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Research Article Existence and Localization Results for p x -Laplacian via Topological Methods | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 120646 7 pages doi 2010 120646 Research Article Existence and Localization Results for p x -Laplacian via Topological Methods B. Cekic and R. A. Mashiyev Department of Mathematics Dicle University 21280 Diyarbakir Turkey Correspondence should be addressed to B. Cekic bilalcekic@ Received 23 February 2010 Revised 16 April 2010 Accepted 20 June 2010 Academic Editor J. Mawhin Copyright 2010 B. Cekic and R. A. Mashiyev. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We show the existence of a week solution in W0fix Q to a Dirichlet problem for -Ap x u f x u in Q and its localization. This approach is based on the nonlinear alternative of Leray-Schauder. 1. Introduction In this work we consider the boundary value problem -Ap x U f x u in Q u 0 on dQ P where Q c RN N 2 is a nonempty bounded open set with smooth boundary dQ x U div Vu p x -2Vu is the so-called p x -Laplacian operator and CAR f Q X R R is a Caratheodory function which satisfies the growth condition If x s a x C s q q x for . x e Q and all s e R with C const. 0 1 q x 1 q x 1 for . x e Q and a e Lq x Q a x 0 for . x e Q. We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces Lp x Q w 1 p x Q and W01 p x Q . In that context we refer to 1 2 for the fundamental properties of these spaces. 2 Fixed Point Theory and Applications Set LTO Q p p e LTO Q ess inf p x 1 . xeQ For p e LTO Q let p1 ess infxeQp x p x p2 ess supxeQp x TO for . x e Q. Let us define by U Q the set of all measurable real functions defined on Q. For any p e LTO Q we define the variable exponent Lebesgue space by Lp x Q u e U Q pp x u u x px dx TO . Q We define a norm the so-called Luxemburg .