Singular limit of lattice graphs

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.