Nonlinear Periodic Systems with the $p$-Laplacian: Existence and Multiplicity Results

Abstract

We study second-order nonlinear periodic systems driven by the vector $p$-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).