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arXiv 2202.01409math.AP
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高阶分数阶Laplacian方程解的分类

Classification of solutions to equations involving Higher-order fractional Laplacian

Zhuoran Du, Zhenping Feng, Jiaqi Hu, Yuan Li

备注：
19 pages. arXiv admin note: text overlap with arXiv:2201.00917
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摘要：

本文研究了高阶分数Lapalacian方程\begin{equation*} \left\{\begin{aligned} &(-Δ)^{p+{\fracα{2}}}u(x)=u_+^γ~~ \mbox{ in }\mathbb{R}^n,\\ &\int_{\mathbb{R}^n}u_+^γdx<+\infty, \end{aligned}\right. \end{equation*}，其中$p\geq 1$为整数，$0<\alp<2$，$n> 2p+α$和$γ\in (1,\frac{n}{n-2p-\alp})$ .对于在无穷远处满足一定增长性的任意非常数经典解，我们建立了一个整表示公式。利用积分形式的移动平面法证明了这些解关于$\R^n$中的某点径向对称，并且在径向上单调递减.

英文摘要：

In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-Δ)^{p+{\fracα{2}}}u(x)=u_+^γ~~ \mbox{ in }\mathbb{R}^n,\\ &\int_{\mathbb{R}^n}u_+^γdx<+\infty, \end{aligned}\right. \end{equation*} where $p\geq 1$ is an integer, $0<\alp<2$, $n> 2p+α$ and $γ\in (1,\frac{n}{n-2p-\alp})$. We establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this we prove that these solutions are radially symmetric about some point in $\R^n$ and monotone decreasing in the radial direction via method of moving planes in integral forms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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