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

TL;DR

This paper establishes the existence of T-periodic solutions for a class of non-local fractional operators using variational methods, under specific growth and periodicity conditions on the nonlinearity.

Contribution

It introduces a novel variational approach to find periodic solutions for fractional non-local operators with polynomial growth conditions.

Findings

Existence of T-periodic solutions proven

Conditions under which solutions exist are characterized

Extension of variational methods to non-local operators

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)).