Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition

Abstract

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\{\begin{array}{ll}\Delta u=0& \text{in Ω,}\\ u=0& \text{on Γ0,}\\ -{\Delta }_{\Gamma }u+{\partial }_{\nu }u=|u{|}^{p-2}u& \text{on Γ1,}\end{array}$$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$ ($N\ge 2$) with $C^1$ boundary $\partial \Omega ={\Gamma }_0\cup {\Gamma }_1$, ${\Gamma }_0\cap {\Gamma }_1=\varnothing$, ${\Gamma }_1$ being nonempty and relatively open on $\Gamma$, ${\mathcal{H}}^{N-1}({\Gamma }_0)>0$ and $p>2$ being subcritical with respect to Sobolev embedding on $\partial \Omega$. We prove that the problem admits nontrivial solutions at the potential--well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.