  We investigate the existence and stability of three-dimensional (3D) solitons
supported by cylindrical Bessel lattices (BLs) in self-focusing media. If the
lattice strength exceeds a threshold value, we show numerically, and using the
variational approximation, that the solitons are stable within one or two
intervals of values of their norm. In the latter case, the Hamiltonian-vs.-norm
diagram has a "swallowtail" shape, with three cuspidal points. The model
applies to Bose-Einstein condensates (BECs) and to optical media with saturable
nonlinearity, suggesting new ways of making stable 3D BEC solitons and "light
bullets" of an arbitrary size.
