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

IF 0.7 4区 数学 Q2 MATHEMATICS
Zhenping Feng, Zhuoran Du
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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 $T\rightarrow+\infty$.
分数阶拉普拉斯方程的周期解:最小周期,轴对称和极限
我们关注分数阶拉普拉斯方程的周期解\ begin{equation*}{(-\partial_{xx})^s}u(x)+F’(u(x。光滑函数$F$是具有$+1$和$-1$阱的双阱势。我们证明了最小正周期的值是$2{\pi}\times({1}/{-F‘'(0)})^{1}/({2s})}$。利用移动平面法得到了奇周期解的轴对称性。我们还证明了奇周期解$u_{T}(x)$收敛于与周期$T\rightarrow+\infty$相同的方程的层解。
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来源期刊
CiteScore
1.00
自引率
0.00%
发文量
57
审稿时长
>12 weeks
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.
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