  We study a model of bosons in the lowest Landau level in a rotating trap
where the confinement potential is a sum of a quadratic and a quartic term. The
quartic term improves the stability of the system against centrifugal
deconfinement and allows to consider rotation frequencies beyond the frequency
of the quadratic part. The interactions between particles are modeled by a
Dirac delta potential. We derive rigorously conditions for ground states of the
system to be strongly correlated in the sense that they are confined to the
kernel of the interaction operator, and thus contain the correlations of the
Bose-Laughlin state. Rigorous angular momentum estimates and trial state
arguments indicate a transition from a pure Laughlin state to a state
containing in addition a giant vortex at the center of the trap (Laughlin
quasi-hole). There are also indications of a second transition where the
density changes from a flat profile in a disc or an annulus to a radial
Gaussian confined to a thin annulus.
