  The renormalization group framework can be applied to Quantum Field Theory on
curved space-time, but there is no proof whether the beta-function of the
gravitational coupling indeed goes to zero in the far infrared or not. In a
recent paper we have shown that the amount of dark matter inside spiral
galaxies may be negligible if a small running of the General Relativity
coupling G is present. Here we extend the proposed model to elliptical galaxies
and present a detailed analysis on the modeling of NGC 4494 (an ordinary
elliptical) and NGC 4374 (a giant elliptical). In order to compare our results
to a well known alternative model to the standard dark matter picture, we also
evaluate NGC 4374 with MOND. In this galaxy MOND leads to a significative
discrepancy with the observed velocity dispersion curve and has a significative
tendency towards tangential anisotropy. On the other hand, the approach based
on the renormalization group and general relativity (RGGR) could be applied
with good results to these elliptical galaxies and is compatible with lower
mass-to-light ratios (of about the Kroupa IMF type).
