Periodic solutions for critical fractional problems

We deal with the existence of $2\pi$-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 $s\in (0,1)$, $N \geq 4s$,$m\geq 0$,$2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $W(x)$ is a positive continuous function, and $f(x,u)$ is a superlinear $2\pi$-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 $(-\pi ,\pi {)}^N\times (0,\mathrm{\infty })$, 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.