EXISTENCE OF WEAK SOLUTIONS FOR p(x)-KIRCHHOFF-TYPE EQUATION

This paper is considerned with the existence of solutions for p(x)Kirchhoff-type problem under with Dirichlet boundary condition. By direct variational method and the Mountain Pass theorem, we establish some conditions that ensure the existence nontrivial weak solutions for the problem. AMS Subject Classification: 35D05, 35J60, 35J70


Introduction
In this paper we are concerned with the following problem where Ω ⊂ R N (N ≥ 2) is a smooth bounded domain, p, r, s ∈ C Ω for any x ∈ Ω ; m is a non-negative measurable real function.
We assume that M and m are satisfy the following conditions specief conditions.
The p(x)−Laplacian operator ∆ p(x) u = div |∇u| p(x)−2 ∇u is a generalization of p−Laplace operator ∆ p u = div |∇u| p−2 ∇u where p > 1 is a real constant.The p(x)−Laplacian possesses more complicated structure than the p−Laplacian; for example, it is not homogeneous.This fact implies some diffculties; for example, we can not use the theory of Sobolev spaces in many problems involving this operator.Some of the nonlinear problems involving p(x)-growth conditions are extremely attractive because those problems can be used to model dynamical phenomenons that arise from the study of electrorheological fluids or elastic mechanics [1,11,15,22,26].Moreover, problems with variable exponent growth conditions also appear in the mathematical modelling of stationary thermo-rheological viscous flows of non-Newtonian fluids, in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium and image processing [3,4,6].The detailed application backgrounds of the p(x)−Laplace operator can be found in [13,14,20,21,17].
In the present paper, by help of the Mountain Pass theorem, we obtain the existence at least one nontrivial weak solution of problem (P).This paper is organized as follows.In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces .In Section 3, we show the existence of weak solutions of problem (P).
In particular, if p(x) = p is constant, then ||u| p | q(x) = |u| p pq(x) .We also consider the weighted variable exponent Lebesgue space L p(x) c(x) (Ω).Let c : Ω → R be a measurable real function such that c(x) > 0 a.e.x ∈ Ω.We define The then L p(x) c(x) (Ω) is a Banach space which has similar properties with the usual variable exponent Lebesgue spaces.The modular of this space is ρ (c(x),p(x)) : Proposition 2.5.(see [13,17] (Ω) are separable and reflexive Banach spaces; (ii) Let q ∈ C + Ω .If q (x) < p * (x) , for all x ∈ Ω, then the embedding Proposition 2.6.(see [14]) Let X be a Banach space and p(x) dx.

Main Results
We say that u ∈ X is a weak solution of (P) if where ϕ ∈ X.

Z. Yucedag
We associate to the problem (P) the energy functional, defined as I : X → R, where In a standart way, it can be shown that I ∈ C 1 (X, R).Moreover, we have for any u, υ ∈ X.Hence, we can infer that critical points of functional I are the weak solutions for problem (P).
The proof of Lemma 3.3 is complete.
Proof.Then by (M 1 ), we have The proof of Lemma 3.4 is complete.
Proof.Let assume that there exists a sequence {u n } ⊂ X such that Firstly, we prove that {u n } is bounded in X. Arguing by contradiction and passing to a subsequence, we have u n → ∞ as n → ∞.Using (M 1 ), (3.2) and considering u n > 1, for n large enough, we have In view of (M 1 ), we conclude that Ω |∇u n | p(x)−2 ∇u n (∇u n − ∇u) → 0.
Proof of Theorem 3.1, from Lemma 3.3, Lemma 3.4, Lemma 3.5 and the fact that I (0) = 0, I satisfies the Mountain Pass theorem [24].Therefore, I has at least one nontrivial weak solution.