  We prove existence and regularity of periodic in time solutions of completely
resonant nonlinear forced wave equations with Dirichlet boundary conditions for
a large class of non-monotone forcing terms. Our approach is based on a
variational Lyapunov-Schmidt reduction. It turns out that the infinite
dimensional bifurcation equation exhibits an intrinsic lack of compactness. We
solve it via a minimization argument and a-priori estimate methods inspired to
regularity theory of Rabinowitz
