Tohoku Mathematical Journal
2023
September
SECOND SERIES VOL. 75, NO. 3

Tohoku Math. J.
75 (2023), 305-328

Title CRITICAL CONDITIONS AND ASYMPTOTICS FOR DISCRETE SYSTEMS OF THE HARDY-LITTLEWOOD-SOBOLEV TYPE

Author Yutian Lei, Yayun Li and Ting Tang

(Received August 16, 2021, revised December 27, 2021)
Abstract. In this paper, we study the Euler-Lagrange system associated with the extremal sequences 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 related to the study of conformal geometry. In discrete case, we show that if the solutions of the system are summable, they must be monotonically decreasing at infinity. Moreover, the decay rates of the solutions 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. In addition, we also obtain analogous properties of the Euler-Lagrange system of the extremal sequences of the discrete reversed Hardy-Littlewood-Sobolev inequality.

Mathematics Subject Classification. Primary 26D15; Secondary 40B05, 47J20.
Key words and phrases. Discrete Hardy-Littlewood-Sobolev inequality, discrete reversed Hardy-Littlewood-Sobolev inequality, critical condition, asymptotic behavior.

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