Mountain-Pass Solutions for Second-Order Ergodic Mean-Field Game Systems

We study the existence of mountain-pass solutions to a potential-free mean-field game system in the whole space $\mathbb R^n$ under the mass-supercritical regime, assuming an aggregating local coupling and a $C^2$ Hamiltonian that is $\gamma$-homogeneous with $\gamma > 1$. Due to the lack of smoothness of the underlying variational structure, the standard deformation lemma and the classical mountain-pass theorem are not directly applicable. To overcome this difficulty, we constrain the nonlinear term and employ a two-stage linearization argument to establish the existence of least-energy solutions to an auxiliary mean-field game problem with general coercive potentials. In the vanishing coercive potential limit, we recover compactness by using maximal regularity for Hamilton-Jacobi equations together with Pohozaev-type identities, and show that the potential-free mean-field game system admits a classical solution, which is also an optimizer of a Gagliardo-Nirenberg type inequality. Finally, we analyze the mountain-pass geometry of the variational structure, which yields that the solution obtained above corresponds to a mountain-pass type solution of the original mean-field game system. These results provide an affirmative answer to the longstanding problem concerning the existence of mountain-pass solutions for mean-field game systems. Furthermore, as a byproduct, we relax the admissible set and provide a unified framework for establishing the optimal Gagliardo-Nirenberg inequality below the Sobolev critical exponent.