Periodic solutions of nonlinear wave equations with non-monotone forcing terms

We present new existence and regularity results 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. The infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness.We solve it via a minimization argument and a-priori estimate methods related to the regularity theory of Rabinowitz.