Solutions for nonlinear variational inequalities with a nonsmooth potential

Abstract

First we examine a resonant variational inequality driven by the $p$-Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving the $p$-Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the form $\varphi ={\varphi }_1+{\varphi }_2$ with ${\varphi }_1$ locally Lipschitz and ${\varphi }_2$ proper, convex, lower semicontinuous.