Neumann problem with a nonlinear $p(x)$-elliptic equation solved by topological Degree methods

Abstract

In this paper, we prove the existence of weak solutions to Neumann boundary value problems for the nonlinear $p(x)$-elliptic equation of the form $$-div\,a(x,u,\nabla u)=b(x)|u|^{p(x)-2}u+\lambda H(x,u,\nabla u).$$
We established the existence by using the topological degree, introduced by Berkovits.