  Essay selected for Honorable mention 2014 by the Gravity Research Foundation.
  We study an isothermal system of semi-degenerate self-gravitating fermions in
General Relativity (GR). The most general solutions present mass density
profiles with a central degenerate compact core governed by quantum statistics
followed by an extended plateau, and ending in a power law behaviour $r^{-2}$.
By fixing the fermion mass $m$ in the keV regime, the different solutions
depending on the free parameters of the model: the degeneracy and temperature
parameters at the center, are systematically constructed along the
one-parameter sequences of equilibrium configurations up to the critical point,
which is represented by the maximum in a central density ($\rho_0$) Vs. core
mass ($M_c$) diagram. We show that for fully degenerate cores, the
Oppenheimer-Volkoff (OV) mass limit $M_{c}^{cr}\propto M_{pl}^3/m^2$ is
obtained, while instead for low degenerate cores, the critical core mass
increases showing the temperature effects in a non linear way. The main result
of this work is that when applying this theory to model the distribution of
dark matter in big elliptical galaxies from miliparsec distance-scales up to
$10^2$ Kpc, we do not find any critical core-halo configuration of
self-gravitating fermions, able to explain both the most super-massive dark
object at their center together with the DM halo simultaneously.
