Critical Conditions And Asymptotic Behavior For Discrete Systems Of The Hardy-Littlewood-Sobolev Type

In this paper,we study the Euler-Lagrange system related to the extremal se-quences of the discrete Hardy-Littlewood-Sobolev inequality with the Sobolev-type critical conditions.This system comes into play in estimating bounds of the Coulomb energy and is associated with the study of conformal geometry.If the solutions of the system are summable,they must be monotonically decreasing at infinity.More-over,the decay rates are obtained.By estimating the infinite series,we prove that the Serrin-type condition is critical for the existence of super-solutions of the system.