Constant Sign and Nodal Solutions for Problems with the <inline-formula> <graphic file="1687-2770-2009-820237-i1.gif"/></inline-formula>-Laplacian and a Nonsmooth Potential Using Variational Techniques

Abstract

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.