一类平均场博弈方程在有界区域上的尖峰解——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1585-1609.

一类平均场博弈方程在有界区域上的尖峰解——献给邓引斌教授 70 寿辰

吴雅婷¹(), 徐良顺²^,^*()

¹ 华中师范大学数学与统计学学院武汉 430079
² 广西大学数学学院南宁 530004

收稿日期:2026-02-25 修回日期:2026-04-07 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 徐良顺 E-mail:ytwu@ccnu.edu.cn;lsxu@gxu.edu.cn
作者简介:吴雅婷,E-mail: ytwu@ccnu.edu.cn
基金资助:
国家自然科学基金(12301243);国家自然科学基金(12301250)

Multiple Spike Solutions for Mean Field Games in Bounded Domains

Yating Wu¹(), Liangshun Xu²^,^*()

¹ School of Mathematics and Statistics, Central China Normal University, Wuhan 430079
² School of Mathematics, Guangxi University, Nanning 530004

Received:2026-02-25 Revised:2026-04-07 Online:2026-08-26 Published:2026-06-10
Contact: Liangshun Xu E-mail:ytwu@ccnu.edu.cn;lsxu@gxu.edu.cn
Supported by:
NSFC(12301243);NSFC(12301250)

摘要/Abstract

摘要：

该文研究了一类出现在经济学与金融学中、定义在二维有界域$ \Omega \subset \mathbb{R}^2$ 上的平均场博弈方程组. 当非线性指数 $p \to \infty$ 时, 运用内-外粘接的技巧, 作者构造了该方程组的一类多峰解. 值得指出的是, 该解的尖峰位置既包含在区域 $\Omega$ 内部, 也出现在边界$ \partial \Omega$ 上.

关键词: 平均场博弈, 尖峰解, 内-外粘接

Abstract:

We consider Mean Field Games system posed on the bounded domain $\Omega \subset \mathbb{R}^2$, which arises in Economics and Finance. In the limit as the nonlinearity exponent $p \to \infty$, we construct a multi-spikes solution via using the so-called inner-outer scheme. Notably, the spike location include both in $\Omega$ and on boundary $\partial \Omega$.

Key words: mean fields games, spiky solution, inner-outer gluing

中图分类号:

O175.23

吴雅婷, 徐良顺. 一类平均场博弈方程在有界区域上的尖峰解——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1585-1609.

Yating Wu, Liangshun Xu. Multiple Spike Solutions for Mean Field Games in Bounded Domains[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1585-1609.

参考文献 18

[1]	Ao W, Lin C, Wei J. On non-topological solutions of the $A_2$ and $B_2$ Chern-Simons system. Mem Amer Math Soc, 2016, 239(1132): Art 88
[2]	Chae D, Imanuvilov O Y. The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. Comm Math Phys, 2000, 215(1): 119-142 doi: 10.1007/s002200000302
[3]	Cirant M. Stationary focusing mean-field games. Comm Partial Differential Equations, 2016, 41(8): 1324-1346 doi: 10.1080/03605302.2016.1192647
[4]	Cesaroni A, Cirant M. Concentration of ground states in stationary mean-field games systems. Anal PDE, 2019, 12(3): 737-787 doi: 10.2140/apde.2019.12.737
[5]	Dávila J, Del Pino M, Dolbeault J, et al. Existence and stability of infinite time blow-up in the Keller-Segel system. Arch Ration Mech Anal, 2024, 248(4): Art 61
[6]	Del Pino M, Wei J. Collapsing steady states of the Keller-Segel system. Nonlinearity, 2006 19(3): 661-684 doi: 10.1088/0951-7715/19/3/007
[7]	Del Pino M, Kowalczyk M, Wei J. Concentration on curves for nonlinear Schrödinger equations. Comm Pure Appl Math, 2007, 60(1): 113-146 doi: 10.1002/cpa.v60:1
[8]	Del Pino M, Esposito P, Musso M. Nondegeneracy of entire solutions of a singular Liouvillle equation. Proc Amer Math Soc, 2012, 140(2): 581-588 doi: 10.1090/proc/2012-140-02
[9]	Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Springer
[10]	Huang M, Caines P, Malhamé R. An invariance principle in large population stochastic dynamic games. J Syst Sci Complex, 2007, 20(2): 162-172
[11]	Huang M, Caines P, Malhamé R. Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst, 2006, 6(3): 221-251 doi: 10.4310/CIS.2006.v6.n3.a5
[12]	Huang M, Caines P, Malhamé R. The nash certainty equivalence principle and McKean-Vlasov systems: an invariance principle and entry adaptation. 46th IEEE Conference on Decision and Control, 2007, 121-126
[13]	Huang M, Caines P, Malhamé R. Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\varepsilon$-Nash equilibria. IEEE Trans Automat Control, 2007, 52(9): 1560-1571 doi: 10.1109/TAC.2007.904450
[14]	Kong F, Wei J, Xu L. Existence of multi-spikes in the Keller-Segel model with logistic growth. Math Models Methods Appl Sci, 2023, 33(11): 2227-2270 doi: 10.1142/S021820252340002X
[15]	Lasry J M, Lions P L. Mean field games. Jpn J Math, 2007, 2(1): 229-260
[16]	Lasry J M, Lions P L. Large investor trading impacts on volatility. Ann Inst H Poincaré C Anal Non Linéaire, 2007, 24(2): 311-323 doi: 10.4171/aihpc
[17]	Lasry J M, Lions P L. Mean field games. II. Finite horizon and optimal control. C R Math Acad Sci Paris, 2006, 343(10): 679-684 doi: 10.1016/j.crma.2006.09.018
[18]	Lasry J M, Lions P L. Mean field games. I. The stationary case. C R Math Acad Sci Paris, 2006, 343(9): 619-625 doi: 10.1016/j.crma.2006.09.019