Periodic solutions for a pseudo-relativistic Schrödinger equation

We study the existence and the regularity of non trivial T-periodic solutions to the following nonlinear pseudo-relativistic Schrödinger equation (√- Δx+m2-m)u(x)=f(x,u(x)) in (0,T) N where T>0, m is a non negative real number, f is a regular function satisfying the Ambrosetti-Rabinowitz condition and a polynomial growth at rate p for some 1<2#-1. We investigate such problem using critical point theory after transforming it to an elliptic equation in the infinite half-cylinder (0,T) N×(0,∞) with a nonlinear Neumann boundary condition. By passing to the limit as m→0 in (0.1) we also prove the existence of a non trivial T-periodic weak solution to (0.1) with m=0.