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

Wei DAI; Zhao LIU

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

Acta mathematica scientia, Series B >

2017 , Vol. 37 >Issue 5: 1415 - 1436

DOI: https://doi.org/10.1016/S0252-9602(17)30082-6

Articles

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

Wei DAI ,
Zhao LIU

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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 date: 2015-11-30

Revised date: 2017-04-16

Online published: 2017-10-25

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).

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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

Cite this article

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 . DOI: 10.1016/S0252-9602(17)30082-6

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