Periodic solutions of fractional Laplace equations: least period, axial symmetry and limit. (English) Zbl 1519.34033

This paper is concerned with the following fractional Laplace equation \[ (-\partial_{xx})^{s}u(x) + F'(u(x)) = 0, \quad x\in \mathbb{R}, \] where \((-\partial_{xx})^{s}\), \(0<s<1\), denotes the usual fractional Laplacian; \(F\) is a smooth double-well potential satisfying \[ \left\{ \begin{array}{ll} F(1)=F(-1)=0< F(u)\text{ for all }-1<u<1;\\ F'(1)=F'(-1)=0 \end{array} \right. \] and \(F\) is nondecreasing in \((-1,0)\) and nonincreasing in \((0,1)\). The authors show that the value of least positive period is \(\frac{2\pi}{(-F''(0))^{1/(25)}}\). The axial symmetry of odd periodic solutions is obtained by moving plane method. The authors also prove that odd periodic solutions \(u_T(x)\) converge to a layer solution of the same equation as periods \(T\rightarrow +\infty\).
Reviewer: Minghe Pei (Jilin)


34C25 Periodic solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
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