Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth

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.