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

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arxiv 2308.05678 v1 pith:HLMQRFS5 submitted 2023-08-10 math.AP

Massimiliano Berti , Beatrice Langella , Diego Silimbani This is my paper
classification math.AP
keywords solutionsklein-gordonmathbbcompletelyequationsobtainedperiodicresonant
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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 $\mathbb{S}^3$.

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Cited by 1 Pith paper

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  1. Time-periodic solutions of the conformally invariant wave equation on the Einstein cylinder

    gr-qc 2025-08 conditional novelty 3.0 of 10

    For the cubic conformal wave equation on the Einstein cylinder, time-periodic solutions form complex trunk-branch bifurcation networks, and a resummed Poincaré-Lindstedt series is claimed to encode these structures.

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