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

¹ Dipartimento di Matematica, Università di Milano, via Cesare Saldini 50, 20133 Milano, Italy
marco.cirant@unimi.it
² Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy
gianmaria.verzini@polimi.it

Received: 14 January 2016
Revised: 11 May 2016
Accepted: 21 May 2016

Abstract

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