Periodic solutions for critical fractional problems

Abstract

We deal with the existence of $2\pi$-periodic solutions to the following non-local critical problem

$$\begin{aligned} \left\{ \begin{array}{ll} [(-\Delta _{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &{}\quad \text{ in } \; (-\pi ,\pi )^{N} \\ u(x+2\pi e_{i})=u(x) &{}\quad \text{ for } \text{ all } \; x \in \mathbb {R}^{N}, \quad i=1, \dots , N, \end{array} \right. \end{aligned}$$

where $s\in (0,1)$, $N \ge 4s$, $m\ge 0$, $2^{*}_{s}=\frac{2N}{N-2s}$ 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 $(-\,\pi ,\pi )^{N}\times (0, \infty )$, 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.