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

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} Δu=0 \qquad &\text{in $Ω$,}\\ u=0 &\text{on $Γ_0$,}\\ -Δ_Γu +\partial_νu =|u|^{p-2}u\qquad &\text{on $Γ_1$,} \end{cases} $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\partialΩ=Γ_0\cupΓ_1$, $Γ_0\capΓ_1=\emptyset$, $Γ_1$ being nonempty and relatively open on $Γ$, $\mathcal{H}^{N-1}(Γ_0)>0$ and $p>2$ being subcritical with respect to Sobolev embedding on $\partialΩ$. 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.