  An interacting cigar-shaped Bose-Einstein condensate (BEC) inside a driven
optical cavity exhibits an intrinsic cross-Kerr (CK) nonlinearity due to the
interaction with the optical mode of the cavity. Although the CK coupling is
much weaker than those of the radiation pressure and the atom-atom
interactions, it can affect the bistability behavior of the system when the
intensity of the laser pump is strong enough. On the other hand, there is a
competition between the CK nonlinearity and the atom-atom interaction so that
the latter can neutralize the effect of the former. Furthermore, the CK
nonlinearity causes the effective frequency of the Bogoliubov mode of the BEC
as well as the quantum fluctuations of the system to be increased by increasing
the cavity driving rate. However, in the dispersive interaction regime the
effect of the CK nonlinearity is negligible. In addition, we show that by
increasing the \textit{s}-wave scattering frequency of atomic collisions one
can generate a strong stationary quadrature squeezing in the Bogoliubov mode of
the BEC.
