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arXiv · 2405.05484

Critical Mass Phenomena and Blow-up behavior of Ground States in stationary second order Mean-Field Games systems with decreasing cost

Marco Cirant·Fanze Kong·Juncheng Wei·Xiaoyu Zeng
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Abstract

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 $M<M^*$. Moreover, the blow-up behavior of energy minimizers are captured as $M\nearrow M^*$. In particular, given the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as $M\nearrow M^*.$ While studying the existence of least energy solutions, we establish new local $W^{2,p}$ estimates of solutions to Hamilton-Jacobi equations with superlinear gradient terms.

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Marco Cirant, Fanze Kong, Juncheng Wei, Xiaoyu Zeng. 2024-05-09. Critical Mass Phenomena and Blow-up behavior of Ground States in stationary second order Mean-Field Games systems with decreasing cost. https://arxiv.org/abs/2405.05484

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