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

TL;DR

This paper establishes the existence of multiple small amplitude time periodic solutions for resonant Klein-Gordon equations on the sphere, using variational methods and Strichartz estimates, relevant as toy models in General Relativity.

Contribution

It introduces a novel variational Lyapunov-Schmidt approach to find time periodic solutions in a resonant setting on $\

Findings

Existence of Cantor families of solutions

Solutions for quadratic, cubic, and quintic nonlinearities

Use of Strichartz estimates for compactness

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 $\mathbb{S}^3$.