  It is well known that attractive potential which is inversely proportional to
the squared distance from the origin gives rise to the critical quantum
collapse in the framework of the three-dimensional (3D) linear Schroedinger
equation. This article summarizes theoretical analysis, chiefly published in
several original papers, which demonstrates suppression of the collapse caused
by this potential, and the creation of the otherwise missing ground state in a
3D gas of bosonic dipoles pulled by the same potential to the central charge,
with repulsive contact interactions between them, represented by the cubic term
in the respective Gross-Pitaevskii equation (GPE). In two dimensions (2D),
quintic self-repulsion is necessary for the suppression of the collapse;
alternatively, this may be provided by the effective quartic repulsion,
produced by the Lee-Huang-Yang correction to the GPE. 3D states carrying
angular momentum are constructed in the model with the symmetry reduced from
spherical to cylindrical by an external polarizing field. Interplay of the
collapse suppression and miscibility-immiscibility transition is considered in
a binary condensate. The consideration of the 3D setting in the form of the
many-body quantum system, with the help of the Monte Carlo method, demonstrates
that, although the quantum collapse cannot be fully suppressed, the
self-trapped states, predicted by the GPE, exist in the many-body setting as
metastable modes protected against the collapse by a tall potential barrier.
