j. differential geometry 63 (2003) 399-473

FROM ONE BUBBLE TO SEVERAL BUBBLES: THE LOW-DIMENSIONAL CASE

OLIVIER DRUET

Abstract

We study in this paper sequences of solutions of elliptic PDE’s with critical Sobolev growth on compact Riemannian manifolds. We prove some compactness results for such sequences which apply in particular to sequences of solutions of the Yamabe equation. We also underline the effect of the dimension and the geometry of the manifold on the blow-up behaviour of such sequences.

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3, and H_{1}^{2}(M) be the standard Sobolev space consisting of functions in L^{2}(M) whose gradient is also in L^{2}(M). We let h be a smooth function on M and consider equations like

(E)  Δ_{g}u + hu = u^{2^{⋆}−1}

where Δ_{g} = −div_{g}∇ is the Laplace-Beltrami operator, and 2^{⋆} = \frac{2n}{n−2} is the critical Sobolev exponent for the embedding of H_{1}^{2}(M) into Lebesgue’s spaces L^{q}(M). Such equations have been the target of investigation for decades. They arise naturally in conformal geometry when h = C(n)S_{g}, C(n) = (n − 2)/4(n − 1), where S_{g} is the scalar curvature of g. In this case, if u is a positive solution of (E), then the conformal metric u^{\frac{4}{n−2}} g has constant scalar curvature. Equation (E) when h = C(n)S_{g} is referred to as the Yamabe equation. Equations like (E) arise also naturally in the study of sharp Sobolev inequalities. Possible surveys on the Yamabe equation, including the final resolution of the Yamabe problem by Schoen [27], are [21, 28, 29]. Possible monographs on sharp Sobolev inequalities are [9] and [18].

Received 01/29/2003.

We let in this paper (h_{ε})_{ε>0} be a sequence of smooth functions on M verifying that there exists h_{0} ∈ C^{2}(M) such that Δ_{g} + h_{0} is coercive and such that

(0.1)  lim_{ε→0} h_{ε} = h_{0} in C^{2}(M)

and we consider (u_{ε})_{ε>0} a sequence of smooth positive solutions of

(E_{ε})  Δ_{g}u_{ε} + h_{ε}u_{ε} = u_{ε}^{2^{⋆}−1} .

We assume that (u_{ε}) is of bounded energy in the sense that there exists Λ > 0 such that

(0.2)  lim sup_{ε→0} ∥u_{ε}∥_{2^{⋆}}^{2^{⋆}} ≤ Λ

where, as in the sequel, ∥ . ∥_{p} denotes the L^{p}-norm. Then, after passing to a subsequence,

(0.3)  lim_{ε→0} u_{ε} = u_{0} weakly in H_{1}^{2}(M)

for some smooth nonnegative function u_{0}, solution of the limit equation

Δ_{g}u_{0} + h_{0}u_{0} = u_{0}^{2^{⋆}−1}.

We assume in what follows that (0.2) and (0.3) hold. Since u_{ε} is positive, the maximum principle gives that either u_{0} ≡ 0 or u_{0} > 0. If (u_{ε}) is bounded in L^{∞}(M), then, thanks to standard elliptic theory,

(0.4)  lim_{ε→0} u_{ε} = u_{0} in C^{2}(M) .

Throughout this paper, we assume that (0.4) is false so that

(0.5)  lim_{ε→0} ∥u_{ε}∥_{∞} = +∞.

Then the u_{ε}’s develop a concentration phenomenon. This concentration phenomenon is well understood in H_{1}^{2}(M), thanks for instance to Struwe [35]. Following Struwe [35], we get that, up to a subsequence,

(0.6)  u_{ε} = u_{0} + ∑_{i=1}^{N} B_{ε}^{i} + R_{ε}

where N ≥ 1 is an integer, the B_{ε}^{i} are bubbles obtained by rescaling fundamental positive solutions of the critical Euclidean equation Δu = u^{2^{⋆}−1}, and the R_{ε}’s are lower order terms in H_{1}^{2}(M), so that R_{ε} → 0 in H_{1}^{2}(M) as ε → 0. A more precise definition of the bubbles is that

B_{ε}^{i}(x) = (\frac{μ_{i,ε}}{μ_{i,ε}^{2} + a_{n}d_{g}(x_{i,ε}, x)^{2}} )^{\frac{n−2}{2}}

where (x_{i,ε}) is a converging sequence of points in M, and (μ_{i,ε}) is a sequence of positive real numbers converging to 0 as ε → 0. Here and in all the sequel,

a_{n} = \frac{1}{n (n − 2)}.

Moreover, the bubbles do not interact at the H_{1}^{2}-level so that

∥u_{ε}∥_{2^{⋆}}^{2^{⋆}} = ∥u_{0}∥_{2^{⋆}}^{2^{⋆}} + NΛ_{min} + o (1)

where the minimum energy Λ_{min} is given by Λ_{min} = K_{n}^{−n/2}, where K_{n} is the sharp constant for the Euclidean Sobolev inequality ∥u∥_{2^{⋆}}^{2} ≤ K∥∇u∥_{2}^{2}. We refer to the above description as the H_{1}^{2}-theory for blowup. The C^{0}-theory, that we will use in this paper, was recently developed by Druet-Hebey-Robert [12, 13]. A special situation is when the u_{ε}’s are of minimal energy, that is when Λ = Λ_{min} in (0.2). In such a case, thanks to the splitting of the energy in the above Struwe decomposition, we easily get that either u_{0} ≡ 0 or u_{ε} → u_{0} in C^{2}(M). In other words, either u_{0} ≡ 0 or blow-up does not occur. A first and naïve question we address in this paper is whether or not such an alternative holds also when the bound on the energy in (0.2) is arbitrary, and, more generally, whether or not the dimension of the manifold has something to do with the vanishing or nonvanishing of u_{0}. An independent natural question when blow-up occurs is to determine the location of geometric concentration points. When the energy of the u_{ε}’s is minimal and n ≥ 4, see for instance Druet-Hebey [9] and Druet-Robert [14], we can prove that h_{0} (\overline{x}) = C(n)S_{g} (\overline{x}) where \overline{x} is the geometric concentration point of the u_{ε}’s. Another question we ask in this paper is whether or not this continues to hold when the bound on the energy in (0.2) is arbitrary. A positive answer to this question would provide another example of the criticality of the Yamabe equation. At last, we address the question of the compactness of solutions of (E_{ε}). This was first handled by Schoen [28] in the case of the Yamabe equation. We refer also to Schoen [29, 30].

We concentrate in this paper on the low-dimensional case, where 3 ≤ n ≤ 5. We say that the u_{ε}’s blow-up if (0.5) holds. We let then S be the set of the geometric concentration points of the u_{ε}’s, defined as the set consisiting of the limits of the x_{i,ε}’s as ε → 0. Independently, we say that compactness holds for the u_{ε}’s if (0.4) holds. Our main result, which answers the above questions for low dimensions, is the following:

Theorem. Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension 3 ≤ n ≤ 5. Let (u_{ε}) be a sequence of positive solutions of (E_{ε}). We assume that (0.1), (0.2), (0.3) hold. If the u_{ε}’s blow-up, then u_{0} ≡ 0, and when n = 4, 5, there exists x_{0} ∈ S such that h_{0} (x_{0}) = C(n)S_{g} (x_{0}), where S is the set of geometric concentration points, and C(n) is as above. In particular, compactness holds for the u_{ε}’s if n = 4, 5 and h_{0} (x) ≠ C(n)S_{g} (x) for all x in M. Compactness holds also for the u_{ε}’s if n = 3, 4, 5 and h_{ε} (x) ≤ C(n)S_{g} (x) for all x in M and all ε, with the additional condition that (M, g) has to be conformally distinct to the unit n-sphere if h_{0}(x) = C(n)S_{g}(x) for all x in M.

The compactness result of the theorem in its last part applies to the Yamabe equation, a situation where we recover the compactness result of Schoen [28]. In particular, if (M, g) is a compact Riemannian manifold of dimension n = 3, 4, 5, conformally distinct to the unit n-sphere, and if (g_{ε} = u_{ε}^{4/(n−2)}g) is a sequence of conformal metrics to g of constant scalar curvature 1 and of bounded volume, then the sequence (u_{ε}) is precompact in C^{2}(M). Our theorem, in its last part, was proved in dimension n = 3 by Li and Zhu [24]. The proof of our theorem relies on the C^{0}-theory for blow-up developed in Druet-Hebey-Robert [12]. The paper is organized as follows: in Section 1, we describe the C^{0}-theory developed in [12]. Section 2 is devoted to the estimate of the distance between concentration points and Section 3 deals with the special case of almost isolated concentration points. The analysis of the distance between concentration points was initiated (in the context of surfaces of constant mean curvature) by Brezis and Coron (see [4, 5, 6]). At last, in Section 4, we prove the theorem and give some results concerning higher dimensions. We also provide some instructive examples of blowing-up sequences of solutions of Equation (E_{ε}) in this last section.

1. A C^{0}-theory for blowing-up sequences of solutions of elliptic PDE’s

In this section, we describe, and give some consequences of, the pointwise version of Struwe’s result (see (0.6)) obtained in Druet-Hebey-Robert [12]. We first recall the result which was proved in [12]. This result and one of its consequence (Claim 1 below) are the starting point for the analysis of Section 2.

Theorem ([12]). Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3, (h_{ε}) be a sequence of smooth functions on M satisfying (0.1), and (u_{ε}) be a sequence of smooth positive solutions of Equation (E_{ε}) satisfying (0.2) − (0.3). Assume that (0.5) holds, that is that ∥u_{ε}∥_{∞} → +∞ as ε → 0. Then there exist N ∈ ℕ^{⋆}, N converging sequences (x_{i,ε}) of points in M and N sequences (μ_{i,ε}) of positive real numbers converging to 0 such that, after passing to a subsequence,

(1 − η_{ε}) u_{0} (x) + \frac{1}{C} ∑_{i=1}^{N} (\frac{μ_{i,ε}}{μ_{i,ε}^{2} + a_{n}d_{g} (x_{i,ε}, x)^{2}} )^{\frac{n−2}{2}}

≤ u_{ε} (x) ≤ (1 + η_{ε}) u_{0} (x) + C ∑_{i=1}^{N} (\frac{μ_{i,ε}}{μ_{i,ε}^{2} + d_{g} (x_{i,ε}, x)^{2}} )^{\frac{n−2}{2}}

for all x ∈ M and all ε where C > 1 is independent of ε and x and (η_{ε}), independent of x, is a sequence of positive real numbers converging to 0 as ε → 0. In particular, the u_{ε}’s are pointwisely controled on both sides by u_{0} and standard bubbles.

This result has important applications when dealing with sharp Sobolev inequalities (see the monographs [9] and [18]). Other directions of research are the study of the energy function (see [19]). The above theorem is proved in Chapters 4 and 6 of Druet-Hebey-Robert [12]. Many asymptotic analysis of this kind are available in the minimal energy case: we cite among others the works of Atkinson-Peletier [2], Brezis-Peletier [7], Robert [25, 26] in the radially symmetric case on the Euclidean ball, the work of Han [15] when dealing with solutions u_{ε} of Δu_{ε} = u_{ε}^{2^{⋆}−1−ε} on arbitrary domains of ℝ^{n}, the work of Hebey-Vaugon [20] on arbitrary Riemannian manifolds with h_{ε} → +∞ as ε → 0. In the case of the standard sphere, we refer also to Chang-Gursky-Yang [8], to Druet-Robert [14], to Li [22, 23] and to Schoen-Zhang [34]. One difficulty to get pointwise estimates when there are several bubbles is that bubbles do interact at a C^{0}-level except in dimension n = 3 where one can prove a priori that the concentration points are isolated.

Let us come back to the above result. We let (x_{i,ε}) and (μ_{i,ε}) be the points in M and the positive real numbers given by the theorem. We refer to Chapters 4 and 6 of [12] for all the following assertions: first, we have that

(1.1) for any i, j ∈ \{1, …, N\} , i ≠ j, \frac{d_{g} (x_{i,ε}, x_{j,ε})^{2}}{μ_{i,ε}μ_{j,ε}} → +∞ as ε → 0.

Then we have that for any i ∈ {1, …, N},

lim_{ε→0} μ_{i,ε}^{\frac{n}{2} −1}u_{ε} (exp_{x_{i,ε}} (μ_{i,ε}x)) = u (x)

in C_{loc}^{2} (ℝ^{n}\S_{i}) where

u(x) = (1 + a_{n}|x|^{2})^{1− \frac{n}{2}}

is a solution of Δ_{ξ}u = u^{2^{⋆}−1} in ℝ^{n}, ξ the Euclidean metric, and

S_{i} = \{lim_{ε→0} \frac{1}{μ_{i,ε}} exp_{x_{i,ε}}^{−1} (x_{j,ε}) , j ≠ i s.t. x_{j,ε} ∈ B_{x_{i,ε}} (\frac{i_{g}(M)}{2})\} .

In this definition, i_{g}(M) is the injectivity radius of M and we assume that the limits exist, which is always the case after passing to a new subsequence.

Note that, as a direct consequence of the above theorem associated to standard elliptic theory, one gets that

lim_{ε→0} u_{ε} = u_{0} in C_{loc}^{2} (M\S)

where

(1.2)  S = \{\overline{x}_{i}, i ∈ \{1, …, N\}\} with \overline{x}_{i} = lim_{ε→0} x_{i,ε}.

As proved in Section 6.3 of [12], the estimate of the theorem may be precised: let us define S_{0} ∈ C^{0} (M × M) by

S_{0} (x, y) = \{ \begin{matrix}1 & if x = y \\ (n − 2) ω_{n−1}d_{g} (x, y)^{n−2} G_{0} (x, y) & if x ≠ y \end{matrix}

where G_{0} is the Green function of Δ_{g} + h_{0}, h_{0} as in (0.1). The fact that S_{0} ∈ C^{0} (M × M) comes from standard property of the Green function. We refer the reader to the appendix of [12] for estimates on Green’s functions of linear elliptic operators on compact manifolds. We let (x_{ε}) be a sequence of points in M such that x_{ε} → x_{0} as ε → 0. Then we have the following asymptotic estimate on u_{ε} (x_{ε}) as ε → 0:

(1.3) u_{ε} (x_{ε}) = u_{0} (x_{ε}) (1 + o (1)) + ∑_{i=1}^{N}(S_{0} (\overline{x}_{i}, x_{0}) + o (1))φ_{i,ε} (x_{ε})

where for i = 1, …, N, φ_{i,ε} is the standard bubble

(1.4)  φ_{i,ε} (x) = (\frac{μ_{i,ε}}{μ_{i,ε}^{2} + a_{n}d_{g} (x_{i,ε}, x)^{2}} )^{\frac{n−2}{2}} .

Thanks to (1.1) and (1.3), we also have that Struwe’s H_{1}^{2}-description holds with the bubbles of the theorem (see section 6.3 of [12]). Namely, we have that

(1.5)  u_{ε} = u_{0} + ∑_{i=1}^{N} φ_{i,ε} + R_{ε}

with ∥R_{ε}∥_{H_{1}^{2}(M)} → 0 as ε → 0. In the following, we shall always consider that the concentration points are ordered such that

(1.6)  μ_{1,ε} ≤ μ_{2,ε} ≤ ⋯ ≤ μ_{N,ε}.

As a last remark, note that standard elliptic theory leads thanks to (1.3) to the following if u_{0} ≡ 0:

(1.7) lim_{ε→0} μ_{N,ε}^{1− \frac{n}{2}} u_{ε} = a_{n}^{− \frac{n−2}{2}} (n − 2) ω_{n−1} ∑_{i=1}^{N} (lim_{ε→0} \frac{μ_{i,ε}}{μ_{N,ε}} )^{\frac{n−2}{2}} G_{0} (\overline{x}_{i}, .)

in C_{loc}^{2} (M\S), S and \overline{x}_{i} as in (1.2).

We derive now from (1.3) an asymptotic estimate (Claim 1 below) we will often use in the sequel. We first set up some notations. We let j ∈ {1, …, N} and we let (δ_{ε}) be a sequence of positive real numbers converging to 0 as ε → 0. We set

(1.8)  A (j, δ_{ε}) = {i ∈ {1, …, N} s.t. d_{g} (x_{i,ε}, x_{j,ε}) = O (δ_{ε})}.

Note that j ∈ A (j, δ_{ε}). For i ∈ A (j, δ_{ε}), we let

(1.9)  z_{i} = lim_{ε→0} \frac{1}{δ_{ε}} exp_{x_{j,ε}}^{−1} (x_{i,ε})

where the limits are assumed to exist (this is always true after passing to a subsequence). We set

(1.10)  Σ (j, δ_{ε}) = {z_{i}, i ∈ A (j, δ_{ε})}.

We let also

(1.11)  λ_{ε} (j, δ_{ε}) = ( sup_{i∈A(j,δ_{ε})} μ_{i,ε})^{1− \frac{n}{2}} δ_{ε}^{n−2}

and

(1.12)  c (j, δ_{ε}) = ∑_{k∉A(j,δ_{ε})} (lim_{ε→0} λ_{ε} (j, δ_{ε}) φ_{k,ε} (x_{j,ε})) S_{0} (\overline{x}_{k}, \overline{x}_{j})

+ lim_{ε→0}(λ_{ε} (j, δ_{ε}) u_{0} (\overline{x}_{j}))

and for k ∈ A (j, δ_{ε}),

(1.13)  λ_{k} (j, δ_{ε}) = a_{n}^{− \frac{n−2}{2}} lim_{ε→0} (λ_{ε} (j, δ_{ε}) δ_{ε}^{2−n}μ_{k,ε}^{\frac{n}{2} −1}) .

In (1.12), we assume that the limits do exist but they may be equal to +∞. By convention, we say that lim_{ε→0}(λ_{ε} (j, δ_{ε}) u_{0} (\overline{x}_{j})) = 0 if u_{0} ≡ 0. We prove the following:

Claim 1. Let j ∈ {1, …, N} and let (δ_{ε}) be a sequence of positive real numbers converging to 0 as ε → 0. We assume that the following holds:

(H1) λ_{ε} (j, δ_{ε}) δ_{ε}^{1− \frac{n}{2}} → +∞ as ε → 0.

(H2) c (j, δ_{ε}) < +∞.

Then, after passing to a subsequence, we have that

lim_{ε→0} λ_{ε} (j, δ_{ε}) u_{ε} (exp_{x_{j,ε}} (δ_{ε}z)) = H (z)

in C_{loc}^{2}(ℝ^{n}\Σ (j, δ_{ε})) where

H (z) = ∑_{k∈A(j,δ_{ε})} \frac{λ_{k} (j, δ_{ε})}{|z − z_{k}|^{n−2}} + c (j, δ_{ε}) .

All the notations of this claim were introduced in (1.8)-(1.13).

Proof. Let j ∈ {1, …, N} and let (δ_{ε})_{ε>0} be a sequence of positive real numbers converging to 0 as ε → 0. Assume that assumptions (H1) and (H2) of Claim 1 hold. We let 0 < δ < \frac{i_{g}(M)}{2} and we set for z ∈ B_{0} (δδ_{ε}^{−1}), the Euclidean ball of center 0 and radius δδ_{ε}^{−1},

w_{ε} (z) = δ_{ε}^{\frac{n}{2} −1}u_{ε} (exp_{x_{j,ε}} (δ_{ε}z)) and (1.14)

g_{ε} (z) = exp_{x_{j,ε}}^{⋆} g (δ_{ε}z) .

Since δ_{ε} → 0 as ε → 0, we have that

(1.15)  lim_{ε→0} g_{ε} = ξ in C_{loc}^{2} (ℝ^{n}) .

Note also that g_{ε} is controled on both sides by ξ in the sense of bilinear forms. Since u_{ε} verifies Equation (E_{ε}), we have that w_{ε} verifies

(1.16)  Δ_{g_{ε}}w_{ε} + δ_{ε}^{2}h_{ε} (exp_{x_{j,ε}} (δ_{ε}z)) w_{ε} = w_{ε}^{2^{⋆}−1}

in B_{0} (δδ_{ε}^{−1}). We claim that for any R > 0, there exists C_{R} > 0 independent of ε such that

(1.17)  λ_{ε} (j, δ_{ε}) δ_{ε}^{1− \frac{n}{2}} ∥w_{ε}∥_{L^{∞}(B_{0}(R)\∪_{k∈A(j,δε)}B_{z_{k}}(\frac{1}{R}))} ≤ C_{R}

for all ε > 0 where A (j, δ_{ε}) is as defined in (1.8), z_{k} is as in (1.9) and λ_{ε} (j, δ_{ε}) is as in (1.11). In order to prove (1.17), we let R > 0 and we let (z_{ε}) be a sequence of points in B_{0} (R) \ ∪_{k∈A(j,δ_{ε})} B_{z_{k}} (\frac{1}{R}). After passing to a subsequence, we may assume that lim_{ε→0} z_{ε} = z_{0}. We let x_{ε} = exp_{x_{j,ε}} (δ_{ε}z_{ε}) and we write thanks to (1.3) and (1.14) that

\begin{aligned}λ_{ε} (j, δ_{ε}) δ_{ε}^{1− \frac{n}{2}} w_{ε} (z_{ε}) = & λ_{ε} (j, δ_{ε}) u_{0} (x_{ε}) (1 + o (1)) \\ & + λ_{ε} (j, δ_{ε}) ∑_{i=1}^{N}(S_{0} (\overline{x}_{k}, \overline{x}_{j}) + o (1))φ_{k,ε} (x_{ε}) . \end{aligned} (1.18)

Thanks to (1.12) and to assumption (H2), we have that, up to a subsequence,

(1.19)  lim_{ε→0} λ_{ε} (j, δ_{ε}) u_{0} (x_{ε}) = lim_{ε→0}(λ_{ε} (j, δ_{ε}) u_{0} (\overline{x}_{j}))

where this limit is finite and is by convention equal to 0 if u_{0} ≡ 0. Let k ∈ A (j, δ_{ε}). By (1.11), we have that

(\frac{μ_{k,ε}}{δ_{ε}} )^{\frac{n}{2} −1} ≤ δ_{ε}^{\frac{n}{2} −1}λ_{ε} (j, δ_{ε})^{−1}

which leads thanks to (H1) to μ_{k,ε} = o (δ_{ε}). For ε > 0 small enough, we

have thanks to (1.15) that d_{g} (x_{ε}, x_{k,ε}) ≥ \frac{1}{2R}δ_{ε} so that \frac{d_{g}(x_{ε},x_{k,ε})}{μ_{k,ε}} → +∞ as ε → 0. This leads with (1.4) to

λ_{ε} (j, δ_{ε}) φ_{k,ε} (x_{ε}) = [a_{n}^{− \frac{n−2}{2}} + o (1)] λ_{ε} (j, δ_{ε}) μ_{k,ε}^{\frac{n}{2} −1}d_{g} (x_{ε}, x_{k,ε})^{2−n}
= λ_{k} (j, δ_{ε}) |z_{0} − z_{k}|^{2−n} + o (1)

where λ_{k} (j, δ_{ε}) is defined by (1.13) and is finite thanks to (1.11). Since k ∈ A (j, δ_{ε}) and δ_{ε} → 0 as ε → 0, we clearly have that \overline{x}_{k} = \overline{x}_{j} so that S_{0} (\overline{x}_{k}, \overline{x}_{j}) = 1. Thus we have obtained that

for any k ∈ A (j, δ_{ε}) ,

lim_{ε→0} λ_{ε} (j, δ_{ε}) (S_{0} (\overline{x}_{k}, \overline{x}_{j}) + o (1))φ_{k,ε} (x_{ε}) = λ_{k} (j, δ_{ε}) |z_{0} − z_{k}|^{2−n} .

(1.20)

Let now k ∉ A (j, δ_{ε}). Since d_{g} (x_{j,ε}, x_{ε}) = O (δ_{ε}) and \frac{d_{g}(x_{j,ε},x_{k,ε})}{δ_{ε}} → +∞ as ε → 0, we have that

lim_{ε→0} \frac{d_{g} (x_{ε}, x_{k,ε})}{d_{g} (x_{j,ε}, x_{k,ε})} = 1

so that

lim_{ε→0} \frac{φ_{k,ε} (x_{ε})}{φ_{k,ε} (x_{j,ε})} = 1.

Thus we obtain that for any k ∉ A (j, δ_{ε}),

lim_{ε→0} λ_{ε} (j, δ_{ε}) φ_{k,ε} (x_{ε}) = lim_{ε→0} λ_{ε} (j, δ_{ε}) φ_{k,ε} (x_{j,ε})

which does exist, after passing to a subsequence, and is finite thanks to (H2). Combining (1.18) with (1.19), (1.20) and this last relation, we get that

lim_{ε→0} λ_{ε} (j, δ_{ε}) δ_{ε}^{1− \frac{n}{2}} w_{ε} (z_{ε}) = H (z_{0})

where H is as in Claim 1. In particular (1.17) is proved. Standard elliptic theory permits then to conclude thanks to (1.14) − (1.17) that Claim 1 holds. q.e.d.

2. Estimating the distance between bubbles

For any j ∈ {1, …, N}, we set

(2.1)  A_{j} = {i ≠ j s.t. μ_{j,ε} = O (μ_{i,ε})}

and (2.2)

r_{j,ε}^{2} = \{ \begin{matrix}min_{i∈A_{j}} (\frac{μ_{j,ε}}{μ_{i,ε}} d_{g} (x_{i,ε}, x_{j,ε})^{2} + μ_{i,ε}μ_{j,ε}) & if u_{0} ≡ 0 \\ min \{min_{i∈A_{j}} (\frac{μ_{j,ε}}{μ_{i,ε}} d_{g} (x_{i,ε}, x_{j,ε})^{2} + μ_{i,ε}μ_{j,ε}); μ_{j,ε}\} & if u_{0} ≢ 0. \end{matrix}

If A_{j} = ∅ (which is possible only for j = N) and u_{0} ≡ 0, we let r_{j,ε} = 1 for all ε > 0. Note that, thanks to (1.1),

(2.3)  lim_{ε→0} \frac{r_{j,ε}}{μ_{j,ε}} = +∞ for all j ∈ \{1, …, N\} .

The aim of this section is to get an estimate of r_{j,ε} in terms of μ_{j,ε}. If r_{j,ε} does not converge to 0 as ε → 0, we say that (x_{j,ε}, μ_{j,ε}) is an almost isolated concentration point. We deal with almost isolated concentration points in Section 3. We treat in the following claim the case when r_{j,ε} → 0 as ε → 0:

Claim 2. If n = 3, there exists δ_{0} > 0 such that lim inf_{ε→0} r_{j,ε} ≥ δ_{0} for all j ∈ {1, …, N}. In other words, the concentration points are isolated in dimension n = 3. Assume now that n ≥ 4. Let j ∈ {1, …, N} be such that r_{j,ε} → 0 as ε → 0. Then the following assertions hold:

(i) After passing to a subsequence, we have that

lim_{ε→0} r_{j,ε}^{n−2}μ_{j,ε}^{1− \frac{n}{2}} u_{ε} (exp_{x_{j,ε}} (r_{j,ε}z)) = a_{n}^{− \frac{n−2}{2}} (\frac{1}{|z|^{n−2}} + h_{j} (z))

in C_{loc}^{2} (ℝ^{n}\Σ_{j}) where

Σ_{j} = {z_{j,k}, k ∈ B_{j}} ∪ {0} ,
B_{j} = {k ≠ j, d_{g} (x_{j,ε}, x_{k,ε}) = O (r_{j,ε})} and
z_{j,k} = lim_{ε→0} \frac{1}{r_{j,ε}} exp_{x_{j,ε}}^{−1} (x_{k,ε}) , k ∈ B_{j}

and where

h_{j} (z) = ∑_{k∈A_{j}∩B_{j}} \frac{λ_{j,k}}{|z − z_{j,k}|^{n−2}} + c_{j}

with

λ_{j,k} = (lim_{ε→0} \frac{μ_{k,ε}}{μ_{j,ε}} )^{\frac{n}{2} −1}

and

a_{n}^{− \frac{n−2}{2}} c_{j} = ∑_{k∈A_{j}\B_{j}} S_{0} (\overline{x}_{k}, \overline{x}_{j}) lim_{ε→0} (r_{j,ε}^{n−2}μ_{j,ε}^{1− \frac{n}{2}} φ_{k,ε} (x_{j,ε}))

+ lim_{ε→0} (r_{j,ε}^{n−2}μ_{j,ε}^{1− \frac{n}{2}} u_{0} (\overline{x}_{j})) .

(ii) We have that

(h_{0} (\overline{x}_{j}) − C(4)S_{g} (\overline{x}_{j}) + o (1))r_{j,ε}^{2} ln (\frac{1}{μ_{j,ε}} ) = 2h_{j} (0)

when n = 4 and that

(h_{0} (\overline{x}_{j}) − C(n)S_{g} (\overline{x}_{j}) + o (1))r_{j,ε}^{n−2}μ_{j,ε}^{4−n}

= a_{n}^{2−n}Kn^{\frac{n}{2}} \frac{(n − 2)^{2} (n − 4)}{8 (n − 1)}ω_{n−1}h_{j} (0)

when n ≥ 5. Here, C(n) = \frac{n−2}{4(n−1)} and a_{n} = \frac{1}{n(n−2)}. Moreover,

h_{j} (0) > 0.

(iii) If h_{0} (\overline{x}_{j}) > C(n)S_{g} (\overline{x}_{j}), we have that ∇h_{j} (0) = 0.

Proof. Let j ∈ {1, …, N} be such that

(2.4)  lim_{ε→0} r_{j,ε} = 0.

We want to apply Claim 1 to (j, r_{j,ε}). We let B_{j} be as in Claim 2 so that B_{j} ∪ {j} = A (j, r_{j,ε}), A (j, r_{j,ε}) as in (1.8). We verify that the assumptions of Claim 1 are satisfied by (j, r_{j,ε}). First, by (2.4), r_{j,ε} → 0 as ε → 0. Let k ∈ A_{j} ∩B_{j}. Then we have that d_{g} (x_{j,ε}, x_{k,ε}) = O (r_{j,ε}) so that, using (1.1), (2.1) and (2.2), we obtain that μ_{k,ε} = O (μ_{j,ε}). Thus

(2.5)  for any k ∈ A_{j} ∩ B_{j}, μ_{k,ε} = O (μ_{j,ε}) .

If k ∉ A_{j}, by the definition (2.1) of A_{j}, we have that μ_{k,ε} = o (μ_{j,ε}). Thus for any k ∈ B_{j}, μ_{k,ε} = O (μ_{j,ε}). This gives that there exists C > 0 such that

(2.6)  C ≤ λ_{ε} (j, r_{j,ε}) μ_{j,ε}^{\frac{n}{2} −1}r_{j,ε}^{2−n} ≤ 1

for all ε > 0 where λ_{ε} (j, r_{j,ε}) is defined by (1.11). By (2.3), we thus get that

lim_{ε→0} λ_{ε} (j, r_{j,ε}) r_{j,ε}^{1− \frac{n}{2}} = +∞

so that assumption (H1) of Claim 1 is satisfied. Let k ∉ B_{j}, k ≠ j, that is k ∈ {1, …, N} such that

\frac{d_{g} (x_{k,ε}, x_{j,ε})}{r_{j,ε}} → +∞ as ε → 0.

We write thanks to (1.4) and (2.6) that

(λ_{ε} (j, r_{j,ε}) φ_{k,ε} (x_{j,ε}))^{\frac{2}{n−2}} ≤ \frac{r_{j,ε}^{2}}{μ_{j,ε}} \frac{μ_{k,ε}}{μ_{k,ε}^{2} + a_{n}d_{g} (x_{j,ε}, x_{k,ε})^{2}} .

If k ∈ A_{j}, we get then by (2.2) that λ_{ε} (j, r_{j,ε}) φ_{k,ε} (x_{j,ε}) = O (1). If k ∉ A_{j}, then μ_{k,ε} = o (μ_{j,ε}) and we write that

(λ_{ε} (j, r_{j,ε}) φ_{k,ε} (x_{j,ε}))^{\frac{2}{n−2}} ≤ \frac{1}{a_{n}} \frac{μ_{k,ε}}{μ_{j,ε}} \frac{r_{j,ε}^{2}}{d_{g} (x_{k,ε}, x_{j,ε})^{2}} = o (1)

since k ∉ B_{j}, k ≠ j. If u_{0} ≢ 0, then r_{j,ε}^{2} ≤ μ_{j,ε} by (2.2) so that (2.6) gives that λ_{ε} (j, r_{j,ε}) ≤ 1. We have thus proved thanks to (1.12) that assumption (H2) of Claim 1 holds. Applying Claim 1 to (j, r_{j,ε}), we get that assertion (i) of Claim 2 holds for j thanks to (2.5) and (2.6). In order to compute c_{j}, note that, as just proved, if k ∉ A_{j}, k ∉ B_{j}, k ≠ j, then r_{j,ε}^{n−2}μ_{j,ε}^{1− \frac{n}{2}} φ_{k,ε} (x_{j,ε}) → 0 as ε → 0. We claim now that

(2.7)  h_{j} (0) > 0.

Let us prove this claim. If A_{j} = ∅, then r_{j,ε} = 1 if u_{0} ≡ 0, a situation which is excluded by (2.4). If u_{0} ≢ 0 and r_{j,ε}^{2} = μ_{j,ε}, we have that

c_{j} ≥ an^{\frac{n−2}{2}} u_{0} (\overline{x}_{j}) > 0

so that h_{j} (0) > 0. Assume now that A_{j} ≠ ∅ and that there exists k ∈ A_{j} such that

r_{j,ε}^{2} = \frac{μ_{j,ε}}{μ_{k,ε}} d_{g} (x_{j,ε}, x_{k,ε})^{2} + μ_{j,ε}μ_{k,ε}.

If k ∈ B_{j}, then h_{j} (0) > 0 since, by (2.1), λ_{j,i} > 0 for all i ∈ A_{j} ∩ B_{j}. If k ∉ B_{j}, we write thanks to (1.4) that

(r_{j,ε}^{n−2}μ_{j,ε}^{1− \frac{n}{2}} φ_{k,ε} (x_{j,ε}))^{\frac{2}{n−2}} = \frac{μ_{k,ε}^{2} + d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{k,ε}^{2} + a_{n}d_{g} (x_{j,ε}, x_{k,ε})^{2}} ≥ 1

so that h_{j} (0) > 0 in this last case. Relation (2.7) is proved. Note that (2.7) is the second part of assertion ii) for i = j.

Let us set

(2.8)  C_{j} = {k ∈ B_{j} s.t. z_{j,k} = 0}

where z_{j,k} is as in Claim 2. Let k ∈ C_{j}. If k ∉ A_{j}, then μ_{k,ε} = o (μ_{j,ε}) and if k ∈ A_{j}, it follows from (1.1), (2.2) and (2.8) that μ_{k,ε} = o (μ_{j,ε}) also. Thus

(2.9)  μ_{k,ε} = o (μ_{j,ε}) for all k ∈ C_{j}.

We let

(2.10)  s_{j,k,ε}^{2} = \frac{μ_{k,ε}}{μ_{j,ε}} d_{g} (x_{k,ε}, x_{j,ε})^{2} + μ_{j,ε}μ_{k,ε}

for k ∈ C_{j}. Note that, by (2.3), (2.8) and (2.9), we have that

(2.11)  s_{j,k,ε} = o (r_{j,ε}) for all k ∈ C_{j}.

We let now D_{j} be a subset of C_{j} and (R_{k})_{k∈D_{j}} be a sequence of positive real numbers such that

(2.12) for any k, k′ ∈ D_{j}, k ≠ k′, \frac{d_{g} (x_{k,ε}, x_{k′,ε})}{s_{j,k,ε}} → +∞ as ε → 0

and such that

(2.13) for any k′ ∈ C_{j}, ∃ a unique k ∈ D_{j} such that

lim sup_{ε→0} \frac{d_{g} (x_{k,ε}, x_{k′,ε})}{s_{j,k,ε}} ≤ \frac{R_{k}}{10} and lim sup_{ε→0} \frac{s_{j,k′,ε}}{s_{j,k,ε}} ≤ \frac{R_{k}}{10} .

We claim that there exists C > 0 independent of ε such that for any k ∈ D_{j},

for any x ∈ B_{x_{k,ε}} (R_{k}s_{j,k,ε}) \B_{x_{k,ε}} (\frac{R_{k}}{4} s_{j,k,ε}) , (2.14)

|∇u_{ε}|_{g}(x) ≤ Cμ_{k,ε}^{\frac{n}{2} −1}s_{j,k,ε}^{1−n}, u_{ε}(x) ≤ Cμ_{k,ε}^{\frac{n}{2} −1}s_{j,k,ε}^{2−n} .

The proof of this claim is based on Claim 1. We check that we can apply Claim 1 to (k, s_{j,k,ε}) for k ∈ D_{j}. First, by (2.4) and (2.11), it is clear that s_{j,k,ε} → 0 as ε → 0. Let i ∈ A (k, s_{j,k,ε}), that is i ∈ {1, …, N} such that d_{g} (x_{i,ε}, x_{k,ε}) = O (s_{j,k,ε}). Since d_{g} (x_{i,ε}, x_{j,ε}) ≤ d_{g} (x_{i,ε}, x_{k,ε}) + d_{g} (x_{k,ε}, x_{j,ε}), we get thanks to (2.8) and (2.11) that i ∈ C_{j}. By (2.13), we then have that s_{j,i,ε} = O (s_{j,k,ε}). By (1.1), (2.9) and (2.10), d_{g} (x_{i,ε}, x_{k,ε}) = O (s_{j,k,ε}) = o(d_{g} (x_{j,ε}, x_{k,ε})). This leads thanks to the definition (2.10) of s_{j,i,ε} and s_{j,k,ε} to μ_{i,ε} = O (μ_{k,ε}). Thus μ_{i,ε} = O (μ_{k,ε}) for all i ∈ A (k, s_{j,k,ε}). This gives the existence of some C > 0 such that

(2.15)  C ≤ λ_{ε} (k, s_{j,k,ε}) μ_{k,ε}^{\frac{n}{2} −1}s_{j,k,ε}^{2−n} ≤ 1

where λ_{ε} (k, s_{j,k,ε}) is defined by (1.11). This implies in particular that

λ_{ε} (k, s_{j,k,ε}) s_{j,k,ε}^{1− \frac{n}{2}} ≥ C (\frac{s_{j,k,ε}^{2}}{μ_{k,ε}^{2}} )^{\frac{n−2}{4}} ≥ C (\frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{k,ε}μ_{j,ε}} )^{\frac{n−2}{4}}

so that λ_{ε} (k, s_{j,k,ε}) s_{j,k,ε}^{1− \frac{n}{2}} → +∞ as ε → 0 thanks to (1.1). This proves that assumption (H1) of Claim 1 holds for (k, s_{j,k,ε}). Let i ∉ A (k, s_{j,k,ε}), that is i ∈ {1, …, N} such that

\frac{d_{g} (x_{i,ε}, x_{k,ε})}{s_{j,k,ε}} → +∞ as ε → 0.

In order to estimate (λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε})), we write thanks to (1.4) and (2.15) that

(2.16) (λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}} ≤ \frac{s_{j,k,ε}^{2}}{μ_{k,ε}} \frac{μ_{i,ε}}{μ_{i,ε}^{2} + a_{n}d_{g} (x_{i,ε}, x_{k,ε})^{2}} .

We distinguish several cases. First, assume that i ∈ C_{j}. Then, by (2.12),

(2.13) and thanks to the fact that \frac{d_{g}(x_{i,ε},x_{k,ε})}{s_{j,k,ε}} → +∞ as ε → 0, we get that

(2.17)  \frac{d_{g} (x_{k,ε}, x_{i,ε})}{s_{j,i,ε}} → +∞ as ε → 0.

This gives that

\frac{d_{g} (x_{i,ε}, x_{k,ε})^{2}}{μ_{i,ε}^{2}} = \frac{d_{g} (x_{k,ε}, x_{i,ε})^{2}}{s_{j,i,ε}^{2}} (\frac{d_{g} (x_{i,ε}, x_{j,ε})^{2}}{μ_{i,ε}μ_{j,ε}} + \frac{μ_{j,ε}}{μ_{i,ε}} )
→ +∞ as ε → 0

thanks to (1.1). Thus (2.16) becomes in this case

(λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}} = O (\frac{μ_{i,ε}}{μ_{k,ε}} \frac{s_{j,k,ε}^{2}}{d_{g} (x_{i,ε}, x_{k,ε})^{2}} ) .

We write now thanks to (2.10) that

s_{j,k,ε}^{2} ≤ 2\frac{μ_{k,ε}}{μ_{j,ε}} (d_{g} (x_{i,ε}, x_{k,ε})^{2} + d_{g} (x_{i,ε}, x_{j,ε})^{2}) + μ_{j,ε}μ_{k,ε}

so that we get that

(λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}}

= O (\frac{μ_{i,ε}}{μ_{j,ε}} ) + O (\frac{μ_{i,ε}}{μ_{j,ε}} \frac{d_{g} (x_{i,ε}, x_{j,ε})^{2}}{d_{g} (x_{i,ε}, x_{k,ε})^{2}} + \frac{μ_{i,ε}μ_{j,ε}}{d_{g} (x_{i,ε}, x_{k,ε})^{2}} )

= O (\frac{μ_{i,ε}}{μ_{j,ε}} ) + O (\frac{s_{j,i,ε}^{2}}{d_{g} (x_{i,ε}, x_{k,ε})^{2}} ) = o (1)

since i ∈ C_{j} and thanks to (2.9) and (2.17). Second, assume that i = j. In this case, (2.16) becomes

(λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}} ≤ \frac{μ_{j,ε}^{2} + d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2} + a_{n}d_{g} (x_{j,ε}, x_{k,ε})^{2}} ≤ n(n − 2).

Third, assume that i ∉ C_{j} and that i ∈ A_{j}. Then we write with (2.16) that

(λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}}

≤ (\frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}} + μ_{j,ε}) \frac{μ_{i,ε}}{μ_{i,ε}^{2} + a_{n}d_{g} (x_{i,ε}, x_{k,ε})^{2}} .

Since i ∉ C_{j} and i ≠ j, we have by (2.8) that

d_{g} (x_{i,ε}, x_{k,ε}) = (1 + o (1))d_{g} (x_{i,ε}, x_{j,ε}) .

Since k ∈ C_{j}, we also have that d_{g} (x_{j,ε}, x_{k,ε}) = o (r_{j,ε}). Since i ∈ A_{j}, this leads with (2.1) and (2.2) to

(λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}} = O (\frac{μ_{i,ε}^{2} + d_{g} (x_{j,ε}, x_{i,ε})^{2}}{μ_{i,ε}^{2} + \frac{d_{g}(x_{i,ε},x_{j,ε})^{2}}{n(n−2)}} ) = O (1) .

At last, assume that i ∉ C_{j}, i ∉ A_{j} and i ≠ j. Since i ∉ C_{j} and i ≠ j, we have as above that

d_{g} (x_{i,ε}, x_{k,ε}) = (1 + o (1))d_{g} (x_{i,ε}, x_{j,ε}) .

Since i ∉ A_{j}, i ≠ j, we know that μ_{i,ε} = o (μ_{j,ε}) so that, by (1.1), \frac{d_{g}(x_{i,ε},x_{j,ε})}{μ_{i,ε}} → +∞ as ε → 0. Thus (2.16) becomes in this case

(λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}))^{\frac{2}{n−2}}

= O (\frac{μ_{i,ε}}{μ_{j,ε}} \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{d_{g} (x_{i,ε}, x_{j,ε})^{2}} ) + O (\frac{μ_{i,ε}μ_{j,ε}}{d_{g} (x_{i,ε}, x_{j,ε})^{2}} )

= o (1)

since μ_{i,ε} = o (μ_{j,ε}) and i ∉ C_{j}, i ≠ j, and thanks to (1.1). Thus we have proved that

for any i ∉ A (k, s_{j,k,ε}) , λ_{ε} (k, s_{j,k,ε}) φ_{i,ε} (x_{k,ε}) = O (1) .

Assume that u_{0} ≢ 0. Then we can write thanks to (2.10) and (2.15) that

λ_{ε} (k, s_{j,k,ε})^{\frac{2}{n−2}} ≤ \frac{s_{j,k,ε}^{2}}{μ_{k,ε}} ≤ \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}} + μ_{j,ε}

= o (\frac{r_{j,ε}^{2}}{μ_{j,ε}} ) + o (1) = o (1)

using (2.2) and (2.8). Thus assumption (H2) of Claim 1 is verified by (k, s_{j,k,ε}). We can apply Claim 1. This proves (2.14) thanks to the choice of R_{k} we made. Indeed, by (2.11), any i ∈ A (k, s_{j,k,ε}) belongs to C_{j} and by (2.13), we get then that

Σ (k, s_{j,k,ε}) ⊂ B_{0} (\frac{R_{k}}{5} ) .

This clearly ends the proof of (2.14) thanks to (2.15).

We let now η : [0, +∞[ ↦ ℝ be a smooth function verifying that 0 ≤ η ≤ 1, η ≡ 1 on [0, \frac{1}{4}] and η ≡ 0 on [\frac{1}{2}, +∞[. We set

(2.18)  σ_{j,ε} = ∏_{k∈D_{j}} (1 − η (\frac{d_{g} (x_{k,ε}, .)}{R_{k}s_{j,k,ε}})) and v_{j,ε} = σ_{j,ε}u_{ε}.

We claim that there exists C > 0 independent of ε such that

(2.19)  v_{j,ε} ≤ Cφ_{j,ε} in B_{x_{j,ε}} (δ_{j}r_{j,ε})

where δ_{j} ∈ ℝ_{+}^{⋆} is fixed such that

(2.20)  δ_{j} ≤ \frac{1}{2} min\{|z_{j,k}|, k ∈ B_{j}\C_{j}, k ≠ j\}.

It is possible to find such a δ_{j} thanks to (2.8). The proof of (2.19) is based on (1.3) which gives that there exists C > 0 independent of ε such that

\frac{u_{ε} (x)}{φ_{j,ε} (x)} ≤ C (1 + \frac{∥u_{0}∥_{∞}}{φ_{j,ε} (x)} + ∑_{k≠j} \frac{φ_{k,ε} (x)}{φ_{j,ε} (x)})

for all x ∈ B_{x_{j,ε}} (δ_{j}r_{j,ε}). As well as r_{j,ε} and δ_{j} have been chosen so that

\frac{∥u_{0}∥_{∞}}{φ_{j,ε} (x)} + ∑_{k∉C_{j}} \frac{φ_{k,ε} (x)}{φ_{j,ε} (x)} ≤ C

for all x ∈ B_{x_{j,ε}} (δ_{j}r_{j,ε}) for some C > 0 independent of ε (see the proof that the assumptions of Claim 1 hold for (j, r_{j,ε})), it is easily checked that s_{j,k,ε} has been defined so that

\frac{φ_{k,ε} (x)}{φ_{j,ε} (x)} ≤ C

for all x ∈ M\B_{x_{k,ε}} (\frac{R_{k}}{4} s_{j,k,ε}). These two assertions, whose proofs are left to the reader, imply (2.19).

from one bubble to several bubbles 417

We set

(2.21) Λ_{j,ε} ={(y, ν, θ) ∈ M × ℝ_{+}^{⋆} × ℝ s.t.

d_{g} (x_{j,ε}, y) ≤ μ_{j,ε} , \frac{1}{2} ≤ \frac{ν}{μ_{j,ε}} ≤ 2 , −\frac{1}{2} ≤ θ ≤ \frac{1}{2}\}

and we let (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) ∈ Λ_{j,ε} be such that

(2.22)  J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = min_{(y,ν,θ)∈Λ_{j,ε}} J_{j,ε} (y, ν, θ)

where

J_{j,ε} (y, ν, θ) = ∫_{M} |∇ (η (\frac{d_{g} (y, .)}{2δ_{j}r_{j,ε}}) (v_{j,ε} − (1 + θ) ψ_{y,ν}))|_{g}^{2} dv_{g}

with

(2.23)  ψ_{y,ν} (x) = (\frac{ν}{ν^{2} + a_{n}d_{g} (x, y)^{2}} )^{\frac{n−2}{2}} .

We claim first that

(2.24)  J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) → 0 as ε → 0.

In order to prove (2.24), we first note that (x_{j,ε}, μ_{j,ε}, 0) ∈ Λ_{j,ε} so that

J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) ≤ J_{j,ε} (x_{j,ε}, μ_{j,ε}, 0) .

We write now that

J_{j,ε} (x_{j,ε}, μ_{j,ε}, 0) = ∫_{M} |∇ (η (\frac{d_{g} (x_{j,ε}, .)}{2δ_{j}r_{j,ε}}) (v_{j,ε} − φ_{j,ε}))|_{g}^{2} dv_{g}
≤ \frac{C}{r_{j,ε}^{2}} ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})\B_{x_{j,ε}}( \frac{δ_{j}}{2} r_{j,ε})} (v_{j,ε} − φ_{j,ε})^{2} dv_{g}
+C ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇(v_{j,ε} − φ_{j,ε})|_{g}^{2} dv_{g}

where C > 0 is some constant independent of ε. Thanks to (2.3) and (2.19), it is easily checked that

lim_{ε→0} \frac{1}{r_{j,ε}^{2}} ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})\B_{x_{j,ε}}( \frac{δ_{j}}{2} r_{j,ε})} (v_{j,ε} − φ_{j,ε})^{2} dv_{g} = 0

418 o. druet

so that we obtain that

(2.25) J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = O (∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (v_{j,ε} − φ_{j,ε})|_{g}^{2} dv_{g}) .

We write with (1.5) that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (v_{j,ε} − φ_{j,ε})|_{g}^{2} dv_{g} (2.26)

≤ C ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇((σ_{j,ε} − 1) φ_{j,ε})|_{g}^{2} dv_{g}

+ C ∑_{i≠j} ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}φ_{i,ε})|_{g}^{2} dv_{g}

+ C ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}u_{0})|_{g}^{2} dv_{g} + C ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}R_{ε})|_{g}^{2} dv_{g}. with ∥R_{ε}∥_{H_{1}^{2}(M)} → 0 as ε → 0. We write that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}R_{ε})|_{g}^{2} dv_{g}

= O (∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇σ_{j,ε}|_{g}^{2} R_{ε}^{2} dv_{g})
+ O (∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇R_{ε}|_{g}^{2} σ_{j,ε}^{2} dv_{g})
= O (∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇σ_{j,ε}|_{g}^{2} R_{ε}^{2} dv_{g}) + o (1)

since R_{ε} → 0 in H_{1}^{2}(M) as ε → 0. Thanks to (2.12) and (2.18), we have that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇σ_{j,ε}|_{g}^{2}R_{ε}^{2} dv_{g}

= O ( ∑_{k∈D_{j}} \frac{1}{s_{j,k,ε}^{2}} ∫_{B_{x_{k,ε}}(\frac{1}{2} R_{k}s_{j,k,ε})} R_{ε}^{2} dv_{g})

= O ( ∑_{k∈D_{j}} \frac{1}{s_{j,k,ε}^{2}} Vol_{g} (B_{x_{k,ε}} (\frac{1}{2}R_{k}s_{j,k,ε}))^{\frac{2}{n}} ∥R_{ε}∥_{2^{⋆}}^{2})

with Hölder’s inequalities. By Sobolev’s inequalities, since R_{ε} → 0 in H_{1}^{2}(M) as ε → 0, we finally obtain that

(2.27)  ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}R_{ε})|_{g}^{2} dv_{g} = o (1) .

In the same way, it is easily checked thanks to (2.12), (2.18) and to the fact that we assumed (2.4) that

(2.28)  ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}u_{0})|_{g}^{2} dv_{g} = o (1) .

Let now i ∈ {1, …, N}, i ≠ j. We write thanks to (2.12) and (2.18) that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}φ_{i,ε})|_{g}^{2} dv_{g}

= O ( ∑_{k∈D_{j}} \frac{1}{s_{j,k,ε}^{2}} ∫_{B_{x_{k,ε}}(\frac{1}{2} R_{k}s_{j,k,ε})\B_{x_{k,ε}}(\frac{1}{4} R_{k}s_{j,k,ε})} φ_{i,ε}^{2} dv_{g})

+ O (∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})\∪_{k∈D_{j}} B_{x_{k,ε}}(\frac{1}{4} R_{k}s_{j,k,ε})} |∇φ_{i,ε}|_{g}^{2} dv_{g})

= O ((∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})\∪_{k∈D_{j}} B_{x_{k,ε}}(\frac{1}{4} R_{k}s_{j,k,ε})} φ_{i,ε}^{2^{⋆}} dv_{g})^{\frac{2}{2^{⋆}}} )

+ O (∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})\∪_{k∈D_{j}} B_{x_{k,ε}}(\frac{1}{4} R_{k}s_{j,k,ε})} |∇φ_{i,ε}|_{g}^{2} dv_{g}) .

If i ∈ C_{j}, (1.1) and (2.10) give that \frac{s_{j,i,ε}}{μ_{i,ε}} → +∞ as ε → 0 so that, by (2.12) and (2.13), we have that

B_{x_{i,ε}} (Rμ_{i,ε}) ⊂ ⋃_{k∈D_{j}} B_{x_{k,ε}} (\frac{1}{4}R_{k}s_{j,k,ε})

for all R > 0 as soon as ε is small enough. Direct computations give then that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})\∪_{k∈D_{j}} B_{x_{k,ε}}(\frac{1}{4} R_{k}s_{j,k,ε})} (φ_{i,ε}^{2^{⋆}} + |∇φ_{i,ε}|_{g}^{2}) dv_{g} = o (1) .

If i ∉ C_{j}, direct computations, distinguishing whether i ∈ A_{j} or not, give the same result thanks to (1.1), (2.1), (2.2) and (2.20). Thus we have that for any i ∈ {1, …, N}, i ≠ j,

(2.29)  ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (σ_{j,ε}φ_{i,ε})|_{g}^{2} dv_{g} = o (1) .

At last, we write thanks to (2.12) and (2.18) that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇((σ_{j,ε} − 1) φ_{j,ε})|_{g}^{2} dv_{g}

= O ( ∑_{k∈D_{j}} \frac{1}{s_{j,k,ε}^{2}} ∫_{B_{x_{k,ε}}( \frac{R_{k}}{2} s_{j,k,ε})} φ_{j,ε}^{2} dv_{g})

+ O ( ∑_{k∈D_{j}} ∫_{B_{x_{k,ε}}( \frac{R_{k}}{2} s_{j,k,ε})} |∇φ_{j,ε}|_{g}^{2} dv_{g}) .

By (1.1), (2.9) and (2.10), we have that

\frac{d_{g} (x_{k,ε}, x_{j,ε})}{s_{j,k,ε}} → +∞ as ε → 0

so that

φ_{j,ε}^{2} = O (μ_{j,ε}^{2−n} (1 + \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2}} )^{2−n}) and

|∇φ_{j,ε}|_{g}^{2} = O (μ_{j,ε}^{−n} \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2}} (1 + \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2}} )^{−n})

in B_{x_{k,ε}} (\frac{R_{k}}{2} s_{j,k,ε}) for all k ∈ D_{j}. Thus we have that

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇((σ_{j,ε} − 1) φ_{j,ε})|_{g}^{2} dv_{g}

= O ( ∑_{k∈D_{j}} (\frac{s_{j,k,ε}}{μ_{j,ε}} )^{n−2} (1 + \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2}} )^{2−n})

+ O ( ∑_{k∈D_{j}} (\frac{s_{j,k,ε}}{μ_{j,ε}} )^{n} \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2}} (1 + \frac{d_{g} (x_{j,ε}, x_{k,ε})^{2}}{μ_{j,ε}^{2}} )^{−n}) .

By (2.9) and (2.10), we have that for any k ∈ D_{j},

\frac{s_{j,k,ε}}{μ_{j,ε}} = o (\frac{d_{g} (x_{j,ε}, x_{k,ε})}{μ_{j,ε}}) + o (1) .

This easily leads to

∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇((σ_{j,ε} − 1) φ_{j,ε})|_{g}^{2} dv_{g} = o (1) .

Coming back to (2.26) with (2.27) − (2.29) and this last relation, we obtain that

(2.30)  lim_{ε→0} ∫_{B_{x_{j,ε}}(δ_{j}r_{j,ε})} |∇ (v_{j,ε} − φ_{j,ε})|_{g}^{2} dv_{g} = 0.

Thanks to (2.25), this proves (2.24). Let us prove now that

(2.31)  lim_{ε→0} θ_{j,ε} = 0 , lim_{ε→0} \frac{μ_{j,ε}}{ν_{j,ε}} = 1 and lim_{ε→0} \frac{d_{g} (x_{j,ε}, y_{j,ε})}{μ_{j,ε}} = 0.

Note that, by the definition (2.21) of Λ_{j,ε}, all these limits exist after passing to a subsequence. We write that

J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = ∫_{M} |∇(η_{j,ε}(v_{j,ε} − (1 + θ_{j,ε}) ψ_{j,ε}))|_{g}^{2} dv_{g}

where

η_{j,ε} = η (\frac{d_{g} (y_{j,ε}, .)}{2δ_{j}r_{j,ε}}) and ψ_{j,ε} = ψ_{y_{j,ε},ν_{j,ε}}.

Then we write that

\begin{aligned}J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = & ∥∇ (η_{j,ε}v_{j,ε}) ∥_{2}^{2} + (1 + θ_{j,ε})^{2} ∥∇ (η_{j,ε}ψ_{j,ε}) ∥_{2}^{2} \\ & − 2 (1 + θ_{j,ε}) ∫_{M}(∇ (η_{j,ε}v_{j,ε}) , ∇ (η_{j,ε}ψ_{j,ε}))_{g} dv_{g}. \end{aligned} (2.32)

This leads in particular to

J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) ≥ (∥∇ (η_{j,ε}v_{j,ε}) ∥_{2} − (1 + θ_{j,ε}) ∥∇ (η_{j,ε}ψ_{j,ε}) ∥_{2})^{2}.

Thanks to (2.3), (2.19) and (2.30), it is easily checked by direct computations that lim_{ε→0} ∥∇ (η_{j,ε}v_{j,ε}) ∥_{2} = K_{n}^{− \frac{n}{4}} . Independently, direct computations give thanks to (2.3) and to the definition (2.21) of Λ_{j,ε} that ∥∇ (η_{j,ε}ψ_{j,ε}) ∥_{2} → K_{n}^{− \frac{n}{4}} as ε → 0. By (2.24), the above relation then gives that θ_{j,ε} → 0 as ε → 0. Coming back to (2.32), we then get that

lim_{ε→0} ∫_{M}(∇ (η_{j,ε}v_{j,ε}) , ∇ (η_{j,ε}ψ_{j,ε}))_{g} dv_{g} = K_{n}^{− \frac{n}{2}}

which leads in turn thanks to (2.19) and (2.30) to

lim_{ε→0} ∫_{M}(∇ (η_{j,ε}φ_{j,ε}) , ∇ (η_{j,ε}ψ_{j,ε}))_{g} dv_{g} = K_{n}^{− \frac{n}{2}} .

But this last relation is easily seen to be possible if and only if \frac{μ_{j,ε}}{ν_{j,ε}} → 1

and \frac{d_{g}(x_{j,ε},y_{j,ε})}{μ_{j,ε}} → 0 as ε → 0. This ends the proof of (2.31).

Let 0 < δ < \frac{i_{g}(M)}{2}. We set for x ∈ B_{0} (δr_{j,ε}^{−1}), the Euclidean ball of

center 0 and radius δr_{j,ε}^{−1},

g_{j,ε} (x) = exp_{y_{j,ε}}^{⋆} g (r_{j,ε}x) , (2.33)

u_{j,ε} (x) = r_{j,ε}^{\frac{n}{2} −1}u_{ε} (exp_{y_{j,ε}} (r_{j,ε}x)) ,
h_{j,ε} (x) = h_{ε} (exp_{y_{j,ε}} (r_{j,ε}x)) ,
σ̃_{j,ε} (x) = σ_{j,ε} (exp_{y_{j,ε}} (r_{j,ε}x)) and
ṽ_{j,ε} (x) = σ̃_{j,ε} (x) u_{j,ε} (x) = r_{j,ε}^{\frac{n}{2} −1}v_{j,ε} (exp_{y_{j,ε}} (r_{j,ε}x)) .

By (2.4), we know that

(2.34)  lim_{ε→0} g_{j,ε} = ξ in C_{loc}^{4} (ℝ^{n}) .

Note also that g_{j,ε} is controled on both sides by the Euclidean metric in the sense of bilinear forms. Since u_{ε} verifies Equation (E_{ε}), u_{j,ε} verifies

(2.35)  Δ_{g_{j,ε}}u_{j,ε} + r_{j,ε}^{2}h_{j,ε}u_{j,ε} = u_{j,ε}^{2^{⋆}−1}

in B_{0} (δr_{j,ε}^{−1}). Independently, using (2.3), (2.31) and (2.33), one gets since assertion (i) of Claim 2 holds for i = j, as proved above, that

(2.36)  lim_{ε→0} (\frac{r_{j,ε}}{ν_{j,ε}} )^{\frac{n}{2} −1} u_{j,ε} = a_{n}^{− \frac{n−2}{2}} (\frac{1}{|z|^{n−2}} + h_{j} (z))

in C_{loc}^{2}(ℝ^{n}\Σ_{j}) where Σ_{j} and h_{j} are as in Claim 2. By (2.20), we have that

Σ_{j} ∩ B_{0} (δ_{j}) = {0} .

At last, (2.19) together with (2.31) gives the existence of some C > 0 such that for any ε > 0,

(2.37)  ṽ_{j,ε} (x) ≤ Cψ̃_{j,ε} in B_{0} (δ_{j})

where

ψ̃_{j,ε} (x) = (\frac{ν_{j,ε}r_{j,ε}}{ν_{j,ε}^{2} + a_{n}r_{j,ε}^{2}|x|^{2}} )^{\frac{n−2}{2}} .

We write

(2.38)  η_{j}ṽ_{j,ε} = (1 + θ_{j,ε}) η_{j}ψ̃_{j,ε} + w_{j,ε}

where w_{j,ε} ∈ C_{c}^{∞}(B_{0} (δ_{j})) and

η_{j} = η (\frac{.}{2δ_{j}} ) .

We express (2.22). Differentiating J_{j,ε} with respect to θ, we obtain that

(2.39)  ∫_{B_{0}(δ_{j})} (∇ (η_{j}ψ̃_{j,ε}) , ∇w_{j,ε})_{g_{j,ε}} dv_{g_{j,ε}} = 0.

Differentiating J_{j,ε} with respect to y, we get thanks to (2.36) and (2.38) that

(2.40) ∫_{B_{0}(δ_{j})} (∇ (η_{j} \frac{∂ψ̃_{j,ε}}{∂x_{i}} ) , ∇w_{j,ε})_{g_{j,ε}} dv_{g_{j,ε}} = O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

for all i = 1, …, n. At last, differentiating J_{j,ε} with respect to ν, we obtain thanks to (2.39) that (2.41)

∫_{B_{0}(δ_{j})} (∇ (η_{j}|x|^{2} (1 + a_{n} \frac{r_{j,ε}^{2}|x|^{2}}{ν_{j,ε}^{2}} )^{− \frac{n}{2}} ) , ∇w_{j,ε})_{g_{j,ε}} dv_{g_{j,ε}} = 0.

The aim is now to estimate ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}. We write first thanks to (2.38) and (2.39) that

∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} = ∫_{B_{0}(δ_{j})}(∇w_{j,ε}, ∇ (η_{j}ṽ_{j,ε}))_{g_{j,ε}} dv_{g_{j,ε}}
= ∫_{B_{0}(δ_{j})} w_{j,ε}Δ_{g_{j,ε}} (η_{j}ṽ_{j,ε}) dv_{g_{j,ε}}.

424 o. druet

Writing thanks to (2.33), (2.35), (2.36) and (2.38) that

w_{j,ε}Δ_{g_{j,ε}} (η_{j}ṽ_{j,ε}) = (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} − r_{j,ε}^{2}h_{j,ε} (η_{j}ṽ_{j,ε}) w_{j,ε}

+ O (|w_{j,ε}| |∇σ̃_{j,ε}|_{g_{j,ε}} |∇u_{j,ε}|_{g_{j,ε}})

+ O (|w_{j,ε}| |Δ_{g_{j,ε}} σ̃_{j,ε}| u_{j,ε}) + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O (|σ̃_{j,ε} − σ_{j,ε}^{2^{⋆}−1}| u_{j,ε}^{2^{⋆}−1} |w_{j,ε}|)

in B_{0} (δ_{j}), we get that

∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} (2.42)

= ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} − r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} h_{j,ε} (η_{j}ṽ_{j,ε}) w_{j,ε} dv_{g_{j,ε}} + O (∫_{B_{0}(δ_{j})} |w_{j,ε}| |∇σ̃_{j,ε}|_{g_{j,ε}} |∇u_{j,ε}|_{g_{j,ε}} dv_{g_{j,ε}})

+ O (∫_{B_{0}(δ_{j})} |w_{j,ε}| |Δ_{g_{j,ε}} σ̃_{j,ε}| u_{j,ε} dv_{g_{j,ε}}) + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O (∫_{B_{0}(δ_{j})} |σ̃_{j,ε} − σ_{j,ε}^{2^{⋆}−1}| u_{j,ε}^{2^{⋆}−1} |w_{j,ε}| dv_{g_{j,ε}}) .

For any k ∈ D_{j}, we set

B_{j,k,ε} = \frac{1}{r_{j,ε}} exp_{y_{j,ε}}^{−1} (B_{x_{k,ε}} (\frac{R_{k}}{2} s_{j,k,ε}) \B_{x_{k,ε}} (\frac{R_{k}}{4} s_{j,k,ε})) .

Using (2.12), (2.14), (2.18), (2.33), Hölder’s and Sobolev’s inequalities,

from one bubble to several bubbles 425

we get that ∫_{B_{0}(δ_{j})} |w_{j,ε}| |∇σ̃_{j,ε}|_{g_{j,ε}} |∇u_{j,ε}|_{g_{j,ε}} dv_{g_{j,ε}}

= O ( ∑_{k∈D_{j}} \frac{r_{j,ε}}{s_{j,k,ε}} ∫_{B_{j,k,ε}} |w_{j,ε}| |∇u_{j,ε}|_{g_{j,ε}} dv_{g_{j,ε}} )

= O ( ∑_{k∈D_{j}} \frac{r_{j,ε}}{s_{j,k,ε}} \frac{rj,ε^{\frac{n}{2}}μ_{k,ε}^{\frac{n}{2} −1}}{s_{j,k,ε}^{n−1}} ∫_{B_{j,k,ε}} |w_{j,ε}| dv_{g_{j,ε}} )

= O ( ∑_{k∈D_{j}} \frac{r_{j,ε}^{\frac{n}{2} +1}μ_{k,ε}^{\frac{n}{2} −1}}{s_{j,k,ε}^{n}} ∥w_{j,ε}∥_{2^{⋆}}Vol_{g_{j,ε}} (B_{j,k,ε})^{\frac{2^{⋆}−1}{2^{⋆}}} )

= O (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2}) .

In the same way, one also gets that ∫_{B_{0}(δ_{j})} |w_{j,ε}| |Δ_{g_{j,ε}} σ̃_{j,ε}| u_{j,ε} dv_{g_{j,ε}}

= O (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2})

and that ∫_{B_{0}(δ_{j})} |σ̃_{j,ε} − σ_{j,ε}^{2^{⋆}−1}| u_{j,ε}^{2^{⋆}−1} |w_{j,ε}| dv_{g_{j,ε}}

= O (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} +1}) ∥∇w_{j,ε}∥_{2})

= o (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2})

since μ_{k,ε} = o (s_{j,k,ε}) by (2.9) and (2.10). At last, we write thanks to (0.1), (2.33), (2.37), Hölder’s and Sobolev’s inequalities that

r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} h_{j,ε} (η_{j}ṽ_{j,ε}) w_{j,ε} dv_{g_{j,ε}}

= O (r_{j,ε}^{2}∥∇w_{j,ε}∥_{2} (∫_{B_{0}(δ_{j})} ψ̃j,ε^{\frac{2^{⋆}}{2^{⋆}−1}} dv_{g_{j,ε}})^{\frac{2^{⋆}−1}{2^{⋆}}} )

which leads with direct computations to r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} h_{j,ε} (η_{j}ṽ_{j,ε}) w_{j,ε} dv_{g_{j,ε}}

= o(∥∇w_{j,ε}∥_{2}^{2}) + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε}^{3}) .

Plugging these estimates into (2.42), we obtain that (1 + o(1)) ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} (2.43)

= ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} + o (ν_{j,ε}^{3})

+ O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + O (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2}) .

Relations (2.24) and (2.38) give that

J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} → 0

as ε → 0 so that Hölder’s and Sobolev’s inequalities permit us to write with (2.38) that ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} (2.44)

= (1 + θ_{j,ε})^{2^{⋆}−1} ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}}
+ (2^{⋆} − 1) (1 + θ_{j,ε})^{2^{⋆}−2} ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} w_{j,ε}^{2} dv_{g_{j,ε}}
+ o (∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}) .

from one bubble to several bubbles 427

Direct computations give now thanks to (2.36) and to the Cartan expansion of the metric g_{j,ε} around 0 that

∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ (η_{j}ψ̃_{j,ε}))_{g_{j,ε}} dv_{g_{j,ε}} = ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}}

+ O (r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} |x|^{2}ψ̃_{j,ε}^{2^{⋆}−1} |w_{j,ε}| dv_{g_{j,ε}})

+ O (r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} |x|^{2} |∇ψ̃_{j,ε}| |∇w_{j,ε}| dv_{g_{j,ε}}) + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

so that, using (2.39), Hölder’s and Sobolev’s inequalities, we get that ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}}

= O (r_{j,ε}^{2}∥∇w_{j,ε}∥_{2} (∫_{B_{0}(δ_{j})} (|x|^{2}ψ̃_{j,ε}^{2^{⋆}−1})^{\frac{2^{⋆}}{2^{⋆}−1}} dv_{g_{j,ε}})^{\frac{2^{⋆}−1}{2^{⋆}}} )

+ O (r_{j,ε}^{2}∥∇w_{j,ε}∥_{2} (∫_{B_{0}(δ_{j})} |x|^{4} |∇ψ̃_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}})^{\frac{1}{2}} )

+ O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) .

After simple computations, we finally obtain that ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} (2.45)

= O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε}^{3}) + o(∥∇w_{j,ε}∥_{2}^{2}).

Coming back to (2.43) with (2.44) and (2.45), we arrive thanks to (2.31)

428 o. druet

to (1 + o (1)) ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} (2.46)

= (2^{⋆} − 1) ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} w_{j,ε}^{2} dv_{g_{j,ε}} + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2}) + o (ν_{j,ε}^{3}) .

Let us now consider the following eigenvalue problem:

(2.47)  \{ \begin{matrix}Δ_{g_{j,ε}}ζ_{i,ε} = τ_{i,ε} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} ζ_{i,ε} in B_{0} (δ_{j}) \\ ζ_{i,ε} = 0 on ∂B_{0} (δ_{j}) \\ ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} ζ_{i,ε}ζ_{k,ε} dv_{g_{j,ε}} = K_{n}^{− \frac{n}{2}} δ_{ik} \end{matrix}

with τ_{1,ε} ≤ ⋯ ≤ τ_{i,ε} ≤ . . .. By the result of Appendix 1, we know that

(2.48) lim_{ε→0} τ_{i,ε} = τ_{i} for all i ∈ ℕ^{⋆}

and that

(2.49) lim_{ε→0} ∫_{B_{0}(δ_{j})} |∇ (ζ_{i,ε} − ζ̃_{i,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}} = 0 for all i ∈ ℕ^{⋆}

where

(2.50) ζ̃_{i,ε} = (\frac{r_{j,ε}}{ν_{j,ε}} )^{\frac{n}{2} −1} ζ_{i} (\frac{r_{j,ε}}{ν_{j,ε}} x)

with (ζ_{i}, τ_{i}) the solutions of the following eigenvalue problem:

\{ \begin{matrix}Δ_{ξ}ζ_{i} = τ_{i}u^{2^{⋆}−2}ζ_{i} in ℝ^{n} \\ ∫_{ℝ^{n}} u^{2^{⋆}−2}ζ_{i}ζ_{k} dv_{ξ} = K_{n}^{− \frac{n}{2}} δ_{ik}. \end{matrix}

from one bubble to several bubbles 429

Thanks to the work of Bianchi-Egnell [3], we know that ζ_{1} = u , τ_{1} = 1, (2.51)

ζ_{i} = λ_{i} \frac{∂u}{∂x_{i−1}} , τ_{i} = 2^{⋆} − 1 for i = 2, …, n + 1,

ζ_{n+2} = λ_{n+2} (u − \frac{2}{n(n − 2)}|x|^{2}u^{\frac{n}{n−2}} ) , τ_{n+2} = 2^{⋆} − 1,

where λ_{2}, …, λ_{n+2} are some positive real numbers and that

(2.52)  τ_{n+3} > 2^{⋆} − 1.

Let us now write that

w_{j,ε} = ∑_{i=1}^{n+2} α_{i,ε}ζ_{i,ε} + R_{j,ε}

with

(2.53)  α_{i,ε} = \frac{∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ζ_{i,ε})_{g_{j,ε}} dv_{g_{j,ε}}}{∫_{B_{0}(δ_{j})} |∇ζ_{i,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}}

so that

∫_{B_{0}(δ_{j})} (∇R_{j,ε}, ∇ζ_{i,ε})_{g_{j,ε}} dv_{g_{j,ε}} = 0

for i = 1, …, n + 2. In particular, we obtain thanks to (2.48) that ∫_{B_{0}(δ_{j})} |∇R_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} (2.54)

≥ (τ_{n+3} + o (1)) ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} R_{j,ε}^{2} dv_{g_{j,ε}}.

We also have that ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} (2.55)

= K_{n}^{− \frac{n}{2}} ∑_{i=1}^{n+2} τ_{i,ε}α_{i,ε}^{2} + ∫_{B_{0}(δ_{j})} |∇R_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

thanks to (2.47). At last, we can write that

∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} w_{j,ε}^{2} dv_{g_{j,ε}} (2.56)

= K_{n}^{− \frac{n}{2}} ∑_{i=1}^{n+2} α_{i,ε}^{2} + ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} R_{j,ε}^{2} dv_{g_{j,ε}}.

We now estimate the α_{i,ε}’s. We write thanks to (2.47), (2.49) and (2.53) that

K_{n}^{− \frac{n}{2}} τ_{i,ε}α_{i,ε} = ∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ (ζ_{i,ε} − ζ̃_{i,ε}))_{g_{j,ε}} dv_{g_{j,ε}}

+ ∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ \binom{}{ζ_{i,ε}})_{g_{j,ε}} dv_{g_{j,ε}}

= ∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ \binom{}{ζ_{i,ε}})_{g_{j,ε}} dv_{g_{j,ε}} + o(∥∇w_{j,ε}∥_{2}).

It is then easily checked that

∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ \binom{}{ζ_{i,ε}})_{g_{j,ε}} dv_{g_{j,ε}} = O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

for i = 1, …, n + 2 thanks to (2.39), (2.40), (2.41), (2.50) and (2.51). Thus we obtain that

α_{i,ε}^{2} = o(∥∇w_{j,ε}∥_{2}^{2}) + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) .

Then (2.55) becomes

∥∇w_{j,ε}∥_{2}^{2}(1 + o (1)) = o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + ∥∇R_{j,ε}∥_{2}^{2}

and (2.56) becomes

∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} w_{j,ε}^{2} dv_{g_{j,ε}} = ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} R_{j,ε}^{2} dv_{g_{j,ε}}

+o(∥∇w_{j,ε}∥_{2}^{2}) + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) .

Using (2.46), (2.52) and (2.54), we thus obtain that (2.57)

∥∇w_{j,ε}∥_{2}^{2} = O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε}^{3}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

We claim now that

(2.58) θ_{j,ε} = O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o(ν_{j,ε}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

In order to prove this claim, we first note that

∫_{B_{0}(δ_{j})} |∇ (η_{j}ṽ_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}} = ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

+ (1 + θ_{j,ε})^{2} ∫_{B_{0}(δ_{j})} |∇ (η_{j}ψ̃_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

thanks to (2.38) and (2.39). Direct computations lead then with the Cartan expansion of the metric g_{j,ε} around 0 to

∫_{B_{0}(δ_{j})} |∇ (η_{j}ψ̃_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

= ∫_{B_{0}(δ_{j})} |∇ψ̃_{j,ε}|^{2} dx + O (r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} |x|^{2} |∇ψ̃_{j,ε}|^{2} dx)

+ O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

= K_{n}^{− \frac{n}{2}} + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o(ν_{j,ε}).

We thus get thanks to (2.31) and (2.57) that

\begin{aligned}∫_{B_{0}(δ_{j})} |∇ (η_{j}ṽ_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}} = & K_{n}^{− \frac{n}{2}} (1 + θ_{j,ε})^{2} + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) \\ & + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) + o(ν_{j,ε}). \end{aligned} (2.59)

432 o. druet

Independently, using (2.33), (2.35) and (2.36), we have that ∫_{B_{0}(δ_{j})} |∇ (η_{j}ṽ_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}} = ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}} dv_{g_{j,ε}} − r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} h_{j,ε} (η_{j}ṽ_{j,ε})^{2} dv_{g_{j,ε}}

+ O (∫_{B_{0}(δ_{j})} |∇u_{j,ε}|_{g_{j,ε}} |∇σ̃_{j,ε}|_{g_{j,ε}} u_{j,ε} dv_{g_{j,ε}}) + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O (∫_{B_{0}(δ_{j})} |Δ_{g_{j,ε}} σ̃_{j,ε}| u_{j,ε}^{2} dv_{g_{j,ε}}) + O (∫_{B_{0}(δ_{j})} |σ_{j,ε}^{2} − σ_{j,ε}^{2^{⋆}}| u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}) .

Following what we did to estimate the different terms of (2.42), we deduce from this equation the following:

∫_{B_{0}(δ_{j})} |∇ (η_{j}ṽ_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

= ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}} dv_{g_{j,ε}} − r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} h_{j,ε} (η_{j}ṽ_{j,ε})^{2} dv_{g_{j,ε}}

+O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Writing thanks to (2.38), Hölder’s and Sobolev’s inequalities that ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}} dv_{g_{j,ε}}

= (1 + θ_{j,ε})^{2^{⋆}} ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}} dv_{g_{j,ε}}
+ 2^{⋆} (1 + θ_{j,ε})^{2^{⋆}−1} ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}}
+ O (∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}})

and thanks to the Cartan expansion of the metric g_{j,ε} around 0 that

∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}} dv_{g_{j,ε}} = K_{n}^{− \frac{n}{2}} + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε}) ,

from one bubble to several bubbles 433

we get thanks to (2.31), (2.45) and (2.57) that

∫_{B_{0}(δ_{j})} (ηṽ_{j,ε})^{2^{⋆}} dv_{g_{j,ε}} = (1 + θ_{j,ε})^{2^{⋆}} K_{n}^{− \frac{n}{2}} + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε})

+ O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Using (2.37), it is easily checked that

r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} h_{j,ε} (η_{j}ṽ_{j,ε})^{2} dv_{g_{j,ε}} = o(ν_{j,ε}).

Thus we arrive to ∫_{B_{0}(δ_{j})} |∇ (η_{j}ṽ_{j,ε})|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

= (1 + θ_{j,ε})^{2^{⋆}} K_{n}^{− \frac{n}{2}} + O ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o(ν_{j,ε})

+ O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Combining (2.59) with this last relation, we get (2.58).

We now apply the Pohozaev identity with test function f = \frac{1}{2}|x|^{2}, to u_{j,ε} in Ω_{j,ε} (see Appendix 2) where

Ω_{j,ε} = B_{0} (δ_{j}) \ ⋃_{k∈D_{j}} Ω_{k,j,ε}

with

Ω_{j,k,ε} = r_{j,ε}^{−1} exp_{y_{j,ε}}^{−1} (B_{x_{k,ε}} (\frac{R_{k}}{2} s_{j,k,ε})) .

We thus have that

∫_{Ω_{j,ε}} (r_{j,ε}^{2}h_{j,ε} + \frac{r_{j,ε}^{2}}{2} (∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f) u_{j,ε}^{2} dv_{g_{j,ε}} (2.60)

= \frac{1}{n} ∫_{Ω_{j,ε}}(Δ_{g_{j,ε}}f + n)u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

+ ∫_{Ω_{j,ε}}(∇^{2}f − g_{j,ε})(∇u_{j,ε}, ∇u_{j,ε}) dv_{g_{j,ε}} + A_{ε} where A_{ε} is the boundary term

A_{ε} =∫_{∂Ω_{j,ε}}(\frac{1}{2} |∇u_{j,ε}|_{g_{j,ε}}^{2}(∇f, ν)_{g_{j,ε}}− (∇u_{j,ε}, ∇f)_{g_{j,ε}}(∇u_{j,ε}, ν)_{g_{j,ε}}) dσ_{g_{j,ε}}

− \frac{n − 2}{2} ∫_{∂Ω_{j,ε}} u_{j,ε}(∇u_{j,ε}, ν)_{g_{j,ε}} dσ_{g_{j,ε}}

− ∫_{∂Ω_{j,ε}}(∇f, ν)_{g_{j,ε}} (\frac{1}{2^{⋆}} u_{j,ε}^{2^{⋆}} − \frac{r_{j,ε}^{2}}{2} h_{j,ε}u_{j,ε}^{2}) dσ_{g_{j,ε}}

+ \frac{1}{2} ∫_{∂Ω_{j,ε}}(Δ_{g_{j,ε}}f + n)u_{j,ε}(∇u_{j,ε}, ν)_{g_{j,ε}} dσ_{g_{j,ε}}

− \frac{1}{4} ∫_{∂Ω_{j,ε}}(∇ (Δ_{g_{j,ε}}f) , ν)_{g_{j,ε}}u_{j,ε}^{2} dσ_{g_{j,ε}}

where ν denotes the unit outer normal of ∂Ω_{j,ε}. Note that, by (2.11) and (2.12),

∂Ω_{j,ε} = ∪_{k∈D_{j}}∂Ω_{j,k,ε} ∪ ∂B_{0} (δ_{j})

and the union is a disjoint one. We denote by A_{k,ε} the part of A_{ε} corresponding to ∂Ω_{j,k,ε}. Noting that, on ∂Ω_{j,k,ε},

r_{j,ε}|∇f| ≤ d_{g} (y_{j,ε}, x_{k,ε}) + \frac{R_{k}}{2} s_{j,k,ε},

we can estimate A_{k,ε} thanks to (0.1), (2.4), (2.10), (2.11), (2.14) and (2.34). This leads to

A_{k,ε} = O ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}))

for all k ∈ D_{j}. In order to estimate the part of A_{ε} corresponding to ∂B_{0} (δ_{j}), we use (0.1), (2.3), (2.4), (2.31), (2.34), (2.36) and the explicit form of h_{j} given in Claim 2, point (i). We finally obtain that

(2.61) A_{ε} = \frac{(n − 2)^{2}}{2} a_{n}^{2−n}ω_{n−1}h_{j} (0) (\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2} + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}})) .

Since (η_{j}ψ̃_{j,ε}) is radially symmetrical, noting that σ̃_{j,ε} = 1 in Ω_{j,ε}, we get with (2.31), (2.34), (2.36), (2.38) and (2.57) that

∫_{Ω_{j,ε}}(∇^{2}f − g_{j,ε})(∇u_{j,ε}, ∇u_{j,ε}) dv_{g_{j,ε}} (2.62)

= o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε}^{3}) + o ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

The Cartan expansion of the metric g_{j,ε} around 0 gives that

Δ_{g_{j,ε}}f + n = \frac{1}{3}Ric_{g_{j,ε}} (0)_{kl} x^{k}x^{l}

+ \frac{1}{2}∂_{klm} (ln \sqrt{|g_{j,ε}|}) (0)x^{k}x^{l}x^{m} + O (r_{j,ε}^{4}|x|^{4})

in B_{0} (δ_{j}) where Ric_{g_{j,ε}} denotes the Ricci curvature of g_{j,ε} and where |g_{j,ε}| is the determinant of the matrix ((g_{j,ε})_{ik}). Since σ̃_{j,ε} = 1 in Ω_{j,ε}, we can write thanks to (2.3), (2.31), (2.36), (2.38) and to the fact that \binom{}{ψ_{j,ε}} is radially symmetrical that

∫_{Ω_{j,ε}}(Δ_{g_{j,ε}}f + n)u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

= \frac{(1 + θ_{j,ε})^{2^{⋆}}}{3} Ric_{g_{j,ε}} (0)_{kl} ∫_{B_{0}(δ_{j})} x^{k}x^{l}ψ̃_{j,ε}^{2^{⋆}} dx + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O (r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} |x|^{2}ψ̃_{j,ε}^{2^{⋆}−1}|w_{j,ε}| dx) + O (r_{j,ε}^{4} ∫_{B_{0}(δ_{j})} |x|^{4}ψ̃_{j,ε}^{2^{⋆}} dx)

+ O (r_{j,ε}^{2} ∑_{k∈D_{j}} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} ) + O (r_{j,ε}^{2}∥w_{j,ε}∥_{2^{⋆}}^{2^{⋆}}) .

Using (2.57) and (2.58), this leads thanks to Hölder’s and Sobolev’s inequalities and to some explicit computations to the following:

∫_{Ω_{j,ε}}(Δ_{g_{j,ε}}f + n)u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} (2.63)

= \frac{n}{3}K_{n}^{− \frac{n}{2}} S_{g} (y_{j,ε}) ν_{j,ε}^{2} + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2})

+ O (r_{j,ε}^{2} ∑_{k∈D_{j}} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} ) + o (ν_{j,ε}^{3}) .

Let k ∈ D_{j}. We write that

r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

= ∫_{B_{x_{k,ε}}( \frac{R_{k}}{2} s_{j,k,ε})} d_{g} (y_{j,ε}, x)^{2} (\frac{ν_{j,ε}}{ν_{j,ε}^{2} + a_{n}d_{g} (y_{j,ε}, x)^{2}} )^{n} dv_{g}.

Let us assume first that d_{g} (y_{j,ε}, x_{k,ε}) = O (s_{j,k,ε}). Then we write that for some R > 0 large and for ε small,

r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

≤ ∫_{B_{y_{j,ε}}(Rs_{j,k,ε})} d_{g} (y_{j,ε}, x)^{2} (\frac{ν_{j,ε}}{ν_{j,ε}^{2} + a_{n}d_{g} (y_{j,ε}, x)^{2}} )^{n} dv_{g} .

Independently, by (1.1), (2.9) and (2.10), we have that s_{j,k,ε} = o(d_{g}(x_{j,ε}, x_{k,ε})). Thanks to (2.31), we then obtain that s_{j,k,ε} = o (ν_{j,ε}). Thus we get that

r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} = O (\frac{s_{j,k,ε}^{n+2}}{ν_{j,ε}^{n}} )

= O ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} \frac{s_{j,k,ε}^{2n}}{ν_{j,ε}^{n}μ_{k,ε}^{n−2}} ) .

Since we assumed that d_{g} (y_{j,ε}, x_{k,ε}) = O (s_{j,k,ε}), we also get thanks to (2.9), (2.10) and (2.31) that s_{j,k,ε}^{2} = O (ν_{j,ε}μ_{k,ε}). Thus, since μ_{k,ε} → 0 as ε → 0, we have obtained in this case that

r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} = o ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Let us assume now that \frac{d_{g}(y_{j,ε},x_{k,ε})}{s_{j,k,ε}} → +∞ as ε → 0. In this case, we can write that

\begin{matrix}r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} \\ = O (d_{g} (y_{j,ε}, x_{k,ε})^{2} (\frac{s_{j,k,ε}}{ν_{j,ε}} )^{n} (1 + a_{n} \frac{d_{g} (y_{j,ε}, x_{k,ε})^{2}}{ν_{j,ε}^{2}} )^{−n}) . \end{matrix}

We write then thanks to (1.1), (2.10) and (2.31) that

(\frac{s_{j,k,ε}}{ν_{j,ε}} )^{n} = O ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} \frac{μ_{k,ε}}{ν_{j,ε}} (1 + a_{n} \frac{d_{g} (y_{j,ε}, x_{k,ε})^{2}}{ν_{j,ε}^{2}} )^{n−1})

so that we get that

r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

= O ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} μ_{k,ε}ν_{j,ε} \frac{d_{g} (y_{j,ε}, x_{k,ε})^{2}}{ν_{j,ε}^{2} + a_{n}d_{g} (y_{j,ε}, x_{k,ε})^{2}} ) .

This gives that the estimate of the first case also holds in this second case. We have thus obtained that

(2.64)  r_{j,ε}^{2} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} = o ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2})

for all k ∈ D_{j}. Let us estimate the left-hand side term of (2.60). Using (0.1) and the Cartan expansion of the metric g_{j,ε} around 0, one gets that

r_{j,ε}^{2}h_{j,ε} + \frac{r_{j,ε}^{2}}{2} (∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f

= r_{j,ε}^{2} (h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε}))

+ (\frac{3}{2}r_{j,ε}^{2}∂_{k}h_{j,ε}(0) + \frac{1}{4}∂_{k} (Δ_{g_{j,ε}}^{2} f) (0)) x^{k} + O (r_{j,ε}^{4}|x|^{2})

in B_{0} (δ_{j}) so that we obtain with (2.4), (2.31), (2.36), (2.38) and the fact that \binom{}{ψ_{j,ε}} is radially symmetrical that

∫_{Ω_{j,ε}} (r_{j,ε}^{2}h_{j,ε} + \frac{r_{j,ε}^{2}}{2} (∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f) u_{j,ε}^{2} dv_{g_{j,ε}}

= r_{j,ε}^{2} (h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε})) (1 + θ_{j,ε})^{2} ∫_{B_{0}(δ_{j})} ψ̃_{j,ε}^{2} dx

+ o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ O (r_{j,ε}^{2} ∑_{k∈D_{j}} ∫_{Ω_{j,k,ε}} ψ̃_{j,ε}^{2} dv_{g_{j,ε}} ) + O (r_{j,ε}^{4} ∫_{B_{0}(δ_{j})} |x|^{2}ψ̃_{j,ε}^{2} dx)

+ O (r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} ψ̃_{j,ε}|w_{j,ε}| dx) + O (r_{j,ε}^{2} ∫_{B_{0}(δ_{j})} |w_{j,ε}|^{2} dx) .

This leads thanks to (2.4), (2.57) and (2.58) and after computations similar to those developed above to prove (2.63) and (2.64) to

∫_{Ω_{j,ε}} (r_{j,ε}^{2}h_{j,ε} + \frac{r_{j,ε}^{2}}{2} (∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f) u_{j,ε}^{2} dv_{g_{j,ε}}

= (h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε})) × \{ \begin{matrix}64ω_{3}ν_{j,ε}^{2} ln (\frac{r_{j,ε}}{ν_{j,ε}} ) & if n = 4 \\ \frac{4(n − 1)}{n − 4}K_{n}^{− \frac{n}{2}} ν_{j,ε}^{2} & if n ≥ 5 \end{matrix}

+ o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + o (ν_{j,ε}^{3}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Coming back to (2.60) with (2.61) − (2.64) and this last estimate, we finally arrive to (2.65)

\frac{3}{2}ω_{2}h_{j} (0) \frac{ν_{j,ε}}{r_{j,ε}} = o (\frac{ν_{j,ε}}{r_{j,ε}} ) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} ) (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}))

when n = 3, to

(h_{ε} (y_{j,ε}) − C(4)S_{g} (y_{j,ε}))r_{j,ε}^{2} ln (\frac{1}{ν_{j,ε}} ) (2.66)

= 2h_{j} (0) + o (1) + O (\frac{r_{j,ε}^{2}}{ν_{j,ε}^{2}} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}))

when n = 4, to

(h_{ε} (y_{j,ε}) − C(5)S_{g} (y_{j,ε}))r_{j,ε}^{3}ν_{j,ε}^{−1} (2.67)

= 9\sqrt{15}\frac{ω_{4}}{ω_{5}} h_{j} (0) + o (1)

+ O (\frac{r_{j,ε}^{3}}{ν_{j,ε}^{3}} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{3} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}))

when n = 5 and to

(h_{ε} (y_{j,ε}) − C(n)S_{g} (y_{j,ε}))r_{j,ε}^{n−2}ν_{j,ε}^{4−n} (2.68)

= \frac{n − 4}{4 (n − 1)} \frac{(n − 2)^{2}}{2} a_{n}^{2−n}Kn^{\frac{n}{2}} ω_{n−1}h_{j} (0) + o (1) + o (r_{j,ε}^{n−2}ν_{j,ε}^{5−n})

+ O ((\frac{r_{j,ε}}{ν_{j,ε}} )^{n−2} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}))

when n ≥ 6. In these relations, C(n) = \frac{n−2}{4(n−1)}.

We prove now point (ii) of Claim 2 by induction on j. Let us start by j = 1. Assume that r_{1,ε} → 0 as ε → 0. Note that, by (1.6), A_{1} = {2, …, N}. We write with (2.2) that

r_{1,ε}^{2} ≤ \frac{μ_{1,ε}}{μ_{k,ε}} d_{g} (x_{1,ε}, x_{k,ε})^{2} + μ_{1,ε}μ_{k,ε}

for all k ∈ {2, …, N} which gives thanks to (1.1) and (1.6) that d_{g}(x_{1,ε}, x_{k,ε}) ≥ r_{1,ε}(1 + o(1)) for all k ∈ {2, …, N}. Then C_{1} = ∅ where C_{1} is as in (2.8) and thus D_{1} = ∅. Then (2.65) leads to a contradiction when n = 3 since h_{1}(0) > 0 by (2.7) and \frac{ν_{1,ε}}{r_{1,ε}} → 0 as ε → 0 thanks to (2.3) and (2.31). Thus, in dimension n = 3, we have that lim inf_{ε→0} r_{1,ε} > 0. This proves that the assertion of Claim 2 in the case n = 3 holds for j = 1. For n ≥ 4, since D_{1} = ∅, (2.66) − (2.68) clearly lead thanks to (0.1) and (2.31) to the assertion (ii) of Claim 2 for j = 1.

Let now j ∈ {2, …, N} and assume that the assertion of Claim 2 for n = 3 holds for i = 1, …, j − 1 and that point (ii) of Claim 2 holds for i = 1, …, j − 1. Assume moreover that r_{j,ε} → 0 as ε → 0. Let k ∈ D_{j}. Thanks to (1.6), (2.1) and (2.9), we have that k < j and that j ∈ A_{k}. Then, thanks to (2.2) and (2.10), we obtain that s_{j,k,ε} ≥ r_{k,ε}. By (2.11), since we assumed that r_{j,ε} → 0 as ε → 0, we get that r_{k,ε} → 0 as ε → 0. If n = 3, since k < j, the induction hypothesis gives a contradiction so that we get that D_{j} = ∅. As above, we deduce then from (2.65) that the assertion of Claim 2 in the case n = 3 holds for j. Assume now that n ≥ 4. Let k ∈ D_{j}. We write with (2.10) and (2.31) that

\frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}} = \frac{d_{g} (x_{j,ε}, x_{k,ε})}{s_{j,k,ε}} + o (\frac{μ_{j,ε}}{s_{j,k,ε}} ) = O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} ) .

Thus, since s_{j,k,ε} ≥ r_{k,ε} and thanks to (2.9),

(\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}) = O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} (\frac{μ_{k,ε}}{r_{k,ε}} )^{n−2}) .

Since k < j and r_{k,ε} → 0 as ε → 0, we can apply point (ii) of Claim 2 to k to obtain that

(\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}(1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}})=\begin{cases}O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} μ_{k,ε}^{2} ln (\frac{1}{μ_{k,ε}} )) & if n = 4 \\ O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} μ_{k,ε}^{2}) & if n ≥ 5. \end{cases}

Thanks to (2.9), this leads to

(\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}) = \{ \begin{matrix}o (μ_{j,ε}^{2} ln (\frac{1}{μ_{j,ε}} )) & if n = 4 \\ o (μ_{j,ε}^{2}) & if n ≥ 5 \end{matrix}

for all k ∈ D_{j}. Then (2.66) − (2.68) together with (0.1), (2.31) and this last estimate clearly lead to point (ii) of Claim 2 for j. This proves that the assertion of Claim 2 for n = 3 holds for all j ∈ {1, …, N} and that point (ii) of Claim 2 holds for all j ∈ {1, …, N} when n ≥ 4.

It remains to prove point (iii) of Claim 2. We assume throughout the end of this section that n ≥ 4. Let j ∈ {1, …, N} and assume that h_{0} (\overline{x}_{j}) > C(n)S_{g} (\overline{x}_{j}). Then, thanks to point (ii) of Claim 2, we have that

r_{j,ε} = O (ln (\frac{1}{μ_{j,ε}} )^{− \frac{1}{2}} ) if n = 4, (2.69)

r_{j,ε} = O (μ_{j,ε}^{− \frac{n−4}{n−2}} ) if n ≥ 5.

We let 0 < δ < δ_{j} and we set

Ω_{j,ε} (δ) = B_{0} (δ) \ ∪_{k∈D_{j}} Ω_{j,k,ε}.

By integration by parts, we have that

(2.70)  ∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε}Δ_{ξ}u_{j,ε} dx = \frac{1}{2} ∫_{∂Ω_{j,ε}(δ)} |∇u_{j,ε}|_{ξ}^{2} \frac{x_{γ}}{|x|} dσ_{ξ}

− ∫_{∂Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} \frac{x^{i}∂_{i}u_{j,ε}}{|x|} dσ_{ξ}

for all γ = 1, …, n. By (2.11) and (2.12), we have that

∂Ω_{j,ε} (δ) = ∪_{k∈D_{j}}∂Ω_{j,k,ε} ∪ ∂B_{0} (δ) ,

the union being disjoint. Using (2.36) and the explicit form of h_{j}, we get that

lim_{ε→0} ((\frac{r_{j,ε}}{ν_{j,ε}} )^{n−2} ∫_{∂B_{0}(δ)} (\frac{1}{2} |∇u_{j,ε}|_{ξ}^{2} \frac{x_{γ}}{|x|} − ∂_{γ}u_{j,ε} \frac{x^{i}∂_{i}u_{j,ε}}{|x|} ) dσ_{ξ}) (2.71)
= a_{n}^{2−n} (n − 2) ω_{n−1}∂_{γ}h_{j} (0) .

Let k ∈ D_{j}. Thanks to (1.6), (2.1), (2.2), (2.9) and (2.10), we get successively that k < j, that j ∈ A_{k}, that r_{k,ε} ≤ s_{j,k,ε} and that r_{k,ε} → 0 as ε → 0. Applying point (ii) of Claim 2 to k, we thus obtain that

(2.72)  \frac{1}{s_{j,k,ε}} = \{ \begin{matrix}O (ln (\frac{1}{μ_{k,ε}} )^{\frac{1}{2}} ) & if n = 4 \\ O (μ_{k,ε}^{− \frac{n−4}{n−2}} ) & if n ≥ 5 \end{matrix}

442 o. druet

for all k ∈ D_{j}. Thanks to (2.14), we get that

(\frac{r_{j,ε}}{ν_{j,ε}} )^{n−2} ∫_{∂Ω_{j,k,ε}} |∇u_{j,ε}|_{ξ}^{2} dσ_{ξ} = O (\frac{r_{j,ε}^{n−1}}{ν_{j,ε}^{n−2}} \frac{μ_{k,ε}^{n−2}}{s_{j,k,ε}^{n−1}} ) .

for all k ∈ D_{j} which gives thanks to (2.9), (2.31), (2.69) and (2.72) that

(\frac{r_{j,ε}}{ν_{j,ε}} )^{n−2} ∫_{∂Ω_{j,k,ε}} |∇u_{j,ε}|_{ξ}^{2} dσ_{ξ} = o (1) .

Coming back to (2.70) with (2.71) and this last estimate, we arrive to

\begin{matrix}lim_{ε→0} & ((\frac{r_{j,ε}}{ν_{j,ε}} )^{n−2} ∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε}Δ_{ξ}u_{j,ε} dx) \\ = a_{n}^{2−n} (n − 2) ω_{n−1}∂_{γ}h_{j} (0) . \end{matrix} (2.73)

We write now that

\begin{aligned}∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε}Δ_{ξ}u_{j,ε} dx = & ∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε}Δ_{g_{j,ε}}u_{j,ε} dx \\ & + ∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (Δ_{ξ}u_{j,ε} − Δ_{g_{j,ε}}u_{j,ε}) dx. \end{aligned} (2.74)

By Equation (2.35) and some integration by parts, we obtain that

∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε}Δ_{g_{j,ε}}u_{j,ε} dx = \frac{r_{j,ε}^{2}}{2} ∫_{Ω_{j,ε}(δ)} ∂_{γ}h_{j,ε}u_{j,ε}^{2} dx + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

using (0.1), (2.3), (2.4), (2.9), (2.14), (2.31), (2.36), (2.69) and (2.72) to estimate the different boundary terms. Noting that σ̃_{j,ε} = 1 in Ω_{j,ε} (δ), we get with (0.1), (2.33) and (2.37) that

\frac{r_{j,ε}^{2}}{2} ∫_{Ω_{j,ε}(δ)} ∂_{γ}h_{j,ε}u_{j,ε}^{2} dx = O (r_{j,ε}^{3} ∫_{B_{0}(δ)} ψ̃_{j,ε}^{2} dx) = o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

thanks to (2.31) and (2.69) so that we arrive to

(2.75)  ∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε}Δ_{g_{j,ε}}u_{j,ε} dx = o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) .

We write that

∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (Δ_{ξ}u_{j,ε} − Δ_{g_{j,ε}}u_{j,ε}) dx (2.76)

= ∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (g_{j,ε}^{αβ} − δ^{αβ}) ∂_{αβ}u_{j,ε} dx

+ ∫_{Ω_{j,ε}(δ)} \frac{∂_{α} (\sqrt{|g_{j,ε}|}g_{j,ε}^{αβ})}{\sqrt{|g_{j,ε}|}}∂_{β}u_{j,ε}∂_{γ}u_{j,ε} dx.

By integration by parts, we get that

∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (g_{j,ε}^{αβ} − δ^{αβ}) ∂_{αβ}u_{j,ε} dx

= \frac{1}{2} ∫_{Ω_{j,ε}(δ)} ∂_{γ} (g_{j,ε}^{αβ}) ∂_{α}u_{j,ε}∂_{β}u_{j,ε} dx

− ∫_{Ω_{j,ε}(δ)} ∂_{α} (g_{j,ε}^{αβ}) ∂_{γ}u_{j,ε}∂_{β}u_{j,ε} dx + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

using (0.1), (2.3), (2.4), (2.9), (2.14), (2.31), (2.36), (2.69) and (2.72) to estimate the boundary terms. Thus we arrive with (2.76) to

∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (Δ_{ξ}u_{j,ε} − Δ_{g_{j,ε}}u_{j,ε}) dx

= \frac{1}{2} ∫_{Ω_{j,ε}(δ)} ∂_{γ} (g_{j,ε}^{αβ}) ∂_{α}u_{j,ε}∂_{β}u_{j,ε} dx + o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2})

+ ∫_{Ω_{j,ε}(δ)} g_{j,ε}^{αβ} \frac{∂_{α} (\sqrt{|g_{j,ε}|})}{\sqrt{|g_{j,ε}|}}∂_{β}u_{j,ε}∂_{γ}u_{j,ε} dx.

With an expansion of the different terms involving the metric g_{j,ε} around 0, noting that

∫_{∂B_{0}(r)} x^{α}x^{β}x^{γ}dσ_{ξ} = 0

for all α, β, γ and all r > 0 and writing thanks to (2.38) that u_{j,ε} = (1 + θ_{j,ε}) ψ̃_{j,ε} + w_{j,ε} in Ω_{j,ε} (δ) for δ > 0 small enough since σ̃_{j,ε} = 1 in Ω_{j,ε} (δ), we obtain that

∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (Δ_{ξ}u_{j,ε} − Δ_{g_{j,ε}}u_{j,ε}) dx

= o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) + O (r_{j,ε}^{3} ∫_{B_{0}(δ)} |x|^{2} |∇ψ̃_{j,ε}|^{2} dx)

+ O (r_{j,ε}^{2} ∫_{B_{0}(δ)} |x| |∇ψ̃_{j,ε}| |∇w_{j,ε}| dx) + O (r_{j,ε}^{2}∥∇w_{j,ε}∥_{2}^{2})

+ O (r_{j,ε}^{2} ∑_{k∈D_{j}} ∫_{Ω_{j,k,ε}} |x| |∇ψ̃_{j,ε}|^{2} dx) .

Using (2.14), (2.57) and (2.69), one easily checks that this leads to

∫_{Ω_{j,ε}(δ)} ∂_{γ}u_{j,ε} (Δ_{ξ}u_{j,ε} − Δ_{g_{j,ε}}u_{j,ε}) dx = o ((\frac{ν_{j,ε}}{r_{j,ε}} )^{n−2}) .

Coming back to (2.73) with (2.74), (2.75) and this last relation, we get that ∂_{γ}h_{j}(0) = 0 for all k = 1, …, n. Thus point (iii) of Claim 2 is proved. q.e.d.

3. Almost isolated concentration points

We consider in this section the case of an almost isolated concentration point. We just sketch the arguments since they mainly follow the lines of those developed in Section 2. We thus refer the reader to the corresponding parts of the previous section for details on some of the assertions below. We let j ∈ {1, …, N} and we assume that

lim_{ε→0} r_{j,ε} = r_{0} > 0.

Thanks to the definition (2.2) of r_{j,ε}, this implies that

(3.1)  u_{0} ≡ 0,

that

(3.2)  μ_{i,ε} = O (μ_{j,ε}) for all i ∈ {1, …, N}

and that

(3.3)  for any i ∈ \{1, …, N\} , i ≠ j, μ_{j,ε} = O (μ_{i,ε}) ⇒ \overline{x}_{i} ≠ \overline{x}_{j}.

It comes then from (1.7) that

(3.4) lim_{ε→0} μ_{j,ε}^{1− \frac{n}{2}} u_{ε} = a_{n}^{− \frac{n−2}{2}} (n − 2) ω_{n−1} ∑_{i=1}^{N} (lim_{ε→0} \frac{μ_{i,ε}}{μ_{j,ε}} )^{\frac{n}{2} −1} G_{0} (\overline{x}_{i}, .)

in C_{loc}^{2} (M\S), S as in (1.2), where G_{0} is the Green function of Δ_{g} + h_{0}. Let us set

(3.5)  C_{j} = \{k ∈ \{1, …, N\} , k ≠ j s.t. \overline{x}_{j} = \overline{x}_{k}\}.

Thanks to (3.2) and (3.3), we have that

(3.6)  for any k ∈ C_{j}, μ_{k,ε} = o (μ_{j,ε}) .

We let now

(3.7)  s_{j,k,ε}^{2} = \frac{μ_{k,ε}}{μ_{j,ε}} d_{g} (x_{k,ε}, x_{j,ε})^{2} + μ_{j,ε}μ_{k,ε}

for k ∈ C_{j}. Note that, thanks to (3.6),

(3.8)  s_{j,k,ε} = o (1) for all k ∈ C_{j}.

We let now D_{j} be a subset of C_{j} and (R_{k})_{k∈D_{j}} be a sequence of positive real numbers such that

(3.9) for any k, k′ ∈ D_{j}, k ≠ k′, \frac{d_{g} (x_{k,ε}, x_{k′,ε})}{s_{j,k,ε}} → +∞ as ε → 0

and such that

for any k′ ∈ C_{j}, ∃ a unique k ∈ D_{j} such that

lim sup_{ε→0} \frac{d_{g} (x_{k,ε}, x_{k′,ε})}{s_{j,k,ε}} ≤ \frac{R_{k}}{10} and lim sup_{ε→0} \frac{s_{j,k′,ε}}{s_{j,k,ε}} ≤ \frac{R_{k}}{10} .

(3.10)

We claim that there exists C > 0 independent of ε such that for any k ∈ D_{j},

\begin{matrix}for any x ∈ B_{x_{k,ε}} (R_{k}s_{j,k,ε}) \B_{x_{k,ε}} (\frac{R_{k}}{4} s_{j,k,ε}) , \\ |∇u_{ε}|_{g}(x) ≤ Cμ_{k,ε}^{\frac{n}{2} −1}s_{j,k,ε}^{1−n} , u_{ε}(x) ≤ Cμ_{k,ε}^{\frac{n}{2} −1}s_{j,k,ε}^{2−n}. \end{matrix} (3.11)

The proof of such a claim is based on Claim 1 and follows exactly the proof of (2.14) in Section 2. We let η : [0, +∞[ ↦ ℝ be a smooth function verifying that η ≡ 1 on [0, \frac{1}{4}] and η ≡ 0 on [\frac{1}{2}, +∞[. We set

(3.12)  σ_{j,ε} = ∏_{k∈D_{j}} (1 − η (\frac{d_{g} (x_{k,ε}, .)}{R_{k}s_{j,k,ε}})) and v_{j,ε} = σ_{j,ε}u_{ε}.

It is easily checked thanks to (1.3) that there exists C > 0 independent of ε such that

(3.13)  v_{j,ε} ≤ Cφ_{j,ε} in B_{x_{j,ε}} (δ_{j})

where δ_{j} ∈ ℝ_{+}^{⋆} is fixed such that

δ_{j} ≤ \frac{1}{2} min\{d_{g} (\overline{x}_{j}, \overline{x}_{k}) , k ∈ \{1, …, N\} \C_{j}, k ≠ j\} and δ_{j} ≤ \frac{1}{6}i_{g}(M).

We set

(3.14) Λ_{j,ε} = { (y, ν, θ) ∈ M × ℝ_{+}^{⋆} × ℝ s.t.

d_{g} (x_{j,ε}, y) ≤ μ_{j,ε} , \frac{1}{2} ≤ \frac{ν}{μ_{j,ε}} ≤ 2 , −\frac{1}{2} ≤ θ ≤ \frac{1}{2}\}

and we let (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) ∈ Λ_{j,ε} be such that

(3.15)  J_{j,ε}(y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = min_{(y,ν,θ)∈Λ_{j,ε}} J_{j,ε}(y, ν, θ)

where

J_{j,ε}(y, ν, θ) = ∫_{M} |∇ (η (\frac{d_{g} (y, .)}{2δ_{j}}) (v_{j,ε} − (1 + θ) ψ_{y,ν})|_{g}^{2} dv_{g}

with ψ_{y,ν} as in (2.23). Let us prove that

(3.16)  J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) → 0 as ε → 0.

Note that (x_{j,ε}, μ_{j,ε}, 0) ∈ Λ_{j,ε} so that

J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) ≤ J_{j,ε} (x_{j,ε}, μ_{j,ε}, 0) .

We write then that

J_{j,ε} (x_{j,ε}, μ_{j,ε}, 0) = ∫_{M} |∇ (η (\frac{d_{g} (x_{j,ε}, .)}{2δ_{j}}) (v_{j,ε} − φ_{j,ε}))|_{g}^{2} dv_{g}
≤ C ∫_{B_{x_{j,ε}}(δ_{j})\B_{x_{j,ε}}( \frac{δ_{j}}{2} )} (v_{j,ε} − φ_{j,ε})^{2} dv_{g}

+C ∫_{B_{x_{j,ε}}(δ_{j})} |∇(v_{j,ε} − φ_{j,ε})|_{g}^{2} dv_{g}

where C > 0 is some constant independent of ε. Thanks to (3.13), it is easily checked that

lim_{ε→0} ∫_{B_{x_{j,ε}}(δ_{j})\B_{x_{j,ε}}( \frac{δ_{j}}{2} )} (v_{j,ε} − φ_{j,ε})^{2} dv_{g} = 0

so that we obtain that

(3.17) J_{j,ε} (y_{j,ε}, ν_{j,ε}, θ_{j,ε}) = O (∫_{B_{x_{j,ε}}(δ_{j})} |∇ (v_{j,ε} − φ_{j,ε}) |_{g}^{2} dv_{g}) .

We write with (1.5) and (3.1) that

∫_{B_{x_{j,ε}}(δ_{j})} |∇ (v_{j,ε} − φ_{j,ε}) |_{g}^{2} dv_{g}

≤ C ∫_{B_{x_{j,ε}}(δ_{j})} |∇((σ_{j,ε} − 1) φ_{j,ε})|_{g}^{2} dv_{g}

+C ∑_{i≠j} ∫_{B_{x_{j,ε}}(δ_{j})} |∇ (σ_{j,ε}φ_{i,ε}) |_{g}^{2} dv_{g}

+C ∫_{B_{x_{j,ε}}(δ_{j})} |∇ (σ_{j,ε}R_{ε}) |_{g}^{2} dv_{g}.

with ∥R_{ε}∥_{H_{1}^{2}(M)} → 0 as ε → 0. Following the proofs of (2.27) − (2.29) and of the relation preceding (2.30), we then estimate all the terms of the right-hand side of this relation. This leads to

(3.18)  lim_{ε→0} ∫_{B_{x_{j,ε}}(δ_{j})} |∇ (v_{j,ε} − φ_{j,ε})|_{g}^{2} dv_{g} = 0.

Thanks to (3.17), this proves (3.16). As in Section 2, we deduce from (3.16) that

(3.19)  lim_{ε→0} θ_{j,ε} = 0, lim_{ε→0} \frac{μ_{j,ε}}{ν_{j,ε}} = 1 and lim_{ε→0} \frac{d_{g} (x_{j,ε}, y_{j,ε})}{μ_{j,ε}} = 0.

Let 0 < 3δ_{j} < δ < \frac{i_{g}(M)}{2}. We set for x ∈ B_{0} (δ), the Euclidean ball of center 0 and radius δ,

g_{j,ε} (x) = exp_{y_{j,ε}}^{⋆} g (x) , (3.20)

u_{j,ε} (x) = u_{ε} (exp_{y_{j,ε}} (x)) ,
h_{j,ε} (x) = h_{ε} (exp_{y_{j,ε}} (x)) ,
σ̃_{j,ε} (x) = σ_{j,ε} (exp_{y_{j,ε}} (x)) and
ṽ_{j,ε} (x) = σ̃_{j,ε} (x) u_{j,ε} (x) = v_{j,ε} (exp_{y_{j,ε}} (x)) .

Note that g_{j,ε} is controled on both sides by the Euclidean metric in the sense of bilinear forms. Since u_{ε} verifies Equation (E_{ε}), u_{j,ε} verifies

(3.21)  Δ_{g_{j,ε}}u_{j,ε} + h_{j,ε}u_{j,ε} = u_{j,ε}^{2^{⋆}−1}

in B_{0} (δ). Note also that (3.13) just becomes

(3.22)  ṽ_{j,ε} (x) ≤ Cψ̃_{j,ε} in B_{0} (δ_{j})

where

ψ̃_{j,ε} (x) = (\frac{ν_{j,ε}}{ν_{j,ε}^{2} + a_{n}|x|^{2}} )^{\frac{n−2}{2}} .

We write

(3.23)  η_{j}ṽ_{j,ε} = (1 + θ_{j,ε}) η_{j}ψ̃_{j,ε} + w_{j,ε}

where w_{j,ε} ∈ C_{c}^{∞}(B_{0} (δ_{j})) and

η_{j} = η (\frac{.}{2δ_{j}} ) .

We express (3.15). Differentiating J_{j,ε} with respect to θ, we obtain that

(3.24)  ∫_{B_{0}(δ_{j})} (∇ (η_{j}ψ̃_{j,ε}) , ∇w_{j,ε})_{g_{j,ε}} dv_{g_{j,ε}} = 0.

Differentiating J_{j,ε} with respect to y, we get that

(3.25)  ∫_{B_{0}(δ_{j})} (∇ (η_{j} \frac{∂ψ̃_{j,ε}}{∂x_{i}} ) , ∇w_{j,ε})_{g_{j,ε}} dv_{g_{j,ε}} = O (ν_{j,ε}^{n−2})

for all i = 1, …, n. At last, differentiating J_{j,ε} with respect to ν, we obtain thanks to (3.24) that

(3.26) ∫_{B_{0}(δ_{j})} (∇ (η_{j}|x|^{2} (1 + a_{n} \frac{|x|^{2}}{ν_{j,ε}^{2}} )^{− \frac{n}{2}} ) , ∇w_{j,ε})_{g_{j,ε}} dv_{g_{j,ε}} = 0.

The aim is to estimate ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}. We write first thanks to (3.23) and (3.24) that

∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} = ∫_{B_{0}(δ_{j})}(∇w_{j,ε}, ∇ (η_{j}ṽ_{j,ε}))_{g_{j,ε}} dv_{g_{j,ε}}
= ∫_{B_{0}(δ_{j})} w_{j,ε}Δ_{g_{j,ε}} (η_{j}ṽ_{j,ε}) dv_{g_{j,ε}}.

Writing thanks to (3.4), (3.19), (3.20), (3.21) and (3.23) that

w_{j,ε}Δ_{g_{j,ε}} (η_{j}ṽ_{j,ε}) = (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} − h_{j,ε} (η_{j}ṽ_{j,ε}) w_{j,ε}
+O (|w_{j,ε}| |∇σ̃_{j,ε}|_{g_{j,ε}} |∇u_{j,ε}|_{g_{j,ε}})
+O (|w_{j,ε}| |Δ_{g_{j,ε}} σ̃_{j,ε}| u_{j,ε}) + O (ν_{j,ε}^{n−2})
+O (|σ̃_{j,ε} − σ_{j,ε}^{2^{⋆}−1}| u_{j,ε}^{2^{⋆}−1} |w_{j,ε}|)

in B_{0} (δ_{j}), we get following the proof of (2.43) and using (0.1), (3.6), (3.7), (3.9), (3.11), (3.12) and (3.22) that

(1 + o(1)) ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} (3.27)

= ∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} + o (ν_{j,ε}^{3}) + O (ν_{j,ε}^{\frac{n}{2} −1}∥∇w_{j,ε}∥_{2})

+ O (ν_{j,ε}^{n−2}) + O (( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2}) .

Relations (3.16) and (3.23) give that ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}} → 0 as ε → 0 so that Hölder’s and Sobolev’s inequalities permit to write with (3.23) that

∫_{B_{0}(δ_{j})} (η_{j}ṽ_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} (3.28)

= (1 + θ_{j,ε})^{2^{⋆}−1} ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}}
+ (2^{⋆} − 1) (1 + θ_{j,ε})^{2^{⋆}−2} ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} w_{j,ε}^{2} dv_{g_{j,ε}}

+ o (∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}) .

Direct computations give then thanks to (3.4) and to the Cartan expansion of the metric g_{j,ε} around 0 that

∫_{B_{0}(δ_{j})} (∇w_{j,ε}, ∇ (η_{j}ψ̃_{j,ε}))_{g_{j,ε}} dv_{g_{j,ε}}

= ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} + O (∫_{B_{0}(δ_{j})} |x|^{2}ψ̃_{j,ε}^{2^{⋆}−1} |w_{j,ε}| dv_{g_{j,ε}})

+ O (∫_{B_{0}(δ_{j})} |x|^{2} |∇ψ̃_{j,ε}| |∇w_{j,ε}| dv_{g_{j,ε}}) + O (ν_{j,ε}^{n−2})

so that, using (3.24), Hölder’s and Sobolev’s inequalities, we get after simple computations that

∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−1} w_{j,ε} dv_{g_{j,ε}} = O (ν_{j,ε}^{n−2}) + o (ν_{j,ε}^{3}) + o(∥∇w_{j,ε}∥_{2}^{2})
+ O (ν_{j,ε}^{\frac{n}{2} −1}∥∇w_{j,ε}∥_{2}) .

Coming back to (3.27) with (3.28) and this last estimate, we arrive thanks to (3.19) to

(1 + o (1)) ∫_{B_{0}(δ_{j})} |∇w_{j,ε}|_{g_{j,ε}}^{2} dv_{g_{j,ε}}

= (2^{⋆} − 1) ∫_{B_{0}(δ_{j})} (η_{j}ψ̃_{j,ε})^{2^{⋆}−2} w_{j,ε}^{2} dv_{g_{j,ε}} + O (ν_{j,ε}^{n−2})

+ O (( ∑_{k∈D_{j}}(\frac{μ_{k,ε}}{s_{j,k,ε}} )^{\frac{n}{2} −1}) ∥∇w_{j,ε}∥_{2})+ o (ν_{j,ε}^{3})+ O (ν_{j,ε}^{\frac{n}{2} −1}∥∇w_{j,ε}∥_{2}) .

Considering an eigenvalue problem like in (2.47), we then follow the lines of Section 2 and use the lemma of Appendix 1 to obtain that

(3.29) ∥∇w_{j,ε}∥_{2}^{2} = O (ν_{j,ε}^{n−2}) + o (ν_{j,ε}^{3}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

We also get, following the proof of (2.58), that

(3.30)  θ_{j,ε} = O (ν_{j,ε}^{n−2}) + o(ν_{j,ε}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

We now apply the Pohozaev identity with test function f = \frac{1}{2}|x|^{2}, to u_{j,ε} (see Appendix 2) in

Ω_{j,ε} (δ) = B_{0} (δ) \ ⋃_{k∈D_{j}} Ω_{j,k,ε}

for some δ > 0 small enough with

Ω_{j,k,ε} = exp_{y_{j,ε}}^{−1} (B_{x_{k,ε}} (\frac{R_{k}}{2} s_{j,k,ε})) .

We thus have that

∫_{Ω_{j,ε}(δ)} (h_{j,ε} + \frac{1}{2}(∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f) u_{j,ε}^{2} dv_{g_{j,ε}} (3.31)

= \frac{1}{n} ∫_{Ω_{j,ε}(δ)}(Δ_{g_{j,ε}}f + n)u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

+ ∫_{Ω_{j,ε}(δ)}(∇^{2}f − g_{j,ε})(∇u_{j,ε}, ∇u_{j,ε}) dv_{g_{j,ε}} + A_{ε}

where A_{ε} is the boundary term

A_{ε} = ∫_{∂Ω_{j,ε}(δ)} (\frac{1}{2} |∇u_{j,ε}|_{g_{j,ε}}^{2} (∇f, ν)_{g_{j,ε}}

− (∇u_{j,ε}, ∇f)_{g_{j,ε}}(∇u_{j,ε}, ν)_{g_{j,ε}}) dσ_{g_{j,ε}}

− \frac{n − 2}{2} ∫_{∂Ω_{j,ε}(δ)} u_{j,ε}(∇u_{j,ε}, ν)_{g_{j,ε}} dσ_{g_{j,ε}}

− ∫_{∂Ω_{j,ε}(δ)}(∇f, ν)_{g_{j,ε}} (\frac{1}{2^{⋆}} u_{j,ε}^{2^{⋆}} − \frac{1}{2}h_{j,ε}u_{j,ε}^{2}) dσ_{g_{j,ε}}

+ \frac{1}{2} ∫_{∂Ω_{j,ε}(δ)}(Δ_{g_{j,ε}}f + n)u_{j,ε}(∇u_{j,ε}, ν)_{g_{j,ε}} dσ_{g_{j,ε}}

− \frac{1}{4} ∫_{∂Ω_{j,ε}(δ)}(∇ (Δ_{g_{j,ε}}f) , ν)_{g_{j,ε}}u_{j,ε}^{2} dσ_{g_{j,ε}}

where ν denotes the unit outer normal of ∂Ω_{j,ε}(δ). Note that, by (3.8) and (3.9),

∂Ω_{j,ε}(δ) = ∪_{k∈D_{j}}∂Ω_{j,k,ε} ∪ ∂B_{0} (δ)

and the union is a disjoint one. We denote by A_{k,ε} the part of A_{ε} corresponding to ∂Ω_{j,k,ε}. Noting that, on ∂Ω_{j,k,ε},

|∇f| ≤ d_{g} (y_{j,ε}, x_{k,ε}) + \frac{R_{k}}{2} s_{j,k,ε},

we get thanks to (0.1), (3.6), (3.7), (3.8) and (3.11) that

A_{k,ε} = O ((\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}))

for all k ∈ D_{j}. In order to estimate the part of A_{ε} corresponding to ∂B_{0} (δ), we use (3.4). We finally obtain that (3.32)

A_{ε} = (A_{j} (δ)+o (1))ν_{j,ε}^{n−2} +O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (x_{k,ε}, y_{j,ε})}{s_{j,k,ε}}))

where

\begin{aligned}A_{j} (δ) = & ∫_{∂B_{\overline{x}_{j}} (δ)} (\frac{1}{2}|∇H_{j}|_{g}^{2} (∇f_{j}, ν)_{g} − (∇H_{j}, ∇f_{j})_{g} (∇H_{j}, ν)_{g}) dσ_{g} \\ & − \frac{n − 2}{2} ∫_{∂B_{\overline{x}_{j}} (δ)} H_{j} (∇H_{j}, ν)_{g} dσ_{g} \\ & + \frac{1}{2} ∫_{∂B_{\overline{x}_{j}} (δ)} (∇f_{j}, ν)_{g} h_{0}H_{j}^{2} dσ_{g} \\ & + \frac{1}{2} ∫_{∂B_{\overline{x}_{j}} (δ)}(Δ_{g}f_{j} + n)H_{j} (∇H_{j}, ν)_{g} dσ_{g} \\ & − \frac{1}{4} ∫_{∂B_{\overline{x}_{j}} (δ)}(∇ (Δ_{g}f_{j}) , ν)_{g}H_{j}^{2} dσ_{g} \end{aligned} (3.33)

with f_{j} = \frac{1}{2}d_{g} (\overline{x}_{j}, .)^{2} and

(3.34) H_{j} (x) = a_{n}^{− \frac{n−2}{2}} (n − 2) ω_{n−1} ∑_{i=1}^{N} (lim_{ε→0} \frac{μ_{i,ε}}{μ_{j,ε}} )^{\frac{n}{2} −1} G_{0} (\overline{x}_{i}, .) .

Since (ψ̃_{j,ε}) is radially summetrical, we have that

(∇^{2}f − g_{j,ε}) (∇ \binom{}{ψ_{j,ε}} , ∇ \binom{}{ψ_{j,ε}}) = 0

so that, noting that σ̃_{j,ε} = 1 and η_{j} = 1 in Ω_{j,ε}(δ) for δ small enough, we can write with (3.4), (3.19), (3.23) and the Cartan expansion of the metric g_{j,ε} around 0 that

∫_{Ω_{j,ε}(δ)}(∇^{2}f − g_{j,ε})(∇u_{j,ε}, ∇u_{j,ε}) dv_{g_{j,ε}}

= O (∫_{Ω_{j,ε}(δ)} |x|^{2} |∇ψ̃_{j,ε}|_{g_{j,ε}} |∇w_{j,ε}|_{g_{j,ε}} dv_{g_{j,ε}}) + O (δ^{2}∥∇w_{j,ε}∥_{2}^{2})

where O (X) means |O (X)| ≤ CX with C independent of ε and δ. This leads by direct computations thanks to (3.29) to

∫_{Ω_{j,ε}(δ)}(∇^{2}f − g_{j,ε})(∇u_{j,ε}, ∇u_{j,ε}) dv_{g_{j,ε}} (3.35)

= O (δ^{\frac{1}{2}} ν_{j,ε}^{n−2}) + o (ν_{j,ε}^{3}) + O (δ^{\frac{1}{2}} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

The Cartan expansion of the metric g_{j,ε} around 0 gives that

Δ_{g_{j,ε}}f + n = \frac{1}{3}Ric_{g_{j,ε}} (0)_{kl} x^{k}x^{l}

+ \frac{1}{2}∂_{klm} (ln \sqrt{|g_{j,ε}|}) (0)x^{k}x^{l}x^{m} + O (|x|^{4})

in B_{0} (δ) where Ric_{g_{j,ε}} denotes the Ricci curvature of g_{j,ε} and where |g_{j,ε}| is the determinant of the matrix ((g_{j,ε})_{ik}). Thus we can write thanks to (3.4), (3.19), (3.23) and to the fact that \binom{}{ψ_{j,ε}} is radially symmetrical that

∫_{Ω_{j,ε}(δ)}(Δ_{g_{j,ε}}f + n)u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}}

= \frac{(1 + θ_{j,ε})^{2^{⋆}}}{3} Ric_{g_{j,ε}} (0)_{kl} ∫_{B_{0}(δ)} x^{k}x^{l}ψ̃_{j,ε}^{2^{⋆}} dx

+ O ( ∑_{k∈D_{j}} ∫_{Ω_{j,k,ε}} |x|^{2}ψ̃_{j,ε}^{2^{⋆}} dx) + O (∥w_{j,ε}∥_{2^{⋆}}^{2^{⋆}}) + O (ν_{j,ε}^{n})

+ O (∫_{B_{0}(δ)} |x|^{2}ψ̃_{j,ε}^{2^{⋆}−1}|w_{j,ε}| dx) + O (∫_{B_{0}(δ)} |x|^{4}ψ̃_{j,ε}^{2^{⋆}} dx) .

Using (3.29) and (3.30), this leads thanks to Hölder’s and Sobolev’s inequalities and to some explicit computations to the following:

(3.36) ∫_{Ω_{j,ε}}(Δ_{g_{j,ε}}f + n)u_{j,ε}^{2^{⋆}} dv_{g_{j,ε}} = \frac{n}{3}K_{n}^{− \frac{n}{2}} S_{g} (y_{j,ε}) ν_{j,ε}^{2} + o (ν_{j,ε}^{n−2})

+ o (ν_{j,ε}^{3}) + O ( ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Let us estimate the last term of (3.31). Using (0.1) and the Cartan expansion of the metric g_{j,ε} around 0, one gets that

h_{j,ε} + \frac{1}{2}(∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f

= (h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε}))

+ (\frac{3}{2}∂_{k}h_{j,ε}(0) + \frac{1}{4}∂_{k} (Δ_{g_{j,ε}}^{2} f) (0)) x^{k} + O (|x|^{2})

in B_{0} (δ) so that we obtain with (3.4), (3.19), (3.23) and the fact that \binom{}{ψ_{j,ε}} is radially symmetrical that

∫_{Ω_{j,ε}(δ)} (h_{j,ε} + \frac{1}{2}(∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f) u_{j,ε}^{2} dv_{g_{j,ε}}

= (h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε})) (1 + θ_{j,ε})^{2} ∫_{B_{0}(δ)} ψ̃_{j,ε}^{2} dx

+ O ( ∑_{k∈D_{j}} ∫_{Ω_{j,k,ε}} ψ̃_{j,ε}^{2} dx) + O (∫_{B_{0}(δ)} |x|^{2}ψ̃_{j,ε}^{2} dx)

+ O (∫_{B_{0}(δ)} ψ̃_{j,ε}|w_{j,ε}| dx) + O (∫_{B_{0}(δ)} |w_{j,ε}|^{2} dx) .

This leads thanks to (3.29) and (3.30) and after direct computations to

∫_{Ω_{j,ε}(δ)} (h_{j,ε} + \frac{1}{2}(∇h_{j,ε}, ∇f)_{g_{j,ε}} + \frac{1}{4} (Δ_{g_{j,ε}})^{2} f) u_{j,ε}^{2} dv_{g_{j,ε}}

= (h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε})) × \{ \begin{matrix}64ω_{3}ν_{j,ε}^{2} ln (\frac{r_{j,ε}}{ν_{j,ε}} ) & if n = 4 \\ \frac{4(n − 1)}{n − 4}K_{n}^{− \frac{n}{2}} ν_{j,ε}^{2} & if n ≥ 5 \end{matrix}

+ O (δ^{\frac{1}{2}} ν_{j,ε}^{n−2}) + o (ν_{j,ε}^{3}) + O (δ^{\frac{1}{2}} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2}) .

Coming back to (3.31) with (3.32), (3.35), (3.36) and this last estimate, we finally arrive to (3.37)

∫_{B_{\overline{x}_{j}} (δ)} (h_{0} + \frac{1}{2} (∇h_{0}, ∇f_{j})_{g} + \frac{1}{4} (Δ_{g})^{2} f_{j}) H_{j}^{2} dv_{g} = A_{j} (δ) + o(1)

with o (1) → 0 as δ → 0 when n = 3. Remember here that, when n = 3, D_{j} = ∅ thanks to Claim 2. When n = 4, we get that

\begin{matrix}(h_{ε} (y_{j,ε}) − \frac{1}{6}S_{g} (y_{j,ε})) ln (\frac{1}{ν_{j,ε}} ) \\ = \frac{A_{j} (δ)}{64ω_{3}} + O (\sqrt{δ}) + O (ν_{j,ε}^{−2} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (x_{k,ε}, y_{j,ε})}{s_{j,k,ε}})) \end{matrix} (3.38)

and when n ≥ 5, we arrive to

(h_{ε} (y_{j,ε}) − C(n)S_{g} (y_{j,ε})) (3.39)

= (\frac{n − 4}{4(n − 1)}Kn^{\frac{n}{2}} A_{j} (δ) + O (\sqrt{δ})) ν_{j,ε}^{n−4}

+ O (ν_{j,ε}^{−2} ∑_{k∈D_{j}} (\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (x_{k,ε}, y_{j,ε})}{s_{j,k,ε}})) .

We assume for the end of this section that n ≥ 4. Let k ∈ D_{j}. We write with (3.7) and (3.19) that

\frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}} = \frac{d_{g} (x_{j,ε}, x_{k,ε})}{s_{j,k,ε}} + o (\frac{μ_{j,ε}}{s_{j,k,ε}} ) = O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} ) .

Thanks to (3.6), j ∈ A_{k} so that, by (2.2), we get that s_{j,k,ε} ≥ r_{k,ε}. In particular, by (3.8), r_{k,ε} → 0 as ε → 0 and we can apply Claim 2, point (ii), to get that

(\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}})

≤ (\frac{μ_{k,ε}}{r_{k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}})

= O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} (\frac{μ_{k,ε}}{r_{k,ε}} )^{n−2})

= \{ \begin{matrix}O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} μ_{k,ε}^{2} ln (\frac{1}{μ_{k,ε}} )) & if n = 4 \\ O (\sqrt{\frac{μ_{j,ε}}{μ_{k,ε}}} μ_{k,ε}^{2}) & if n ≥ 5. \end{matrix}

Thanks to (3.6)and (3.19), this leads to (3.40)

(\frac{μ_{k,ε}}{s_{j,k,ε}} )^{n−2} (1 + \frac{d_{g} (y_{j,ε}, x_{k,ε})}{s_{j,k,ε}}) = \{ \begin{matrix}o (ν_{j,ε}^{2} ln (\frac{1}{ν_{j,ε}} )) & if n = 4 \\ o (ν_{j,ε}^{2}) & if n ≥ 5 \end{matrix}

for all k ∈ D_{j}.

4. Proof of the results and examples of blow-up

4.1 Proof of the theorem

Let us prove our theorem as stated in the introduction. We assume that 3 ≤ n ≤ 5. If u_{0} ≢ 0, we know by (2.2) that r_{1,ε}^{2} ≤ μ_{1,ε}. In particular, r_{1,ε} → 0 as ε → 0 and claim 2 holds. Plugging the estimate r_{1,ε}^{2} ≤ μ_{1,ε} into the estimate of Claim 2, point (ii), we get a contradiction for n = 4, 5. For n = 3, since Claim 2 says that r_{1,ε} does not go to 0 as ε → 0, we also get a contradiction. This proves that for 3 ≤ n ≤ 5, u_{0} ≡ 0. We assume now that n = 4, 5. We claim that

(4.1)  h_{0} (\overline{x}_{N}) = C(n)S_{g} (\overline{x}_{N}) ,

an assertion which clearly implies the second part of the theorem. Let us prove (4.1). We distinguish two cases. The easiest one is when (x_{N,ε}, μ_{N,ε}) is an almost isolated concentration point, that is when lim inf_{ε→0} r_{N,ε} > 0. In this case, we can apply the results of Section 3: relations (3.38) and (3.39) together with (3.40) clearly give, since ν_{N,ε} → 0 as ε → 0, that h_{ε} (y_{N,ε}) − C(n)S_{g} (y_{N,ε}) → 0 as ε → 0. By (0.1) and (3.19), this proves (4.1) in this case. Let us now consider the case when r_{N,ε} → 0 as ε → 0. Applying point (ii) of Claim 2, we first get that h_{0} (\overline{x}_{N}) ≥ C(n)S_{g} (\overline{x}_{N}). Assume by contradiction that

(4.2)  h_{0} (\overline{x}_{N}) > C(n)S_{g} (\overline{x}_{N}) .

Since r_{N,ε} → 0 as ε → 0 and u_{0} ≡ 0, (2.2) gives the existence of some i ∈ A_{N} such that

\frac{μ_{N,ε}}{μ_{i,ε}} d_{g} (x_{i,ε}, x_{N,ε})^{2} → 0 as ε → 0.

By (1.6), μ_{N,ε} ≥ μ_{i,ε} so that \overline{x}_{i} = \overline{x}_{N}. This proves that

C = \{i ∈ A_{N} s.t. \overline{x}_{i} = \overline{x}_{N}\} ∪ \{N\}

possesses at least two elements. We let k ∈ C be such that, after passing to a subsequence,

r_{k,ε} = min_{i∈C} r_{i,ε}

and we let

D = (A_{k} ∩ B_{k}) ∪ {k}

with B_{k} as in Claim 2. It is easily checked with the above choice of k that for any i ∈ D, D = (A_{i} ∩ B_{i}) ∪ {i} and that D possesses at least two elements. Moreover, it is clear that λ_{ij} > 0 for all i, j ∈ D, i ≠ j, λ_{ij} as in Claim 2. We let now R_{ε} → 0 as ε → 0 be such that {x_{i,ε}, i ∈ D} ⊂ B_{x_{k,ε}} (R_{ε}) and such that there exists j ∈ D such that x_{j,ε} ∈ ∂B_{x_{k,ε}} (R_{ε}). It is then easily checked that all the z_{ji}’s of Claim 2, i ∈ D, lie in an Euclidean ball whose boundary contains 0. This implies that ∇h_{j}(0) ≠ 0 since A_{j} ∩ B_{j} ≠ ∅ and λ_{ji} > 0 for all i ∈ A_{j} ∩ B_{j}. Assuming that (4.2) holds, point (iii) of Claim 2 gives that ∇h_{j}(0) = 0. This is a contradiction. Thus (4.2) is false. This proves (4.1) so that, if n = 4, 5, there exists at least one geometric concentration point x_{0} ∈ S such that h_{0} (x_{0}) = C(n)S_{g} (x_{0}). As a consequence, compactness holds, that is (0.4) holds, if h_{0} ≠ C(n)S_{g} everywhere in M.

Let us prove the last part of the theorem. We assume that h_{ε} (x) ≤ C(n)S_{g} (x) for all x ∈ M and all ε > 0. In this situation, we clearly get from (2.66) and (2.67) that lim inf_{ε→0} r_{1,ε} > 0 if n = 4, 5. Remember here that h_{1} (0) > 0 by (2.7) and that D_{1} = ∅. Note that, by Claim 2, we also have that lim inf_{ε→0} r_{1,ε} > 0 if n = 3 without any assumption, except (0.1), on h_{ε}. Thus, in this situation, for n = 3, 4, 5, (x_{1,ε}, μ_{1,ε}) is an isolated concentration point in the sense that C_{1} = ∅, C_{1} as in (3.5). Applying the results of Section 3 and in particular (3.37) − (3.39), we get that

∫_{B_{\overline{x}_{1}}(δ)} (h_{0} + \frac{1}{2} (∇h_{0}, ∇f_{1})_{g} + \frac{1}{4} (Δ_{g})^{2} f_{1}) H_{1}^{2} dv_{g} = A_{1} (δ) + o(1)

when n = 3 and that A_{1} (δ) ≤ O (δ) if n = 4, 5 for all δ > 0. Here, A_{1} (δ) is as in (3.33). Moreover, when n = 4, 5, since ν_{1,ε} → 0 as ε → 0 thanks to (3.19), we also get that h_{0} (\overline{x}_{1}) = C(n)S_{g} (\overline{x}_{1}). Let us now write that

G_{0} (\overline{x}_{1}, .) = G̃_{0} (\overline{x}_{1}, .) + α_{0} ( .)

where G_{0} and G̃_{0} are the Green functions of Δ_{g} + h_{0} and Δ_{g} + C(n)S_{g} respectively. We have that

Δ_{g}α_{0} + h_{0}α_{0} = (C(n)S_{g} − h_{0}) G̃_{0} (\overline{x}_{1}, .) .

By standard properties of the Green function, we know that

G̃_{0} (\overline{x}_{1}, x) ≤ Cd_{g} (\overline{x}_{1}, x)^{2−n}

for some C > 0. If n = 4, 5, since h_{0} ≤ C(n)S_{g} and h_{0} (\overline{x}_{1}) = C(n)S_{g} (\overline{x}_{1}), we also have that

|h_{0}(x) − C(n)S_{g}(x)| ≤ Cd_{g} (\overline{x}_{1}, x)^{2}

for some C > 0. This permits to prove that Δ_{g}α_{0}+h_{0}α_{0} ∈ L^{p}(M) for all p < n. By Sobolev’s embedding theorem, we then get that α_{0} ∈ C^{0}(M). Thanks to the maximum principle, we get that α_{0} > 0 in M except if h_{0} ≡ C(n)S_{g} in which case α_{0} ≡ 0. Up to change from the beginning g into φ^{\frac{4}{n−2}} g, u_{ε} into u_{ε}φ^{−1} and h_{ε} into

h_{ε} := C(n)S_{φ^{\frac{4}{n−2}} g} + (h_{ε} − C(n)S_{g}) φ^{− \frac{4}{n−2}} ,

we may assume without loss of generality that for all the geometric concentration points \overline{x}_{i} ∈ S (which are the same for u_{ε} and u_{ε}φ^{−1}), the Green function of Δ_{g} + C(n)S_{g} writes as

G̃_{0} (\overline{x}_{i}, x) = \frac{1}{(n − 2)ω_{n−1}d_{g} (\overline{x}_{i}, x)^{n−2}} + M̃(\overline{x}_{i}) + o (1)

for x close to \overline{x}_{i}. For the existence of a conformal change of metric having this property in dimensions n = 3, 4, 5, we refer to Lee-Parker [21]. Moreover, thanks to the positive mass theorem of Schoen-Yau [31, 32] (see also Schoen [28], Schoen-Yau [33] and Witten [36]), we know that M̃(\overline{x}_{i}) > 0 except if (M, g) is conformally diffeomorphic to the standard sphere. Since α_{0} ∈ C^{0}(M) and α_{0} > 0 except if h_{0} ≡ C(n)S_{g}, we get that

G_{0} (\overline{x}_{1}, x) = \frac{1}{(n − 2)ω_{n−1}d_{g} (\overline{x}_{1}, x)^{n−2}} + M (\overline{x}_{1}) + o (1)

for x close to \overline{x}_{1} with M (\overline{x}_{1}) > 0 except if (M, g) is conformally diffeomorphic to the standard sphere and h_{0} ≡ C(n)S_{g}. Coming back to the definition (3.34) of H_{1}, we also have that

H_{1} (x) = a_{n}^{− \frac{n−2}{2}} (\frac{1}{d_{g} (\overline{x}_{1}, x)^{n−2}} + M_{1}) + o (1)

for x close to \overline{x}_{1} with M_{1} > 0 except if (M, g) is conformally diffeomorphic to the standard sphere, h_{0} ≡ C(n)S_{g} and there is only one concentration point. Some computations, which can be carried in conformal normal coordinates (see Lee-Parker [21]) up to the above conformal change of metric, lead then to

lim_{δ→0} A_{1} (δ) = \frac{(n − 2)^{2}}{2} a_{n}^{2−n}ω_{n−1}M_{1}

for n = 3, 4, 5 and to

lim_{δ→0} ∫_{B_{\overline{x}_{1}}(δ)} (h_{0} + \frac{1}{2} (∇h_{0}, ∇f_{1})_{g} + \frac{1}{4} (Δ_{g})^{2} f_{1}) H_{1}^{2} dv_{g} = 0

if n = 3. Thus we get a contradiction when n = 3, 4, 5 and h_{0} ≤ C(n)S_{g} except if (M, g) is conformally diffeomorphic to the standard sphere and h_{0} ≡ C(n)S_{g}. This ends the proof of the theorem. q.e.d.

Let us now state some results in higher dimensions. We claim that if n ≥ 7, then for any x ∈ S, h_{0} (x) = C(n)S_{g} (x). Of course, this implies again that compactness holds, that is that (0.4) holds, if h_{0} ≠ C(n)S_{g} everywhere in M and n ≥ 7. As shown by the examples below (see Section 4.2), the situation in dimension n = 6 is more intricate. However, the above statement continues to hold for n = 6 if u_{0} ≡ 0. Let us prove this statement. We assume that n = 6 and u_{0} ≡ 0 or that n ≥ 7. We let i ∈ \{1, …, N\}. We need to prove that h_{0} (\overline{x}_{i}) = C(n)S_{g} (\overline{x}_{i}). Up to change i, we may assume that μ_{i,ε} ≥ μ_{j,ε} for all j ∈ {1, …, N} such that \overline{x}_{i} = \overline{x}_{j}. If (x_{i,ε}, μ_{i,ε}) is an almost isolated concentration point, then it is a direct consequence of (3.38)–(3.40) together with (0.1) and (3.19) that h_{0} (\overline{x}_{i}) = C(n)S_{g} (\overline{x}_{i}). We assume now that r_{i,ε} → 0 as ε → 0. Then point (ii) of Claim 2 implies that h_{0} (\overline{x}_{i}) ≥ C(n)S_{g} (\overline{x}_{i}). Assume by contradiction that

(4.3)  h_{0} (\overline{x}_{i}) > C(n)S_{g} (\overline{x}_{i}) .

We get then by point (ii) of Claim 2 that

(4.4)  r_{i,ε}^{n−2} = O (μ_{i,ε}^{n−4}) .

In particular, we obtain for n ≥ 7 that r_{i,ε} = o (\sqrt{μ_{i,ε}}). Thus, by (2.2), there exists l ∈ A_{i} such that

r_{i,ε}^{2} = \frac{μ_{i,ε}}{μ_{l,ε}} d_{g} (x_{l,ε}, x_{i,ε})^{n−2} + μ_{i,ε}μ_{l,ε}.

If \overline{x}_{l} ≠ \overline{x}_{i}, we then get that r_{i,ε}^{2} ≥ C \frac{μ_{i,ε}}{μ_{l,ε}} for some C > 0, a contradiction with (4.4) for n ≥ 6 since μ_{i,ε} → 0 and μ_{l,ε} → 0 as ε → 0. Thus r_{i,ε} is achieved for some l ∈ A_{i} with \overline{x}_{i} = \overline{x}_{l}. This proves that

C = \{k ∈ A_{i} s.t. \overline{x}_{i} = \overline{x}_{k}\} ∪ \{i\}.

possesses at least two elements. We let k ∈ C be such that, after passing to a subsequence,

r_{k,ε} = min_{i∈C} r_{i,ε}

and we let

D = A_{k} ∪ B_{k} ∪ {k}

with B_{k} as in Claim 2. Arguing as above in the low-dimensional case, one can conclude that (4.3) is false and thus end the proof of the above statement. Namely: if n = 6 and u_{0} ≡ 0 or if n ≥ 7, then h_{0} (x) = C(n)S_{g} (x) for all x ∈ S.

As a remark, the compactness result we obtain in the theorem and in dimensions n ≥ 7 answers a question asked in Hebey [19] (question Q10). In this paper, the author is mainly interested by the energy function, defined as the minimal energy a solution of Δ_{g}u + αu = u^{2^{⋆}−1}, α ∈ ℝ, can have. It follows also from the above results that this energy function is lower semi-continuous with respect to α for α > C(n) max_{M} S_{g} in dimensions n ≥ 4 (see question Q6 of [19]).

4.2 Examples of blowing-up sequences

We provide in this section some blowing up sequences (u_{ε}) of solutions of (E_{ε}). We follow Druet-Hebey [11]. We consider (S^{n}, h) the unit sphere of ℝ^{n+1} equipped with its round metric. Its scalar curvature is n(n − 1) so that C(n)S_{h} = \frac{n(n−2)}{4}. It is well-known that all the solutions of

Δ_{h}u + C(n)S_{h}u = u^{2^{⋆}−1}

are given by

u = (\frac{n(n − 2)}{4} (β^{2} − 1))^{\frac{n−2}{4}} (β − cos d_{h} (x_{0}, x))^{1− \frac{n}{2}}

where β > 1 and x_{0} ∈ S^{n}. For i = 1, 2, we let (x_{i,ε})_{ε>0} be two sequences of points in S^{n} and (β_{i,ε}) be two sequences of positive real numbers such that β_{i,ε} > 1. We let in the following

B_{i,ε} = (\frac{n(n − 2)}{4} (β_{i,ε}^{2} − 1))^{\frac{n−2}{4}} (β_{i,ε} − cos d_{h} (x_{i,ε}, x))^{1− \frac{n}{2}}

for i = 1, 2 and

u_{ε} = λB_{1,ε} + B_{2,ε}

where λ ∈ ℝ^{+}. Clearly, u_{ε} ∈ C^{∞} (S^{n}) and u_{ε} > 0 for all ε > 0. It is easily checked that u_{ε} verifies

Δ_{h}u_{ε} + h_{ε}u_{ε} = u_{ε}^{2^{⋆}−1}

in S^{n} where the linear term h_{ε} is given by

h_{ε} = C(n)S_{h} + Φ_{ε}

with

Φ_{ε} = \frac{(λB_{1,ε} + B_{2,ε})^{2^{⋆}−1} − λB_{1,ε}^{2^{⋆}−1} − B_{2,ε}^{2^{⋆}−1}}{λB_{1,ε} + B_{2,ε}} .

We assume in the sequel that n ≥ 6.

First, we fix x_{1,ε} = x_{0} and we let β_{1,ε} = β_{2,ε} = β_{ε} verifying that β_{ε} → 1 as ε → 0. We choose x_{2,ε} ∈ S^{n} such that d_{h} (x_{2,ε}, x_{0}) ≥ (β_{ε} − 1)^{\frac{1}{20}} and such that d_{h} (x_{0}, x_{1,ε}) → 0 as ε → 0. One can check by direct computations that h_{ε} converges to C(n)S_{h} in C^{0}(M), with the additional property that the convergence of h_{ε} is C^{1} for n = 6. This provides examples of blowing-up sequences of solutions u_{ε} of Equation (E_{ε}) for which there is one geometric concentration point carrying two minimal energies. Namely, the concentration points are not isolated.

Second, we choose x_{1,ε} = x_{2,ε} = x_{0} in S^{n}, β_{1,ε} = β_{1} and we let β_{2,ε} → 1 as ε → 0. We let also λ = 1. One checks then by direct computations that the sequence (h_{ε}) is bounded in L^{∞} (S^{n}) and that h_{ε} → C(n)S_{h} in L^{p}(M) for all p > 1. Moreover, we clearly have that lim_{ε→0} u_{ε} = u_{0} weakly in H_{1}^{2} (S_{n}) where u_{0} ≢ 0. This provides examples of blowing-up sequences of solutions u_{ε} of Equation (E_{ε}) which does not converge weakly to 0 in dimensions n ≥ 6.

At last, we choose x_{1,ε} = x_{2,ε} = x_{0} in S^{n}, β_{1,ε} = β_{1} and we let β_{2,ε} → 1 as ε → 0. We let also λ > 1. One checks then by direct computations that the sequence (h_{ε}) is bounded in L^{∞} (S^{n}) and that h_{ε} → h_{0} in L^{p}(M) for all p > 1 for some h_{0} ∈ C^{0}(M). Moreover, when n ≥ 7, inf_{S^{n}} |h_{ε} − C(n)S_{h}| → 0 as ε → 0 but when n = 6, inf_{S^{n}} |h_{ε} − C(n)S_{h}| ↛ 0 as ε → 0. Independently, we clearly have that lim_{ε→0} u_{ε} = u_{0} weakly in H_{1}^{2} (S_{n}) where u_{0} ≢ 0. This provides examples of blowing-up sequences of solutions u_{ε} of Equation (E_{ε}) which does not converge weakly to 0 and for which there exists ε_{0} > 0 such that h_{ε} > C(n)S_{h} + ε_{0} for all ε > 0 in dimension n = 6.

We refer to [12] for other examples of blowing-up sequences of solutions of equations like (E).

Appendix 1: An eigenvalue problem

We study in this appendix an eigenvalue problem we used in a crucial way in Sections 2 and 3. The limiting eigenvalue problem, that is the Euclidean one, was studied by Bianchi-Egnell [3] and was used first by Adimurthi-Pacella-Yadava [1] in the study of blow-up problems in the Euclidean space. The lemma below, or more precisely a variant of it, was used in blow-up problems in Riemannian geometry by Druet-Hebey [10] (see also [17]).

Lemma. We let (g_{ε})_{ε>0} be a sequence of Riemannian metrics in B_{0} (3δ) ⊂ ℝ^{n}, δ > 0, verifying that there exists λ > 1 such that

(A1)  λ^{−1}ξ ≤ g_{ε} ≤ λξ

in the sense of bilinear forms with ξ the Euclidean metric, that

(A2)  g_{ε} (0)_{ij} = δ_{ij}

for all i, j ∈ {1, …, n} and that

(A3)  (g_{ε}) is bounded in C^{2}(B_{0} (2δ)) .

We let also (μ_{ε})_{ε>0} be a sequence of positive real numbers converging to 0 as ε → 0 and (φ_{ε})_{ε>0} be a sequence of smooth functions with compact support in B_{0} (δ) verifying that

(A4)  μ_{ε}^{\frac{n}{2} −1}φ_{ε} (μ_{ε}x) → u as ε → 0

strongly in D^{1,2} (ℝ^{n}) where

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

We consider (τ_{i,ε}, ζ_{i,ε}) the solutions of the following eigenvalue problem:

(A5)  \{ \begin{matrix}Δ_{g_{ε}}ζ_{i,ε} = τ_{i,ε}φ_{ε}^{2^{⋆}−2}ζ_{i,ε} in B_{0} (δ) \\ ζ_{i,ε} = 0 on ∂B_{0} (δ) \\ ∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}ζ_{i,ε}ζ_{j,ε} dv_{g_{ε}} = K_{n}^{− \frac{n}{2}} δ_{ij} \end{matrix}

with τ_{1,ε} ≤ ⋯ ≤ τ_{i,ε} ≤ … Then, after passing to a subsequence, we have that

lim_{ε→0} τ_{i,ε} = τ_{i} as ε → 0

and that

lim_{ε→0} μ_{ε}^{\frac{n}{2} −1}ζ_{i,ε} (μ_{ε}x) → ζ_{i} as ε → 0

strongly in D^{1,2} (ℝ^{n}) for all i ≥ 1 where (ζ_{i}, τ_{i}) satisfy

(A6)  \{ \begin{matrix}Δ_{ξ}ζ_{i} = τ_{i}u^{2^{⋆}−2}ζ_{i} in ℝ^{n} \\ ∫_{ℝ^{n}} u^{2^{⋆}−2}ζ_{i}ζ_{j} dv_{ξ} = K_{n}^{− \frac{n}{2}} δ_{ij} \end{matrix}

and τ_{1} ≤ ⋯ ≤ τ_{i} ≤ . . ..

Proof. We prove the result by induction on i. We mainly follow [10] (see also [17]) where a similar result was used. When i = 1,

τ_{1,ε} = inf_{φ∈C_{c}^{∞}(B_{0}(δ)), φ≢0} \frac{∫_{B_{0}(δ)} |∇φ|_{g_{ε}}^{2} dv_{g_{ε}}}{∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}φ^{2} dv_{g_{ε}}} .

Taking φ = φ_{ε}, we get by (A1)–(A4) that

lim sup_{ε→0} τ_{1,ε} ≤ \frac{∫_{ℝ^{n}} |∇u|_{ξ}^{2} dx}{∫_{ℝ^{n}} u^{2^{⋆}} dx} = 1.

Thus, up to a subsequence, we have that lim_{ε→0} τ_{1,ε} = τ̂_{1} ≤ 1. Thanks to (A1) and (A5), we then get that ζ̂_{1,ε} = μ_{ε}^{\frac{n}{2} −1}ζ_{1,ε} (μ_{ε}x) is bounded in D^{1,2} (ℝ^{n}). Thus, up to a subsequence, we know that ζ̂_{1,ε} ⇀ ζ̂_{1} as ε → 0 weakly in D^{1,2} (ℝ^{n}). Thanks to (A2)–(A5), it is easily checked that

Δ_{ξ}ζ̂_{1} = τ̂_{1}u^{2^{⋆}−2}ζ̂_{1}

in ℝ^{n} and that

(A7)  ∫_{ℝ^{n}} u^{2^{⋆}−2}ζ̂_{1}^{2} dx = K_{n}^{− \frac{n}{2}} .

It comes from the Euclidean Sobolev inequality and the equation verified by ζ̂_{1} that

(∫_{ℝ^{n}} ζ̂_{1}^{2^{⋆}} dx)^{\frac{2}{2^{⋆}}} ≤ K_{n} ∫_{ℝ^{n}} |∇ζ̂_{1}|_{ξ}^{2} dx
= K_{n}τ̂_{1} ∫_{ℝ^{n}} u^{2^{⋆}−2}ζ̂_{1}^{2} dx

≤ K_{n}τ̂_{1} (∫_{ℝ^{n}} u^{2^{⋆}} dx)^{\frac{2^{⋆}−2}{2^{⋆}}} (∫_{ℝ^{n}} ζ̂_{1}^{2^{⋆}} dx)^{\frac{2}{2^{⋆}}}

= K_{n}τ̂_{1} (∫_{ℝ^{n}} ζ̂_{1}^{2^{⋆}} dx)^{\frac{2}{2^{⋆}}} .

Since τ̂_{1} ≤ 1 and ζ̂_{1} ≢ 0 thanks to (A7), we thus get that τ̂_{1} = 1 and that all the above inequalities are equalities: this implies that ζ̂_{1} = u. Since, by Bianchi-Egnell [3], ζ_{1} = u and τ_{1} = 1, the lemma is proved for i = 1.

Let p ≥ 2 and assume that the lemma holds for all 1 ≤ i ≤ p − 1. We write that

τ_{p,ε} = inf_{φ∈H_{p,ε}} \frac{∫_{B_{0}(δ)} |∇φ|_{g_{ε}}^{2} dv_{g_{ε}}}{∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}φ^{2} dv_{g_{ε}}}

where

H_{p,ε} = {φ ∈ C_{c}^{∞} (B_{0} (δ)) s.t.

∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}ζ_{i,ε}φ dv_{g_{ε}} = 0 for all i = 1, …, p − 1}.

We claim first that

(A8)  lim sup_{ε→0} τ_{p,ε} ≤ τ_{p}.

Let us prove this claim. First, note that we have

τ_{p} = inf_{φ∈H_{p}} \frac{∫_{ℝ^{n}} |∇φ|_{ξ}^{2} dx}{∫_{ℝ^{n}} u^{2^{⋆}−2}φ^{2} dx}

where

H_{p} = {φ ∈ C_{c}^{∞} (ℝ^{n}) s.t. ∫_{ℝ^{n}} u^{2^{⋆}−2}ζ_{i}φ dx = 0 for all i = 1, …, p − 1}.

Let f ∈ H_{p}. We set

f_{ε} (x) = μ_{ε}^{1− \frac{n}{2}} f (\frac{x}{μ_{ε}} )

and

f̂_{ε} (x) = f_{ε} − Kn^{\frac{n}{2}} ∑_{i=1}^{p−1} (∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}ζ_{i,ε}f_{ε} dv_{g_{ε}}) ζ_{i,ε}.

By (A5), it is clear that, for ε > 0 small enough, f̂_{ε} ∈ H_{p,ε}. It is easily checked thanks to (A5) that

∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}f̂_{ε}^{2} dv_{g_{ε}}

= ∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}f_{ε}^{2} dv_{g_{ε}} − Kn^{\frac{n}{2}} ∑_{i=1}^{p−1} (∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}ζ_{i,ε}f_{ε} dv_{g_{ε}})^{2}

and that

∫_{B_{0}(δ)} |∇f̂_{ε}|_{g_{ε}}^{2} dv_{g_{ε}}

= ∫_{B_{0}(δ)} |∇f_{ε}|_{g_{ε}}^{2} dv_{g_{ε}} − Kn^{\frac{n}{2}} ∑_{i=1}^{p−1} τ_{i,ε} (∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}ζ_{i,ε}f_{ε} dv_{g_{ε}})^{2} .

Thanks to (A2), (A3) and to the fact that the lemma holds for i = 1, …, p − 1, we easily get that

∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}ζ_{i,ε}f_{ε} dv_{g_{ε}} → 0

as ε → 0 for all i = 1, …, p − 1 and then thanks to (A4) that

∫_{B_{0}(δ)} φ_{ε}^{2^{⋆}−2}f̂_{ε}^{2} dv_{g_{ε}} → ∫_{ℝ^{n}} u^{2^{⋆}−2}f^{2} dx

as ε → 0 and

∫_{B_{0}(δ)} |∇f̂_{ε}|_{g_{ε}}^{2} dv_{g_{ε}} → ∫_{ℝ^{n}} |∇f|_{ξ}^{2} dx.

Since it holds for all f ∈ H, this clearly proves (A8). In particular, (τ_{p,ε}) is bounded and, up to a subsequence, we have that lim_{ε→0} τ_{p,ε} = τ̂_{p} ≤ τ_{p}. Thanks to (A1) and (A5), we then get that ζ̂_{p,ε} = μ_{ε}^{\frac{n}{2} −1}ζ_{p,ε} (μ_{ε}x) is bounded in D^{1,2} (ℝ^{n}). Thus, up to a subsequence, we know that ζ̂_{p,ε} ⇀ ζ̂_{p} as ε → 0 weakly in D^{1,2} (ℝ^{n}). Thanks to (A2)–(A5), it is easily checked that

Δ_{ξ}ζ̂_{p} = τ̂_{p}u^{2^{⋆}−2}ζ̂_{p}

in ℝ^{n} and that

∫_{ℝ^{n}} u^{2^{⋆}−2}ζ_{i}ζ̂_{p} dx = K_{n}^{− \frac{n}{2}} δ_{ip}

for all i = 1, …, p. Since τ_{p,ε} ≥ τ_{p−1,ε}, we also know that τ̂_{p} ≥ τ_{p−1}. This clearly implies that τ̂_{p} = τ_{p} and that ζ̂_{p} = ζ_{p}. This ends the proof of the lemma. q.e.d.

Appendix 2: A Pohozaev identity

We prove in this appendix the Pohozaev identity we repeatedly used in this paper. We let (M, g) be a complete Riemannian manifold and Ω be a compact subset of M with smooth boundary. We let x_{0} ∈ M and R > 0 be such that Ω ⊂ B_{x_{0}} (R) and we assume that u is a smooth positive function verifying that

Δ_{g}u + hu = u^{2^{⋆}−1}

in B_{x_{0}} (R) for some h ∈ C^{∞}(B_{x_{0}} (R)). At last, we let f ∈ C^{∞}(B_{x_{0}} (R)). Integrating by parts, we have that

∫_{Ω} (∇u, ∇f)_{g} Δ_{g}u dv_{g} = ∫_{Ω} (∇ ((∇u, ∇f)_{g}) , ∇u)_{g} dv_{g}
− ∫_{∂Ω} (∇u, ∇f)_{g} (∇u, ν)_{g} dσ_{g}

where ν denotes the unit outer normal of ∂Ω and dσ_{g} is the induced Riemannian measure on ∂Ω. Noting that

(∇ ((∇u, ∇f)_{g}) , ∇u)_{g} = ∇^{2}f(∇u, ∇u) + \frac{1}{2}(∇f, ∇ (|∇u|_{g}^{2}))_{g},

468 o. druet

we obtain by integration by parts that

∫_{Ω} (∇u, ∇f)_{g} Δ_{g}u dv_{g} = \frac{1}{2} ∫_{Ω} Δ_{g}f |∇u|_{g}^{2} dv_{g} + ∫_{Ω} ∇^{2}f(∇u, ∇u) dv_{g}

+ \frac{1}{2} ∫_{∂Ω} (∇f, ν)_{g} |∇u|_{g}^{2} dσ_{g}

− ∫_{∂Ω} (∇u, ∇f)_{g} (∇u, ν)_{g} dσ_{g}

so that

∫_{Ω} (∇u, ∇f)_{g} Δ_{g}u dv_{g} + \frac{n − 2}{2} ∫_{Ω} |∇u|_{g}^{2} dv_{g}

= \frac{1}{2} ∫_{∂Ω} (∇f, ν)_{g} |∇u|_{g}^{2} dσ_{g} − ∫_{∂Ω} (∇u, ∇f)_{g} (∇u, ν)_{g} dσ_{g}

+ \frac{1}{2} ∫_{Ω} (Δ_{g}f + n) |∇u|_{g}^{2} dv_{g} + ∫_{Ω} (∇^{2}f − g) (∇u, ∇u) dv_{g}.

Now we use the equation satisfied by u to get that

∫_{Ω} |∇u|_{g}^{2} dv_{g} = ∫_{∂Ω} u (∇u, ν)_{g} dσ_{g} + ∫_{Ω} u^{2^{⋆}} dv_{g} − ∫_{Ω} hu^{2} dv_{g} and that

∫_{Ω} (∇u, ∇f)_{g} Δ_{g}u dv_{g}

= ∫_{Ω} Δ_{g}f (\frac{1}{2^{⋆}} u^{2^{⋆}} − \frac{1}{2}hu^{2}) dv_{g} + \frac{1}{2} ∫_{Ω} (∇f, ∇h)_{g} u^{2} dv_{g}

+ ∫_{∂Ω} (∇f, ν)_{g} (\frac{1}{2^{⋆}} u^{2^{⋆}} − \frac{1}{2}hu^{2}) dσ_{g}

which gives that

∫_{Ω} (∇u, ∇f)_{g} Δ_{g}u dv_{g} + \frac{n − 2}{2} ∫_{Ω} |∇u|_{g}^{2} dv_{g}

= ∫_{∂Ω} (∇f, ν)_{g} (\frac{1}{2^{⋆}} u^{2^{⋆}} − \frac{1}{2}hu^{2}) dσ_{g} + \frac{n − 2}{2} ∫_{∂Ω} (∇u, ν)_{g} u dσ_{g}

+ ∫_{Ω}(Δ_{g}f + n) (\frac{1}{2^{⋆}} u^{2^{⋆}} − \frac{1}{2}hu^{2}) dv_{g}

+ ∫_{Ω} (h + \frac{1}{2} (∇h, ∇f)_{g}) u^{2} dv_{g}.

from one bubble to several bubbles 469

Thus we have obtained that

∫_{B} (h + \frac{1}{2} (∇h, ∇f)_{g}) u^{2} dv_{g}

= ∫_{Ω}(Δ_{g}f + n) (\frac{1}{2}|∇u|_{g}^{2} + \frac{1}{2}hu^{2} − \frac{1}{2^{⋆}} u^{2^{⋆}}) dv_{g}

+ ∫_{Ω} (∇^{2}f − g) (∇u, ∇u) dv_{g} − ∫_{∂Ω} (∇f, ν)_{g} (\frac{1}{2^{⋆}} u^{2^{⋆}} − \frac{1}{2}hu^{2}) dσ_{g}

+ \frac{1}{2} ∫_{∂Ω} (∇f, ν)_{g} |∇u|_{g}^{2} dσ_{g} − ∫_{∂Ω} (∇u, ∇f)_{g} (∇u, ν)_{g} dσ_{g}

− \frac{n − 2}{2} ∫_{∂Ω} u (∇u, ν)_{g} dσ_{g}.

Integrating by parts and using the equation satisfied by u, we have that ∫_{Ω}(Δ_{g}f + n)|∇u|_{g}^{2} dv_{g}

= ∫_{Ω}(∇((Δ_{g}f + n) u), ∇u)_{g} dv_{g} − \frac{1}{2} ∫_{Ω}(∇ (Δ_{g}f) , ∇u^{2})_{g} dv_{g}

= ∫_{∂Ω}(Δ_{g}f + n) (∇u, ν)_{g} u dσ_{g} − \frac{1}{2} ∫_{∂Ω}(∇ (Δ_{g}f) , ν)_{g}u^{2} dσ_{g}

+ ∫_{Ω}(Δ_{g}f + n) (u^{2^{⋆}} − hu^{2}) dv_{g} − \frac{1}{2} ∫_{Ω} (Δ_{g}^{2}f) u^{2} dv_{g}.

Thus we get that

∫_{Ω}(Δ_{g}f + n) (\frac{1}{2}|∇u|_{g}^{2} + \frac{1}{2}hu^{2} − \frac{1}{2^{⋆}} u^{2^{⋆}}) dv_{g}

= \frac{1}{n} ∫_{Ω}(Δ_{g}f + n)u^{2^{⋆}} dv_{g} − \frac{1}{4} ∫_{Ω} (Δ_{g}^{2}f) u^{2} dv_{g}

+ \frac{1}{2} ∫_{∂Ω}(Δ_{g}f + n) (∇u, ν)_{g} u dσ_{g} − \frac{1}{4} ∫_{∂Ω}(∇ (Δ_{g}f) , ν)_{g}u^{2} dσ_{g}.

This finally leads to the following:

∫_{Ω} (h + \frac{1}{2} (∇f, ∇h)_{g} + \frac{1}{4} (Δ_{g}^{2}) f) u^{2} dv_{g}

= \frac{1}{n} ∫_{Ω}(Δ_{g}f + n)u^{2^{⋆}} dv_{g} + ∫_{Ω} (∇^{2}f − g) (∇u, ∇u) dv_{g} + A

where A is the boundary term

A = \frac{1}{2} ∫_{∂Ω} (∇f, ν)_{g} |∇u|_{g}^{2} dσ_{g} − ∫_{∂Ω} (∇u, ∇f)_{g} (∇u, ν)_{g} dσ_{g}

− \frac{n − 2}{2} ∫_{∂Ω} (∇u, ν)_{g} u dσ_{g} − ∫_{∂Ω} (∇f, ν)_{g} (\frac{1}{2^{⋆}} u^{2^{⋆}} − \frac{1}{2}hu^{2}) dσ_{g}

+ \frac{1}{2} ∫_{∂Ω}(Δ_{g}f + n) (∇u, ν)_{g} u dσ_{g} − \frac{1}{4} ∫_{∂Ω}(∇ (Δ_{g}f) , ν)_{g}u^{2} dσ_{g}.

This is the relation we referred to as the Pohozaev identity, with test function f, applied in Ω to a function u which verifies the above equation.

Acknowledgements. The author thanks Emmanuel Hebey for his constant support during the preparation of this work. The author thanks also Emmanuel Hebey and Frédéric Robert for useful discussions and comments on this work.

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