  In this paper we study second order non-linear periodic systems driven by the
ordinary vector $p$-Laplacian with a non-smooth, locally Lipschitz potential
function. Our approach is variational and it is based on the non-smooth
critical point theory. We prove existence and multiplicity results under
general growth conditions on the potential function. Then we establish the
existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to
be the first such result (even for smooth problems) for systems monitored by
the $p$-Laplacian. In the last section of the paper we examine the scalar
\hbox{non-linear} and semilinear problem. Our approach uses a generalized
Landesman--Lazer type condition which generalizes previous ones used in the
literature. Also for the semilinear case the problem is at resonance at any
eigenvalue.
