  Exciton-polaritons can condense to a macroscopic quantum state through a
non-equilibrium process of pumping and decay. In recent experiments, polariton
condensates are used to observe, for a short time, nonlinear Josephson
phenomena by coupling two condensates. However, it is still not clear how these
phenomena are affected by the pumping and decay at long times and how the
coupling alters the polariton condensation. Here, we consider a polariton
Josephson junction pumped on one side and study its dynamics within a
mean-field theory. The Josephson current is found to give rise to
multi-stability of the stationary states, which are sensitive to the initial
conditions and incoherent noises. These states can be attributed to either the
self-trapping effect or the parity-time (PT) symmetry of the system. These
results can be used to explain the emission spectra and the $\pi$-phase locking
observed in recent experiments. We further predict that the multi-stability can
reduce to the self-trapped state if the PT symmetry is broken. Moreover, the
polaritons can condense even below the threshold, exhibiting hysteresis.
