Abstract
We are concerned with the existence of time periodic solutions of nonlinear wave equations on n-dimensional spheres. The existence of nontrivial periodic solution in case 0 is a solution is proved. the existence of multiple, especially infinitely many, time periodic solutions is established for several classes f nonlinear terms which satisfy some symmetry properties such as time translation invariance or oddness. Furhermore, this paper also studies the effect of perturbations which are not small and which destroy the symmetry, and shows how multiple solutions persist despite the nonsymmetric perturbations if the growth of nonlinear term at infinity is suitable controlled.
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