Universal Journal of Applied Mathematics 1(3): 192-197, 2013 DOI: 10.13189/ujam.2013.010306 http://www.hrpub.org Existence of Weak Solutions for a Nonlocal Problem Involving the p(x)-Laplace Operator Mustafa Avci Faculty of Economics and Administrative Sciences, Batman University, 72000 Batman, Turkey * Corresponding Author: avcixmustafa@gmail.com Copyright c ⃝2013 Horizon Research Publishing All rights reserved. Abstract This paper deals with the existence of weak solutions for some nonlocal problem involving the p (x)- Laplace operator. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions. Keywords p(x)-Laplace operator, p (x)-Kirchhoff-type equations, variable exponent Sobolev spaces, variational method, mountain pass theorem, Ekeland variational principle MSC: 35D05; 35J60; 35J70 1 Introduction In the present paper, we are concerned with the Dirichlet boundary value problem    −M ( ∫ Ω |∇u| p(x) p(x) dx)Δ p(x) u = λa (x) |u| q(x)−2 u in Ω, u =0 on ∂ Ω, (P) where Ω ⊂ R N , N ≥ 3, is a smooth bounded domain, λ> 0; p, q ∈ C ( Ω ) and a is a non-negative measurable real-valued function, M is a continuous function which obey some specific conditions. The purpose of the present paper is to find a nontrivial weak solution for a p (x)-Kirchhoff-type equation (P) in the variable exponent Sobolev spaces. The main tool is variational approach. By help of the well-known theorems mountain pass theorem and Ekeland variational principle, the existence result is obtained. Problem (P) is related to the stationary version of a model, the so-called Kirchhoff equation, introduced by Kirchhoff [14]. To be more precise, Kirchhoff established a model given by the equation ρ ∂ 2 u ∂t 2 − ( P 0 h + E 2L ∫ L 0     ∂u ∂x     2 dx ) ∂ 2 u ∂x 2 =0, where ρ, P 0 , h, E, L 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. For p (x)-Kirchhoff-type equations see, for example, [4, 6, 12]. The importance of problem (P) arises mainly from the existence of the p (x)-Laplacian △ p(x) u = div ( |∇u| p(x)−2 ∇u ) . Obviously, when p (x) = 2, △ 2 = △ is the usual Laplace operator. However, in case p (x) ̸= 2 the situation is very crucial, as for example, one encounters the lack of the homogeneity, and a result of this, some classical theories, such as the theory of Sobolev spaces, is not applicable. For the papers involving the p(x)-Laplacian operator we refer the readers to [3, 5, 11, 13, 16, 17, 18] and references therein. Moreover, the nonlinear problems involving the p (x)-Laplacian extremely attractive because they can be used to model dynamical phenomena 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 mathe- matical 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, 2, 19, 20, 22] and references therein.