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<M*. Moreover, the blow-up behavior of energy minimizers is characterized as M NE arrow M*. In particular, by considering the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as M NE arrow M*. While studying the existence of least-energy solutions, we establish new local W-2,W-p estimates for solutions to Hamilton-Jacobi equations with superlinear gradient terms. (c) 2025 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).