Acta mathematica scientia, Series B >
OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM
Received date: 2013-09-23
Revised date: 2014-06-18
Online 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).
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
{uj = ∑k∈Znvqk/(1 + |j|) α(1 + |k − j|)λ(1 + |k|) β,
vj = ∑k∈Znupk/(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 ∈lp+1(Zn) and v ∈ lq+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.
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 . DOI: 10.1016/S0252-9602(14)60117-X
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