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. As applications, we first show that extensions of action ($2<p<2^*$) and energy ($2<p<2+\frac{4}{d}$) ground states of the nonlinear Schr\"{o}dinger (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 of lattice graphs tends to zero. As a by-product of the arguments developed for the singular limit problem of lattice graphs, we obtain multiplicity results for fixed-mass critical points of energy functional on lattice graphs. Furthermore, employing a strategy analogous to the analysis of the singular limit problem of lattice graphs, we investigate the optimal constants of Gagliardo--Nirenberg type inequalities on lattice graphs for $2<p<2^*$. A distinctive feature of this paper is that, going beyond the classical subcritical framework, we also establish novel results on the singular limit problem of action ($d \geq 3$ and $p>2^*$) and energy ($p>2+\frac{4}{d}$) ground states on lattice graphs, and on the optimal constants of Gagliardo--Nirenberg type inequalities for $d \geq 3$ and $p=2^*$ on lattice graphs, thereby substantially extending the existing literature. Notably, we settle an open problem posed by Dovetta [Adv. Math. 444 (2024), 109633] by establishing a new Gagliardo--Nirenberg type inequality.