{"title":"Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit","authors":"Zhenping Feng, Zhuoran Du","doi":"10.12775/tmna.2022.016","DOIUrl":null,"url":null,"abstract":"We are concerned with periodic solutions of the fractional Laplace equation\n\\begin{equation*}\n{(-\\partial_{xx})^s}u(x)+F'(u(x))=0 \\quad \\mbox{in }\\mathbb{R},\n\\end{equation*}\nwhere $0< s< 1$. The smooth function $F$ is a double-well potential with wells at\n$+1$ and $-1$. We show that the value of least positive period is\n$2{\\pi}\\times({1}/{-F''(0)})^{{1}/({2s})}$.\n The axial symmetry of odd periodic solutions is obtained by moving plane method.\nWe also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution\n of the same equation as periods $T\\rightarrow+\\infty$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topological Methods in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.12775/tmna.2022.016","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 $T\rightarrow+\infty$.
期刊介绍:
Topological Methods in Nonlinear Analysis (TMNA) publishes research and survey papers on a wide range of nonlinear analysis, giving preference to those that employ topological methods. Papers in topology that are of interest in the treatment of nonlinear problems may also be included.