摘要
In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/(1 + |j|)^α(1 + |k- j|)^λ(1 + |k|)^β,(0.1)vj =∑ k ∈Zn uk^p/(1 + |j|)^β(1 + |k- j|)^λ(1 + |k|)^α,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(Z^n) and v ∈ l^q+1(Z^n). Then we show that problem(0.1) has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{(n-λ^-)(q+1)/pq-1,(n-λ^-)(p+1)/pq-1} ≥λ^-.
In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/(1 + |j|)^α(1 + |k- j|)^λ(1 + |k|)^β,(0.1)vj =∑ k ∈Zn uk^p/(1 + |j|)^β(1 + |k- j|)^λ(1 + |k|)^α,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(Z^n) and v ∈ l^q+1(Z^n). Then we show that problem(0.1) has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{(n-λ^-)(q+1)/pq-1,(n-λ^-)(p+1)/pq-1} ≥λ^-.
基金
supported by NNSF of China(11261023,11326092),NNSF of China(11271170)
Startup Foundation for Doctors of Jiangxi Normal University
GAN PO 555 Program of Jiangxi
NNSF of Jiangxi(20122BAB201008)