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.

References in corpus (1)

• Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems