  There are several issues to do with dwarf galaxy predictions in the standard
$\Lambda$CDM cosmology that have suscitated much recent debate about the
possible modification of the nature of dark matter as providing a solution. We
explore a novel solution involving ultra-light axions that can potentially
resolve the missing satellites problem, the cusp-core problem, and the `too big
to fail' problem. We discuss approximations to non-linear structure formation
in dark matter models containing a component of ultra-light axions across four
orders of magnitude in mass, $10^{-24}\text{ eV}\lesssim m_a \lesssim
10^{-20}\text{ eV}$, a range too heavy to be well constrained by linear
cosmological probes such as the CMB and matter power spectrum, and too
light/non-interacting for other astrophysical or terrestrial axion searches. We
find that an axion of mass $m_a\approx 10^{-21}\text{ eV}$ contributing
approximately $85\%$ of the total dark matter can introduce a significant kpc
scale core in a typical Milky Way satellite galaxy in sharp contrast to a
thermal relic with a transfer function cut off at the same scale, while still
allowing such galaxies to form in significant number. Therefore ultra-light
axions do not suffer from the \emph{Catch 22} that applies to using a warm dark
matter as a solution to the small scale problems of cold dark matter. Our model
simultaneously allows formation of enough high redshift galaxies to allow
reconciliation with observational constraints, and also reduces the maximum
circular velocities of massive dwarfs so that baryonic feedback may more
plausibly resolve the predicted overproduction of massive MWG dwarf satellites.
