Time-periodic solutions of completely resonant Klein–Gordon equations on

  • Massimiliano Berti

    International School for Advanced Studies (SISSA), Trieste, Italy
  • Beatrice Langella

    International School for Advanced Studies (SISSA), Trieste, Italy
  • Diego Silimbani

    International School for Advanced Studies (SISSA), Trieste, Italy
Time-periodic solutions of completely resonant Klein–Gordon equations on $\mathbb{S}^{3}$ cover
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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 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 .

Cite this article

Massimiliano Berti, Beatrice Langella, Diego Silimbani, Time-periodic solutions of completely resonant Klein–Gordon equations on . Ann. Inst. H. Poincaré C Anal. Non Linéaire (2024), published online first

DOI 10.4171/AIHPC/125