  Recent experiments have revealed the formation of stable droplets in dipolar
Bose-Einstein condensates. This surprising result has been explained by the
stabilization given by quantum fluctuations. We study in detail the properties
of a BEC in the presence of quantum stabilization. The ground-state phase
diagram presents three main regimes: mean-field regime, in which the quantum
correction is perturbative, droplet regime, in which quantum stabilization is
crucial, and a multi-stable regime. In the absence of a multi-stable region,
the condensate undergoes a crossover from the mean-field to the droplet
solution marked by a characteristic growth of the peak density that may be
employed to clearly distinguish quantum stabilization from other stabilization
mechanisms. Interestingly quantum stabilization allows for three-dimensionally
self-bound condensates. We characterized these self-bound solutions, and
discuss their realization in experiments. We conclude with a discussion of the
lowest-lying excitations both for trapped condensates, and for self-bound
solutions.
