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

TL;DR

This paper proves the existence of at least three solutions for a quasilinear elliptic equation involving the p-Laplacian with critical Sobolev exponent, using variational methods and concentration compactness.

Contribution

It establishes multiple solutions for a critical p-Laplacian problem, extending previous results with new variational techniques.

Findings

Existence of at least three solutions proven.

Application of variational and concentration compactness methods.

Addresses critical growth nonlinearities in elliptic equations.

Abstract

In this note we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-\Delta_p u = |u|^{p^*-2}u + \lambda f(x,u)$ in a smooth bounded domain $\Omega$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partial\Omega$, where $p^*=Np/(N-p)$ 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.