Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

We investigate the following Dirichlet problem with variable exponents: \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u=\lambda \alpha (x)\left\vert u\right\vert ^{\alpha (x)-2}u\left\vert v\right\vert ^{\beta (x)}+F_{u}(x,u,v),\text{ in }\Omega , \\ -\bigtriangleup _{q(x)}v=\lambda \beta (x)\left\vert u\right\vert ^{\alpha (x)}\left\vert v\right\vert ^{\beta (x)-2}v+F_{v}(x,u,v),\text{ in }\Omega , \\ u=0=v,\text{ on }\partial \Omega. \end{array} \right. \end{equation*} We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-known Ambrosetti--Rabinowitz type growth condition. More precisely, we manage to show that the problem admits four, six and infinitely many solutions respectively.


Introduction
In this paper, we consider the existence of multiple solutions to the following Dirichlet problem for an elliptic system with variable exponents: pP q $ & %´△ ppxq u " λαpxq |u| αpxq´2 u |v| βpxq`F u px, u, vq, in Ω, △ qpxq v " λβpxq |u| αpxq |v| βpxq´2 v`F v px, u, vq, in Ω, where △ ppxq u :" divp|∇u| ppxq´2 ∇uq is called ppxq-Laplacian which is nonlinear and nonhomogeneous, Ω Ă R N is a bounded domain, and pp¨q, qp¨q, αp¨q, βp¨q ą 1 are in the space C 1 pΩq which consists of differentiable functions with continuous first order derivatives on Ω.
Elliptic equations and systems of elliptic equations with variable exponent growth as in problem pP q arise from applications in electrorheological fluids and image restoration. We refer the readers to [1], [5], [24], [39] and the references therein for more details in applications. In particular, see [5] for a model with variable exponent growth and its important applications in image denoising, enhancement, and restoration. Problems with variable exponent growth also brought challenging pure mathematical problems. Compared with the classical Laplacian ∆ " ∆ 2 which is linear and homogeneous and the p -Laplacian ∆ p¨: " divp|∇¨| p´2 ∇¨q which is nonlinear but homogeneous for constant p, the ppxq-Laplacian is both nonlinear and inhomogeneous. Due to the nonlinear and inhomogeneous nature of the ppxq-Laplacian, nonlinear (systems of) elliptic equations involving ppxq-Laplacian and nonlinearities with variable growth rates are much more difficult to deal with. Driven by the real-world applications and mathematical challenges, the study of elliptic equations and systems with variable exponent growth has attracted many researchers with different backgrounds, and become a very attracting field. We refer the readers to [3], [9], [10], [12], [15], [20], [23], [25], [35], [36], [37] and the related references to tract the rapid development of the field.
In this paper, our main goal is to obtain some existence results for the problem pP q, a Dirichlet problem for elliptic systems with variable exponents, without the famous Ambrosetti-Rabinowitz condition via critical point theory. For this purpose, we propose a new set of growth conditions for the nonlinearities in the current system of elliptic equations setting. Our new set of growth conditions involve only variable growths which naturally match the variable nature of the problem under investigation. Under our growth conditions, we can use a novel method to verify that the corresponding functional to the problem pP q satisfies the Cerami compactness condition which is a weaker compactness condition yet is still sufficient to yield critical points of the functional. See the details of proofs in Section 3. The current study generalizes in particular our former investigations [32] and [37]. However, this generalization from a single elliptic equation to the current system of elliptic equations is by no means trivial. Besides technical complexities, the assumptions in the current study are more involved. In particular, though we still do not need any monotonicity on the nonlinear terms, we do need impose certain monotonicity assumptions on the variable exponents to close our argument in the system setting.
When we utilize variational argument to obtain existence of weak solutions to elliptic equations, typically we impose the famous Ambrosetti-Rabinowitz growth condition on the nonlinearity to guarantee the boundedness of Palais-Samle sequence. Under the Ambrosetti-Rabinowitz growth condition, one then tries to verify the Palais-Smale condition. However, the Ambrosetti-Rabinowitz type growth condition excludes a number of interesting nonlinearities. In view of this fact, a lot of efforts were made to show the existence of weak solutions in the variational framework without this type of growth condition, especially for the usual p-Laplacian and a single nonlinear elliptic partial differential equation (see, in particular, [14], [17], [18], [19], [21], [29] and the references therein). Our results can be regarded as extensions of the corresponding results for the p-Laplacian problems. There are also some related earlier works which dealt with elliptic variational problems in the variable exponent spaces framework, see [2], [13], [33], [37], [27] and related works. These earlier studies were mainly focused on a single elliptic equation. In the interesting earlier study [33], the author considered the existence of solutions of the following variable exponent differential equations without Ambrosetti-Rabinowitz condition on bounded domain,

"´△
ppxq u " f px, uq, in Ω, u " 0, on BΩ. (1.1) However, in some aspects the assumption is even stronger than the Ambrosetti-Rabinowitz condition. In a recent study [2], the authors considered the variable exponent equation in the whole space R N under the following assumptions: p1 0 q there exists a constant θ ě 1, such that θF px, tq ě F px, stq for any px, tq P R NˆR and s P r0, 1s, where F px, tq " f px, tqt´p`F px, tq; p2 0 q f P CpR NˆR , Rq satisfies lim |t|Ñ8 F px,tq |t| p`" 8. In [13], the authors considered the problem (1.1) in a bounded domain under the condition p2 0 q. In [27], the authors studied a variable exponent differential equation with a potential term in the whole space R N . The authors proposed conditions under which they could show the existence of infinitely many high energy solutions without the Ambrosetti-Rabinowitz condition. In the above mentioned works, the growths conditions involved either the supremum or the infimum of the variable exponents. In our current study, we are able to provide a number of existence results in the system setting under assumptions that only involve variable exponent growths which match naturally the variable growth of the problem under study.
We point out here that the growth conditions we use here and the method to check the Cerami compactness condition are different from all the above mentioned works. Due to the differences between the p-Laplacian and ppxq -Laplacian mentioned as above, it is usually challenging to judge whether or not results about p-Laplacian can be generalized to ppxq-Laplacian. Meanwhile, some new methods and techniques are needed to study elliptic equations involving the non-standard growth, as the commonly known methods and techniques to study elliptic equations involving standard growth may fail. The main reason, as mentioned earlier, is that the principal elliptic operators in the elliptic equations involving the non-standard growth is not homogeneous anymore. To see some new features associated with the ppxq -Laplacian, we first point out that the norms in variable exponent spaces are the so-called Luxemburg norms |u| pp¨q (see Section 2) and the integral ş Ω |upxq| ppxq dx does not have the usual constant power relation as in the spaces L p for constants p. Another subtle feature is on the principal Dirichlet eigenvalue. As invetigated in [10], even for a bounded smooth domain Ω Ă R N , the principle eigenvalue λ pp¨q defined by the Rayleigh quotient is zero in general, and only under some special conditions λ pp¨q ą 0 holds. For example, when Ω Ă R (N " 1) is an interval, results show that λ pp¨q ą 0 if and only if pp¨q is monotone. This feature on the ppxq-Laplacian Dirichlet principle eigenvalue plays an important role for us in proposing the assumptions on the variable exponents and in our proofs of the main results. Now we shall first list the assumptions on the nonlinearity F and variable exponents involved in the current system setting. Our assumptions are as follows.
pH p,q q There are vectors l p , l q P R N zt0u such that for any x P Ω, φ p ptq " ppx`tl p q is monotone for t P I x,p plq " tt | x`tl p P Ωu, and φ q ptq " qpx`tl q q is monotone for t P I x,q plq " tt | x`tl q P Ωu.
To gain a first understanding, we briefly comment on some of the above assumptions. pH 0 q means that the nonlinearity F has a subcritical variable growth rate in the sense of variable exponent Sobolev embedding and in the current system of elliptic equations setting. pH 1 q and pH 2 q describe the far and near field behaviors of the nonlinearity F . Notice that in the current setting, the far field behavior pH 1 q is more involved. We emphasize that pH p,q q is crucial for our later arguments, for it guarantees that the Rayleigh quotients for´∆ ppxq and´∆ qpxq (see (1.2) for´∆ ppxq ) are positive. Finally, the assumptions pH 0 q-pH 4 q on the nonlinearity F are consistent, which can be seen by the following example: F px, u, vq " |u| ppxq rlnp1`|u|qs apxq`| v| qpxq rlnp1`|v|qs bpxq`| u| θ 1 pxq |v| θ 2 pxq lnp1`|u|q lnp1`|v|q, where 1 ă θ 1 pxq ă ppxq, 1 ă θ 2 pxq ă qpxq, θ 1 pxq ppxq`θ 2 pxq qpxq " 1, @x P Ω. In addition, F does not satisfy the Ambrosetti-Rabinowitz condition. Now we are in a position to state our main results. Theorem 1.1. If λ is small enough and the assumptions (H α,β ), pH 0 q, (H 2 )-(H 3 ) and (H p,q ) hold, then the problem pP q has at least four nontrivial solutions each with constant sign respectively. Theorem 1.2. If λ is small enough and the assumptions pH α,β q, pH 0 q-pH 3 q and pH p,q q hold, then the problem pP q has at least six nontrivial solutions each with constant sign respectively. Theorem 1.3. If the assumptions pH α,β q, pH 0 q, pH 1 q and pH 4 q hold, then there are infinitely many pairs of solutions to the problem pP q. This rest of the paper is organized as follows. In Section 2, we do some functionalanalytic preparations. In Section 3, we give the proofs of our main results.

Functional-analytic Preliminary
Throughout this paper, we will use letters c, c i , C, C i , i " 1, 2, ... to denote generic positive constants which may vary from line to line, and we will specify them whenever it is necessary.
In order to discuss problem pP q, we shall discuss the functional analytic framework. First, we present some results about space W 1,pp¨q 0 pΩq which we call variable exponent Sobolev space. These results on the variable exponent spaces will be used later (for details, see [6], [7], [9], [16], [26]). We denote CpΩq the space of continuous functions on Ω with the usual uniform norm, and C`pΩq " hˇˇh P CpΩq, hpxq ą 1 for x P Ω ( .
For h " hp¨q P CpΩq, we denote h`:" max xPΩ hpxq and h´:" min xPΩ hpxq. For p " pp¨q P C`pΩq, we introduce When equipped with the Luxemberg norm  [6], [7], [9]). i) The space pL pp¨q pΩq, |¨| pp¨q q is a separable, uniform convex Banach space, and its conjugate space is L ppp¨qq 0 pΩq, where ppp¨qq 0 :" pp¨q pp¨q´1 is the conjugate function of pp¨q. For any u P L pp¨q pΩq and v P L ppp¨qq 0 pΩq, we havěˇˇˇż ii) If p 1 , p 2 P C`pΩq, p 1 pxq ď p 2 pxq for any x P Ω, then L p 2 p¨q pΩq Ă L p 1 p¨q pΩq, and the imbedding is continuous.
where p i pxq P C`pΩq, i " 1,¨¨¨, m, then Y is a Banach space. The following proposition can be regarded as a vectorial generalization of the classical proposition on the Nemytsky operator.
If there exist ηpxq, p 1 pxq,¨¨¨, p k pxq P C`pΩq, hpxq P L ηp¨q pΩq and positive constant c ą 0 such that then the Nemytsky operator from Y to pL ηp¨q pΩqq m defined by pN f uqpxq " f px, upxqq is a continuous and bounded operator.
Proof. Similar to the proof of [4], we omit it here.
The following two propositions concern the norm-module relations in the variable exponent Lebesgue spaces. Unlike in the usual Lebesgue spaces setting, the norm and module of a function in the variable exponent spaces do not enjoy the usual power equality relation.
Proposition 2.4. (see [9]). If u, u n P L pp¨q pΩq, n " 1, 2,¨¨¨, then the following statements are equivalent to each other.
The spaces W 1,pp¨q pΩq and W 1,qp¨q pΩq are defined by and be endowed with the following norm We denote by W 1,pp¨q 0 pΩq the closure of C 8 0 pΩq in W 1,pp¨q pΩq. Then we have in particular the following Sobolev embedding relation and Poincaré type inequality.
We know from iii) of Proposition 2.5 that |∇u| pp¨q and }u} pp¨q are equivalent norms on W 1,pp¨q 0 pΩq. From now on we will use |∇u| pp¨q to replace }u} pp¨q as the norm on W Under the assumption (H p,q ), λ pp¨q defined in (1.2 ) is positive, i.e., we have the following proposition.
For any pu, vq and pφ, ψq in X, let From Proposition 2.2, Proposition 2.5 and condition pH 0 q, it is easy to see that Φ 1 , Φ 2 , Φ, Ψ P C 1 pX, Rq and then The integral functional associated with the problem pP q is ϕpu, vq " Φpu, vq´Ψpu, vq.
Without loss of generality, we may assume that F px, 0, 0q " 0, @x P Ω. Obviously, we have where B j denotes the partial derivative of F with respect to its j-th variable, then the condition pH 0 q holds |F px, u, vq| ď cp|u| γpxq`| v| δpxq`1 q, @x P Ω.
(2.1) From Proposition 2.2, Proposition 2.5 and condition pH 0 q, it is easy to see that ϕ P C 1 pX, Rq and satisfies We say pu, vq P X is a critical point of ϕ if The dual space of X will be denoted as X˚, then for any H P X˚, there exists f P pW 1,pp¨q 0 pΩqq˚, g P pW 1,qp¨q 0 pΩqq˚such that Hpu, vq " f puq`gpvq. We denote }¨}˚, }¨}˚, pp¨q and }¨}˚, qp¨q the norms of X˚, pW Therefore It's easy to see that Φ is a convex functional, and we have the following proposition.
Remark 2.8. A proof of a simple version of the above proposition can be found in the references [9], [15]. In the system setting here, the idea of proof is essentially the same. For readers' convenience and for completeness, we present it here.
Proof. i) It follows from Proposition 2.2 that Φ 1 is continuous and bounded. For any ξ, η P R N , we have the following inequalities (see [9]) from which we can get the strict monotonicity of Φ 1 : We claim that ∇u n pxq Ñ ∇upxq in measure. Denote In view of (2.2), we have Without loss of generality, we may assume that 0 ă ş From (2.4) and the bounded property of tu n u in X, we have Thus t∇u n u converges in measure to ∇u in Ω, so we have by Egorov's Theorem that ∇u n pxq Ñ ∇upxq a.e. x P Ω up to a subsequence. From pu n , v n q á pu, vq in X, we have lim nÑ`8 pΦ 1 pu n , v n q, pu n´u , v n´v qq " lim nÑ`8 pΦ 1 pu n , v n q´Φ 1 pu, vq, pu n´u , v n´v qq " 0.
If f n , f P X˚, f n Ñ f in X˚, let pu n , v n q " pΦ 1 q´1pf n q, pu, vq " pΦ 1 q´1pf q, then Φ 1 pu n , v n q " f n , Φ 1 pu, vq " f . So tpu n , v n qu is bounded in X. Without loss of generality, we can assume that pu n , v n q á pu 0 , v 0 q in X. Since f n Ñ f in X˚, we have lim nÑ`8 (2.16) Since Φ 1 is of type pS`q, pu n , v n q Ñ pu 0 , v 0 q, we conclude that pu n , v n q Ñ pu, vq in X, so pΦ 1 q´1 is continuous. The proof of Proposition 2.7 is complete.
When ε is small enough, (2.17) is valid. Since p P C 1 pΩq, there exist a small enough positive ε such that where y " x 0`τ px´x 0 q and τ P p0, 1q, op1q P R N is a function and op1q Ñ 0 uniformly as |x´x 0 | Ñ 0.
To complete the proof of this lemma, it is sufficient to show that It is easy to see the following two inequalities hold: To proceed, we shall use polar coordinates. Let r " |x´x 0 |. Since p P C 1 pΩq, it follows from (2.17) that there exist positive constants c 1 and c 2 such that ppε, ωq´c 2 pε´rq ď ppr, ωq ď ppε, ωq´c 1 pε´rq, @pr, ωq P Bpx 0 , ε, δ, θq.

Proofs of main results
With the preparations in the last section, we will in this section give our proofs of the main results. To be rigorous, we first give the definition of a weak solution to the problem pP q.
The corresponding functional of pP q is given by ϕ " ϕpu, vq defined below on X: As compactness is crucial in showing the existence of weak solutions via critical point theory. We shall introduce the type of compactness which we shall use in the current study, i.e., the Cerami compactness condition.
Definition 3.2. We say ϕ satisfies Cerami condition in X, if any sequence tu n u Ă X such that tϕpu n , v n qu is bounded and }ϕ 1 pu n , v n q} p1`}pu n , v n q}q Ñ 0 as n Ñ`8 has a convergent subsequence.
It is well-known that the Cerami condition is weaker than the usual Palais-Samle condition. Under our new growth condition for the system under investigation, we manage to show that the corresponding functional ϕ satisfies the above Cerami type compactness condition which is sufficient to yield critical points. More specifically, we have the following lemma.
Proof. Let tpu n , v n qu Ă X be a Cerami sequence such that ϕpu n , v n q Ñ c. From Definition 3.2, we know that }ϕ 1 pu n , v n q} p1`}pu n , v n q}q Ñ 0 as n Ñ`8. We first claim that to show ϕ satisfies the Cerami condition, it is sufficient to show that the Cerami sequence tpu n , v n qu is bounded in X. Indeed, suppose tpu n , v n qu is bounded, then tpu n , v n qu admits a weakly convergent subsequence in X. Without loss of generality, we assume that pu n , v n q á pu, vq in X, then Ψ 1 pu n , v n q Ñ Ψ 1 pu, vq in X˚. Since ϕ 1 pu n , v n q " Φ 1 pu n , v n q´Ψ 1 pu n , v n q Ñ 0 in X˚, we have Φ 1 pu n , v n q Ñ Ψ 1 pu, vq in X˚.
Since Φ 1 is a homeomorphism, we have pu n , v n q Ñ pu, vq, hence ϕ satisfies Cerami condition. Therefore, our claim holds.
Noticing that ap¨q ą pp¨q and bp¨q ą qp¨q on Ω, we can conclude that "ż Ω |u n | ppxq rlnpe`|u n |qs apxq´1`| v n | qpxq rlnpe`|v n |qs bpxq´1 dx * is bounded, which further yields that Let ε ą 0 satisfy ε ă mint1, p´´1, q´´1, 1 p˚`, 1 q˚`, p pγ q´´1, p qδ q´´1u. Since }ϕ 1 pu n , v n q} }pu n , v n q} Ñ 0 , we have ż Ω |∇u n | ppxq`| ∇v n | qpxq dx " ż Ω F u px, u n , v n qu n`Fv px, u n , v n qv n dx`ż Ω λpαpxq`βpxqq |u n | αpxq |v n | βpxq dx`op1q " ż Ω |F u px, u n , v n qu n | ε rlnpe`|u n |qs 1´εˇF u px, u n , v n q u n lnpe`|u n |qˇˇˇˇ1´ε dx The above inequality contradicts with (3.1). Therefore, we can conclude that tpu n , v n qu is bounded, and the proof of Lemma 3.3 is complete. Now we are in a position to give a proof of Theorem 1.1.
Denote F``px, u, vq " F px, Spuq, Spvqq, where Sptq " maxt0, tu. For any pu, vq P X, we say pu, vq belong to the first, the second, the third or the fourth quadrant of X, if u ě 0 and v ě 0, u ď 0 and v ě 0, u ď 0 and v ď 0, u ě 0 and v ď 0, respectively.
Proof of Theorem 1.1.
As noticed above, λ pp¨q , λ qp¨q ą 0 and we have also by the choice of σ that Next, we shall use spatial decomposition technique. We divide the underlying domain Ω into disjoint subsets Ω 1 ,¨¨¨, Ω n 0 such that min xPΩ j p˚pxq ą max In the following, we denote fj " min xPΩ j f pxq, fj " max xPΩ j f pxq, j " 1,¨¨¨, n 0 , @f P CpΩq, Denote also }u} pp¨q,Ω i the norm of u on Ω i , i.e.
It is easy to see that }u} pp¨q,Ω i ď C }u} pp¨q , and there exist ξ i , η i P Ω i such that When }u} pp¨q is small enough, we have Similarly, when }v} qp¨q is small enough, we have Cpσq ż Ω |v| δpxq dx ď 1 4 ż Ω 1 qpxq |∇v| qpxq dx.
Thus, when }pu, vq} is small enough, we have When λ is small enough, for any pu, vq P X with small enough norm, we have ϕ``pu, vq " Φpu, vq´Ψ``pu, vq Therefore, when λ is small enough, there exist r ą 0 and ε ą 0 such that ϕpu, vq ě ε ą 0 for every pu, vq P X and }pu, vq} " r.
Let Ω 0 Ă Ω be an open ball with radius ε. Notice that pH α,β q holds. Let ε ą 0 be small enough such that Now we pick two functions u 0 , v 0 P C 2 0 pΩ 0 q that are positive in Ω 0 . From pH α,β q, it is easy to see that Thus, ϕ``pu, vq has at least one nontrivial critical point pu1, v1 q with ϕ``pu1, v1 q ă 0.
From assumption pH 3 q, it is easy to see that pu1, v1 q lies in the first quadrant of X. It is easy to see that Sp´u1q P W 1,pp¨q 0 pΩq. Choosing Sp´u1q as a test function, we have Thus, u1 ě 0. Similarly, we have v1 ě 0. Therefore, pu1, v1 q is a nontrivial solution with constant sign of pP q and such that ϕpu1, v1 q ă 0. By the former discussion and (3.5), we can see that u1 and v1 are both nontrivial. Similarly, we can see that pP q has a nontrivial pui , vi q with constant sign in the i-th quadrant of X, such that ϕpui , vi q ă 0, i " 2, 3, 4. Thus pP q has at least four nontrivial solutions with constant sign. By now, we finished the proof of Theorem 1.1.
Proof of Theorem 1.2.
According to the proof of Theorem 1.1, when λ is small enough, there exist r ą 0 and ε ą 0 such that ϕ``pu, vq ě ε ą 0 for every pu, vq P X and }pu, vq} " r.
We may assume that there exists two different points x 1 , x 2 P Ω such that ∇ppx 1 q ‰ 0, ∇qpx 2 q ‰ 0. Now we define h 1 P C 0 pBpx 1 , εqq h 2 P C 0 pBpx 2 , εqq as follows: From Lemma 2.10, we may let ε ą 0 be small enough such that ε ă 1 2 |x 2´x1 | and ż which imply that ϕ``pth 1 , th 2 q Ñ´8 pas t Ñ`8q. Since ϕ``p0, 0q " 0, ϕ`s atisfies the conditions of the Mountain Pass Lemma. From Lemma 3.3, we know that ϕ``satisfies Cerami condition. Therefore, we conclude that ϕ``admits at least one nontrivial critical point pu 1 , v 1 q with ϕ``pu 1 , v 1 q ą 0. From assumption pH 3 q , we can easily see that pu 1 , v 1 q lies in the first quadrant of X. Thus, pu 1 , v 1 q is a nontrivial solution with constant sign to the problem pP q in the first quadrant of X satisfying ϕpu 1 , v 1 q ą 0.
Similarly, we can see that pP q has a nontrivial solution pu 2 , v 2 q in the third quadrant in X, which satisfy ϕpu 2 , v 2 q ą 0, and ( u 1 , v 1 ), (u 2 , v 2 ) are all nontrivial. From Theorem 1.1, pP q has nontrivial solutions with constant sign pui , vi q in the i-th quadrant of X (i " 1, 2, 3, 4), which satisfies ϕpui , vi q ă 0. Thus, pP q has at least six nontrivial solutions with constant sign. By far, we have finished the proof of Theorem 1.2.
Now we proceed to prove Theorem 1.3. For this purpose, we need to do some preparations. Noticing that X is a reflexive and separable Banach space (see [38], Section 17, Theorem 2-3), then there are te j u Ă X and ej ( Ă X˚such that X " spante j , j " 1, 2,¨¨¨u, X˚" span W˚t ej , j " 1, 2,¨¨¨u, and ă ej , e j ą" For convenience, we write X j " spante j u, Y k " Lemma 3.4. If γ, δ P C``Ω˘, γpxq ă p˚pxq and δpxq ă q˚pxq for any x P Ω, denote Proof. Obviously, 0 ă β k`1 ď β k , so β k Ñ β ě 0. Let u k P Z k satisfy }pu k , v k q} " 1, 0 ď β k´| u k | γp¨q´| v k | δp¨q ă 1 k . Then there exists a subsequence of tpu k , v k qu (which we still denote by pu k , v k q) such that pu k , v k q á pu, vq, and ă ej , pu, vq ą" lim kÑ8 @ ej , pu k , v k q D " 0, @ej , which implies that pu, vq " p0, 0q, and so pu k , v k q á p0, 0q. Since the imbedding from W 1,pp¨q 0 pΩq to L γp¨q pΩq is compact, then u k Ñ 0 in L γp¨q pΩq. Similarly, we have v k Ñ 0 in L δp¨q pΩq. Hence we get β k Ñ 0 as k Ñ 8. Proof of Lemma 3.4 is complete.
In odder to prove Theorem 1.3, we need the following lemma (see in particular, [40,Theorem 4.7]). For a version of this lemma with the Palais-Samle condition, the (P.S.)condition, see [4, P 221, Theorem 3.6].
Proof of Theorem 1.3.
According to pH α,β q, pH 0 q, pH 1 q and pH 4 q, we know that ϕ is an even functional and satisfies the Cerami condition. Let Vk " Z k , it is a closed linear subspace of X and Vk ' Y k´1 " X.
We may assume that there exists different points x n , y n P Ω such that ∇ppx n q ‰ 0, ∇qpy n q ‰ 0. We then define h n P C 0 pBpx n , ε n qq and hn P C 0 pBpy n , ε n qq as follows: h n pxq " " 0, |x´x n | ě ε n ε n´| x´x n | , |x´x n | ă ε n , hnpxq " " 0, |x´y n | ě ε n ε n´| x´y n | , |x´y n | ă ε n .
Without loss of generality, we may assume that supp h i X supp hi " ∅ for i " 1, 2,¨¨ä nd supp h i X supp h j " ∅, supp hi X supp hj " ∅, @i ‰ j.
Set Vḱ " spantph 1 , h1q,¨¨¨, ph k , hkqu. We will prove that there are infinitely many pairs of Vk and Vḱ , such that ϕ satisfies the conditions of Lemma 3.5 and that the corresponding critical value ̟ k :" inf gPΓ sup pu,vqPVḱ ϕpgpu, vqq Ñ`8 when k Ñ`8, which implies that there are infinitely many pairs of solutions to the problem pP q.
Next we show that (A 2 ) holds. From the definition of ph n , hnq, it is easy to see that ϕpth, th˚q Ñ´8 as t Ñ`8, for any ph, h˚q P Vḱ " spantph 1 , h1q,¨¨¨, ph k , hkqu with ph, h˚q " 1. Therefore, pA 2 q also holds. Now, applying Lemma 3.5, we finish the proof of Theorem 1.3.