  We investigate the multistability of exciton-polariton condensates excited by
a nonresonant pump. An increase in pumping power moves the system away from
non-Hermitian spectral degeneracy towards spectrum splitting through an
exceptional point, which induces a transition from monostability to
multistability. In the region of multistability, the system contains one steady
and two metastable states. The analyses of stability show that metastable
states maintain a finite lifetime and eventually evolve to steady states. A
steady state with multi-peak soliton different from general single-peak soliton
is discovered for attractive polariton-polariton interaction. Moreover, we
depict the diagram of the multistability in full parameter space to accurately
manipulate the multistability. Our results open up exciting possibilities for
controlling non-Hermitian quantum multistable states, which may be useful to
designing polariton-based devices exploiting optical multistability.
