  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.
