Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz

Abstract

A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation uu ‐ uxx + 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). Copyright © 1980 Wiley Periodicals, Inc., A Wiley Company