TY - JOUR
T1 - Cantor families of periodic solutions for completely resonant nonlinear wave equations
JF - Duke Math. J. 134 (2006) 359-419
Y1 - 2006
A1 - Massimiliano Berti
A1 - Philippe Bolle
AB - We prove the existence of small amplitude, $2\\\\pi \\\\slash \\\\om$-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.
UR - http://hdl.handle.net/1963/2161
U1 - 2083
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -