Bifurcation and segregation in quadratic two-populations Mean Field Games systems

We search for non-constant normalized solutions to the semilinear elliptic system $$\{\begin{array}{ll}-\nu \mathrm{\Delta }{v}_i+g_i(v_j^2)v_i={\lambda }_i{v}_i,v_i>0& \text{in }\mathrm{\Omega }\\ {\mathrm{\partial }}_n{v}_i=0& \text{on }\mathrm{\partial }\mathrm{\Omega }\\ {\int }_{\mathrm{\Omega }}{v}_i^{2}dx=1,& 1\le i,j\le 2,j\mathrm{\ne }i,\end{array}$$ where $\nu >0$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters ${\lambda }_i$ 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 $g_i(s)=s$, 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. $${\int }_{\mathrm{\Omega }}{v}_1{v}_2\to 0\text{as }\nu \to 0.$$