ON DIFFERENTIAL SYSTEMS WITH STRONGLY INDEFINITE VARIATIONAL STRUCTURE

CYRIL JOEL BATKAM, FABRICE COLIN, AND TOMASZ KACZYNSKI

Abstract. We prove some multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. The nonlinear terms are not assumed to satisfy the Ambrosetti-Rabinowitz superquadratic condition. To establish the existence of solutions we apply a version of the symmetric mountain pass theorem for strongly indefinite functionals, also established in this paper, which covers the situations where all the Palais-Smale sequences of the energy functional may be unbounded.

1. Introduction

This paper is concerned with the existence and the multiplicity of critical points of strongly indefinite even functionals which appear in the study of periodic solutions of Hamiltonian systems and of solutions of certain class of elliptic systems. We recall that a functional Φ : X ℝ defined on a Banach space is said to be strongly indefinite if it is neither bounded from above nor from below, even modulo subspaces of finite dimension or codimension. Its study then gives rise to an interesting and challenging variational problem, because the usual powerful critical point theorems in [1, 3, 8, 33] cannot be applied directly.

Critical Point Theory for strongly indefinite even functionals was studied in [4, 10, 12, 6, 13, 18, 21]. In his seminal paper [13], Benci introduced various index and pseudo-index theories whose definitions, however, depended on the topology of the sublevel sets of the functional, and he required a certain dimension property which severely restricts the class of problems for which his results apply. In [4, 18, 21], the Galerkin type approximations were used to reduce the study of strongly indefinite functionals to a semidefinite situation where the basic idea of Lusternik-Schnirelmann theory applies. However, in order to control the critical points of the reduced functional it is required in theses papers that the original functional satisfies a strong version of the usual compactness condition in Critical Point Theory. In the recent papers [10, 12], the first two authors of this paper generalized the well known fountain theorems of Bartsch and Willem to the case of strongly indefinite functionals. In contrast with [4, 18, 21], they used no reduction method and carried out the proofs directly in the infinite-dimensional setting.

The first goal of this paper is to extend the results in [10, 12] to more general situations for which the Palais-Smale sequences of the functionals may be unbounded. This will have a crucial importance in applications, since we can free ourselves from the Palais-Smale type assumptions such as the Ambrosetti-Rabinowitz superlinear condition and its variations, which are extensively used in literature in the study of superlinear problems. We stress that in contrast with the common feature in the study of symmetric strongly indefinite functionals, our approach does not use any reduction method. The main ingredient is a weak-strong topology introduced by Kryszewski and Szulkin in [20]. Our critical point theorems are stated in section 2.

2010 Mathematics Subject Classification. Primary 35K20; 35J45 Secondary 47J30; 35B10; 35J60; 35J57.

Key words and phrases. Critical point theorems, Strongly indefinite functionals, unbounded Hamiltonian systems, Elliptic systems, Periodic solutions, Large energy solutions.

In section 3, we consider the applications to the problem of finding infinitely many large energy periodic solutions of the following first-order Hamiltonian system

tu Δ_{x}u V x u H_{v} t, u, v in ℝ Ω, _{t}v Δ_{x}v V x v H_{u} t, u, v in ℝ Ω,  (HS)

where Ω ℝ^{N} (N 1) is a bounded smooth domain, and H is a superquadratic C^{1}-function which is T -periodic (T 0) with respect to the t-variable. By periodic solution, we mean a solution z u, v : ℝ Ω ℝ^{2M} of (HS) satisfying the conditions

z t, x  z t T, x  t, x ℝ Ω
z t, x  0 t, x ℝ Ω.

Setting

J  \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ,  J_{0}  0 1 1 1 ,  z u, v ,

A J_{0} Δ_{x} V , and H \frac{1}{2} Az, z H t, x, z ,

system (HS) simply reads

J _{t}z H_{z}, t, x ℝ Ω,

which is the form of unbounded Hamiltonian systems or infinite-dimensional Hamiltonian systems in L^{2} Ω, ℝ^{2M} . This kind of systems were studied under various assumptions by Brezis and Nirenberg [14], and by Clément et al. [16].

Usually, in the study of superquadratic Hamiltonian systems, the nonlinear terms satisfy the following condition introduced by Ambrosetti and Rabinowitz [1]

μ 2, R 0 ; 0 μH t, z H_{z} t, z z for z R,  (AR)

see, for instance, [2, 5, 7, 29] and references therein. It is well known that this condition is mainly used to verify the boundedness of the energy functional and without it the problem then becomes very complicated. Moreover, without (AR) it may happen that all the Palais-Smale sequences of the functional are unbounded.

Let

H t, u, v : \frac{1}{2}H_{z} t, x, z z H t, x, z , S^{1} : ℝ T ℤ , and Θ S^{1} Ω.

Our assumptions on the potential V are the following:

V_{1} V C \overline{Ω}, ℝ .

V_{2} 0 σ S , S Δ_{x} V. For the Hamiltonian H we make the following assumptions:

H_{1} H C^{1} Θ ℝ^{2M}, ℝ is T -periodic with respect to t.

H_{2} H t, x, 0 0 and H_{z} t, x, z z as z 0, uniformly in t, x .

H_{3} H t, x, z 0 for z 0 and \frac{H t,x,z}{z ^{2}} as z , uniformly in t, x .

H_{4} R 0 such that

1 H t, x, z a_{1} z ^{2} if z R,
2 \frac{H_{z} t,x,z}{z} ^{σ} a_{2}H t, x, z if z R,
where a_{1}, a_{2} 0, σ 1 if N 1 and σ \frac{N}{2} 1 if N 2.

Under these assumptions, Mao et al. [26] obtained a nontrivial periodic solution for (HS) by using a local linking theorem. We will prove here that in case where the Hamiltonian is even with respect to z, the number of periodic solutions is in fact infinite. This fact has already been observed by Bartsch and Ding in [5] where they studied (HS) in both the cases where Ω is bounded, or Ω ℝ^{N}. However, they used condition (AR) to verify the boundedness of the Palais-Smale sequences of the energy functional associated to (HS), which was crucial for their argument. Moreover, one does not know whether the solutions they obtained there are large energy solutions. We would like to emphasize that in our situation we do not know whether the Palais-Smale sequences of that energy functional are bounded. Our result reads as follows:

Theorem 1.1. Assume that V_{1} , V_{2} , and H_{1} - H_{4} are satisfied. If in addition H in even in z, then (HS) has infinitely many pairs z_{k} of T -periodic solutions such that z_{k} _{}as k .

Finally, as another application of our abstract result, we consider in section 4 the following elliptic system of Hamiltonian type

 ;  ; Δu g x, v in Ω, Δv f x, u in Ω, u v 0 on Ω,  (ES)

where Ω is a bounded smooth domain in ℝ^{N}, N 3. The solutions of this problem describe steady states of some reaction-diffusion systems which are derived from several applications, such as mathematical biology or chemical reactions.

We study this problem under the following assumptions:

E_{1} f, g C Ω ℝ and there is a constant C 0 such that

f x, u C 1 u ^{p 1} and g x, u C 1 u ^{q 1} x, u ,

where p, q 2 satisfy \frac{1}{p} \frac{1}{q} 1 \frac{2}{N} .

Furthermore, in case N 5 we impose

\frac{1}{p} \frac{1}{2} \frac{2}{N} and \frac{1}{q} \frac{1}{2} \frac{2}{N} .

E_{2} f x, u  u and g x, u  u as u 0, uniformly in x.

E_{3} F x, u u^{2} and G x, u u^{2} uniformly in x as u .

E_{4} u f x, u u and u g x, u u are increasing in , 0 0, .

E_{5} f x, u f x, u and g x, u g x, u for all x, u .

Before we state our result on this problem, we recall the following definition.

Definition 1.2. We say that u, v is a strong solution of (ES) if u W^{2,p p 1} Ω W_{0}^{1,p p 1} Ω , v W^{2,q q 1} Ω W_{0}^{1,q q 1} Ω and u, v satisfies

Δu g x, v a.e. in Ω,
Δv f x, u a.e. in Ω.

We will also prove the following

Theorem 1.3. Under assumptions E_{1} - E_{5} , (ES) has infinitely many pairs of strong solutions u_{k}, v_{k} such that u_{k}, v_{k} as k , where represents the norm in the space

W^{2,p p 1} Ω W_{0}^{1,p p 1} Ω W^{2,q q 1} Ω W_{0}^{1,q q 1} Ω .

System (ES) has been already studied from the variational point of view by many authors. Hulshof and van der Vorst [19], and de Figueiredo and Felmer [17] obtained positive solutions by requiring among others that the Hamiltonian H u, v F u G v satisfies (AR). In [18], Felmer and Wang considered the case where the full superquadratic range is not reached, and by using a variation of (AR) they obtained infinitely many solutions in case where H is even in u, v by using the Galerkin approximation. Recently, Szulkin and Weth [31] improved the above results by reducing the energy functional to the Nehari-Pankov manifold. To circumvent the difficulty that the Nehari-Pankov manifold is not necessary of class C^{1}, they required the maps in assumption E_{4} to be strictly increasing. In [9], the first author of this paper showed that it is sufficient that these maps are only increasing. He applied the generalized fountain theorem obtained in [11] to a family of perturbed functionals. However, working with a family of modified functionals makes things unnecessary complicated. We considerably simplify this approach in the present paper.

Throughout the paper we denote by _{r} the norm of the Lebesgue space L^{r}.

2. Critical point theorems for strongly indefinite functionals

Let X be a separable Hilbert space which admits an orthogonal decomposition X X^{}X^{}, where X^{}is closed and X^{}X^{}^{}. We denote ⟨, ⟩ the inner product of X. Let a_{j} _{j 0} be an orthonormal basis of X^{}. We define on X a new norm by setting

u : max _{j 0}^{}\frac{1}{2^{j 1}} ⟨P^{}u, a_{j}⟩ , P^{}u , u X,

where P^{}: X X^{}are orthogonal projections, and we denote by τ the topology generated by this norm. This topology was introduced by Kryszewski and Szulkin in [20] and is related to the topology on X which is strong on X^{}and weak on X^{}. More precisely, if u_{n} is a bounded sequence in X then

u_{n} ^{τ} u P^{}u_{n} P^{}u and P^{}u_{n} P^{}u.

Now we recall some standard notations in Critical Point Theory: Let Φ : X ℝ, a, b ℝ, S X, and α 0. We denote

dist u, S : u S , dist_{τ} u, S : u S , S_{α} : u X ; dist u, S α ,

Φ^{b} : u X ; Φ u a , Φ_{a} : u X; Φ u a , and Φ_{a}^{b} : Φ_{a} Φ^{b}.

We say that a functional Φ C^{1} X, ℝ satisfies the Palais-Smale condition at the level c ℝ if the following holds: P S _{c} Any sequence u_{n} X such that Φ u_{n} c and Φ^{}u_{n} 0 Palais-Smale

sequence at level c possesses a convergent subsequence. This is the famous compactness condition in Critical Point Theory introduced by Palais and Smale [27]. Since we would like to consider some situations where this condition does not hold, we will use its following generalization due to Cerami [15]: Ce _{c} Any sequence u_{n} X such that Φ u_{n} c and 1 u_{n} Φ^{}u_{n} 0

Cerami sequence at level c possesses a convergent subsequence. If Ce _{c} holds we say that the functional Φ satisfies the Cerami condition at the level c Ce _{c}-condition for short .

We now consider the class of C^{1} functionals Φ : X ℝ which satisfy: KS Φ u \frac{1}{2} P^{}u ^{2} \frac{1}{2} P^{}u ^{2} Ψ u , where Ψ C^{1} X, ℝ is bounded below,

weakly sequentially lower semicontinuous, and Ψ^{}is weakly sequentially continuous.

The following deformation lemma will play a key role in the proof of our abstract results.

Lemma 2.1 (Deformation lemma). Assume that Φ satisfies KS . Let S X with S S if Φ is even , d b and ε, θ 0 such that

u Φ^{1} b 2ε, d 2ε S_{2θ}, 1 u Φ^{}u  \frac{8ε}{θ}.  (2.1)

Then there exists η C 0, 1 Φ^{d 2ϵ} , X such that:

(i) η t, u u if t 0 or if u Φ^{1} b 2ε, d 2ε S_{2θ}, (ii) η 1, Φ^{d ε} S Φ^{b ε} , (iii) Φ η , u is non increasing, u Φ^{d 2ε} , (iv) each point t, u 0, 1 Φ^{d 2ε} has a τ-neighborhood N_{t,u} such that v

η s, v s, v N_{t,u} 0, 1 Φ^{d 2ε} is contained in a finite-dimensional subspace of X,

(v) η is τ-continuous, (vi) if Φ is even then η t, is odd t 0, 1 .

Proof. We define the vector field

w u : 2∇Φ u Φ^{}u ^{2}, u Φ^{1} b 2ε, d 2ε .

By assumption KS we know that Φ^{}is weakly sequentially semicontinuous, and this implies that the function

v Φ_{b 2ε}^{d 2ε} ⟨Φ^{}v , w u ⟩ ℝ

is τ-continuous. Hence every u Φ_{b 2ε}^{d 2ε} has a τ-neighborhood N_{u} such that

⟨Φ^{}v , w u ⟩ 1 v N_{u},  (2.2)
u 2 v v N_{u},  (2.3)

where (2.3) holds because the set z X; z α is τ-closed for any α 0. Now since KS implies that Φ is τ-upper semicontinuous, the set N : Φ^{1} , b 2ε is τ-open. It follows that N : N N_{v}; b 2ε Φ v d 2ε is a τ-open covering of the metric space Φ^{d 2ε} , τ . We can extract a τ-locally finite τ-open covering M : M_{i}; i I of Φ^{c 2ϵ} finer than N. Let

V : _{i I} M_{i}.

For every i I we have either M_{i} N_{v} for some v or M_{i} N. In the first case we define v_{i} : w v and in the second case v_{i} : 0. Consider a τ-Lipschitz continuous partition of unity λ_{i}; i I subordinated to M and define on V the vector field

h u : _{i I} λ_{i} u v_{i}.

Let us define

A : Φ^{1} b 2ϵ, d 2ϵ S_{2δ}, B : Φ^{1} b ϵ, d ϵ ,
ψ u : dist_{τ} u, V A dist_{τ} u, V A dist_{τ} u, B ^{1} on V,

and

f u : ψ u h u , u V.

Clearly, the vector field f is locally Lipschitz continuous and τ-locally Lipschitz continuous. By using (2.1), (2.3) and the definition of the vector field h we see that

f u  \frac{θ}{4ε} 1 2 u  u V.

It follows from Corollary 7.6 in [30] that the problem

\begin{matrix}\frac{d}{dt}σ t, u f σ t, u \\ σ 0, u u Φ^{d 2ε} \end{matrix}

has a unique solution σ , u defined on ℝ^{}. We define η : 0, 1 Φ^{d 2ε} by setting

η t, u : σ d b 2ε t, u .

An argument similar to that in the proof of Lemma 8 in [10] shows that η satisfies i - v . If Φ is even, then we replace h u by \frac{1}{2} h u h u in such a way that f is odd. vi is then a consequence of the existence and the uniqueness of the solution of the above Cauchy problem. □

Now we can state our first critical point theorem, which extends the generalized saddle point theorem in [24] to the case where the Palais-Smale sequences of Φ may be unbounded.

Theorem 2.2 (Saddle point theorem). Assume that Φ satisfies KS . If there exists R 0 such that

A_{0}  b : inf_{u X^{}}Φ u sup_{u X^{}, u R} Φ u and d : sup_{u X^{}, u R} Φ u ,

then for some c b, d , there is a sequence u_{n} X such that

Φ u_{n} c and 1 u_{n} Φ^{}u_{n} 0 as n .

Moreover, if Φ satisfies the Ce _{α} condition for all α b, d , then Φ has a critical value in b, d .

The proof follows the lines of [24].

Proof. We assume by contradiction that for every c b, d there is no Ce _{c} sequence for Φ. Then there exists ε 0 such that

u Φ^{1} b 2ε, d 2ε 1 u Φ^{}u ε.

Let

M : u X^{}; u R .

We apply Lemma 2.1 with S X and θ 8, and we define μ : 0, 1 M X by setting μ t, u : P^{}η t, u , where η is given by Lemma 2.1. Clearly, iv and v of Lemma 2.1 imply that μ is τ-continuous and each t, u 0, 1 Φ^{d 2ε} has a τ-neighborhood N_{t,u} such that v μ s, v s, v N_{t,u} 0, 1 Φ^{d 2ε} is contained in a finite-dimensional subspace of X. We claim that 0 μ 0, 1 M . Indeed, if there exists t_{0}, u_{0} 0, 1 M such that μ t_{0}, u_{0} 0, then η t_{0}, u_{0} X^{}and by iii of Lemma 2.1 and assumption A_{0} we have

Φ u_{0} Φ η 0, u_{0} Φ η t_{0}, u_{0} b sup_{M}Φ,

which contradicts the fact that u_{0} M. μ is then an admissible homotopy in the sense of Kryszewski and Szulkin [20] such that 0 μ 0, 1 M . It follows from Theorem 2.4- iii of [20] that the Kryszewski-Szulkin’s degree (see [20]) deg_{KS} μ t, , int M , 0 is well defined and does not depend on t 0, 1 . Hence

deg_{KS} μ 1, , int M , 0 deg_{KS} μ 0, , int M , 0 deg_{KS} id, int M , 0 .

It follows from Theorem 2.4- i of [20] that there is u int M such that μ 1, u 0, which implies that η 1, u X^{}. By the definition of b we have b Φ η 1, u . But ii of Lemma 2.1 implies, since M Φ^{d ε} , that Φ η 1, u b ε. This gives a contradiction. □

In order to obtain a multiplicity result we need to introduce some notations. We consider an orthonormal basis e_{j} _{j 0} of X^{}and we set

X_{k}^{}: X^{}_{j 0}^{k} ℝe_{j} , X_{k}^{}: \overline{_{j k}^{}ℝe_{j}} , and B_{k}^{}ρ : u X_{k}^{}u ρ .

Theorem 2.3 (Fountain theorem). Assume that Φ satisfies KS , that Φ is even, and that there exist ρ_{k} r_{k} 0 such that:

A_{1}  a_{k} : sup_{u X_{k}^{}; u ρ_{k}} Φ u 0, d_{k} : sup_{u X_{k}^{}; u ρ_{k}} Φ u .
A_{2}  b_{k} : inf_{u X_{k}^{}; u r_{k}} Φ u , k .

Then, there exist a sequence u_{k}^{n} _{n} X and a number c_{k} b_{k} such that

Φ u_{k}^{n} c_{k} and 1 u_{k}^{n} Φ^{}u_{k}^{n} 0 as n .

Moreover, if Φ satisfies the Ce _{c} condition for any c 0, then Φ has an unbounded sequence of critical values.

Theorem 2.3 generalized Theorem 12 in [10] where the sequence u_{k}^{n} was a Palais-Smale sequence.

Proof. Let Γ_{k} be the set of maps γ : B_{k}^{}ρ_{k} X such that:

(a) γ is odd, τ-continuous and γ _{B_{k}^{}ρ_{k}} id, (b) each u int B^{}k ρ_{k} has a τ-neighborhood N_{u} in X_{k}^{}such that id γ N_{u} int B_{k}^{}ρ_{k} is contained in a finite dimensional subspace of X,

(c) Φ γ u Φ u u B_{k}^{}ρ_{k} .

A_{1} and A_{2} imply that there is k_{0} big enough such that b_{k} a_{k} for every k k_{0}. We define for k k_{0}

c_{k} : inf_{γ Γ_{k}} sup_{u B_{k}^{}ρ_{k}} Φ γ u .

It follows from Lemma 10 in [10] that if γ Γ_{k} then

γ B_{k}^{}ρ_{k}  u X_{k}^{}; u r_{k}  ,

which implies that c_{k} b_{k}. We assume by contradiction that there is no Cerami sequence of Φ at level c_{k}. Then there exist ε 0, \frac{c_{k} a_{k}}{2} and θ 0 such that

u Φ^{1} c_{k} 2ε, c_{k} 2ε γ B_{k}^{}ρ_{k} _{2θ}  1 u Φ^{}u  \frac{8ε}{θ},

where γ Γ_{k} is such that

sup_{B_{k}^{}ρ_{k}}Φ γ c_{k} ε.  (2.4)

We apply Lemma 2.1 with b d c_{k} and S γ B_{k}^{}ρ_{k} . We assume that

c_{k} 2ϵ a_{k}.  (2.5)

We define on B_{k}^{}ρ_{k}

β u : η 1, γ u .

It follows from i , ii , iv , v , vi of Lemma 2.1 and (2.5) that β Γ_{k}. Now by using (2.4) and ii of Lemma 2.1 we obtain

sup_{\begin{matrix}u X_{k}^{} \\ u ρ_{k} \end{matrix}} Φ β u sup_{\begin{matrix}u X_{k}^{} \\ u ρ_{k} \end{matrix}} Φ η 1, γ u  c_{k} ε

giving a contradiction with the definition of c_{k}. It then follows that Φ has a Cerami sequence at level c_{k}. □

Using the same argument as in the proof of Theorem 6 in [12], we can prove the following dual version of Theorem 2.3.

Theorem 2.4 (Dual fountain theorem). Let Φ u \frac{1}{2} u^{}^{2} \frac{1}{2} u^{}^{2} Ψ u , where Ψ C^{1} X, ℝ is even, bounded below, and weakly sequentially lower semicontinuous, and Ψ^{}is weakly sequentially continuous. If k k_{0}, ρ_{k} r_{k} 0 such that

B_{1} a^{k} : inf_{u X_{k}^{}, u ρ_{k}} Φ u 0, b^{k} : sup_{u X_{k}^{}, u r_{k}} Φ u 0,

B_{2} d^{k} : inf_{u X_{k}^{}, u ρ_{k}} Φ u 0, k ,

Then there exist c^{k} d^{k}, 0 and a sequence u_{k}^{n} X such that

Φ u_{k}^{n} c_{k} and 1 u_{n} Φ^{}u_{k}^{n} 0, as n .

Moreover, if Φ satisfies the Ce _{c} condition for any c d^{k_{0}}, 0 , then Φ has a sequence u_{k} of critical points such that Φ u_{k} 0^{}as k .

3. Periodic solutions of superquadratic Hamiltonian systems

In this section, we apply our multiplicity result above to the Hamiltonian system

tu Δ_{x}u H_{v} t, u, v in ℝ Ω, _{t}v Δ_{x}v H_{u} t, u, v in ℝ Ω.  (HS)

We assume here that assumptions V_{1} , V_{2} , and H_{1} - H_{4} are satisfied, and that H is even in z.

By V_{1} the operator S Δ_{x} V acting on L^{2} Ω, ℝ^{2M} is self-adjoint with domain D S H^{2} Ω, ℝ^{2M} H_{0}^{1} Ω, ℝ^{2M} . Let L : J _{t} A. Observing that he operators A and J A acting on L^{2} Ω, ℝ^{2M} with domains D A D J A H^{2} Ω, ℝ^{2M} H_{0}^{1} Ω, ℝ^{2M} are self-adjoint, then the operator L acting on L^{2} S^{1}, L^{2} Ω, ℝ^{2M} is also self-adjoint. We know by [5] that

L^{2} S^{1}, L^{2} Ω, ℝ^{2M} L^{2} S^{1} Ω, ℝ^{2M} , 0 σ A σ J A , 0 σ L ,

so there is an orthogonal decomposition

L^{2} Ω, ℝ^{2M} E^{}E^{},

such that L is negative on E^{}and positive on E^{}. The space X D L ^{\frac{1}{2}} is a Hilbert space with the inner product

⟨w, w⟩ ⟨ L ^{\frac{1}{2}} w, L ^{\frac{1}{2}} w⟩_{L^{2}}

and norm z ^{2} ⟨z, z⟩. Moreover, we have an orthogonal decomposition

X X^{}X^{}with X^{}X E^{}.

We have the following

Lemma 3.1 ([5], Lemma 4.6). The embedding of X in L^{r} S^{1} Ω, ℝ^{2M} is compact for r 2 if N 1, and for r 2, 2 N 2 2 if N 2.

Define the functional Φ : X ℝ,

Φ z \frac{1}{2} z^{}^{2} \frac{1}{2} z^{}^{2} Ψ z , where Ψ z _{Θ} H t, x, z .  (3.1)

Critical points of Φ are weak solutions of (HS).

Lemma 3.2. (1) Φ C^{1} X, ℝ .

(2) Ψ is bounded below and is weakly sequentially lower semicontinuous.

(3) Ψ^{}is weakly sequentially continuous.

Proof. (1) By H_{4} - 2 we have for z R and for t, x Θ

H_{z} t, x, z ^{σ} a_{2}H t, x, z z ^{σ}

a_{2} \frac{1}{2}H_{z} t, x, z z H t, x, z z ^{σ}

\frac{a_{2}}{2} H_{z} t, x, z z ^{σ 1} .

Hence

H_{z} t, x, z a_{3} z ^{\frac{σ 1}{σ 1}}  z R,  t, x Θ,

where a_{3} 0 is constant. On the other hand, since H_{z} is continuous, we have H_{z} t, x, z a_{4} for z R and t, x Θ. We deduce that

H_{z} t, x, z a_{4} a_{3} z ^{\frac{σ 1}{σ 1}} z ℝ^{2M},  t, x Θ.  (3.2)

Let ε 0. H_{2} implies that there is δ 0 such that

H_{z} t, x, z ε z pour tout z δ.

If z δ, then we have in view of (3.2)

H_{z} t, x, z a_{4} a_{3} z ^{\frac{σ 1}{σ 1}} a_{4} \frac{z ^{p}}{δ^{p}} a_{3} z ^{p} \frac{a_{4}}{δ^{p}} a_{3} z ^{p} with p \frac{σ 1}{σ 1} .

Hence

ε 0, C ε 0 ; H_{z} t, x, z ε z C ε z ^{p} z ℝ^{2M}.  (3.3)

Now one can easily verify that

\begin{matrix}σ 1 if N 1 \\ σ 1 \frac{N}{2} if N 2 \end{matrix} \begin{matrix}p 1 2 si N 1 \\ 2 p 1 2 N 2 N if N 2. \end{matrix}

It then follows from Lemma 3.1 that Ψ is well defined on X. We can now use a standard argument to show that Φ is of class C^{1} on X and

⟨Φ^{}z , w⟩ ⟨z^{}z^{}, w⟩ _{Θ} wH_{z} t, x, z , z, w X.  (3.4)

(2) Since H 0, the functional Ψ is bounded below. Let z_{n} z and c ℝ such that Ψ z_{n} c. By Lemma 3.1 we have z_{n} z in L^{2} and, up to a subsequence, z_{n} t, x z x, t a.e. in Θ. Using Fatou’s lemma we obtain Ψ z c, which shows that ψ is weakly lower semicontinuous.

(3) Let z_{n} z. By Lemma 3.1 we have z_{n} z_{n} in L^{p 1} where p \frac{σ 1}{σ 1} which implies, using (3.3) and Theorem A.2 in [32], that H t, x, z_{n} H t, x, z _{\frac{p 1}{p}} 0. It is then easy to deduce, using Fatou’s lemma, that Ψ^{} is weakly sequentially continuous.

□

We now prove that the functional Φ satisfies the Cerami condition.

Lemma 3.3. Let z_{n} X be such that

d sup Φ z_{n} and 1 z_{n} Φ^{}z_{n} 0.

Then z_{n} possesses a convergent subsequence.

Proof. Let us first show that the sequence z_{n} is bounded. Arguing by contradiction we suppose that z_{n} is unbounded. Then, up to a subsequence, we have z_{n} . Note that

⟨Φ^{}z_{n} , z_{n}^{}z_{n}^{}⟩ z_{n}^{}^{2} z_{n}^{}^{2}  _{Θ} z_{n}^{}z_{n}^{}H_{z} t, x, z_{n}

z_{n} ^{2} 1 _{Θ} \frac{z_{n}^{}z_{n}^{}H_{z} t, x, z_{n}}{z_{n} ^{2}} .

Since ⟨Φ^{}z_{n} , z_{n}^{}z_{n}^{}⟩ Φ^{}z_{n} z_{n}^{}z_{n}^{}2 z_{n} Φ^{}z_{n} 0, one necessary has

Θ \frac{z_{n}^{}z_{n}^{}H_{z} t, x, z_{n}}{z_{n} ^{2}} 1.  (3.5)

Set w_{n} z_{n} z_{n} . Assumption H_{4} - 1 implies that H t, x, z a_{1} z ^{2} c_{1} for all t, x, z . We deduce that for n big enough one has

1 d Φ z_{n}  \frac{1}{2}⟨Φ^{}z_{n} , z_{n}⟩  _{Θ} H t, x, z_{n}  (3.6)
a_{2} z_{n} ^{2} c_{1} Θ ,  (3.7)

where Θ denotes the Lebesgue measure of Θ. (3.7) shows that z_{n} _{2} is bounded. Hence

w_{n} _{2} \frac{z_{n} _{2}}{z_{n}} 0.  (3.8)

On the other hand we have

 ;  ; _{Θ} \frac{z_{n}^{}z_{n}^{}H_{z} t, x, z_{n}}{z_{n} ^{2}} ;  ; _{Θ} \frac{z_{n}^{}z_{n}^{}}{z_{n}} \frac{H_{z} t, x, z_{n}}{z_{n}}

Θ w_{n}^{}w_{n}^{}w_{n} \frac{H_{z} t, x, z_{n}}{z_{n}}

Θ \frac{H_{z} t, x, z_{n}}{z_{n}}^{σ} ^{1 σ} _{Θ} w_{n}^{}w_{n}^{}w_{n} ^{σ^{}}^{1 σ^{}},

with \frac{1}{σ} \frac{1}{σ^{}}1. Since

w_{n}^{}w_{n}^{}w_{n} w_{n}^{}w_{n}^{}w_{n}^{}w_{n}^{}w_{n}^{}w_{n}^{}^{2} 2 w_{n}^{}^{2} w_{n}^{}^{2} ,

we have

Θ w_{n}^{}w_{n}^{}w_{n} ^{σ^{}}c_{2} w_{n}^{}_{2σ^{}}^{2σ^{}}w_{n}^{}_{2σ^{}}^{2σ^{}}.

By using the fact that 2σ^{}p 1 2, 2 N 2 N , we have

w_{n}^{}_{2σ^{}}w_{n}^{}_{2}^{α} w_{n}^{}_{2 N 2 N}^{1 α} interpolation inequality ,

where α 0, 1 is such that \frac{α}{2} \frac{1 α}{2 N 2 N} 1. Since X embeds continuously in L^{2 N 2 N} , we have w_{n}^{}_{2 N 2 N} c_{3} w_{n}^{}c_{3} (because w_{n}^{}w_{n} 1). Now, since the decomposition X X^{}X^{}is also orthogonal with respect to the L^{2}-norm, we have w_{n}^{}_{2} w_{n} _{2}. Hence w_{n}^{}_{2σ^{}}c_{3} w_{n} _{2}^{α}, which implies

 ;  ; _{Θ} \frac{z_{n}^{}z_{n}^{}H_{z} t, x, z_{n}}{z_{n} ^{2}} ;  ; c_{4} _{Θ} \frac{H_{z} t, x, z_{n}}{z_{n}}^{σ} ^{1 σ} w_{n} _{2}^{2α}.  (3.9)

Let ε 0. By H_{2} there exists δ 0 such that H_{z} t, x, z ε z for z δ. Then

Θ \frac{H_{z} t, x, z_{n}}{z_{n}} ^{σ}  _{Θ z_{n} δ}  \frac{H_{z} t, x, z_{n}}{z_{n}} ^{σ}  _{Θ δ z_{n} R}  \frac{H_{z} t, x, z_{n}}{z_{n}} ^{σ}

Θ z_{n} R \frac{H_{z} t, x, z_{n}}{z_{n}} ^{σ}

Θ ε^{σ} sup_{Θ δ z_{n} R}  \frac{H_{z} t, x, z_{n}}{z_{n}} ^{σ} Θ a_{2} _{Θ z_{n} R} H t, x, z_{n} ,

where we have used H_{4} - 2 for the last term in the r.h.s. of the equality. We deduce from (3.6) that _{Θ z_{n} R} H t, x, z_{n} is bounded. Hence _{Θ} \frac{H_{z} t,x,z_{n}}{z_{n}} ^{σ} is bounded. It then follows from (3.9) and (3.8) that

Θ \frac{z_{n}^{}z_{n}^{}H_{z} t, x, z_{n}}{z_{n} ^{2}} 0,

which contradicts (3.5). With this contradiction, we conclude that z_{n} is bounded. We may now suppose that z_{n} z in X. Since

z_{n}^{}z^{}^{2} ⟨Φ^{}z_{n} Φ^{}z , z_{n}^{}z^{}⟩ _{Θ} z_{n}^{}z^{}H_{z} t, x, z_{n} ,

a standard argument using Lemma 3.1 shows that z_{n} z. □ Proof of Theorem 1.1. Let e_{i} be an orthonormal basis of X^{}. We recall that

X_{k}^{}X^{}_{j 0}^{k} ℝe_{j} and X_{k}^{}\overline{_{j k}^{}ℝe_{j}} .

Let z X_{k}^{}. By H_{3} we have

δ 0 c_{δ} 0 ; H t, x, z δ z ^{2} c_{δ}.

Then

Φ z \frac{1}{2} z^{}^{2} \frac{1}{2} z^{}^{2} _{Θ} H t, x, z \frac{1}{2} z^{}^{2} \frac{1}{2} z^{}^{2} δ z _{2}^{2} c_{δ} Θ .

Since the decomposition X X^{}X^{}is orthogonal with respect to L^{2}-norm, we have z^{}_{2} z _{2}. Since all norms are equivalent on _{j 0}^{k} ℝe_{j}, there is c_{1} 0 such that c_{1} z^{}^{2} z _{2}^{2}. Hence

Φ z \frac{1}{2} c_{1}δ z^{}^{2} \frac{1}{2} z^{}^{2} c_{1} Θ .

Let us choose δ such that c_{1}δ 1. Then we have

Φ z  \frac{1}{2} z ^{2} c_{δ} Θ  as z .  (3.10)

Let z X_{k}^{}. Then for every ε 0 we have

Φ z \frac{1}{2} z ^{2}  _{Θ} H t, x, z

\frac{1}{2} z ^{2} \frac{ε}{2} z _{2}^{2} \frac{c ε}{p 1} z _{p 1}^{p 1} by (3.3)

\frac{1}{2} z ^{2} c_{2}ε z _{2}^{2} \frac{c ε}{p 1} z _{p 1}^{p 1} because X L^{2}

\frac{1}{2} c_{2}ε z ^{2} \frac{c ε}{p 1} β_{k}^{p 1} z ^{p 1} ,

where β_{k} sup\begin{matrix}w X_{k}^{} \\ w 1 \end{matrix} w _{p 1} . By choosing ε \frac{1}{2c_{2}} we obtain

Φ z \frac{1}{2} \frac{1}{2} u ^{2} c_{3}β_{k}^{p 1} z ^{p 1} .

Hence, since β_{k} 0 as k (see [32], Lemma 3.8), we have

z r_{k} : c_{3} p 1 β_{k}^{p 1} ^{1 1 p} Φ z \frac{1}{2} \frac{1}{2} \frac{1}{p 1} r_{k}^{2} 0, k .  (3.11)

(3.10) and (3.11) show that the conditions A_{1} and A_{2} of Theorem 2.3 are satisfied. In view of Lemmas 3.2 and 3.3, we can apply Theorem 2.3 and get the result. □

4. Solutions of superquadratic elliptic systems

Now we consider the problem

 ;  ; Δu g x, v in Ω, Δv f x, u in Ω, u v 0 on Ω,  (ES)

and we assume in the sequel that the assumptions E_{1} - E_{5} are satisfied. Consider the Laplacian as the operator

Δ : H^{2} Ω H_{0}^{1} Ω L^{2} Ω L^{2} Ω ,

and let φ_{j} _{j 1} a corresponding system of orthogonal and L^{2} Ω -normalized eigenfunctions, with eigenvalues λ_{j} _{j 1} . Then, writing

u _{j 1}^{}a_{j}φ_{j}, with a_{j} _{Ω} uφ_{j},

we set, for 0 s 2

E^{s} : u L^{2} Ω _{j 1}^{}λ_{j}^{s} a_{j} ^{2}

and

A^{s} u : _{j 1}^{}λ_{j}^{s 2}a_{j}φ_{j}, u D A^{s} E^{s}.

One can verify easily that A^{s} is an isomorphism onto L^{2} Ω . We denote A^{s} : A^{s} ^{1} . It is well known (see [22]) that the space E^{s} is a fractional Sobolev space with the inner product

⟨u, v⟩_{s} _{Ω} A^{s}uA^{s}v.

Lemma 4.1 ([28]). E^{s} embeds continuously in L^{r} Ω for s 0 and r 1 satisfying \frac{1}{r} \frac{1}{2} \frac{s}{N} . Moreover, the embedding is compact in the case of strict inequality.

By assumption E_{1} , there exist s, t 0 such that s t 2 and

\frac{1}{p} \frac{1}{2} \frac{s}{N} and \frac{1}{q} \frac{1}{2} \frac{t}{N} .  (4.1)

We consider the functional

Φ u, v : _{Ω} A^{s}uA^{t}v Ψ u, v , u, v E^{s} E^{t}.

where

Ψ u, v _{Ω} F x, u G x, v .

It follows from Lemma 4.1 that the inclusions E^{s} L^{p} Ω and E^{t} L^{q} Ω are continuous. This, together with the estimate

; ; _{Ω} A^{s}uA^{t}v ; ; A^{s}u _{2} A^{t}u _{2} u _{s} v _{t},

imply that the functional Φ above is well defined on E : E^{s} E^{t}. Now a standard argument shows that if assumption E_{1} holds, then the functional Φ is of class C^{1} on E and its critical points are weak solutions of (ES). We recall that u, v E^{s} E^{t} is a weak solution of (ES) if

Ω A^{s}uA^{t}k A^{s}hA^{t}v _{Ω} hf x, u kg x, v  0,  h, k E^{s} E^{t}.

The following regularity result is due to de Figueiredo and Felmer [17].

Lemma 4.2. If u, v E^{s} E^{t} is a weak solution of (ES), then u, v is a strong solution of (ES).

We endow E E^{s} E^{t} with the inner product

⟨ u, v , ϕ, φ ⟩_{s t} ⟨u, ϕ⟩_{s} ⟨v, φ⟩_{t}, u, v , ϕ, φ E,

and the associated norm u, v _{s t}^{2} ⟨ u, v , u, v ⟩_{s t} .

In the following we assume without loss of generality that s t. One can easily verify that E has the orthogonal decomposition with respect to ⟨ , ⟩_{s t} E E^{}E^{}, where

E^{}: u, A^{s t} u u E^{s} and E^{}: u, A^{s t} u u E^{s} .  (4.2)

If we denote by P^{}: E E^{}the orthogonal projections, then a direct calculation yields

P^{}u, v \frac{1}{2} u A^{t s} v, v A^{s t} u ,  u, v E,  (4.3)

and

Φ u, v \frac{1}{2} P^{}u, v _{s t}^{2} \frac{1}{2} P^{}u, v _{s t}^{2}  _{Ω} F x, u G x, v .  (4.4)

Let us recall the following technical result, which will play a crucial role in the verification of the Cerami condition.

Lemma 4.3 ([23], Lemma 3.2). Let x ℝ^{N}, u, v, s ℝ such that s 1 and su v 0. If E_{2} - E_{4} are satisfied then

f x, u s \frac{s}{2}u 1 s v  F x, u F x, u v 0,
g x, u s \frac{s}{2}u 1 s v  G x, u G x, u v 0.

Lemma 4.4. Let z_{n} u_{n}, v_{n} E and c 0 such that

Φ z_{n} c and 1 z_{n} _{s t} Φ^{}z_{n} 0.

Then z_{n} has a convergent subsequence.

Proof. Let us first show that z_{n} is bounded. If z_{n} is unbounded, then we can assume that z_{n} _{s t} . Set w_{n} z_{n} z_{n} _{s t} . Then, up to a subsequence, we have w_{n} w in E. Since F, G 0, we have for n big enough

c 1 Φ z_{n}  \frac{1}{2} z_{n}^{}_{s t}^{2} \frac{1}{2} z_{n}^{}_{s t}^{2}  \frac{1}{2} z_{n} _{s t}^{2} w_{n}^{}_{s t}^{2} w_{n}^{}_{s t}^{2} .

Recall that w_{n}^{}_{s t}^{2} w_{n}^{}_{s t}^{2} 1. Hence w_{n}^{}_{s t}^{2} \frac{1}{2} \frac{c 1}{z_{n} _{s t}^{2}} . But since

z_{n} _{s t}^{2} , we have for n big enough \frac{c 1}{z_{n} _{s t}^{2}}  \frac{1}{6}. It follows that w_{n}^{}_{s t}^{2}  \frac{1}{3}.
Now fix r 0 and set s_{n}  \frac{r}{z_{n} _{s t}} 1 and y_{n}  \frac{r}{z_{n} _{s t}} z_{n}^{}rw_{n}^{}. Then

s_{n} 1, y_{n} E^{}and the sequences s_{n} and y_{n} are bounded. By setting y_{n} a_{n}, b_{n} , where a_{n} E^{s} and b_{n} E^{t}, we have

⟨Φ^{}z_{n} , s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}⟩ ⟨z_{n}^{}z_{n}^{}, s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}⟩

Ω s_{n} \frac{s_{n}}{2} 1 u_{n} 1 s_{n} a_{n} f x, u_{n}

Ω s_{n} \frac{s_{n}}{2} 1 v_{n} 1 s_{n} b_{n} g x, v_{n} .

Since s_{n} and y_{n} are bounded, we obtain

⟨Φ^{}z_{n} , s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}⟩ Φ^{}z_{n} s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}
C_{1} Φ^{}z_{n} z_{n} _{s t} C_{2} Φ^{}z_{n} 0.

We deduce that for n big enough

⟨z_{n}^{}z_{n}^{}, s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}⟩  1 _{Ω} s_{n} \frac{s_{n}}{2} 1 u_{n} 1 s_{n} a_{n} f x, u_{n}
Ω s_{n} \frac{s_{n}}{2} 1 v_{n} 1 s_{n} b_{n} g x, v_{n} .

Noting that rw_{n}^{}z_{n} s_{n}z_{n} y_{n} and

⟨z_{n}^{}z_{n}^{}, s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}⟩ \frac{1}{2} z_{n}^{}_{s t}^{2} \frac{1}{2} z_{n}^{}_{s t}^{2} \frac{1}{2}r^{2} w_{n}^{}_{s t}^{2} \frac{1}{2}r^{2} w_{n}^{}_{s t}^{2} ,

we have

Φ rw_{n}^{}Φ z_{n}  \frac{1}{2} w_{n}^{}_{s t}^{2} ⟨z_{n}^{}z_{n}^{}, s_{n} \frac{s_{n}}{2} 1 z_{n} 1 s_{n} y_{n}⟩
Ω F x, u_{n} F x, 1 s_{n} u_{n} a_{n}
Ω G x, u_{n} G x, 1 s_{n} u_{n} a_{n} .

We deduce by using that for n sufficiently large

Φ rw_{n}^{}Φ z_{n} 1
Ω s_{n} \frac{1}{2}s_{n} 1 u_{n} 1 s_{n} a_{n} f x, u_{n} F x, u_{n} F x, 1 s_{n} u_{n} a_{n}
Ω s_{n} \frac{1}{2}s_{n} 1 v_{n} 1 s_{n} b_{n} g x, u_{n} G x, u_{n} G x, 1 s_{n} v_{n} b_{n} .

We conclude by using Lemma 4.3 that for n big enough, Φ rw_{n}^{}1 Φ z_{n} . If w^{}0, then w_{n}^{}0 in L^{p} Ω L^{q} Ω by Lemma 4.1. By E_{1} and Theorem A.2 in [32] we have _{Ω} F x, 1 s_{n} u_{n} a_{n} 0 and _{Ω} G x, 1 s_{n} v_{n} b_{n} 0.

Since Φ z_{n} c, we then obtain for n sufficiently large

c 1 Φ z_{n}  Φ rw_{n}^{}1
\frac{1}{2}r^{2} w_{n} _{s t}^{2}  _{Ω} F x, 1 s_{n} u_{n} a_{n}  _{Ω} G x, 1 s_{n} v_{n} b_{n} 1

\frac{1}{2}r^{2} w_{n} _{s t}^{2} 2

\frac{r^{2}}{6} 2 because w_{n}^{}_{s t}^{2} \frac{1}{3} .

This leads to a contradiction if we take r arbitrary large. Hence w^{}0 and then w 0. Setting w_{n} w_{n}^{1}, w_{n}^{2} E^{s} E^{t}, we have

\frac{Φ z_{n}}{z_{n} _{s t}^{2}}  \frac{1}{2} w_{n}^{}_{s t}^{2} \frac{1}{2} w_{n}^{}_{s t}^{2}  _{Ω} \frac{F x, u_{n}}{z_{n} _{s t}^{2}} _{Ω} \frac{G x, u_{n}}{z_{n} _{s t}^{2}}

\frac{1}{2} w_{n}^{}_{s t}^{2} \frac{1}{2} w_{n}^{}_{s t}^{2} _{w_{n}^{1} 0} \frac{F x, w_{n}^{1} z_{n} _{s t}}{w_{n}^{1} z_{n} _{s t} ^{2}} _{w_{n}^{2} 0} \frac{G x, w_{n}^{2} z_{n} _{s t}}{w_{n}^{1} z_{n} _{s t} ^{2}}

C _{w_{n}^{1} 0} \frac{F x, w_{n}^{1} z_{n} _{s t}}{w_{n}^{1} z_{n} _{s t} ^{2}} _{w_{n}^{2} 0} \frac{G x, w_{n}^{2} z_{n} _{s t}}{w_{n}^{1} z_{n} _{s t} ^{2}}.

Since Φ z_{n} is bounded and z_{n} _{s t} , the l.h.s. tends to 0. By applying Fatou’s lemma and using E_{3} , we see that the r.h.s. goes to , which is a contradiction. Consequently the sequence z_{n} is bounded.

Now a standard argument shows that the bounded sequence z_{n} has a convergent subsequence. □

Proof of Theorem 1.3. Let a_{j} _{j 0} be an orthonormal basis of E^{s}. We define an orthonormal basis e_{j} _{j 0} of E^{}by setting

e_{j} : \frac{1}{\overline{2}} a_{j}, A^{s t} a_{j} .

Let

E_{k}^{}E^{}_{j 0}^{k} ℝe_{j} and E_{k}^{}\overline{_{j k}^{}ℝe_{j}} .

E_{3} implies that for every δ 0 there is C_{δ} 0 such that

F x, u δ u ^{2} C_{δ} and G x, u δ u ^{2} C_{δ},  x, u .  (4.5)

Let z E_{k}^{}. Then z u, A^{s t} u v, A^{s t} v , where v E^{s} and u E_{k}^{s} : _{j 0}^{k} ℝa_{j}. We obtain by using (4.5), the fact that E^{s t} embeds continuously in L^{2} Ω , and the parallelogram identity

Φ z u _{s}^{2} v _{s}^{2}  _{Ω} F x, u v G x, A^{s t} u v
u _{s}^{2} v _{s}^{2} 2Cδ u _{2}^{2} v _{2}^{2} 2C_{δ} Ω .

Since all the norms are equivalent on the finite-dimensional subspace E_{k}^{s}, there is a constant c_{1} 0 such that c_{1} u _{s} u _{2}. Hence

Φ z 1 c_{2}δ u _{s}^{2} v _{s}^{2} 2C_{δ} Ω .

Choose δ \frac{1}{c_{2}} . Hence Φ z as z _{s t} , and consequently assumption A_{1} of Theorem 2.3 is satisfied.

Let z E_{k}^{}. Then z u, A^{s t} u with u \overline{_{j k}^{}ℝa_{j}} , and

Φ z \frac{1}{2} z _{s t}^{2}  _{Ω} F x, u G x, A^{s t} u

\frac{1}{2} z _{s t}^{2} 2 _{Ω} F x, u G x, A^{s t} u .

By E_{1} there is a constant C_{1} 0 such that

F x, u C_{1} 1 u ^{p} and G x, A^{s t} u C_{1} 1 A^{s t} u ^{q} .

Hence

Φ z u _{s}^{2} 2C_{1} u _{p}^{p} 2C_{1} A^{s t} u _{q}^{q} 4C_{1} Ω .

We define

β_{1,k} : sup_{\begin{matrix}u \overline{_{j k}^{}ℝa_{j}} \\ u _{s} 1 \end{matrix}} u _{p}, β_{2,k} : sup_{\begin{matrix}v \overline{_{j k}^{}ℝ A^{s t} a_{j}} \\ v _{t} 1 \end{matrix}} v _{q},

and β_{k} max β_{1,k}; β_{2,k} . Then

Φ z u _{s}^{2} 2C_{1}β_{k}^{p} u _{s}^{p} 2C_{1}β_{k}^{q} u _{s}^{q} 4C_{1} Ω .

We assume without loss of generality that q p and we set

r_{k} : C_{1}pβ_{k}^{p} ^{\frac{1}{2 p}} .

Then for z _{s t} \overline{2} u _{s} r_{k} we have

Φ z { \atop b_{k}} : Kβk^{\frac{2p}{2 p}}  \frac{1}{4} \frac{1}{p \overline{2} ^{p}} Aβk^{\frac{2 q p}{2 p}} 4C_{1} Ω ,  (4.6)

where K, A 0 are constant. We know by Lemma 3.8 in [32] that β_{1,k} 0 and β_{2,k} 0 as k . This implies that { \atop b_{k}} as k , and then that assumption A_{2} of Theorem 2.3 is satisfied.

An argument similar to that in the proof of Lemma 3.2 shows that Ψ is weakly sequentially lower semicontinuous and that Ψ^{}is weakly sequentially continuous. Since we have shown above that the functional Φ satisfies the Cerami condition at every critical level, we can apply Theorem 2.3 and get the desired result. □

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Cyril Joel Batkam Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, CANADA.

E-mail address: cyril.joel.batkam@usherbrooke.ca

Fabrice Colin Department of Mathematics and Computer Science, Laurentian University, Ramsey Lake Road, Sudbury, Ontario, P3E 2C6, CANADA.

E-mail address: fcolin@cs.laurentian.ca

Tomasz Kaczynski Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, CANADA.

E-mail address: t.kaczynski@usherbrooke.ca