Periodic solutions of nonlinear wave equations with general nonlinearities

We prove the existence of small amplitude periodic solutions, with strongly irrational frequency $\om$ close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For $\om$ close to one we prove the existence of a large number $N_\om$ of $2 \pi \slash \om$-periodic in time solutions $u_1, ..., u_n, ..., u_N$: $N_\om \to + \infty$ as $\om \to 1$. The minimal period of the $n$-th solution $u_n$ is proved to be $2 \pi \slash n \om$. The proofs are based on a Lyapunov-Schmidt reduction and variational arguments.