Existence results for a Dirichlet boundary value problem involving the $p(x)$-Laplacian operator

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 topological degree theory for a class of demicontinuous operators of generalized $(S_+)$ type. We also use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, and specially properties of $p(x)$-Laplacian operator. In order to use this theory, we will transform our problem into an abstract Hammerstein equation of the form $v+S\circ Tv=0$ in the reflexive Banach 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 a nonlinear eigenvalue problem of the form$Au=\lambda u,$ where $Au:=-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)-f(x,u)$. When this problem admits a non-zero weak solution $u$, $\lambda$ is an eigenvalue of it and $u$ is an associated eigenfunction.