  We establish that the ability of a localized trapping potential to bind
weakly-interacting bosons is dramatically enhanced in the vicinity of the
threshold of formation of the single-particle bound-state of the trap.
Specifically, for repulsive particles and a super-threshold trapping potential
the equilibrium number of bound bosons and the size of the ground state diverge
upon approaching the single-particle threshold from above. For attractive
interactions and a sub-threshold trap a collective bound state always forms for
a sufficiently large number of bosons despite the inability of interparticle
attraction alone to form a two-body bound state.
