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 (0.1) ( − Δ x + m 2 − 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 < 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 × ( 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 . [ABSTRACT FROM AUTHOR]