## Communications in Mathematical Analysis

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

### Nonlinear Eigenvalue Problem for the p-Laplacian

Najib Tsouli, Omar Chakrone, Omar Darhouche, and Mostafa Rahmani

#### 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

**Subjects**

Primary: 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 47H11: Degree theory [See also 55M25, 58C30]

**Keywords**

Nonlinear eigenvalue problem p-Laplacian Ljusternik-Schnirelman theory Variational methods

#### 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