The quantum-mechanical collapse (alias fall onto the center  of particles attracted by potential -r⁻²) is a well-known issue in the quantum theory. It is closely related to the quantum anomaly, i.e., breaking of the scaling invariance of the respective Hamiltonian by the quantization. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem different from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge, and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation, resulting in a redifinition of the scattering length which accounts for the contact repulsion between the bosons. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schrödinger equation. The addition of the harmonic trap gives rise to a tristability, in the case when the Schrödinger equation still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counter-intuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential r⁻². The 2D vortical modes avoid the phase singularity at the pivot (r=0) by having the amplitude diverging at r→0, instead of the usual situation with the amplitude of the vortical mode vanishing at r→0 (the norm of the mode converges despite of the singularity of the amplitude at r→0). In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential r⁻² gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form, and also in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.