In this paper we consider a nonlinear periodic boundary value problem with a discontinuous forcing term. Assuming that the partial differential operator satisfies the Leray-Lions conditions, that the discontinuous perturbation term is locally of bounded variation and that there exist an upper solution $\phi$ and a lower solution $\psi$ such that $\psi \le \phi$, we prove the existence of a maximal and a minimal periodic solution within the order interval $[\psi, \phi]$ of an appropriately defined multivalued problem. Our approach is based on a Jordan-type decomposition for the discontinuous 

perturbation term due to Stuart [21] and on a fixed point theorem for monotone maps in order structures.