OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

doi:10.1016/S0252-9602(14)60117-X

数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (6): 1720-1730.doi: 10.1016/S0252-9602(14)60117-X

OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

陈晓莉|郑维军

Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China

收稿日期:2013-09-23 修回日期:2014-06-18 出版日期:2014-11-20 发布日期:2014-11-20
基金资助:
The first author was supported by NNSF of China (11261023, 11326092), Startup Foundation for Doctors of Jiangxi Normal University. The second au-thor was supported by NNSF of China (11271170), GAN PO 555 Program of Jiangxi and NNSF of Jiangxi(20122BAB201008).

OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

CHEN Xiao-Li, ZHENG Wei-Jun

Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China

Received:2013-09-23 Revised:2014-06-18 Online:2014-11-20 Published:2014-11-20
Supported by:
The first author was supported by NNSF of China (11261023, 11326092), Startup Foundation for Doctors of Jiangxi Normal University. The second au-thor was supported by NNSF of China (11271170), GAN PO 555 Program of Jiangxi and NNSF of Jiangxi(20122BAB201008).

摘要/Abstract

摘要：

In this paper, we are concerned with properties of positive solutions of the follow-ing Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form
{u_j= ∑_k∈Zⁿv^q_k/(1 + |j|) ^α(1 + |k − j|)^λ(1 + |k|) ^β,
v_j= ∑_k∈Zⁿu^p_k/(1 + |j|)^β (1 + |k − j|)^λ(1 + |k|)^α, (0.1)
where u, v > 0, 1 < p, q < ∞, 0 < λ < n, 0 ≤α +β ≤n − λ, 1/p+1 < λ+ α/n and 1/p+1 + 1/q+1 ≤λ+α +β /n := λ/n . We first show that positive solutions of (0.1) have the optimal summation interval under assumptions that u ∈l^p⁺¹(Zⁿ) and v ∈ l^q⁺¹(Zⁿ). Then we show that problem (0.1) has no positive solution if 0 < pq ≤ 1 or pq > 1 and max{(n−¯λ)(q+1)/pq−1 , (n−¯λ)(p+1) /pq−¹ } ≥¯λ.

关键词: summation, optimal interval, nonexistence, weighted Hardy-Littlewood-Sobolev inequality

Abstract:

Key words: summation, optimal interval, nonexistence, weighted Hardy-Littlewood-Sobolev inequality

中图分类号:

45E10

陈晓莉, 郑维军. OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM[J]. 数学物理学报(英文版), 2014, 34(6): 1720-1730.

CHEN Xiao-Li, ZHENG Wei-Jun. OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM[J]. Acta mathematica scientia,Series B, 2014, 34(6): 1720-1730.

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OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

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