  We study an isothermal system of semi-degenerate self-gravitating fermions in
general relativity. Such systems present mass density solutions with a central
degenerate core, a plateau and a tail which follows a power law behaviour
$r^{-2}$. The different solutions are governed by the free parameters of the
model: the degeneracy and temperature parameters at the center, and the
particle mass $m$. We then analyze in detail the free parameter space for a
fixed $m$ in the keV regime, by studying 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 known expression for the critical
core mass $M_c^{cr}\propto m_{pl}^3/m^2$ 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 galaxies from the very
center up to the outer halos, we do not find any critical core-halo
configuration of self-gravitating fermions, which be able to explain the super
massive dark object in their centers together with an outer halo
simultaneously.
