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arXiv · 2606.03150

Singular limit of lattice graphs and its application to critical Lane--Emden equations on lattice graphs

Zhentao He·Chao Ji
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Abstract

In this paper, we establish new connections between lattice graphs and metric grids, providing a unified framework for the study of singular limit problems and Gagliardo--Nirenberg type inequalities on lattice graphs. The main technical ingredients are restriction and extension estimates, which enable us to compare variational problems posed on lattice graphs, metric grids and \(\mathbb R^d\). As applications, we first prove that extensions of action ($2 2^*$), the singular limit of energy ground states in the mass-supercritical regime ($p>2+\frac{4}{d}$) on lattice graphs, and the optimal constants in Gagliardo-Nirenberg type inequalities in the Sobolev critical case $d \geq 3$ and $p=2^*$ on lattice graphs. Notably, we settle an open problem posed by Dovetta [Adv. Math. 444 (2024), 109633] by establishing a new Gagliardo-Nirenberg type inequality.

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Zhentao He, Chao Ji. 2026-06-02. Singular limit of lattice graphs and its application to critical Lane--Emden equations on lattice graphs. https://arxiv.org/abs/2606.03150

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