Concentration of ground states in stationary mean-field games systems

Cesaroni, Annalisa; Cirant, Marco

doi:10.2140/apde.2019.12.737

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Concentration of ground states in stationary mean-field games systems

Annalisa Cesaroni and Marco Cirant

Vol. 12 (2019), No. 3, 737–787

DOI: 10.2140/apde.2019.12.737

Abstract

We provide the existence of classical solutions to stationary mean-field game systems in the whole space $ℝ^{N}$ , with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the nonconvex energy associated to the system. Finally, we show that in the vanishing viscosity limit, mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e., classical solutions to mean-field game systems in the whole space without potential, and with aggregating coupling.

Keywords

ergodic mean-field games, semiclassical limit, concentration-compactness method, mass concentration, elliptic systems, variational methods

Mathematical Subject Classification 2010

Primary: 35J50

Secondary: 49N70, 35J47, 91A13, 35B25

Milestones

Received: 16 August 2017

Revised: 25 May 2018

Accepted: 29 June 2018

Published: 7 October 2018

Authors

Annalisa Cesaroni
Dipartimento di Scienze Statistiche Università di Padova Padova Italy

Marco Cirant
Dipartimento di Matematica “Tullio Levi-Civita” Università di Padova Padova Italy

Vol. 12, No. 3, 2019