Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities

By Massimiliano Berti, P. Bolle. 2002

86 citations recorded by the source corpus.

AbstractWe present a variational principle for small amplitude periodic solutions, with fixed frequency, of a completely resonant nonlinear wave equation. Existence and multiplicity results follow by min-max variational arguments.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/762a51785bd515f2867c0908946da1a9d6e22c03

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In Applied Mathematics

In Control and Systems Engineering

Citations in the corpus

Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions (2004, background): “The case of irrational periods, which in principle could provide a large measure of values, has been mostly studied only under strong Diophantine conditions (as the ones introduced in [2]) which essentially remove the small divisors problem, leaving in fact again a zero-measure set of values [30, 3,”
An existence and stability result for standing waves of nonlinear Schrödinger equations (2006, methodology): “To overcome this di ﬃ culty we develop an approach, inspired by [3], which consists in truncating the “rest” term of ˜ S λ outside of a ball centered at the origin and to showing that, as λ > 0 goes to zero, all Palais-Smale sequences at the mountain-pass level lie in this ball.”
Cantor families of periodic solutions for completely resonant nonlinear wave equations (2004, background): “The novelty of [4]-[5] was to solve the bifurcation equation via a variational principle at fixed frequency which, jointly with min-max arguments, enables to find solutions of (1) as critical points of the Lagrangian action functional.”
Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity (2005, methodology): “If, as in this paper, attention is restricted to the question of bifurcation from the trivial solution we find that the preceding observations make it difficult to see how to apply recent local variational methods, such as those developed by Berti & Bolle [5, 6] for problems with complete resonance.”