Multiplicity of periodic solutions of nonlinear wave equations

We prove multiplicity of small amplitude periodic solutions, with fixed frequency ω, of completely resonant wave equations with general nonlinearities. As ω→1 the number Nω of 2π/ω-periodic solutions u1,...,un,...,uNω tends to +∞. The minimal period of the nth solution un is 2π/nω. The proofs are based on the variational Lyapunov–Schmidt reduction (Comm. Math. Phys., to appear) and minimax arguments.