Open Access Research Article

Constant Sign and Nodal Solutions for Problems with the -Laplacian and a Nonsmooth Potential Using Variational Techniques

RaviP Agarwal1*, MichaelE Filippakis2, Donal O'Regan3 and NikolaosS Papageorgiou4

Author Affiliations

1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA

2 Department of Mathematics, Hellenic Army Academy, Vari, 16673 Athens, Greece

3 Department of Mathematics, National University of Ireland, Galway, Ireland

4 Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece

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Boundary Value Problems 2009, 2009:820237 doi:10.1155/2009/820237

Published: 3 March 2009

Abstract

We consider a nonlinear elliptic equation driven by the -Laplacian with a nonsmooth potential (hemivariational inequality) and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing (nodal solution). Our hypotheses on the nonsmooth potential incorporate in our framework of analysis the so-called asymptotically -linear problems.