Boundary Value Problems
Volume 2009 (2009), Article ID 820237, 32 pages
doi:10.1155/2009/820237
Constant Sign and Nodal Solutions for Problems with the p-Laplacian and a Nonsmooth Potential Using Variational Techniques
Ravi P. Agarwal1
, Michael E. Filippakis2
, Donal O'Regan3
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
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.