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

Abstract

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 ZN:(0.1)‖u‖ℓq≤Cp,q‖u‖D1,p,∀u∈D1,p(ZN), where N≥3,1≤p<N,q>p⁎=NpN−p,Cp,q is a constant depending on N,p and q;(0.2)∑i,j∈ZNi≠jf(i)g(j)|i−j|λ≤Cr,s,λ‖f‖ℓr‖g‖ℓs,∀f∈ℓr(ZN),g∈ℓs(ZN), where r,s>1, 0<λ<N, 1r+1s+λN>2, Cr,s,λ is a constant depending on N,r,s and λ.We introduce the discrete Concentration-Compactness principle, and prove the existence of extremal functions (pairs) for the best constants in the supercritical cases q>p⁎ and 1r+1s+λN>2, respectively.