  We elaborate a new solution for the problem of stable propagation of
transversely localized vortex beams in homogeneous optical media with
self-focusing Kerr nonlinearity. Stationary nonlinear Bessel-vortex states are
stabilized against azimuthal breakup and collapse by multiphoton absorption,
while the respective power loss is offset by the radial influx of the power
from an intrinsic reservoir. A linear stability analysis and direct numerical
simulations reveal a region of stability of these vortices. Beams with multiple
vorticities have their stability regions too. These beams can then form robust
tubular filaments in transparent dielectrics as common as air, water and
optical glasses at sufficiently high intensities. We also show that the
tubular, rotating and speckle-like filamentation regimes, previously observed
in experiments with axicon-generated Bessel beams, can be explained as
manifestations of the stability or instability of a specific nonlinear
Bessel-vortex state, which is fully identified.
