S^{n−2}| (\frac{n(n−2)}{K(q_{j})})^{\frac{n−2}{4}} \\ · (\frac{n(n−2)}{K(q_{k})})^{\frac{n−2}{4}} |q_{j} − q_{k}|^{−n+2} & if j ≠ k. \end{matrix}

Now we can state our main theorem.

Theorem 10.3. Assume that K satisfies (K0) ∼ (K1) with β_{j} > \frac{n−2}{2} for any q_{j} ∈ Γ^{−}. For any admisible set A, assume η(A) ≠ 1 for case (i) and the first eigenvalue of η(A) is not zero for case (ii). Then there is a constant c > 0 such that for any solution w of Equation (1.1),

c^{−1} ≤ w(p) ≤ c

holds for any p ∈ S^{n}.

Proof. Suppose u_{i}(x) blows up at some point. Let

A = {q_{1}, …, q_{l}, …, q_{m}}

be the blowup set of u_{i}. Two cases are discussed separately.

Case 1. If l = 1, by (10.7), we can solve M_{i,j}^{−1} in term of M_{i,1}^{−1} for 2 ≤ j ≤ l_{1}, and substitute it into (10.6). If β_{1} = n, then the additional term log M_{i,1} makes two sides of (10.6) unbalanced. Thus, β_{1} ≠ n. Also, it is easy to see that the exponent of M_{i,1}^{−1} of the right hand side

of (10.6) is equal to 1 + (\frac{2β_{2}}{n − 2} − 1)^{−1} . Hence, we have

(10.14)  \frac{2β_{1}^{∗}}{n − 2} = 1 + \frac{1}{\frac{2β_{2}}{n−2} − 1},

which implies

(10.15)  \frac{1}{β_{1}^{∗}} + \frac{1}{β_{2}} = \frac{2}{n − 2}.

Then A is admissible. Applying equality (10.14) and comparing the coefficients of both sides of (10.6) with each other, we have

η = 1 + o(1),

where η is given by (10.12) with A = {q_{1}, q_{2}, …, q_{m}}.

Case 2. l ≥ 2. Since lim_{i→+∞} \frac{M_{i,1}}{M_{i,j}} = λ_{j} > 0 for 1 ≤ j ≤ l, by (10.8),

we have

∑_{k=1}^{l} η_{jk}λ_{k} = 0,

where η_{jk} is given by (10.13) and A = {q_{1}, …, q_{l}, …, q_{m}}. Therefore, the first eigenvalue of (η_{jk}) is equal to 0.

Since both cases yield a contradiction to the assumptions, the apriori bound is established. q.e.d.

We note that the assumptions of Theorem 1.1 imply there exist no admissible subsets of Γ^{−}. Hence, Theorem 1.1 is special case of Theorem 10.3. The asymptotic formulas (10.6) ∼ (10.9) will be very helpful when we come to compute the degree for the nonlinear Equation (1.1).

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