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

Cirant, Marco; Verzini, Gianmaria

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 1145-1177.

We search for non-constant normalized solutions to the semilinear elliptic system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\nu \Delta v_i+g_i\left(v_j^2\right)v_i={\lambda }_i{v}_i,v_i>0\hfill & \text{in}\Omega \hfill \\ {\partial }_n{v}_i=0\hfill & \text{on}\partial \Omega \hfill \\ {\int }_{\Omega }{v}_i^2\mathrm{d}x=1,\hfill & 1\le i,j\le 2,j\ne i,\hfill \end{array}\right.\end{array}$$

where $\nu >0$, $\Omega \subset R^N$ 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 $g_i\left(s\right)=s$, 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.

$$\begin{array}{c}\hfill {\int }_{\Omega }{v}_1{v}_2\to 0\text{as}\nu \to 0.\end{array}$$

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