Periodic solutions for a pseudo-relativistic Schrödinger equation

We study the existence and the regularity of non trivial TT-periodic solutions to the following nonlinear pseudo-relativistic Schrödinger equation equation(0.1)(−Δx+m2−m)u(x)=f(x,u(x))  in  (0,T)N where T>0T>0, mm is a non negative real number, ff is a regular function satisfying the Ambrosetti–Rabinowitz condition and a polynomial growth at rate pp for some 1