  Weak potential wells (or traps) in one and two dimensions, and the potential
wells slightly deeper than the critical ones in three dimensions, feature
shallow bound states with localization length much larger than the well radii.
We address a simple fundamental question of how many repulsively interacting
bosons can be localized by such traps. We find that under rather generic
conditions, for both weakly and strongly repulsive particles, in two and three
dimensions--but not in one-dimension!--the potential well can trap infinitely
many bosons. For example, even hard-core repulsive interactions do not prevent
this "trapping collapse" phenomenon from taking place. For the weakly
interacting/dilute regime, the effect can be revealed by the mean-field
argument, while in the case of strong correlations the evidence comes from
path-integral simulations. We also discuss the possibility of having a
transition between the infinite and finite number of trapped particles when
strong repulsive inter-particle correlations are increased.
