Multiple solutions for the $p-$laplace operator with critical growth

Abstract

In this note we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-Δ_p u = |u|^{p^*-2}u + λf(x,u)$ in a smooth bounded domain $Ω$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partialΩ$, where $p^*=Np/(N-p)$ is the critical Sobolev exponent and $Δ_p u =div(|\nabla u|^{p-2}\nabla u)$ is the $p-$laplacian. The proof is based on variational arguments and the classical concentrated compactness method.