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 c
ondensates. 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.
