Critical mass phenomena and blow-up behaviors of ground states in stationary second order mean-field games systems with decreasing cost

Abstract

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 $M\nearrow M^{⁎}$. In particular, by considering 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 for solutions to Hamilton-Jacobi equations with superlinear gradient terms.