  We investigate the resonant enhancement of nonlinear interactions in a
three-mode cavity optomechanical system with two mechanical oscillators. By
using the Keldysh Green's function technique we find that nonlinear effects on
the cavity density of states can be greatly enhanced by the resonant scattering
of two phononic polaritons, due to their small effective dissipation. In the
large detuning limit and taking into account an upper bound on the achievable
dressed coupling, the optimal point for probing the nonlinear effect is
obtained, showing that such three-mode system can exhibit prominent nonlinear
features also for relatively small values of $g/\kappa$.
