数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (6): 2589-2596.doi: 10.1007/s10473-023-0616-3

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ON A SUPER POLYHARMONIC PROPERTY OF A HIGHER-ORDER FRACTIONAL LAPLACIAN*

Meiqing XU   

  1. School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, China
  • 收稿日期:2022-04-29 修回日期:2023-06-02 发布日期:2023-12-08
  • 作者简介:Meiqing XU, E-mail: xmq157@sjtu.edu.cn
  • 基金资助:
    The work was supported by the NSFC (12031012, 11831003).

ON A SUPER POLYHARMONIC PROPERTY OF A HIGHER-ORDER FRACTIONAL LAPLACIAN*

Meiqing XU   

  1. School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, China
  • Received:2022-04-29 Revised:2023-06-02 Published:2023-12-08
  • About author:Meiqing XU, E-mail: xmq157@sjtu.edu.cn
  • Supported by:
    The work was supported by the NSFC (12031012, 11831003).

摘要: Let $0<\alpha<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider the positive solution $u$ of the PDE
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (-\Delta)^{\frac{\alpha}{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0.1) $
In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition $u\in\mathcal{L}_\alpha$, (0.1) possesses a super polyharmonic property $(-\Delta)^{k+\frac{\alpha}{2}}u\geq 0$ for $k=0,1,\cdots ,m-1$. In this paper, we show another kind of super polyharmonic property $(-\Delta)^k u> 0$ for $k=1,\cdots ,m-1$, under the conditions $(-\Delta)^mu\in\mathcal{L}_\alpha$ and $(-\Delta)^m u\geq 0$. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation $u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-\alpha}}{\rm d}y$. One can classify solutions to (0.1) following the work of [2] and [3] by Chen, Li, Ou.

关键词: super polyharmonic, fractional Laplacian, equivalence, classification

Abstract: Let $0<\alpha<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider the positive solution $u$ of the PDE
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (-\Delta)^{\frac{\alpha}{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0.1) $
In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition $u\in\mathcal{L}_\alpha$, (0.1) possesses a super polyharmonic property $(-\Delta)^{k+\frac{\alpha}{2}}u\geq 0$ for $k=0,1,\cdots ,m-1$. In this paper, we show another kind of super polyharmonic property $(-\Delta)^k u> 0$ for $k=1,\cdots ,m-1$, under the conditions $(-\Delta)^mu\in\mathcal{L}_\alpha$ and $(-\Delta)^m u\geq 0$. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation $u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-\alpha}}{\rm d}y$. One can classify solutions to (0.1) following the work of [2] and [3] by Chen, Li, Ou.

Key words: super polyharmonic, fractional Laplacian, equivalence, classification

中图分类号: 

  • 35R11