Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 3 (2019), 737-787.

Concentration of ground states in stationary mean-field games systems

Annalisa Cesaroni and Marco Cirant

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We provide the existence of classical solutions to stationary mean-field game systems in the whole space  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.

Article information

Anal. PDE, Volume 12, Number 3 (2019), 737-787.

Received: 16 August 2017
Revised: 25 May 2018
Accepted: 29 June 2018
First available in Project Euclid: 25 October 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J50: Variational methods for elliptic systems
Secondary: 49N70: Differential games 35J47: Second-order elliptic systems 91A13: Games with infinitely many players 35B25: Singular perturbations

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


Cesaroni, Annalisa; Cirant, Marco. Concentration of ground states in stationary mean-field games systems. Anal. PDE 12 (2019), no. 3, 737--787. doi:10.2140/apde.2019.12.737.

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