Isolation and simplicity for the first eigenvalue of the $p$-Laplacian with a nonlinear boundary condition

Abstract

We prove the simplicity and isolation of the first eigenvalue for the problem ${\Delta }_{p}u={\left|u\right|}^{p-2}u$ in a bounded smooth domain $\Omega \subset {ℝ}^N$, with a nonlinear boundary condition given by ${\left|\nabla u\right|}^{p-2}\partial u/\partial v=\lambda {\left|u\right|}^{p-2}u$ on the boundary of the domain.