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

We search for non-constant normalized solutions to the semilinear elliptic system \[ \begin{cases} - νΔv_i + g_i(v_j^2) v_i = λ_i v_i,\quad v_i>0 & \text{in $Ω$} \\ \partial_n v_i = 0 & \text{on $\partial Ω$}\\ \int_Ωv_i^2\,dx = 1, & 1\leq i,j\leq 2, \quad j\neq i, \end{cases} \] where $ν>0$, $Ω\subset \mathbb{R}^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters $λ_i$ 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. \[ \int_Ω v_1 v_2 \to 0 \qquad \text{as }ν\to0. \]