Singular limit of periodic metric grids

Abstract

We investigate the asymptotic behavior of nonlinear Schrödinger 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 H1(Rd) to the corresponding ground states on Rd. 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 Rd and on grids. For L2-critical and supercritical powers, we show that the value of such constants on grids is strictly related to that on Rd but, contrary to Rd, constants on grids are not attained. The proofs of these results combine purely variational arguments with new Gagliardo–Nirenberg inequalities on grids.