1. Introduction

In the present paper we are concerned with the following problem

(P)

Where is a smooth bounded domain, with for any is a continuous function and is a Carathéodory function.

Problem (P) is related to the stationary version of a model, the so-called Kirchhoff equation, introduced by[12]. To be more precise, Kirchhoff established a model given by the equation

where are constants which extends the classical D'Alambert's wave equation, by considering the effects of the changes in the length of the strings during the vibrations. There are papers[4,11,14] in which the authors give the existence of solutions of Kirchhoff-type and p-Kirchhoff-type equations. Moreover, for p(x)-Kirchhoff-type equations see, for example,[2,5-7].

The p(x)-Laplace operator is a natural generalization of the p-Laplacian operator

where is a real constant. The main difference between them is that p-Laplacian operator is (p-1)-homogenous, but the p(x)-Laplacian operator, when p(x) is not constant, is not homogeneous. This causes many problems, some classical theories and methods, such as the Lagrange multiplier theorem and the theory of Sobolev spaces, are not applicable. For p(x)- Laplacian operator, we refer the readers to[9,10,15,16] and references there in. Moreover, the nonlinear problems involving the p(x)-Laplacian operator are extremely attractive because they can be used to model dynamical phenomenons which arise from the study of electrorheological fluids or elastic mechanics. Problems with variable exponent growth conditions also appear in the modelling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. The detailed application backgrounds of the p(x)-Laplacian can be found in[1,3,17,19] and the references there in.

2. Preliminaries

First, we recall some basic properties of spaces and (for details, see e.g.,[8,14]).

Set

For any , denote

and define the variable exponent Lebesgue space by

We define a norm, the so-called Luxemburg norm, on by the formula

and then becomes a Banach space.

Define the variable exponent Sobolev space by

then it can be equipped with the norm

The space is defined as the closure of with respect to the norm . For , we can define an equivalent norm since the well-known Poincaré inequality holds.

We say that is a weak solution of (P) if

where

We associate to the problem (P) the energy functional, defined as

where and . We know that from (m0) and (f0) (see Section 3) is well defined and in a standard way we can prove that and that the critical points of I are solutions of (P). Moreover, the derivative of is given by

for all

3. The Main Results

Now, we are ready to set and prove the main result of the present paper.

The proof of Theorem 7 follows from the following two Lemmas.

Proof. Let us assume that there exists a sequence such that

Therefore,

Therefore, is bounded in . From this bound estimate, going to a subsequence if necessary, there exists a in such that . Thanks to the compact embedding , we get

Since, , we have

From (f0) and Proposition 1, it follows

By the embeddings, we get

Hence,

From (m0), it follows

(3.1)

Since the functional (3.1) is of type , we get satisfies (PS) condition.

Proof.

(i) Let us assume . Then by (m0), we have

By the continuous embeddings

(3.2)

Further, using (f0) and (f1), we get

Therefore,

Let us define the function

Since it is clear that there exists a such that

. Hence, for a fixed small enough, there exist two positive real numbers and such that

(ii) From (AR), we have

Hence, for and we have

From (AR), it can be obtained that . Therefore,

Proof of Theorem 7. From Lemma 8, Lemma 9 and the fact that , satisfies the Mountain Pass theorem (see e.g.,[18]). Therefore, has at least one nontrivial critical point, i.e., (P) has a nontrivial weak solution. We are done.