Periodic solutions of nonlinear wave equations with general nonlinearities

Abstract

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 π\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 π\slash n \om $. The proofs are based on a Lyapunov-Schmidt reduction and variational arguments.

29 pages

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