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 $L^2, abla^2rho$ proportional to the Laplacian of the DM density $rho$ itself. We further adopt a general power-law equation of state $ppropto rho^{Gamma}, r^alpha$ 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 Burkerts 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 $rho_0times r_0$ among different galaxies.
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