  We consider a magnetic insulator in contact with a normal metal. We derive a
self-consistent Keldysh effective action for the magnon gas that contains the
effects of magnon-magnon interactions and contact with the metal to lowest
order. Self-consistent expressions for the dispersion relation, temperature and
chemical potential for magnons are derived. Based on this effective action, we
study instabilities of the magnon gas that arise due to spin-current flowing
across the interface between the normal metal and the magnetic insulator. We
find that the stability phase diagram is modified by an interference between
magnon-magnon interactions and interfacial magnon-electron coupling. These
effects persist at low temperatures and for thin magnetic insulators.
