  We investigate self-gravitating equilibria of halos constituted by dark
matter (DM) non-minimally coupled to gravity. In particular, we consider a
theoretically motivated non-minimal coupling which may arise when the
averaging/coherence length $L$ associated to the fluid description of the DM
collective behavior is comparable to the local curvature scale. In the
Newtonian limit, such a non-minimal coupling amounts to a modification of the
Poisson equation by a term $L^2\,\nabla^2\rho$ proportional to the Laplacian of
the DM density $\rho$ itself. We further adopt a general power-law equation of
state $p\propto \rho^{\Gamma}\, r^\alpha$ relating the DM dynamical pressure
$p$ to density $\rho$ and radius $r$, as expected by phase-space density
stratification during the gravitational assembly of halos in a cosmological
context. We confirm previous findings that, in absence of the non-minimal
coupling, the resulting density $\rho(r)$ features a steep central cusp and an
overall shape mirroring the outcomes of $N-$body simulations in the standard
$\Lambda$CDM cosmology, as described by the classic NFW or Einasto profiles.
Most importantly, we find that the non-minimal coupling causes the density
distribution to develop an inner core and a shape closely following, out to
several core scale radii, the Burkert profile. In fact, we highlight that the
resulting mass distributions can fit, with an accuracy comparable to the
Burkert's one, the co-added rotation curves of dwarf, DM-dominated galaxies.
Finally, we show that non-minimally coupled DM halos are consistent with the
observed scaling relation between the core radius $r_0$ and core density
$\rho_0$, in terms of an universal core surface density $\rho_0\times r_0$
among different galaxies.
