Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities
Abstract
We prove the existence of small amplitude periodic solutions, with strongly irrational frequency $ \om $ close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For $ \om $ close to one we prove the existence of a large number $ N_\om $ of $ 2 \pi \slash \om $periodic in time solutions $ u_1, ..., u_n, ..., u_N $: $ N_\om \to + \infty $ as $ \om \to 1 $. The minimal period of the $n$th solution $u_n $ is proved to be $2 \pi \slash n \om $. The proofs are based on a LyapunovSchmidt reduction and variational arguments.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2003
 DOI:
 10.1007/s0022000309728
 arXiv:
 arXiv:math/0211310
 Bibcode:
 2003CMaPh.243..315B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis;
 35L05;
 37K50;
 58E05
 EPrint:
 29 pages