# Multiple solutions for the $p-$laplace operator with critical growth - Mathematics > Analysis of PDEs

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Abstract: In this note we show the existence of at least three nontrivial solutions tothe following quasilinear elliptic equation $-\Delta p u = |u|^{p^*-2}u +\lambda fx,u$ in a smooth bounded domain $\Omega$ of $\R^N$ with homogeneousDirichlet boundary conditions on $\partial\Omega$, where $p^*=Np-N-p$ is thecritical Sobolev exponent and $\Delta p u =div| abla u|^{p-2} abla u$ isthe $p-$laplacian. The proof is based on variational arguments and theclassical concentrated compactness method.

Author: Pablo L. De Nápoli, Julián Fernández Bonder, Analía Silva

Source: https://arxiv.org/