Critical Mass Phenomena and Blow-up behavior of Ground States in stationary second order Mean-Field Games systems with decreasing cost
- Resource Type
- Working Paper
- Authors
- Cirant, Marco; Kong, Fanze; Wei, Juncheng; Zeng, Xiaoyu
- Source
- Subject
- Mathematics - Analysis of PDEs
- Language
This paper is devoted to the study of Mean-field Games (MFG) systems in the mass critical exponent case. We firstly establish the optimal Gagliardo-Nirenberg type inequality associated with the potential-free MFG system. Then, under some mild assumptions on the potential function, we show that there exists a critical mass $M^*$ such that the MFG system admits a least energy solution if and only if the total mass of population density $M$ satisfies $MComment: 58 pages