## Differential and Integral Equations

- Differential Integral Equations
- Volume 24, Number 11/12 (2011), 1047-1062.

### Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth

Claudianor O. Alves, Sérgio H M. Soares, and Luciana Rôze de Freitas

#### Abstract

This paper is concerned with the existence of solutions for the quasilinear problem \[ \left \{ \begin{array}{lll} -div( \left|\nabla u\right| ^{N-2}\nabla u ) + \left| u\right|^{N-2}u= a(x)g(u)\quad \mbox{in}\ \Omega\\ \quad u=0\quad \mbox{on}\ \partial\Omega , \end{array} \right. \] where $\Omega \subset \mathbb{R}^N$ ($N\geq 2$) is an exterior domain; that is, $\Omega = \mathbb{R}^{N} \setminus \omega $, where $w \subset \mathbb{R}^{N}$ is a bounded domain, the nonlinearity $g(u)$ has an exponential critical growth at infinity and $a(x)$ is a continuous function and changes sign in $\Omega$. A variational method is applied to establish the existence of a nontrivial solution for the above problem.

#### Article information

**Source**

Differential Integral Equations, Volume 24, Number 11/12 (2011), 1047-1062.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012875

**Mathematical Reviews number (MathSciNet)**

MR2866010

**Zentralblatt MATH identifier**

1249.35085

**Subjects**

Primary: 35J35: Variational methods for higher-order elliptic equations 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations

#### Citation

Alves, Claudianor O.; de Freitas, Luciana Rôze; Soares, Sérgio H M. Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth. Differential Integral Equations 24 (2011), no. 11/12, 1047--1062. https://projecteuclid.org/euclid.die/1356012875