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

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

戴蔚1, 刘招2   

  1. 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. 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:

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 < pτ1:=(n+α-2t1)/n-α,1 < qτ2:=(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 Gα(x,ξ)=(cn,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 Rk and some point z0 in Rn-k.In the subcritical case (n-t1)/p+1 + (n-t2)/q+1 > n-α, 1 < pτ1 and 1 < qτ2,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 < pτ1:=(n+α-2t1)/n-α,1 < qτ2:=(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 Gα(x,ξ)=(cn,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 Rk and some point z0 in Rn-k.In the subcritical case (n-t1)/p+1 + (n-t2)/q+1 > n-α, 1 < pτ1 and 1 < qτ2,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