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We investigate the existence of ground states for the focusing NonlinearSchr\"odinger Equation on the infinite three-dimensional cubic grid. We extendthe result found for the analogous two-dimensional grid by proving anappropriate Sobolev inequality giving rise to a family of criticalGagliardo-Nirenberg inequalities that hold for every nonlinearity power from$10/3$ and $6$, namely, from the $L^2$-critical power for the same problem in$\mathbb{R}^3$ to the critical power for the same problem in $\mathbb{R}$.Given the Gagliardo-Nirenberg inequality, the problem of the existence ofground state can be treated as already done for the two-dimensional grid.

One-dimensional versions of three-dimensional system: Ground states for the NLS on the spatial grid