  We investigate the existence of solutions to viscous ergodic Mean Field Games
systems in bounded domains with Neumann boundary conditions and local, possibly
aggregative couplings. In particular we exploit the associated variational
structure and search for constrained minimizers of a suitable functional.
Depending on the growth of the coupling, we detect the existence of global
minimizers in the mass subcritical and critical case, and of local minimizers
in the mass supercritical case, notably up to the Sobolev critical case.
