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<α<2, p1, mN+. Consider the positive solution u of the PDE
                        (Δ)α2+mu(x)=up(x)in Rn.                        (0.1)
In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition uLα, (0.1) possesses a super polyharmonic property (Δ)k+α2u0 for k=0,1,,m1. In this paper, we show another kind of super polyharmonic property (Δ)ku>0 for k=1,,m1, under the conditions (Δ)muLα and (Δ)mu0. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation u(x)=Rnup(y)|xy|n2mαdy. 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<α<2, p1, mN+. Consider the positive solution u of the PDE
                        (Δ)α2+mu(x)=up(x)in Rn.                        (0.1)
In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition uLα, (0.1) possesses a super polyharmonic property (Δ)k+α2u0 for k=0,1,,m1. In this paper, we show another kind of super polyharmonic property (Δ)ku>0 for k=1,,m1, under the conditions (Δ)muLα and (Δ)mu0. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation u(x)=Rnup(y)|xy|n2mαdy. 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