1. Introduction

Consider the following nonlinear eigenvalue problem

It is well known that elliptic  equations  involving the non-standard growth are not trivial general- izations of similar problems studied in the constant case since the non-standard growth operator is not homogeneous and, thus, some techniques which can be applied in the case of the constant growth operators will fail in this new situation, such as the La- grange multiplier theorem, see, e.g [1, 2]. Problems

In our case by using the Ljusternik-Schnirelmann theory we obtain the existence of infinitely many solitons for the problem (1.1).

The outline of the rest of the paper is as follows. In Section 2 we present some definitions and basic results that are necessary. In Section 3 we give the proof of our main result about existence of solutions for problem (1.1).

2. Preliminaries and Useful results

We state some basic properties of the variable expo- nent Lebesgue-Sobolev spaces L^p(.)(Ω) and W^m,p(.)(Ω). We refer the reader to the monograph by [5] and to the references therein. Define the generalized Lebesgue

Proposition 2.3  Let I(u) =       (| ∆u |p(x) + | ∇u |p(x))dx, for

Proposition 2.1 (6) Under the hypothesis (1.2), the space (L^p(x)(Ω), | · |_p(x)) is separable, uniformly convex, re-

is a Banach, separable and reflexive space. For more details, we refer the reader to [6, 7, 8] and [ 9]. We  denote  by  W^m,p(x)(Ω)  the  closure  of  C^∞(Ω)

Note that the weak solutions of the problem (1.1) are where the infimum is taken over X \ {0}.

The proof of this proposition is similar to the proof of [6, Theorem 1.3] . Recall that our main result of this work is to show that problem (1.1) has at least one non-decreasing sequence of nonnegative eigenvalues (k)k1. To attain this objective we will use a variational technique based on Ljusternick-Schnirelmann theory on C1-manifolds [11]. In fact, we give a direct characterization of k involving a mini-max argument over sets of genus greater than k.
We set

Definition 2.4 Let X be a real reflexive Banach space and let X stand for its dual with respect to the pairing h:; :i. We shall deal with mappings T acting from X into X. The strong convergence in X (and in X) is denoted by!and the weak convergence by*. T is said to belong to the class (S+), if for any sequence un in X converging weakly to u 2 X and limsup
n!+1 hT ;un ? ui  0, it follows that un converges strongly to u in X. We write T 2 (S+). Consider the following two functionals defined on X:

Proof. It is clear that ‘ and  are even and of class C1 on X andM= ‘?1f1g. ThereforeMis closed. The derivative operator ‘0 satisfies ‘0(u) , 0 8u 2M (i.e., ‘0(u) is onto for all u 2M). Hence ‘ is a submersion, which proves that M is a C1-manifold. Let as split  on two functionals.

Proof.
(i) We recall the following well-known inequalities, which hold for any three real a; b and p

On the set where 1 < p(
) < 2, we employ (2.3) as follows:

Since un is bounded in X, implies that “(1 n )!0 as n ! 1. Hence, sending n to 1 in (2.4) and (2.5), we obtain

Note that the strict monotonicity of T1 implies that T1 is into operator. Moreover, T1 is a coercive operator. Indeed, from Proposition 2.3 and since p? ? 1 > 0, for each u 2W 2;p(x)
0 such that kuk  1, we have

Note that the strict monotonicity of T1 implies that T1 is into operator. Moreover, T1 is a coercive operator. Indeed, from Proposition 2.3 and since p? ? 1 > 0, for each u 2 W 2;p(x)
0  such that kuk  1, we have

3. Existence results

Set?j = fK M: K symmetric, compact and (K)  jg ; where (K) = j being the Krasnoselskii’s genus of set K, i.e., the smallest integer j, such that there exists an odd continuous map from K to Rj n f0g. Now, let us establish some useful properties of Krasnoselskii genus proved by Szulkin [12]. Lemma 3.1 Let X be a real Banach space and A, B be symmetric subsets of E n f0g which are closed in X. Then (a) If there exists an odd continuous mapping

Proof We only need to prove that for any j 2N, ?j , ; and the last assertion. Indeed, since W 2;p(:) 0  is separable, there exists (ei )i1 linearly dense in W 2;p(:) 0 such that

Thus, Vj is a symmetric bounded neighborhood of 0 2 Fj . Moreover, Fj \M is a compact set. By (f ) of Lemma 2.7, we conclude that (Fj \M) = j and then we obtain finally that ?j , ;. This completes the proof of first part of the theorem.
Now, we claim that

for some M > 0 independent of k. Thus kukkp(:)  M. This implies that (uk)k is bounded in X. For a subsequence of fukg if necessary, we can assume that fukg converges weakly in X and strongly in Lp By our choice of F?
k?1, we have uk * 0 weakly in X, because hen ; eki = 0, for any k > n. This contradicts the fact that jukjp(:) = 1 for all k. Since k  tk the claim is proved. Corrolary 3.3 we have the following statements:

For all i  j, we have ?i  ?j and in view of definition of i ; i 2N, we get i  j . As regards to n !1, it is proved before in Theorem 3.2. (iii) Let  2 . Thus there exists u an eigenfunction of  such thatZ

1. Ferrero and G. Warnaut, On a solution of second and fourth order elliptic with power type nonlinearities, Nonlinear Analysis TMA, (70), 2889–2902, 2009.
2. G. Myers, Thin films with high surface tension, SIAM Review, 40 (3), 441–462, 1998.
3. K. Giri, D. Choudhuri and Sh. Pradhan, Existence and Concentration of solutions for a class of elliptic PDEs involving p-biharmonic operator, [math.AP] 8 Jun 2016. http://arxiv.org/abs/ 1606.02512.
4. Lieu and C.Chen
5. Diening, P. Harjulehto, P. H¨ast¨o and M. Ruzicka, Lecture Notes in Mathematics, issue 2017, Springer Science and business Media, 2011.
6. L. Fan and D. Zhao, On the spaces Lp(x)() and Wm;p(x)(), J. Math. Anal. Appl., (263), 424–446, 2001.
7. L. Fan and X. Fan, A Knobloch-type result for p(t) Laplacian systems, J. Math. Anal. Appl., (282), 453–464, 2003.
8. Mihailescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal. T. M. A., (67), 1419–1425, 2007.
9. H. Yao, Solution for Neumann boundary problems involving the p(x)-Laplace operators, Nonlinear Anal. T.M.A., (68), 1271–1283, 2008.
10. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. (69), 3629-3636, 2008.
11. Szulkin, Ljusternick-Schnirelmann theory on C1- manifolds, Ann. Inst. Henri Poincar´e Anal. Non. (5), 119– 39, 1988.
12. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990.
13. L. Fan and Q.H. Zhang, Existence of solutions for p(x)-laplacian Dirichlet problem, Nonlinear Anal. (52), 18431852, 2003.

(Click to view)

(Click to view)

No. of Downloads Per Month

No. of Downloads Per Country