Duke Mathematical Journal

Cantor families of periodic solutions for completely resonant nonlinear wave equations

Massimiliano Berti and Philippe Bolle

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We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω 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

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Duke Math. J., Volume 134, Number 2 (2006), 359-419.

First available in Project Euclid: 8 August 2006

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Zentralblatt MATH identifier

Primary: 35L05: Wave equation 37K50: Bifurcation problems
Secondary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Berti, Massimiliano; Bolle, Philippe. Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math. J. 134 (2006), no. 2, 359--419. doi:10.1215/S0012-7094-06-13424-5. https://projecteuclid.org/euclid.dmj/1155045505

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