Abstract. In this paper we consider a periodic problem driven by the one dimensional $p$-Laplacian and with a discontinuous right hand side. We pass to a multivalued problem, by filling in the gaps at the discontinuity points. Then for the multivalued problem, using the nonsmooth critical point theory, we establish the existence of at least three distinct periodic solutions.
Keywords. Multiple solutions, periodic problem, one-dimensional $p$-Laplacian, discontinuous vector field, nonsmooth Palais-Smale condition, locally Lipschitz function, generalized subdifferential, critical point, Saddle Point Theorem, Ekeland variational principle.