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 $-{\Delta }_{p}u=|u{|}^{p^{\ast }-2}u+\lambda f(x,u)$ in a smooth bounded domain $\Omega$ of ${\mathbb{R}}^N$ with homogeneous Dirichlet boundary conditions on $\partial \Omega$, where $p^{\ast }=Np/(N-p)$ is the critical Sobolev exponent and ${\Delta }_{p}u=div(|\nabla u{|}^{p-2}\nabla u)$ is the $p-$laplacian. The proof is based on variational arguments and the classical concentrated compactness method.

arXiv commentResults improved, hypotheses removed

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