Mean-field games (MFG) systems serve as paradigms to qualitatively describe the game among a huge number of rational players. In this talk, some intricate connections between MFGs and Schrodinger equations are mentioned, then the existence and asymptotic profiles of ground states to MFG systems in the mass critical exponent case are extensively discussed. First of all, we 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 is less than M*. Moreover, the blow-up behavior of energy minimizers are captured as M increases and converges to M*. In particular, given the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states. While studying the existence of least energy solutions, we analyze the maximal regularities of solutions to Hamilton-Jacobi equations with superlinear gradient terms. This is a joint work with Marco Cirant, Fanze Kong and Juncheng Wei.