  The field of cavity optomechanics has achieved groundbreaking photonic
control and detection of mechanical oscillators, based on their coupling to
linear electromagnetic modes. Lately, however, there is an uprising interest in
exploring cavity nonlinearities as a powerful new resource in
radiation-pressure interacting systems. Here, we present a flux-mediated
optomechanical device combining a nonlinear Josephson-based superconducting
quantum interference cavity with a mechanical nanobeam. We demonstrate how the
intrinsic Kerr nonlinearity of the microwave circuit can be used for a
counter-intuitive blue-detuned sideband-cooling scheme based on multi-tone
cavity driving and intracavity four-wave-mixing. Based on the large
single-photon coupling rate of the system of up to $g_0 = 2\pi\cdot 3.6\,$kHz
and a high mechanical quality factor $Q_\mathrm{m} \approx 4\cdot 10^{5}$, we
achieve an effective four-wave cooperativity of $\mathcal{C}_\mathrm{fw} > 100$
and demonstrate four-wave cooling of the mechanical oscillator close to its
quantum groundstate, achieving a final occupancy of $n_\mathrm{m} \sim 1.6$.
Our results significantly advance the recently developed platform of
flux-mediated optomechanics and demonstrate how cavity Kerr nonlinearities can
be utilized for novel control schemes in cavity optomechanics.
