  It is known that solutions of the parabolic elliptic Keller-Segel equations
in the two dimensional plane decay, as time goes to infinity, provided the
initial data admits sub-critical mass and finite second moments, while such
solution concentrate, as $t\rightarrow\infty$, in the critical mass. In the
sub-critical case this decay can be resolved by a steady, self-similar
solution, while no such self similar solution is known to exist for the
concentration in the critical case. This paper is motivated by the Keller-Segel
system of several interacting populations, under the existence of an additional
drift for each component which decays in time at the rate $O(1/\sqrt{t})$. We
show that self-similar solutions always exists in the sub-critical case, while
the existence of such self-similar solution in the critical case depends on the
gap between the decaying drifts for each of the components. For this, we study
the conditions for existence/non existence of solutions for the corresponding
Liouville's systems, which, in turn, is related to the existence/non existence
of minimizers to a corresponding Free Energy functional.
