Strongly nonlinear elliptic problem by topological degree in generalized Sobolev spaces

Abstract

In this paper, we study the strongly nonlinear elliptic problem with a Dirichlet condition \begin{equation*} \left\{\begin{array}{ccc} -div\;a(x,u,\nabla u)=\lambda |u|^{q(x)-2}u+f(x,u,\nabla u) & \mbox{in}\; \Omega,\\ u=0 &\mbox{on}\; \partial\Omega, \end{array}\right. \end{equation*} in the functional framework of Sobolev spaces with variable exponent. We prove that this problem admits at least a weak solution by using the Berkovits topological degree and the properties of the generalized Sobolev spaces.