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

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} .