EXISTENCE RESULTS FOR A QUASILINEAR BOUNDARY VALUE PROBLEM INVESTIGATED VIA DEGREE THEORY

Abstract

In this article we prove the existence of at least one weak solution for the quasilinear problem
$$\left\{\begin{array}{ll} -\Delta_p u(x)=\lambda|u(x)|^{p-2}u(x)+h(x,u(x)) & \textrm{ in } \Omega,\\
u=0 & \textrm{ on } \partial \Omega \end{array}\right.$$
where $\Delta_p u:=\textrm{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$, $\Omega\subset \mathbb{R}^N$ is a non-empty bounded domain with Lipschitz boundary $(\Omega\in C^{0,1})$, $\lambda$ is a positive parameter and $h:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is a bounded Carath\'{e}odory function. The approach is fully based on the degree theory.