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Existence of positive solutions for an approximation of stationary mean-field games. (English) Zbl 1354.49082

Summary: Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a-priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

MSC:

49N70 Differential games and control
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49M30 Other numerical methods in calculus of variations (MSC2010)
91A23 Differential games (aspects of game theory)
91A07 Games with infinitely many players
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
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