  We theoretically explore the phase transition in inhomogeneous
exciton-polariton condensates with variable pumping conditions. Through
Bogoliubov excitations to the radial-symmetric solutions of complex
Gross-Pitaevskii equation, we determine not only the bifurcation of stable and
unstable modes by the sign of fluid compressibility but also two distinct
stable modes which are characterized by the elementary excitations and the
stability of singly quantized vortex. One state is the quasi-condensate BKT
phase with Goldstone flat dispersion; the other state is the localized-BEC
phase which exhibits linear-type dispersion and has an excitation energy gap at
zero momentum.
