Not yet reviewed by Pith; the record is open.

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.