Department of Mathematics, School of Applied Mathematics and Natural Science, National Technical University, Zografou Campus, Athena 15780, Greece
In this paper we consider nonlinear periodic systems driven by the one-dimensional $p$-Laplacian and having a non smooth locally Lipschitz potential. Using a variational approach based on the nonsmooth Critical Point Theory, we establish the existence of a solution. We also prove a multiplicity result based on a nonsmooth extension of the result of Brezis-Nirenberg  due to Kandilakis-Kourogenis-Papageorgiou .
Keywords. Locally linking Lipschitz function, generalized subdifferential, nonsmooth critical point theory, nonsmooth Palais-Smale condition, $p$-Laplacian, periodic system.