Constant Sign and Nodal Solutions for Problems with the
-Laplacian and a Nonsmooth Potential Using Variational Techniques !()
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
Boundary Value Problems 2009, 2009:820237 doi:10.1155/2009/820237Published: 3 March 2009
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.