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

TL;DR

This paper investigates the existence and multiplicity of nontrivial solutions for a Laplace equation with a nonlinear boundary condition, establishing the presence of solutions at minimal energy and infinitely many at higher energies.

Contribution

It introduces new results on the existence and multiplicity of solutions for a Laplace equation with a nonlinear Goldstein-Wentzell boundary condition, including solutions at the minimal energy level.

Findings

Solutions exist at the potential-well depth energy level.

Infinitely many solutions are found at higher energy levels.

The problem admits nontrivial solutions under specified boundary conditions.

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.