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 φ=φ1+φ2 with φ1 locally Lipschitz and φ2 proper, convex, lower semicontinuous.