An extended variational characterization of the Fučík Spectrum for the p-Laplace operator

Abstract

We consider the boundary value problem $$\begin{aligned} \begin{array}{rcl} -\Delta _p u &{} = &{} \alpha |u|^{p-2}u^+-\beta |u|^{p-2}u^- \text{ in } \Omega ,\\ u &{} = &{} 0 \text{ on } \partial \Omega , \end{array} \end{aligned}$$where $$\Delta _p u:=\nabla \cdot (|\nabla u|^{p-2}\nabla u)$$ for $$1<p<\infty $$, $$\Omega $$ is a smooth bounded domain in $$\mathbb {R}^N,u^{\pm }:=\max \{\pm u,0\}$$, and $$(\alpha ,\beta )\in \mathbb {R}^2$$. If this problem has a nontrivial weak solution, then $$(\alpha ,\beta )$$ is an element of the Fucik Spectrum, $$\Sigma \subset \mathbb {R}^2$$. Our main result is to provide a variational characterization for a class of curves in $$\Sigma $$.