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 $$\begin{cases} \Delta u=0 \qquad &\text{in $\Omega$,}\\ u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =|u|^{p-2}u\qquad &\text{on $\Gamma_1$,} \end{cases} $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\partial\Omega=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$, $\Gamma_1$ 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.