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.

Keywords

• laplace equation
• laplace-beltrami operator
• existence and multiplicity for nontrivial solutions
• wentzell boundary conditions
• ventcel boundary conditions
• mountain pass theorem