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

We search for non-constant normalized solutions to the semilinear elliptic system \begin{eqnarray*} \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\,{\rm d}x = 1, & 1\leq i,j\leq 2, \quad j\neq i, \end{cases} \end{eqnarray*}where ν > 0, Ω ⊂ R N is smooth and bounded, the functions g i are positive and increasing, and both the functions v i and the parameters λ 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. \begin{eqnarray*} \int_{\Omega} v_1 v_2 \to 0 \qquad \text{as }\nu\to0. \end{eqnarray*}∫Ωv1v2→0 asν→0.