## Differential and Integral Equations

- Differential Integral Equations
- Volume 19, Number 9 (2006), 1035-1046.

### An optimization problem for the first Steklov eigenvalue of a nonlinear problem

Leandro Del Pezzo, Julián Fernández Bonder, and Julio D. Rossi

#### Abstract

In this paper we study the first (nonlinear) Steklov eigenvalue, $\lambda$, of the following problem: $-\Delta_{p}u + |u|^{p-2}u + \alpha\phi|u|^{p-2}u = 0$ in a bounded smooth domain $\Omega$ with $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu} = \lambda |u|^{p-2}u$ on the boundary $ \partial\Omega$. We analyze the dependence of this first eigenvalue with respect to the weight $\phi$ and with respect to the parameter $\alpha$. We prove that for fixed $\alpha$ there exists an optimal $\phi_\alpha$ that minimizes $\lambda$ in the class of uniformly bounded measurable functions with fixed integral. Next, we study the limit of these minima as the parameter $\alpha$ goes to infinity and we find that the limit is the first Steklov eigenvalue in the domain with a hole where the eigenfunctions vanish.

#### Article information

**Source**

Differential Integral Equations, Volume 19, Number 9 (2006), 1035-1046.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050331

**Mathematical Reviews number (MathSciNet)**

MR2262095

**Zentralblatt MATH identifier**

1212.35339

**Subjects**

Primary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory

Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 47J30: Variational methods [See also 58Exx] 49R50

#### Citation

Del Pezzo, Leandro; Fernández Bonder, Julián; Rossi, Julio D. An optimization problem for the first Steklov eigenvalue of a nonlinear problem. Differential Integral Equations 19 (2006), no. 9, 1035--1046. https://projecteuclid.org/euclid.die/1356050331