Time periodic solutions of completely resonant Klein-Gordon equations on \mathbbS³

2023/08/10 by Berti, Massimiliano, Langella, Beatrice, Silimbani, Diego
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.05678

Abstract

We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere \mathbbS³ 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 \mathbbS³.

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