2022 Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit
Zhenping Feng, Zhuoran Du
Topol. Methods Nonlinear Anal. 60(2): 633-651 (2022). DOI: 10.12775/TMNA.2022.016

Abstract

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 $T\rightarrow+\infty$.

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Zhenping Feng. Zhuoran Du. "Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit." Topol. Methods Nonlinear Anal. 60 (2) 633 - 651, 2022. https://doi.org/10.12775/TMNA.2022.016

Information

Published: 2022
First available in Project Euclid: 10 December 2022

MathSciNet: MR4563251
zbMATH: 1519.34033
Digital Object Identifier: 10.12775/TMNA.2022.016

Keywords: Axial symmetry , fractional Laplacian , layer solution , least positive period , periodic solutions

Rights: Copyright © 2022 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.60 • No. 2 • 2022
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