Boundary Value Problems
Volume 2009 (2009), Article ID 820237, 32 pages
doi:10.1155/2009/820237
Research Article

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

Ravi P. Agarwal,1 Michael E. Filippakis,2 Donal O'Regan,3 and Nikolaos S. Papageorgiou4

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA
2Department of Mathematics, Hellenic Army Academy, Vari, 16673 Athens, Greece
3Department of Mathematics, National University of Ireland, Galway, Ireland
4Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece

Received 10 December 2008; Revised 21 January 2009; Accepted 23 January 2009

Academic Editor: Juan J. Nieto

Copyright © 2009 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a nonlinear elliptic equation driven by the p-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 p-linear problems.