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Cantor families of periodic solutions for completely resonant nonlinear wave equations. (English) Zbl 1103.35077
Summary: 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.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B10 Periodic solutions to PDEs
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