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EnglishWe present recent existence results of periodic solutions for completely resonant nonlinear wave equations in which both `small divisor' difficulties and infinite-dimensional bifurcation phenomena occur. These results can be seen as generalizations of the classical finite-dimensional resonant center theorems of Weinstein-Moser and Fadell-Rabinowitz. The proofs are based on variational bifurcation theory: after a Lyapunov-Schmidt reduction, the small divisor problem in the range equation is overcome with a Nash-Moser implicit function theorem for a Cantor set of non-resonant parameters. Next, the infinite-dimensional bifurcation equation, variational in nature, possesses minimax mountain-pass critical points. The big difficulty is to ensure that they are not in the `Cantor gaps'. This is proved under weak non-degeneracy conditions. Finally, we also discuss the existence of forced vibrations with rational frequency. This problem requires variational methods of a completely different nature, such as constrained minimization and a priori estimates derivable from variational inequalities.
Variational Methods for Hamiltonian PDEs / Berti, Massimiliano. - (2008), pp. 391-420.
| Titolo: | Variational Methods for Hamiltonian PDEs |
| Autori: | Berti, Massimiliano |
| Serie: | |
| Titolo del libro: | Hamiltonian dynamical systems and applications |
| Data di pubblicazione: | 2008 |
| Pagina iniziale: | 391 |
| Pagina finale: | 420 |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/978-1-4020-6964-2_16 |
| Appare nelle tipologie: | 2.1 Book chapter |
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http://hdl.handle.net/20.500.11767/15126
