  We investigate stability of non-equilibrium steady states of Bose-Einstein
condensates with a local one-body loss in the presence of double potential
barriers. We construct an exactly solvable mean-field model, in which the local
loss and the potential barriers take the form of a delta function. Using the
exact solutions of our model, we show that there are parameter regions in which
two steady-state solutions are dynamically stable, i.e., the model exhibits
bistability. We also find that unidirectional hysteresis phenomena appear when
the local-loss rate is varied in some parameter region.
