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

TL;DR

This paper investigates bifurcation and segregation phenomena in quadratic two-population Mean Field Games systems, analyzing existence of solutions and the emergence of segregation as viscosity vanishes.

Contribution

It introduces a variational and bifurcation analysis of a semilinear elliptic system derived from Mean Field Games, highlighting segregation in the zero-viscosity limit.

Findings

Existence of nontrivial solutions via variational methods and bifurcation theory.

Segregation occurs as viscosity tends to zero, with populations separating spatially.

The system models Nash equilibria in large-player differential games.

Abstract

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. \[ \int_{\Omega} v_1 v_2 \to 0 \qquad \text{as }\nu\to0. \]