Nodal and multiple constant sign solutions for resonant [formula omitted]-Laplacian equations with a nonsmooth potential

Abstract

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p -Laplacian and with a nonsmooth potential (hemivariational inequality). Using a variational approach combined with suitable truncation techniques and the method of upper–lower solutions, we prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal. Our hypotheses on the nonsmooth potential allow resonance at infinity with respect to the principal eigenvalue λ 1 > 0 of ( − Δ p , W 0 1 , p ( Z ) ) .