  We consider stationary viscous Mean-Field Games systems in the case of local,
decreasing and unbounded coupling. These systems arise in ergodic mean-field
game theory, and describe Nash equilibria of games with a large number of
agents aiming at aggregation. We show how the dimension of the state space, the
behavior of the coupling and the Hamiltonian at infinity affect the existence
and non-existence of regular solutions. Our approach relies on the study of
Sobolev regularity of the invariant measure and a blow-up procedure which is
calibrated on the scaling properties of the system. In very special cases we
observe uniqueness of solutions. Finally, we apply our methods to obtain new
existence results for MFG systems with competition, namely when the coupling is
local and increasing.
