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

We investigate the existence of ground states for the focusing Nonlinear Schrödinger Equation on the infinite three-dimensional cubic grid. We extend the result found for the analogous two-dimensional grid by proving an appropriate Sobolev inequality giving rise to a family of critical Gagliardo-Nirenberg inequalities that hold for every nonlinearity power from 10/3 to 6, namely, from the L2-critical power for the same problem in R3 to the critical power for the same problem in R. Given the Gagliardo-Nirenberg inequality, the problem of the existence of ground state can be treated as already done for the two-dimensional grid.