Cantor families of periodic solutions for completely resonant nonlinear wave equations

Abstract

We prove the existence of small amplitude, ($2\pi /\omega$)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency $\omega$ belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite-dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity