## Communications in Mathematical Analysis

### 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.

#### Article information

Source
Commun. Math. Anal., Volume 20, Number 1 (2017), 69-82.

Dates
First available in Project Euclid: 15 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.cma/1500084077

Mathematical Reviews number (MathSciNet)
MR3665390

Zentralblatt MATH identifier
1371.35136

#### Citation

Tsouli, Najib; Chakrone, Omar; Darhouche, Omar; Rahmani, Mostafa. Nonlinear Eigenvalue Problem for the p-Laplacian. Commun. Math. Anal. 20 (2017), no. 1, 69--82. https://projecteuclid.org/euclid.cma/1500084077