Families of periodic orbits for some PDEs in higher dimensions

In this paper we consider the following nonlinear plate equations: u_tt +ΔΔu + mu = ψ(x, u) , ψ(x, u) = ±u^3 + O(u^5), ψ(−x,−u) = −ψ(x, u), (1) with Navier boundary conditions in a n–dimensional cube, here ψ is a C∞ function, and m is a positive parameter. For this equation we construct some Cantor families of periodic orbits. Our proof is very simple and is based on contraction mapping principle and on a suitable correspondence between Lyapunov Schmidt decomposition and averaging theory.