Topological degree theory for a class of nonlinear degenerate elliptic problems in weighted Sobolev spaces

Abstract

This article is devoted to study the existence of weak solutions to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic problems of the type:
$$-{\rm{div}}\Big[\mathcal{B}(x,u,\nabla u)+\mathcal{A}(x,\nabla u)\Big]=\lambda \mathcal{G}(x,u,\nabla u)+a(x)|u|^{q-2}u,$$
where $\mathcal{A}$ and $\mathcal{B}$ are Carat\'eodory functions that satisfy some conditions, and $\mathcal{G}(x,s,\eta)$ is a nonlinear term satisfying only the growth condition on $\eta$. our method consists in transforming this Dirichlet boundary value problem with nonlinearity into a new one governed by a Hammerstein equation. Then, we use a topological degree theory developed by Berkovits for operators of generalized monotone type.