  We present a quantitative study of the nonlinear optical response of layered
perovskites placed inside planar photonic microcavities in the regime of strong
light matter coupling, when excitonic and photonic modes hybridize and give
rise to cavity polaritons. Two sources of nonlinearity are specified, the
saturation of the excitonic transition with increase of the optical pump and
Coulomb interaction between the excitons. It is demonstrated, that peculiar
form of the interaction potential, specific to multilayer structure of organic
perovskites, is responsible for substantial increase of the exciton binding
energy and Rabi splitting with respect to conventional semiconductor systems.
This results in dominant contribution of the Rabi splitting quench effect in
the nonlinear optical response. Moreover, due to the tightly bound character of
excitons, the density of Mott transition is essentially higher, allowing to
reach extremely large polariton blueshifts of about 50 meV, which is order of
magnitude higher than in conventional semiconductors.
