Nonhomogeneous equations with critical exponential growth and lack of compactness

Abstract

We study the existence and multiplicity of positive solutions for the following class of quasilinear problems $-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,$ where $\epsilon$ is a positive parameter. We assume that $V:\mathbb{R}^N \to \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\to\mathbb{R}$ is a smooth reaction term with critical exponential growth.