## Analysis & PDE

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

### Concentration of ground states in stationary mean-field games systems

#### 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.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1540432869

Digital Object Identifier
doi:10.2140/apde.2019.12.737

Mathematical Reviews number (MathSciNet)
MR3864209

Zentralblatt MATH identifier
06986452

#### Citation

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. https://projecteuclid.org/euclid.apde/1540432869

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