Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit - Feng, Zhenping; Du, Zhuoran (Topological Methods in Nonlinear Analysis, 2022)
We are concerned with periodic solutions of the fractional Laplace equation\begin{equation*}{(-\partial_{xx})^s}u(x)+F'(u(x))=0 \quad 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$2π×({1}/{-F''(0)})^{{1}/({2s})}$. 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 $T→+∞$.