Singular limit of periodic metric grids

We investigate the asymptotic behaviour of nonlinear Schr\"odinger ground states on $d$-dimensional periodic metric grids in the limit for the length of the edges going to zero. We prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\mathbb{R}^d)$ to the corresponding ground states on $\mathbb{R}^d$. As an application of such convergence results, qualitative properties of ground states and multiplicity results for fixed mass critical points of the energy on grids are derived. Moreover, we compare optimal constants in $d$-dimensional Gagliardo-Nirenberg inequalities on $\mathbb{R}^d$ and on grids. For $L^2$-critical and supercritical powers, we show that the value of such constants on grids is strictly related to that on $\mathbb{R}^d$ but, contrary to $\mathbb{R}^d$, constants on grids are not attained. The proofs of these results combine purely variational arguments with new Gagliardo-Nirenberg inequalities on grids.