  We report on the first mathematically rigorous proofs of a transition to a
giant vortex state of a superfluid in rotating anharmonic traps. The analysis
is carried out within two-dimensional Gross-Pitaevskii theory at large coupling
constant and large rotational velocity and is based on precise asymptotic
estimates on the ground state energy. An interesting aspect is a significant
difference between 'soft' anharmonic traps (like a quartic plus quadratic
trapping potential) and traps with a fixed boundary. In the former case
vortices persist in the bulk until the width of the annulus becomes comparable
to the size of the vortex cores. In the second case the transition already
takes place in a parameter regime where the size of vortices is very small
relative to the width of the annulus. Moreover, the density profiles in the
annulus are different in the two cases. In both cases rotational symmetry of
the density in a true ground state is broken, even though a symmetric
variational ansatz gives an excellent approximation to the energy.
