Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit

We are concerned with periodic solutions of the fractional Laplace equation \begin{equation*} {(-\partial_{xx})^s}u(x)+F'(u(x))=0 \quad \mbox{in }\mathbb{R}, \end{equation*} where $0< s< 1$. The smooth function $F$ is a double-well potential with wells at $+1$ and $-1$. We show that the value of least positive period is $2{\pi}\times({1}/{-F''(0)})^{{1}/({2s})}$. The axial symmetry of odd periodic solutions is obtained by moving plane method. We also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution of the same equation as periods $T\rightarrow+\infty$.