Existence Results for a Dirichlet Boundary Value Problem Involving the p(x)-Laplacian Operator

Abstract

The aim of this paper is to establish the existence of weak solutions, in$W_0^{1,p(x)}(\Omega)$, for a Dirichlet boundary value problem involving the$p(x)$-Laplacian operator. Our technical approach is based on the Berkovits topologicaldegree theory for a class of demicontinuous operators of generalized $(S_+)$ type. Wealso use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, andspecially properties of $p(x)$-Laplacian operator. In order to usethis theory, we will transform our problem into an abstractHammerstein equation of the form $v+S\circ Tv=0$ in the reflexiveBanach space $W^{-1,p'(x)}(\Omega)$ which is the dual space of$W_0^{1,p(x)}(\Omega)$. Note also that the problem can be seen as anonlinear eigenvalue problem of the form$Au=\lambda u,$ where$Au:=-\Div(|\nabla u|^{p(x)-2}\nabla u)-f(x,u)$. When this problemadmits a~non-zero weak solution $u$, $\lambda$ is an eigenvalue ofit and $u$ is an associated eigenfunction.