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

Mohamed El Ouaarabi, Chakir Allalou, Said Melliani, Abderrazak Kassidi


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.

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