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

@article{Hua2021TheEO,
title={The existence of extremal functions for discrete Sobolev inequalities on lattice graphs},
author={Bobo Hua and Ruowei Li},
journal={Journal of Differential Equations},
year={2021}
}
• Published 30 June 2021
• Mathematics
• Journal of Differential Equations
6 Citations

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In this paper, we study the semilinear elliptic equation of the form −∆u+ a(x)|u|u− b(x)|u|u = 0 on lattice graphs Z , where N ≥ 2 and 2 ≤ p < q < +∞. By the Brézis-Lieb lemma and concentration

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. In this paper, we study the nonlinear Choquard equation ∆ 2 u − ∆ u + (1 + λa ( x )) u = ( R α ∗ | u | p ) | u | p − 2 u on a Cayley graph of a discrete group of polynomial growth with the

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A discrete version of the generalized Riesz inequality on Z d is proved, which will derive the extended Hardy-Littlewood and P´olya-Szeg¨o inequalities and establish cases of equality in the latter.

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