Bifurcation and segregation in quadratic two-populations mean field games systems

We search for non-constant normalized solutions to the semilinear elliptic system equation represented where ν > 0, Ω RN is smooth and bounded, the functions gi are positive and increasing, and both the functions vi and the parameters λi are unknown. This system is obtained, via the Hopf-Cole transformation, from a two-populations ergodic Mean Field Games system, which describes Nash equilibria in differential games with identical players. In these models, each population consists of a very large number of indistinguishable rational agents, aiming at minimizing some long-time average criterion. Firstly, we discuss existence of nontrivial solutions, using variational methods when gi(s) = s, and bifurcation ones in the general case; secondly, for selected families of nontrivial solutions, we address the appearing of segregation in the vanishing viscosity limit, i.e.(equation represented).