Critical mass phenomena and blow-up behaviors 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 first derive 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