  We study static configurations of dark matter coupled to a scalar field
responsible for the dark energy of the Universe. The dark matter is modelled as
a Fermi gas within the Thomas-Fermi approximation. The mass of the dark matter
particles is a function of the scalar field. We analyze the profile of the dark
matter halos in galaxies. In this case our framework is equivalent to the model
of the isothermal sphere. In the presence of a scalar field, the velocity of a
massive object orbiting the galaxy is not of the order of the typical velocity
of the dark matter particles, as in the conventional picture. Instead, it is
reduced by a factor that quantifies the dependence of the dark matter mass on
the scalar field. This has implications for dark matter searches. We derive new
solutions of the Einstein equations which describe compact objects composed of
dark matter. Depending on the scale of the dark matter mass, the size of these
objects can vary between microscopic scales and cosmological distances. We
determine the mass to radius relation and discuss the similarities with
conventional neutron stars and exotic astrophysical objects.
