Time periodic solutions of completely resonant Klein-Gordon equations on ${\mathbb{S}}^3$

Abstract

We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere ${\mathbb{S}}^3$ with quadratic, cubic and quintic nonlinearity, regarded as toy models in General Relativity. The solutions are obtained by a variational Lyapunov- Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on ${\mathbb{S}}^3$.

53 pages