Optimal summation interval and nonexistence of positive solutions to a discrete system

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 equation(0.1){uj = ∑k∈Znυkq(1+|j|)α(1+|k-j|)λ(1+|k|)β,uj = ∑k∈Znυkp(1+|j|)β(1+|k-j|)λ(1+|k|)α,where u,υ > 0,1 < p,q < ∞, 0 < λ < n, 0 ≤ α + β ≤ n-λ, 1p+1 < λ+αn and 1p+1 + 1q+1 ≤ λ+α+βn : = λ¯n. We first show that positive solutions of (0.1) have the optimal summation interval under assumptions that u ∈ lp+1(Zn)u ∈ lp+1(Zn) and ∈ lq+1(Zn)∈ 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¯.