Periodic solutions for the non-local operator (-Delta + m^2)^s - m^(2s) with m>=0

Abstract

By using variational methods we investigate the existence of T-periodic solutions to [(-Delta_x + m^2)^s -m^(2s)]u= f(x,u) in (0,T)^N u(x+Te_i)=u(x) for all x in R^N, i=1,...,N where s in (0,1), N>2s, T>0, m>=0 and f(x,u) is a continuous function, T-periodic in x, verifying the Ambrosetti-Rabinowitz condition and a polynomial growth at rate p in (1, (N+2s)/(N-2s)).