Nontrivial solutions for a class of semilinear elliptic equations with a nonlinear Goldstein-Wentzell boundary condition

Abstract

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} -Δu=f(u) \qquad &\text{in $Ω$,}\\ \phantom{-}u=0 &\text{on $Γ_0$,}\\ -Δ_Γu +\partial_νu =g(u)\qquad &\text{on $Γ_1$,} \end{cases} $$ where $Ω$ is a bounded open domain of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $Γ=\partialΩ$, with $Γ=Γ_0\cupΓ_1$, $Γ_0\capΓ_1=\emptyset$, $Γ_1$ being nonempty and relatively open on $Γ$, $\mathcal{H}^{N-1}(Γ_0)>0$. The terms $f$ and $g$ are subcritical with respect to Sobolev embeddings, respectively in $Ω$ and on $\partialΩ$. We prove that, under suitable assumptions, the problem admits nontrivial solutions at the depth of the potential well energy level, which is the minimum energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.