  In this paper, we study theoretically the optomechanical interaction of an
almost pure condensate of photons with an oscillating mechanical membrane in a
micro-cavity. We show that in the Bogoliubov approximation, due to the large
number of photons in the condensate phase, there is a linear strong effective
coupling between the Bogoliubov mode of the photonic Bose-Einstein condensate
(BEC) and the mechanical motion of the membrane which depends on the nonlinear
photon-photon scattering potential. This coupling leads to the cooling of the
mechanical motion, the normal mode splitting (NMS), the squeezing of the output
field and the entanglement between the excited mode of the cavity and the
mechanical mode. We show that, in one hand, the nonlinearity of the photon gas
increases the degree of the squeezing of the output field of the micro-cavity
and the efficiency of the cooling process at high temperatures. In the other
hand, it reduces NMS in the displacement spectrum of the oscillating membrane
and the degree of the optomechanical entanglement. In addition, the temperature
of the photonic BEC can be used to control the above-mentioned phenomena.
