Cantor families of periodic solutions for completely resonant nonlinear wave equations

M. Berti; P. Bolle

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Cantor families of periodic solutions for completely resonant nonlinear wave equations

M. Berti·P. Bolle

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Abstract

We prove existence of small amplitude, $2π\slash \om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \om $ belonging to a Cantor-like set of positive 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 {\it 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 to deal also with nonlinearities which are not odd and with finite spatial regularity.

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BibTeX RIS

M. Berti, P. Bolle. 2004-10-29. Cantor families of periodic solutions for completely resonant nonlinear wave equations. https://arxiv.org/abs/math/0410618

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