  We discuss astrophysical implications of the modified gravity model in which
the two matter components, ordinary and dark, couple to separate gravitational
fields that mix to each other through small mass terms. There are two spin-2
eigenstates: the massless graviton that induces universal Newtonian attraction,
and the massive one that gives rise to the Yukawa-like potential which is
repulsive between the ordinary and dark bodies. As a result the distances much
smaller than the Yukawa radius $r_m$ the gravitation strength between the two
types of matter becomes vanishing. If $r_m \sim 10$ kpc, a typical size of a
galaxy, there are interesting implications for the nature of dark matter. In
particular, one can avoid the problem of the cusp that is typical for the cold
dark matter halos. Interestingly, the flat shape of the rotational curves can
be explained even in the case of the collisional and dissipative dark matter
(as e.g. mirror matter) that cannot give the extended halos but instead must
form galactic discs similarly to the visible matter. The observed rotational
curves for the large, medium-size and dwarf galaxies can be nicely reproduced.
We also briefly discuss possible implications for the direct search of dark
matter.
