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

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 $M^{\ast }$ such that the MFG system admits a least energy solution if and only if the total mass of population density $M$ satisfies $M<M^{\ast }$. Moreover, the blow-up behavior of energy minimizers are captured as $M\nearrow M^{\ast }$. In particular, given the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as $M\nearrow M^{\ast }\mathrm{.}$ 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.