Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems

Abstract

Using variational methods we establish existence and multiplicity of positive solutions for the following class of quasilinear problems $$ -\Delta_{p}u + \lambda V(x)|u|^{p-2}u= \mu |u|^{p-2}u+|u|^{p^{*}-2}u \quad\text{in } {\mathbb R}^{N} $$ where $\Delta_{p}u$ is the $p$-Laplacian operator, $2 \leq p < N$, $p^{*}={pN}/(N-p)$, $\lambda, \mu \in (0, \infty)$ and $V\colon {\mathbb R}^{N} \rightarrow {\mathbb R}$ is a continuous function verifying some hypothesis.