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^* 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 M^*. In particular, given the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as M M^*. While studying the existence of least energy solutions, we establish new local W2,p estimates of solutions to Hamilton-Jacobi equations with superlinear gradient terms.