  We study the existence of energy minimizers for a Bose-Einstein condensate
with dipole-dipole interactions, tightly confined to a plane. The problem is
critical in that the kinetic energy and the (partially attractive) interaction
energy behave the same under mass-preserving scalings of the wave-function. We
obtain a sharp criterion for the existence of ground states, involving the
optimal constant of a certain generalized Gagliardo-Nirenberg inequality.
