Mathematics

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