Nonlinear Eigenvalue Problem for the p-Laplacian

Abstract

This article is devoted to the study of the nonlinear eigenvalue problem $$-\Delta_{p} u \quad=\quad \lambda |u|^{p-2}u \;\mbox{in}\; \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\quad+\quad\beta |u|^{p-2}u=\lambda |u|^{p-2}u \;\mbox{on}\quad\partial\Omega,$$ where $ν$ denotes the unit exterior normal, $1 \lt p \lt ∞ \,\mathrm {and} ∆_{p}u = div(|∇u|^{p−2}∇u)$ denotes the p-laplacian. $Ω ⊂ \mathbb{R}^{N}$ is a bounded domain with smooth boundary where $N ≥ 2$ and $β \in L^{∞}(∂Ω) \,\mathrm{with}\, β^{−} := \mathrm{inf}_{x∈∂Ω}β(x) > 0$. Using Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of positive eigenvalues and the first eigenvalue is simple and isolated. Moreover, we will prove that the second eigenvalue coincides with the second variational eigenvalue obtained via the Ljusternik-Schnirelman theory.