New York Journal of Mathematics

New York J. Math. 6 (2000) 95–106.

An Eigenvalue Problem for Elliptic Systems

Marco Squassina

Abstract. By means of non-smooth critical point theory we prove existence of weak solutions for a general nonlinear elliptic eigenvalue problem under natural growth conditions .

Contents

1. Introduction 95 2. Recollections from non-smooth critical point theory 98 3. The Palais–Smale condition 100 4. Final remarks 104 References 106

1. Introduction

Let Ω be a bounded and open subset of ℝ^{n} and N ≥ 1 . Existence and multiplicity results for quasilinear eigenvalue problems of the type :

\begin{cases}− ∑_{i,j=1}^{n} \frac{∂}{∂x_{j}} (a_{ij}(x, u)\frac{∂u_{k}}{∂x_{i}}) + \frac{1}{2} ∑_{i,j=1}^{n} ∑_{h=1}^{N} \frac{∂a_{ij}}{∂u_{k}} (x, u)\frac{∂u_{h}}{∂x_{i}} \frac{∂u_{h}}{∂x_{j}} = λ\frac{∂G}{∂u_{k}} (x, u) & in Ω \\ k = 1, …, N (u, λ) ∈ M × ℝ \end{cases}

on the submanifold of H_{0}^{1}(Ω, ℝ^{N})

M = {u ∈ H_{0}^{1}(Ω, ℝ^{N}) : ∫_{Ω} G(x, u) dx = 1} ,

have been firstly studied in 1983 by M. Struwe [15] and recently by G. Arioli [1] via techniques of non-smooth critical point theory.

The goal of this paper is to study the following more general eigenvalue problem

−div(∇_{ξ}L(x, u, ∇u)) + ∇_{s}L(x, u, ∇u) = λ∇_{s}G(x, u) (u, λ) ∈ M × ℝ , (1)

95^{ISSN 1076-9803/00}

Received February 23, 2000.

Mathematics Subject Classification. 35J50, 47J10, 58E05.

Key words and phrases. Nonlinear eigenvalue problems, Non-smooth critical point theory.

on the submanifold of W_{0}^{1,p}(Ω, ℝ^{N})

M = {u ∈ W_{0}^{1,p}(Ω, ℝ^{N}) : ∫_{Ω} G(x, u) dx = 1} . (2)

We shall consider functionals f : W_{0}^{1,p}(Ω, ℝ^{N}) → ℝ defined by

f(u) = ∫_{Ω} L(x, u, ∇u) dx , (3)

with 1 < p < n. In general, f is not even locally Lipschitzian unless L does not depend on u or n = 1, so that classical critical point theory fails.

To overcome this difficulty, we shall use the non-smooth critical point theory developed in [8, 9, 10, 11, 12] and also the subdifferential for continuous functions recently introduced in [6].

We shall prove that problem (1) admits a nontrivial weak solution in M × ℝ by restricting f to M and looking for constrained critical points .

We assume that M ≠ ∅, that L : Ω × ℝ^{N} × ℝ^{nN} → ℝ is measurable in x for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN}, of class C^{1} in (s, ξ) for a.e. x ∈ Ω and L(x, s, ·) is strictly convex. Moreover, we shall assume that:

[L_{1}] There exist ν > 0 such that for each ε > 0 there is a_{ε} ∈ L^{1}(Ω) and b_{ε} ∈ ℝ with

ν|ξ|^{p} ≤ L(x, s, ξ) ≤ a_{ε}(x) + ε|s|^{p^{∗}} + b_{ε}|ξ|^{p} (4)

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN}, where p^{∗} denotes the critical Sobolev’s exponent.

[L_{2}] There exists b ∈ ℝ such that for each ε > 0 there exists a_{ε} ∈ L^{1}(Ω) with

|∇_{s}L(x, s, ξ)| ≤ a_{ε}(x) + ε|s|^{p^{∗}} + b|ξ|^{p} (5)

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} . Moreover there is a_{1} ∈ L^{p′}(Ω) with

|∇_{ξ}L(x, s, ξ)| ≤ a_{1}(x) + b|s|^{\frac{p^{∗}}{p′}} + b|ξ|^{p−1} (6)

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} .

[L_{3}] For a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN}

∇_{s}L(x, s, ξ) · s ≥ 0 . (7)

[L_{4}] If N > 1, there exists a bounded Lipschitz function ψ : ℝ → ℝ such that

∇_{s}L(x, s, ξ) · exp_{σ}(r, s) + ∇_{ξ}L(x, s, ξ) · ∇ exp_{σ}(r, s, ξ) ≤ 0 (8)

for a.e. x ∈ Ω, for all ξ ∈ ℝ^{nN}, σ ∈ {−1, 1}^{N} and r, s ∈ ℝ^{N} where

(exp_{σ}(r, s))_{h} := σ_{h} exp[σ_{h}(ψ(r_{h}) − ψ(s_{h}))]

and

[∇ exp_{σ}(r, s, ξ)]_{hi} := − exp[σ_{h}(ψ(r_{h}) − ψ(s_{h}))]ψ′(s_{h})ξ_{i}^{h}

for each h = 1, …, N and i = 1, …, n.

[G_{1}] G(x, s) is measurable in x and of class C^{1} in s with G(x, 0) = 0 a.e. in Ω. If g(x, s) denotes ∇_{s}G(x, s), for every ε > 0 there exists a_{ε} ∈ L^{\frac{np}{n(p−1)+p}} (Ω) such that

|g(x, s)| ≤ a_{ε}(x) + ε|s|^{p^{∗}−1} (9)

for a.e. x ∈ Ω and all s ∈ ℝ^{N}.

[G_{2}] For a.e. x ∈ Ω and for each s ≠ 0 we have g(x, s) · s > 0 .

Under the preceding assumptions, the following is our main result.

Theorem 1. The eigenvalue problem

−div(∇_{ξ}L(x, u, ∇u)) + ∇_{s}L(x, u, ∇u) = λg(x, u) (u, λ) ∈ M × ℝ , (10)

has at least one nontrivial weak solution (u, λ) ∈ M × ℝ.

In the vectorial case (N > 1), to my knowledge, problem (1) has only been considered in [15] and in [1] in the particular case

L(x, s, ξ) = \frac{1}{2} ∑_{i,j=1}^{n} ∑_{h,k=1}^{N} a_{ij}^{hk}(x, s)ξ_{i}^{h}ξ_{j}^{k} , (11)

for coefficients a_{ij}^{hk} : Ω × ℝ^{N} → ℝ^{n^{2}} of the type a_{ij}^{hk}(x, s) = δ^{hk}α_{ij}(x, s).

In [15, Theorem 3.2] the statement is essentially of perturbative nature, since it says that if for each k ∈ ℕ there exists a ϱ_{k} > 0 with

(12) |∇_{s}α_{ij}(x, s)| < ϱ_{k} for a.e. x ∈ Ω, for all s ∈ ℝ^{N},

then the problem has at least k distinct weak solutions:

(u_{ℓ}, λ_{ℓ}) ∈ H_{0}^{1}(Ω, ℝ^{N}) × ℝ, ℓ = 1, …, k .

In other words, the less the coefficients α_{ij}(x, s) vary in s, the more solutions we get.

In [1] a new technical condition is introduced to be compared with (8). It is assumed that there exist K > 0 and an increasing bounded Lipschitz function ψ from [0, +∞[ to [0, +∞[ with ψ(0) = 0, ψ′ non-increasing, ψ(s) → K as s → +∞ and such that

∑_{i,j=1}^{n} ∑_{k=1}^{N} |\frac{∂α_{ij}}{∂s_{k}} (x, s)ξ_{i}ξ_{j}| ≤ 2e^{−4K}ψ′(|s|) ∑_{i,j=1}^{n} α_{ij}(x, s)ξ_{i}ξ_{j} (13)

for a.e. x ∈ Ω, for all ξ ∈ ℝ^{n} and for all r, s ∈ ℝ^{N} .

The proof itself of [2, Lemma 6.1] shows that this condition implies assumption (8) in the case of integrands L like (11). On the other hand, if N ≥ 2, the two conditions look quite similar. However, our condition (8) seems to be preferable, because when N = 1 and L is given by (11), it reduces to the inequality

| ∑_{i,j=1}^{n} \frac{∂α_{ij}}{∂s} (x, s)ξ_{i}ξ_{j}| ≤ 2ψ′(s) ∑_{i,j=1}^{n} α_{ij}(x, s)ξ_{i}ξ_{j},

which is not so restrictive in view of the ellipticity of α_{ij}, while (13) is in this case much stronger. For a general Lagrangian L, in the case N = 1, condition (8) reduces to

|D_{s}L(x, s, ξ)| ≤ ψ′(s)∇_{ξ}L(x, s, ξ) · ξ

for a.e. x ∈ Ω and all (s, ξ) ∈ ℝ×ℝ^{n}. This assumption has already been considered in literature in jumping problems (see e.g. [7]) .

In Remarks 2 and 3 we will show examples of L not of the form (11) and satisfying (8). Finally, we point out that (13) and (8) are not easily comparable to (12).

2. Recollections from non-smooth critical point theory

In this section, we want to recall the relationship between weak solutions to (1) and constrained critical points of f to M. Let a_{0} ∈ L^{1}(Ω), b_{0} ∈ ℝ, a_{1} ∈ L_{loc}^{1}(Ω) and b_{1} ∈ L_{loc}^{∞}(Ω) be such that for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN}

|L(x, s, ξ)| ≤ a_{0}(x) + b_{0}|s|^{\frac{np}{n−p}} + b_{0}|ξ|^{p} , (14)

|∇_{s}L(x, s, ξ)| ≤ a_{1}(x) + b_{1}(x)|s|^{\frac{np}{n−p}} + b_{1}(x)|ξ|^{p} , (15)

|∇_{ξ}L(x, s, ξ)| ≤ a_{1}(x) + b_{1}(x)|s|^{\frac{np}{n−p}} + b_{1}(x)|ξ|^{p} . (16)

Conditions (15) and (16) imply that for every u ∈ W_{0}^{1,p}(Ω, ℝ^{N}) we have

∇_{ξ}L(x, u, ∇u) ∈ L_{loc}^{1}(Ω, ℝ^{nN}), ∇_{s}L(x, u, ∇u) ∈ L_{loc}^{1}(Ω, ℝ^{N}) .

Therefore for every u ∈ W_{0}^{1,p}(Ω, ℝ^{N}) we have

−div (∇_{ξ}L(x, u, ∇u)) + ∇_{s}L(x, u, ∇u) ∈ D′(Ω, ℝ^{N}) .

We shall now recall two definitions from [6], where a new notion of subdifferential for continuous functionals on normed spaces has been recently introduced by M. Degiovanni and I. Campa.

Definition 1. Let X be a real normed space and C ⊆ X. For each u ∈ C, we denote with T_{C} (u) the set of all v ∈ X such that for each ε > 0, there exist δ > 0 and

ν : (B(u, δ) ∩ C) ×]0, δ] → B(v, ε)

continuous with

ξ + tν(ξ, t) ∈ C,

when ξ ∈ B(u, δ) ∩ C and t ∈]0, δ[. T_{C} (u) is said the cone tangent to C at u.

Definition 2. For each u ∈ X, set

∂f(u) := {α ∈ X^{∗} : (α, −1) ∈ N_{epif} (u, f(u))} ,

where

N_{epif} (u) := {ν ∈ X^{∗} : ⟨ν, v⟩ ≤ 0 for all v ∈ T_{epif} (u)} .

∂f(u) is said to be the subdifferential of f at u.

Via ∂f(u) we shall connect critical points for functionals of calculus of variations (3) constrained to M, with weak solutions to the related eigenvalue problem. Define the submanifold of W_{0}^{1,p}(Ω, ℝ^{N})

M = {u ∈ W_{0}^{1,p}(Ω, ℝ^{N}) : Φ(u) = 1} ,

where Φ : W_{0}^{1,p}(Ω, ℝ^{N}) → W^{−1,p′}(Ω, ℝ^{N}) is of class C^{1}, M ≠ ∅, 0 ∉ M and moreover ∇Φ(u) ≠ 0 for each u ∈ M.

We now recall the fundamental definition of weak slope (see, [8, 9, 10, 11, 12]) .

Definition 3. Let (X, d) be a metric space, f : X → ℝ a continuous function and u ∈ X. We denote by |df|(u) the supremum of σ ∈ [0, +∞[ such that there exist δ > 0 and a continuous map

H : B_{δ}(u) × [0, δ] ⟶ X

such that for all (v, t) ∈ B_{δ}(u) × [0, δ]

d(H(v, t), v) ≤ t, f(H(v, t)) ≤ f(v) − σt.

We say that the extended real number |df|(u) is the weak slope of f at u.

It is easy to prove that the map {u ↦ |df|(u)} is lower semicontinuous .

Definition 4. Let (X, d) be a metric space, f : X → ℝ a continuous function and u ∈ X. We say that u is a critical point of f if |df|(u) = 0 .

Definition 5. Let (X, d) be a metric space, c ∈ ℝ and f : X → ℝ a continuous function. A sequence (u_{h}) ⊆ X is said to be a Palais–Smale sequence at level c ((P S)_{c}–sequence, in short) for f, if f(u_{h}) → c and |df|(u_{h}) → 0 . We say that f satisfies the Palais–Smale condition ((P S)_{c} in short) at level c if every (P S)_{c}– sequence admits a convergent subsequence.

We now come to the case when X = W_{0}^{1,p}(Ω, ℝ^{N}) and f : W_{0}^{1,p}(Ω, ℝ^{N}) → ℝ given by (3) . From (14) it follows that f is well defined and continuous.

Since M is metric space endowed with the metric of W_{0}^{1,p}(Ω, ℝ^{N}), the weak slope |df_{|_{M}} | (u) and the (P S)_{c}–condition for f_{|_{M}} may of course be defined .

Theorem 2. For every u ∈ W_{0}^{1,p}(Ω, ℝ^{N}) there exists λ ∈ ℝ such that

|df_{|_{M}} | (u) ≥ sup {∇f(u)(v) − λ∇Φ(u)(v) : v ∈ C_{c}^{∞}(Ω, ℝ^{N}), ∥v∥_{1,p} ≤ 1} .

In particular, for each (P S)_{c}–sequence (u_{h}) for f_{|_{M}} there exists (λ_{h}) ⊆ ℝ such that

lim_{h} sup {∇f(u_{h})(v) − λ_{h}∇Φ(u_{h})(v) : v ∈ C_{c}^{∞}(Ω, ℝ^{N}), ∥v∥_{1,p} ≤ 1} = 0.

Proof. By conditions (15) and (16), for every u ∈ M and v ∈ C_{c}^{∞}(Ω, ℝ^{N}) there exists

f′(u)(v) = ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u, ∇u) · v dx

and the function {u ↦ f′(u)(v)} is continuous from M into ℝ. Now, let us extend f_{|_{M}} to the functional f^{∗} : W_{0}^{1,p}(Ω, ℝ^{N}) → ℝ ∪ {+∞} given by

f^{∗}(u) = \begin{cases}f(u) & if u ∈ M \\ +∞ & if u ∉ M. \end{cases} (17)

We may assume that |df_{|_{M}} |(u) < +∞. Consequently |df^{∗}|(u) = |df_{|_{M}} |(u), so that by [6, Theorem 4.13] there exists ω ∈ ∂f^{∗}(u) with |df^{∗}|(u) ≥ ∥ω∥_{−1,p′}. Moreover, by [6, Corollary 5.4] we have

∂f^{∗}(u) ⊆ ∂f(u) + ℝ∇Φ(u) .

Finally, by [6, Theorem 6.1], we get ∂f(u) = {η} where

⟨η, v⟩ = ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u, ∇u) · v dx = ∇f(u)(v)

for each v ∈ C_{c}^{∞}(Ω, ℝ^{N}) and the proof is complete. □

By the preceding result, each critical point u ∈ W_{0}^{1,p}(Ω, ℝ^{N}) of f_{|_{M}} is a weak solution to the eigenvalue problem :

∇f(u) = λ∇Φ(u) (u, λ) ∈ M × ℝ .

3. The Palais–Smale condition

Recall first a very useful conseguence of Brezis–Browder’s Theorem [5].

Proposition 1. Let T ∈ L_{loc}^{1}(Ω, ℝ^{N}) ∩ W^{−1,p′}(Ω, ℝ^{N}), v ∈ W_{0}^{1,p}(Ω, ℝ^{N}) and η ∈ L^{1}(Ω) with T · v ≥ η. Then T · v ∈ L^{1}(Ω) and

⟨T, v⟩ = ∫_{Ω} T · v dx

Proof. Argue as in [13, Lemma 3] . □

As a consequence of assumption [L_{1}] and convexity of L(x, s, ·), for each ε > 0 there exists a_{ε} ∈ L^{1}(Ω) such that

∇_{ξ}L(x, s, ξ) · ξ ≥ ν|ξ|^{p} − a_{ε}(x) − ε|s|^{p^{∗}} (18)

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} .

We now come to one of the main results of this paper, i.e., the local compactness property for (P S)_{c}–sequences.

Theorem 3. Let (u_{h}) be a bounded sequence in W_{0}^{1,p}(Ω, ℝ^{N}) and set

⟨w_{h}, v⟩ = ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · v dx, (19)

for all v ∈ C_{c}^{∞}(Ω, ℝ^{N}). If (w_{h}) is strongly convergent to some w in W^{−1,p′}(Ω, ℝ^{N}), then (u_{h}) admits a strongly convergent subsequence in W_{0}^{1,p}(Ω, ℝ^{N}).

Proof. Since (u_{h}) is bounded in W_{0}^{1,p}(Ω, ℝ^{N}), we find a u in W_{0}^{1,p}(Ω, ℝ^{N}) such that, up to subsequences,

∇u_{h} ⇀ ∇u in L^{p}(Ω, ℝ^{N}), u_{h} → u in L^{p}(Ω, ℝ^{N}), u_{h}(x) → u(x) a.e. x ∈ Ω.

By [4, Theorem 2.1], up to a subsequence, we have

∇u_{h}(x) → ∇u(x) a.e. x ∈ Ω .

Therefore, by (6) we get

∇_{ξ}L(x, u_{h}, ∇u_{h}) ⇀ ∇_{ξ}L(x, u, ∇u) in L^{p′}(Ω, ℝ^{nN}).

An Eigenvalue Problem for Elliptic Systems 101

We now want to prove that we have ⟨w, u⟩ = ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇u dx + ∫_{Ω} ∇_{s}L(x, u, ∇u) · u dx. (20) Let ψ be as in [L_{4}] and test equation (19) with the following functions

v_{h} = φ(σ_{1} exp{σ_{1}(ψ(u^{1}) − ψ(u_{h}^{1}))}, …, σ_{N} exp{σ_{N}(ψ(u^{N}) − ψ(u_{h}^{N}))}), where φ ∈ C_{c}^{∞}(Ω), φ ≥ 0 and σ_{i} = ±1 for all i = 1, …, N. By direct computation we obtain for a.e. x ∈ Ω

D_{j}v_{hi} = (σ_{i}D_{j}φ + (ψ′(u_{i})D_{j}u_{i} − ψ′(u_{hi})D_{j}u_{hi})φ) exp[σ_{i}(ψ(u_{i}) − ψ(u_{hi}))] for each i = 1, …, N and j = 1, …, n. Therefore, with the notation

[exp{σ(ψ(u) − ψ(u_{h}))}ψ′(u_{h})∇u_{h}]_{ij} := exp{σ_{i}(ψ(u_{i}) − ψ(u_{hi}))}ψ′(u_{hi})D_{j}u_{hi} for each i = 1, …, N and j = 1, …, n, we get

∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · [σ∇φ + ψ′(u)∇uφ] exp{σ(ψ(u) − ψ(u_{h}))} dx
− ⟨w_{h}, φσ exp{σ(ψ(u) − ψ(u_{h}))}⟩
+ ∫_{Ω} {∇_{s}L(x, u_{h}, ∇u_{h}) · σ exp{σ(ψ(u) − ψ(u_{h}))} dx

− ∇_{ξ}L(x, u_{h}, ∇u_{h}) · exp{σ(ψ(u) − ψ(u_{h}))}ψ′(u_{h})∇u_{h}}φ dx = 0 .

Observe that if v = (σ_{1}φ, …, σ_{N}φ) we have

lim_{h} ⟨w_{h}, φσ exp{σ(ψ(u) − ψ(u_{h}))}⟩ = ⟨w, v⟩.

Since u_{h} ⇀ u in W_{0}^{1,p}(Ω, ℝ^{N}), we have

lim_{h} ∫_{Ω} ∇_{ξ}L(x, u_{h},∇u_{h}) · [σ∇φ + ψ′(u)∇uφ] exp{σ(ψ(u) − ψ(u_{h}))} dx

= ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇v dx + ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ψ′(u)∇uφ dx.

Note now that by assumption (8) for each h ∈ ℕ we have

∇_{s}L(x, u_{h},∇u_{h}) · σ exp{σ(ψ(u) − ψ(u_{h}))}

− ∇_{ξ}L(x, u_{h}, ∇u_{h}) · exp{σ(ψ(u) − ψ(u_{h}))}ψ′(u_{h})∇u_{h} ≤ 0.

Therefore, Fatou’s Lemma implies that

lim sup_{h} { ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · σ exp{σ(ψ(u) − ψ(u_{h}))}φ dx

− ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · exp{σ(ψ(u) − ψ(u_{h}))}ψ′(u_{h})∇u_{h}φ dx} ≤ ∫_{Ω} ∇_{s}L(x, u, ∇u) · v dx − ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ψ′(u)∇uφ dx.

Combining the previous inequalities we get ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u, ∇u) · v dx ≥ ⟨w, v⟩ for each v = (σ_{1}φ, …, σ_{N}φ), with φ ∈ C_{c}^{∞}(Ω), φ ≥ 0. Since we may exchange v with −v, we obtain

∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u, ∇u) · v dx = ⟨w, v⟩. (21)

for each v = (σ_{1}φ, …, σ_{N}φ) with φ ∈ C_{c}^{∞}(Ω) and φ ≥ 0. Since each v ∈ C_{c}^{∞}(Ω, ℝ^{N}) is a linear combination of such functions, taking into account Proposition 1, we obtain relation (20). The final step is to prove that (u_{h}) goes to u in W_{0}^{1,p}(Ω, ℝ^{N}). To this aim, let us first get the following inequality

(22)  lim sup_{h} ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇u_{h} dx ≤ ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇u dx .

Because of (7) Fatou’s Lemma yields

(23) ∫_{Ω} ∇_{s}L(x, u, ∇u) · u dx ≤ lim inf_{h} ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · u_{h} dx .

Combining this fact with (20) and taking into account that

⟨w_{h}, u_{h}⟩ → ⟨w, u⟩ as h → +∞,

we deduce

lim sup_{h} ∫_{Ω} ∇_{ξ}L(x, u_{h},∇u_{h}) · ∇u_{h} dx

= lim sup_{h} [− ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · u_{h} dx + ⟨w_{h}, u_{h}⟩]

≤ [− ∫_{Ω} ∇_{s}L(x, u, ∇u) · u dx + ⟨w, u⟩]

= ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇u dx .

In particular, again by Fatou’s Lemma, we have

∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇u dx ≤ lim inf_{h} ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇u_{h} dx

≤ lim sup_{h} ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇u_{h} dx

≤ ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇u dx,

that is

lim_{h} ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇u_{h} dx = ∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇u dx,

which gives convergence in L^{1}(Ω). Therefore, by (18) we conclude that :

lim_{h} ∫_{Ω} |∇u_{h}|^{p} dx = ∫_{Ω} |∇u|^{p} dx,

which gives convergence of (u_{h}) to u in W_{0}^{1,p}(Ω, ℝ^{N}) . □

Corollary 1. Let (u_{h}) be a bounded sequence in W_{0}^{1,p}(Ω, ℝ^{N}), (λ_{h}) a sequence in ℝ and set for all v ∈ C_{c}^{∞}(Ω, ℝ^{N})

⟨λ_{h}w_{h}, v⟩ = ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · v dx .

If (w_{h}) converges to some w ≠ 0 in W^{−1,p′}(Ω, ℝ^{N}) then (u_{h}, λ_{h}) admits a strongly convergent subsequence in W_{0}^{1,p}(Ω, ℝ^{N}) × ℝ.

Proof. By density, we can find η ∈ C_{c}^{∞}(Ω, ℝ^{N}) such that

lim_{h} ⟨w_{h}, η⟩ = ⟨w, η⟩ > 0.

Since of course the sequence

{∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇η dx + ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · η dx}

is bounded, (λ_{h}) is also bounded and the assertion follows by Theorem 3. □

In the next result we prove that f satisfies (P S)_{c}–condition.

Lemma 1. Let c ∈ ℝ. Then, for each (P S)_{c}–sequence (u_{h}) for f_{|_{M}} there exists u ∈ M and λ ∈ ℝ such that, up to subsequences, u_{h} → u in W_{0}^{1,p}(Ω, ℝ^{N}) and λ_{h} → λ in ℝ. In particular, we have

∫_{Ω} ∇_{ξ}L(x, u, ∇u) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u, ∇u) · v dx = λ ∫_{Ω} g(x, u) · v dx

for each v ∈ C_{c}^{∞}(Ω, ℝ^{N}).

Proof. Let (u_{h}) be a (P S)_{c}–sequence for f_{|_{M}} . Since by (4) (u_{h}) is bounded in W_{0}^{1,p}(Ω, ℝ^{N}), up to a subsequence (u_{h}) weakly goes to a u ∈ M . Moreover, since by [G_{1}], g is completely continuous as mapping from W_{0}^{1,p}(Ω, ℝ^{N}) to W^{−1,p′}(Ω, ℝ^{N}), up to a further subsequence, we have

g(x, u_{h}) → g(x, u) in W^{−1,p′}(Ω).

Now, by Theorem 2, there exists a sequence (λ_{h}) ⊆ ℝ with

sup { ∫_{Ω} ∇_{ξ}L(x, u_{h}, ∇u_{h}) · ∇v dx + ∫_{Ω} ∇_{s}L(x, u_{h}, ∇u_{h}) · v dx
− λ_{h} ∫_{Ω} g(x, u_{h}) · v dx : v ∈ C_{c}^{∞}(Ω, ℝ^{N}), ∥v∥_{1,p} ≤ 1} → 0

as h → +∞. Hence, by applying Corollary 1 to

w_{h} = g(x, u_{h}) + Λ_{h}, Λ_{h} → 0 in W^{−1,p′}(Ω, ℝ^{N}),

up to subsequences (u_{h}, λ_{h}) converges to (u, λ) in W_{0}^{1,p}(Ω, ℝ^{N}) × ℝ. □

We may now prove the main result of this paper.

Proof of Theorem 1. By assumption [G_{2}] and G(x, 0) = 0 we easily see that 0 ∉ M and g(x, u) ≢ 0 for each u ∈ M. Since f is bounded from below, there exists a (P S)_{c}–sequence (u_{h}) for f_{|_{M}} at the level

c = inf_{u∈M} f(u) .

Indeed, let (u_{h}) a sequence of minimizers for f in M . Of course we have f(u_{h}) → c. Moreover, if |df|(u_{h}) ↛ 0, we would find a σ > 0 such that |df|(u_{h}) ≥ σ. Then by [8, Theorem 1.1.11] there exists a continuous deformation

η : M × [0, δ] → M

for some δ > 0 such that for all t ∈ [0, δ] and h ∈ ℕ

f(η(u_{h}, t)) ≤ f(u_{h}) − σt .

This easily yields the contradiction f(η(u_{h}, t)) < c for sufficiently large values of h ∈ ℕ. Thus (u_{h}) is a (P S)_{c}–sequence for f_{|_{M}} . Lemma 1 now provides a weak solution (u, λ) ∈ M × ℝ to (1). Of course u ≢ 0 . □

4. Final remarks

We refer the reader to [2] for some concrete examples where the condition (8) is fulfilled for an integrand L like (11) .

Remark 1. Assume that there exists R > 0 such that

|s| ≥ R ⟹ ∇_{s}L(x, s, ξ) = 0

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} and

|s| ≤ R ⟹ ∑_{k=1}^{N} |D_{s_{k}}L(x, s, ξ)| ≤ \frac{1}{4eR}∇_{ξ}L(x, s, ξ) · ξ

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} . Then (8) holds for a ψ defined by

ψ(s) = \begin{cases}\frac{s}{4R} & if 0 ≤ s ≤ R \\ \frac{1}{4} & if s ≥ R . \end{cases} (24)

Indeed, (8) is implied by the following condition:

There exist K > 0 and an increasing bounded Lipschitz function ψ : [0, +∞[→ [0, +∞[ with ψ(0) = 0, ψ′ non-increasing, ψ(t) → K as t → +∞ and such that

∑_{k=1}^{N} |D_{s_{k}}L(x, s, ξ)| ≤ e^{−4K}ψ′(|s|)∇_{ξ}L(x, s, ξ) · ξ (25)

for a.e. x ∈ Ω, for all ξ ∈ ℝ^{n} and for all r, s ∈ ℝ^{N}.

It is easy to verify that the ψ defined in (24) satisfies (25) .

We now exhibit an example of L satisfying (8) and not of quadratic type.

Remark 2. Let L : Ω × ℝ^{N} × ℝ^{nN} → ℝ be defined by

L(x, s, ξ) = \frac{1}{p}(ν + arctan |s|^{2})|ξ|^{p}, ν ≥ e\sqrt{N}(\sqrt{3} + π),

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} . By following [2, Example 9.2] it is possible to show that there exist K > 0 and an increasing bounded Lipschitz function ψ : [0, +∞[→ [0, +∞[ with ψ(0) = 0, ψ′ non-increasing, ψ(s) → K as s → +∞ given by

ψ(s) = \frac{\sqrt{N}e^{4K}}{ν} \begin{cases}\frac{3^{3/4}}{4} s & if s ∈ [0, 3^{−1/4}] \\ \frac{\sqrt{3}}{4} + ∫_{3^{−1/4}}^{s} \frac{τ}{1+τ^{4}} dτ & if s ∈ [3^{−1/4}, +∞[ \end{cases}

such that L satisfies (25). Therefore (8) is fulfilled.

Remark 3. Since condition (8) does not look very nice and it is not clear how to describe the class of systems that satisfy this assumption, it seems natural to look for some classes of quasilinear systems with a more particular structure but requiring a simpler hypothesis . To this aim, consider the eigenvalue problem

−div (A_{1}(u_{1})|∇u_{1}|^{p−2}∇u_{1}) + \frac{1}{p}A′_{1}(u_{1})|∇u_{1}|^{p} = λg_{1}(x, u)  in Ω
⋮  ⋮  ⋮  ⋮
−div (A_{N}(u_{N})|∇u_{N}|^{p−2}∇u_{N}) + \frac{1}{p}A′_{N}(u_{N})|∇u_{N}|^{p} = λg_{N}(x, u) in Ω .

(26)

In a variational setting, the weak solutions u = (u_{1}, ⋯, u_{N}) of (26) are the critical points of f|_{M} where f : W_{0}^{1,p}(Ω, ℝ^{N}) → ℝ is given by

f(u) = \frac{1}{p} ∑_{k=1}^{N} ∫_{Ω} A_{k}(u_{k})|∇u_{k}|^{p} dx .

and M is as in (2). Consider the following assumptions (k = 1, …, N):

[A_{1}] A_{k} ∈ C^{1}(ℝ) with \underline{a}_{k} ≤ A_{k} ≤ \overline{a}_{k} for some \underline{a}_{k}, \overline{a}_{k} > 0 ;

[A_{2}] A′_{k}(s)s ≥ 0 for each s ∈ ℝ;

[A_{3}] there exists a bounded Lipschitz function ψ : ℝ → ℝ such that

A_{k}(s)e^{−p(ψ(t)−ψ(s))} ≤ A_{k}(t) ≤ A_{k}(s)e^{p(ψ(t)−ψ(s))} (27)

for each s, t ∈ ℝ with s ≤ t.

Under the previous assumptions, by exploiting the proof of Theorem 3 it is possible to see that for system (26) assumption (8) may be replaced by (27). Indeed,

(27) immediately implies that

∀k = 1, …, N, ∀s ∈ ℝ : |A′_{k}(s)| ≤ pA_{k}(s)ψ′(s) .

Of course (27) looks much simpler and more understandable. In some sense, this condition says that for each s ∈ ℝ and k fixed, A_{k}(t) must remain within the “exponential cone” determined by

{t ↦ A_{k}(s)e^{−p(ψ(t)−ψ(s))}} and {t ↦ A_{k}(s)e^{p(ψ(t)−ψ(s))}}

for each t ≥ s.

Remark 4. Condition (8) in only needed when N > 1, since in the case N = 1 Theorem 3 may be substituted by [14, Theorem 3.4] where no condition like (8) is requested in order to get the compactness property of (P S)_{c}–sequences .

Secondly, we remark that in the case N = 1 condition (7) can be assumed only for large values of |s|, that is, there exists R > 0 such that

|s| ≥ R ⟹ D_{s}L(x, s, ξ) · s ≥ 0 (28)

for a.e. x ∈ Ω and for all (s, ξ) ∈ ℝ^{N} × ℝ^{nN} (see again [14, Theorem 3.4]) .

106 Marco Squassina

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Dipartimento di Matematica, Università degli studi di Milano, Via C. Saldini 50, I 20133 Milano, Italy.

squassin@mat.unimi.it http://www.dmf.bs.unicatt.it/˜ squassin

This paper is available via http://nyjm.albany.edu:8000/j/2000/6-6.html.