Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

The purpose of this paper is to study T-periodic solutions to { [(-Δx + m2)s - m2s]u = f(x; u) in (0; T)N u(x + Tei) = u(x) for all x Σ RN; i = 1; ... ;N (1) where s 2 (0; 1), N > 2s, T > 0, m > 0 and f(x; u) is a continuous function, T-periodic in x and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator (-Δx + m2)s can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder ST = (0, T)N × (0;∞). By using a variant of the Linking Theorem, we show that the extended problem in ST admits a nontrivial solution v(x; ξ) which is T-periodic in x. Moreover, by a procedure of limit as m ! 0, we prove the existence of a nontrivial solution to (1) with m = 0.