Periodic solutions of an infinite-dimensional Hamiltonian system

We consider the existence of periodic solutions of the unbounded Hamiltonian system(H)∂tu-Δxu=Hv(t,x,u,v)-∂tv-Δxv=Hu(t,x,u,v)for(t,x)∈R×Ω.Unlike previous work, in our case the energy functional does not satisfy the Palais-Smale conditions.