Volume 23, Number 3, July-September 2017
|Page(s)||1145 - 1177|
|Published online||12 May 2017|
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
2 Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy
Received: 14 January 2016
Revised: 11 May 2016
Accepted: 21 May 2016
We search for non-constant normalized solutions to the semilinear elliptic system where ν > 0, Ω ⊂ RN is smooth and bounded, the functions gi are positive and increasing, and both the functions vi and the parameters λi 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 gi(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.
Mathematics Subject Classification: 35J47 / 49N70 / 35B25 / 35B32
Key words: Singularly perturbed problems / normalized solutions to semilinear elliptic systems / multi-population differential games
© EDP Sciences, SMAI 2017
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