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

We search for non-constant normalized solutions to the semilinear elliptic system $$\begin{cases} - \nu \Delta v_i + g_i(v_j^2) v_i = \lambda_i v_i,\quad v_i>0 & \text{in $\Omega$} \\ \partial_n v_i = 0 & \text{on $\partial \Omega$}\\ \int_\Omega v_i^2\,dx = 1, & 1\leq i,j\leq 2, \quad j\neq i, \end{cases}$$ where $\nu>0$, $\Omega \subset \mathbb{R}^N$ 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. $$\int_{\Omega} v_1 v_2 \to 0 \qquad \text{as }\nu\to0.$$