  We demonstrate gallium phosphide (GaP) microdisk optical cavities with
intrinsic quality factors $ > 2.8\times10^{5}$ and mode volumes $< 10
(\lambda/n)^3$, and study their nonlinear and optomechanical properties. For
optical intensities up to $8.0\times10^4$ intracavity photons, we observe
optical loss in the microcavity to decrease with increasing intensity,
indicating that saturable absorption sites are present in the GaP material, and
that two-photon absorption is not significant. We observe optomechanical
coupling between optical modes of the microdisk around 1.5 $\mu$m and several
mechanical resonances, and measure an optical spring effect consistent with a
theoretically predicted optomechanical coupling rate $g_0/2\pi \sim 30$ kHz for
the fundamental mechanical radial breathing mode at 488 MHz.
