Calculus of Variations and Geometric Measure Theory

C. Bernardini - A. Cesaroni

Ergodic Mean-Field Games with aggregation of Choquard-type

created by cesaroni on 17 Aug 2022
modified by bernardini on 17 May 2023


Published Paper

Inserted: 17 aug 2022
Last Updated: 17 may 2023

Journal: Journal of Differential Equations
Volume: 364
Pages: 296-335
Year: 2023
Doi: 10.1016/j.jde.2023.03.045

ArXiv: 2208.08177 PDF


We consider second-order ergodic Mean-Field Games systems in the whole space $\mathbb{R}^N$ with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be $+\infty$.