Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form

{A(u)=finΩu=0on∂Ω\left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right.

where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and f ∈ W^−1,p′^(.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.