A new and extended variational characterization of the Fučík Spectrum with application to nonresonance and resonance problems

Abstract

We consider the boundary value problem $$\begin{aligned} -\Delta u= & {} \alpha u^{+}-\beta u^{-} \quad \hbox { in } \Omega ,\\ u= & {} 0 \quad \hbox { on }\partial \Omega , \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^N,u^{\pm }:=\max \{\pm \,u,0\}$$ , and $$(\alpha ,\beta )\in {\mathbb {R}}^2$$ . When this problem has a nontrivial solution, then $$(\alpha ,\beta )$$ is an element of the Fucik Spectrum, $$\Sigma $$ . Our main result is to extend the variational characterization, due to Castro and Chang in 2010, of several curves in $$\Sigma $$ . As an application we prove existence theorems for nonresonance and resonance problems relative to these extended spectral curves.