The existence of extremal functions for discrete Sobolev inequalities on lattice graphs

By Bobo Hua, Ruowei Li. Journal of Differential Equations, 2021

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In this paper, we study the existence of extremal functions (pairs) of the following discrete Sobolev inequality (0.1) and Hardy-Littlewood-Sobolev inequality (0.2) in the lattice Z : (0.1) ‖u‖lq ≤ Cp,q‖u‖D1,p , ∀u ∈ D (Z ), where N ≥ 3, 1 ≤ p < N, q > p∗ = Np N − p ,Cp,q is a constant depending on N, p and q;

Subfields: Applied Mathematics

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