Free Vibrations for a Nonlinear Wave Equation and a Theorem of P. Rabinowitz* zy Dedicated to Philip Hartman on his 65th birthday HAIM BREZIS, zyxw Universitk Paris 6 and Courant Institute JEAN-MICHEL CORON, IRIA LOUIS NIRENBERG Courant Institute Abstract A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation utt zyxwvu - zyx ux, + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u+m as zyx Iu(+w. The proof uses a modified form (PS), of the Palais-Smale condition (PS). 0. Introduction Let g : R-+R be a continuous nondecreasing function such that g(0) = 0. Set G(t) = g(s) ds. We seek a nontrivial solution of the equation zy 6' under the boundary conditions zyxwv (2) u(0, t) = zyxw U(T, t) = zyxw 0 and periodicity condition (3) u(x, t+27r)= u(x, t). * The first and third authors were partially supported by the Army Research Grant No. DAH 29-78-6-0127. The third author was also partially supported by the National Science Foundation Grant No. NSF-MCS-79-00813. Reproductions in whole or in part is permitted for any purposes of the United States Government. Communications on Pure and Applied Mathematics, Vol. XXXIII, 667-689 (1980) @ 1980 John Wiley & sons, Inc. 00 10-3640/80/0033-0667$01.80