  We study the dipolar Gross-Piteavskii functional with the Lee-Huang-Yang
(LHY) correction term without trapping potential and in the regime where the
dipole-dipole interaction dominates the repulsive short-range interaction. We
show that, above a critical mass, the functional admits minimizers and we prove
their regularity and exponential decay. We also estimate the critical mass in
terms of the parameters of the system.
