Degree theoretic methods in the study of nonlinear periodic problems with nonsmooth potentials

Abstract

In this paper we study periodic problems driven by the scalar ordinary $p$-Laplacian and with a nonsmooth potential. Using degree theoretic methods based on a fixed-point index for nonconvex-valued multifunctions, we prove two existence theorems. In the first we employ nonuniform nonresonance conditions between two successive eigenvalues of the negative $p$-Laplacian with periodic boundary conditions. In the second we use Landesman-Lazer conditions.