CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

doi:10.1016/S0252-9602(17)30082-6

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (5): 1415-1436.doi: 10.1016/S0252-9602(17)30082-6

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

戴蔚¹, 刘招²

1. School of Mathematics and Systems Science, Beihang University(BUAA), Beijing 100191, China;
2. School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, China

收稿日期:2015-11-30 修回日期:2017-04-16 出版日期:2017-10-25 发布日期:2017-10-25
通讯作者: Zhao LIU,E-mail:liuzhao@mail.bnu.edu.cn E-mail:liuzhao@mail.bnu.edu.cn
作者简介:Wei DAI,E-mail:weidai@buaa.edu.cn
基金资助:
The research was supported by the NNSF of China (11371056). The first author was also partly supported by the NNSF of China (11501021) and the China Postdoctoral Science Foundation (2013M540057). The second author was also partly supported by Scientific Research Fund of Jiangxi Provincial Education Department (GJJ160797).

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

Wei DAI¹, Zhao LIU²

1. School of Mathematics and Systems Science, Beihang University(BUAA), Beijing 100191, China;
2. School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, China

Received:2015-11-30 Revised:2017-04-16 Online:2017-10-25 Published:2017-10-25
Contact: Zhao LIU,E-mail:liuzhao@mail.bnu.edu.cn E-mail:liuzhao@mail.bnu.edu.cn
Supported by:
The research was supported by the NNSF of China (11371056). The first author was also partly supported by the NNSF of China (11501021) and the China Postdoctoral Science Foundation (2013M540057). The second author was also partly supported by Scientific Research Fund of Jiangxi Provincial Education Department (GJJ160797).

摘要/Abstract

摘要：

In this paper,we are concerned with the following Hardy-Sobolev type system

where 0 < α < n,0 < t₁,t₂ < min{α,k},and 1 < p ≤ τ₁:=(n+α-2t₁)/n-α,1 < q ≤ τ₂:=(n+α-α2t₂)/n-α. We first establish the equivalence of classical and weak solutions between PDE system (0.1) and the following integral equations (IE) system

where G_α(x,ξ)=(c_n,α)/|x-ξ|^n-α is the Green's function of (-△)^（α）/2 in Rⁿ.Then,by the method of moving planes in the integral forms,in the critical case p=τ₁ and q=τ₂,we prove that each pair of nonnegative solutions (u,v) of (0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z₀ in R^n-k.In the subcritical case （n-t₁）/p+1 + （n-t₂）/q+1 > n-α, 1 < p ≤ τ₁ and 1 < q ≤ τ₂,we derive the nonexistence of nontrivial nonnegative solutions for (0.1)

关键词: Hardy-Sobolev type systems, systems of fractional Laplacian, systems of integral equations, method of moving planes in integral forms, radial symmetry, nonexistence

Abstract:

In this paper,we are concerned with the following Hardy-Sobolev type system

where 0 < α < n,0 < t₁,t₂ < min{α,k},and 1 < p ≤ τ₁:=(n+α-2t₁)/n-α,1 < q ≤ τ₂:=(n+α-α2t₂)/n-α. We first establish the equivalence of classical and weak solutions between PDE system (0.1) and the following integral equations (IE) system

where G_α(x,ξ)=(c_n,α)/|x-ξ|^n-α is the Green's function of (-△)^（α）/2 in Rⁿ.Then,by the method of moving planes in the integral forms,in the critical case p=τ₁ and q=τ₂,we prove that each pair of nonnegative solutions (u,v) of (0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z₀ in R^n-k.In the subcritical case （n-t₁）/p+1 + （n-t₂）/q+1 > n-α, 1 < p ≤ τ₁ and 1 < q ≤ τ₂,we derive the nonexistence of nontrivial nonnegative solutions for (0.1)

Key words: Hardy-Sobolev type systems, systems of fractional Laplacian, systems of integral equations, method of moving planes in integral forms, radial symmetry, nonexistence

戴蔚, 刘招. CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS[J]. 数学物理学报(英文版), 2017, 37(5): 1415-1436.

Wei DAI, Zhao LIU. CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS[J]. Acta mathematica scientia,Series B, 2017, 37(5): 1415-1436.

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CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

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