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

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 ($2<p<2^{\ast }$) and energy ($2<p<2+\frac{4}{d}$) ground states of the nonlinear Schrödinger (NLS) equation on $d$-dimensional lattice graphs converge strongly in $H^1({\mathbb{R}}^d)$ to the corresponding ground states on ${\mathbb{R}}^d$ as the edge length tends to zero. As a by-product of the arguments developed for the singular limit problem on lattice graphs, we obtain multiplicity results for fixed-mass critical points of the energy functional on lattice graphs. Furthermore, employing a strategy analogous to that used in the singular limit analysis, we investigate the optimal constants of Gagliardo-Nirenberg type inequalities on lattice graphs for $2<p<2^{\ast }$. Beyond the classical subcritical framework, we also study the singular limit of action ground states in the Sobolev supercritical regime ($d\ge 3$ and $p>2^{\ast }$), 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 $d\ge 3$ and $p=2^{\ast }$ 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.

53 pages,6 figures. Comments and suggestions are most welcome