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

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

2008-08-22

arxiv.org/abs/0808.3143v2

Nonlinear Anal. TMA., 71 (2009), 6283--6289.

Mathematics

Analysis of PDEs

Results improved, hypotheses removed

Scientific paper

10.1016/j.na.2009.06.036

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.

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