Periodic solutions for a pseudo-relativistic Schrodinger equation

We study the existence and the regularity of non trivial T-periodic solutions to the following nonlinear pseudo-relativistic Schrodinger equation (root-Delta(x) + m(2) - m)u(x) - f(x, u(x)) in (0, T)(N) (0.1) where T > 0, m is a non negative real number, integral is a regular function satisfying the Ambrosetti-Rabinowitz condition and a polynomial growth at rate p for some 1 < p < 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) x (0, infinity) 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. (C) 2015 Elsevier Ltd. All rights reserved.