We deal with the existence of 2π-periodic solutions to the following
non-local critical problem \begin{equation*} \left\{\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in}
(-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N},
\quad i=1, \dots, N, \end{array} \right. \end{equation*} where s∈(0,1), N≥4s, m≥0, 2s∗=N−2s2N is the fractional critical
Sobolev exponent, W(x) is a positive continuous function, and f(x,u) is a
superlinear 2π-periodic (in x) continuous function with subcritical
growth. When m>0, the existence of a nonconstant periodic solution is
obtained by applying the Linking Theorem, after transforming the above
non-local problem into a degenerate elliptic problem in the half-cylinder
(−π,π)N×(0,∞), with a nonlinear Neumann boundary
condition, through a suitable variant of the extension method in periodic
setting. We also consider the case m=0 by using a careful procedure of limit.
As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018