We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form \(-\sigma m^\alpha \),\(\alpha \ge 2/N\), where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that \(\sigma \) is small enough. For large \(\sigma \), we show that existence may fail whenever the time horizon T is large.

中文翻译：

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具有强聚合的时间相关平均场博弈的存在与不存在

我们研究了二阶二次平均场博弈系统的经典解的存在性，该系统具有\(-\sigma m^\al​​pha \) , \(\alpha \ge 2/N\)形式的局部和强烈递减耦合，其中m是人口密度，N是状态空间的维度。我们证明了在\(\sigma \)足够小的假设下存在解。对于大\(\sigma \)，我们表明只要时间范围T大，存在就可能失败。