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Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz
Author(s) -
Brézis Haïm,
Coron JeanMichel,
Nirenberg Louis
Publication year - 1980
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160330507
Subject(s) - mathematics , dirichlet boundary condition , dirichlet distribution , nonlinear system , boundary value problem , interval (graph theory) , mathematical analysis , wave equation , vibration , string (physics) , boundary (topology) , mountain pass theorem , pure mathematics , mathematical physics , combinatorics , physics , quantum mechanics
A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation u u ‐ u xx + g( u ) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g( u )/ u →∞ as | u |→∞. The proof uses a modified form (PS) c of the Palais‐Smale condition (PS).