  We study non-equilibrium microcavity-polariton condensates (MPCs) in a
harmonic potential trap theoretically. We calculate and analyze the steady
state, collective-excitation modes and instability of MPCs. Within excitation
modes, there exist Nambu-Goldstone modes that can reveal the pattern of the
spontaneous symmetry breaking of MPCs. Bifurcation of the stable and unstable
modes is identified in terms of the pumping power and spot size. The unstable
mechanism associated with the inward supercurrent flow is characterized by the
existence of a supersonic region within the condensate.
