  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.
