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

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

Abstract

Abstract:

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.

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

CLC Number:

45E10

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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