  We demonstrate the formation and trapping of different stationary solutions,
oscillatory solutions, and rotating solutions of a polariton condensate in a
planar semiconductor microcavity with a built-in ring-shaped potential well.
Multistable ring shaped solutions are trapped in shallow potential wells. These
solutions have the same ring shaped density distribution but different
topological charges, corresponding to different orbital angular momentum (OAM)
of the emitted light. For stronger confinement potentials, besides the
fundamental modes, higher excited (dipole) modes can also be trapped. If two
modes are excited simultaneously, their beating produces a complex oscillation
and rotation dynamics. When the two modes have the same OAM, a double-ring
solution forms for which the density oscillates between the inner and the outer
ring. When the two modes have different OAM, a rotating solution with a
crescent-shaped density and fractional OAM is created.
