Cantor families of periodic solutions for completely resonant wave equations

Abstract

We present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods.

References

1. Ambrosetti A Rabinowitz P Dual variational methods incritical point theory and applicationsJFunc Anal 1973 14349381. doi:10.1016/0022‐1236(73)90051‐7
2. Baldi P Berti M Periodic solutions of waveequations for asymptotically full measure sets of frequenciesRend Mat Acc Naz Lincei 2006 17257277
3. Bambusi D Paleari S Families of periodic solutionsof resonant PDEsJ Nonlinear Sci 2001 116987. doi:10.1007/s003320010010
4. Berti M Bolle P Periodic solutions of nonlinearwave equations with general nonlinearitiesComm Math Phys 2003 243315328. doi:10.1007/s00220‐003‐0972‐8
5. Berti M Bolle P Multiplicity of periodic solutionsof nonlinear wave equationsNonlinear Anal 2004 5610111046. doi:10.1016/j.na.2003.11.001
6. Berti M Bolle P Cantor families of periodicsolutions for completely resonant nonlinear wave equationsDuke Math J 2006 134359419. doi:10.1215/S0012‐7094‐06‐13424‐5
7. Berti M Bolle P Cantor families of periodicsolutions for wave equations via a variational principleAdvances in Mathematics 2008 21716711727. doi:10.1016/j.aim.2007.11.004
8. Berti M Bolle P Cantor families of periodicsolutions of wave equations with C^k nonlinearitiesNonlinear Differential Equations and Applications(in press)
9. Bourgain J Periodicsolutions of nonlinear wave equationsIn: Harmonic Analysis and Partial Differential Equations. Chicago Lecturesin MathChicagoUniv Chicago Press 1999 6997
10. Craig W Wayne E Newton's method and periodicsolutions of nonlinear wave equationCommPure Appl Math 1993 4614091498. doi:10.1002/cpa.3160461102
11. Fadell E R Rabinowitz P Generalized cohomological indextheories for Lie group actions with an application to bifurcationquestions for Hamiltonian systemsInv Math 1978 45139174. doi:10.1007/BF01390270
12. Gentile G Mastropietro V Procesi M Periodic solutions for completely resonant nonlinear waveequationsComm Math Phys 2005 256437490. doi:10.1007/s00220‐004‐1255‐8
13. Lidskij B V Shulman E Periodic solutions of the equation u_tt - u_xx + u³ = 0Funct Anal Appl 1988 22332333. doi:10.1007/BF01077432
14. Moser J Periodicorbits near an equilibrium and a theorem by Alan WeinsteinComm Pure Appl Math 1976 29724747. doi:10.1002/cpa.3160290613
15. Struwe M Theexistence of surfaces of constant mean curvature with free boundariesActa Math 1988 1601964. doi:10.1007/BF02392272
16. Weinstein A Normalmodes for nonlinear Hamiltonian systemsInv Math 1973 204757. doi:10.1007/BF01405263
17. Yuan X Quasi-periodicsolutions of completely resonant nonlinear wave equationsJ Diff Equations 2006 230213274. doi:10.1016/j.jde.2005.12.012