Stability of mountain-pass solutions in potential-free mass-supercritical MFG systems

Determine the stability properties of the mountain-pass solutions constructed for the potential-free viscous stationary mean-field game system with aggregating local coupling in the mass-supercritical regime, specifically for the solution to equation (\ref{L^1_constrain_equation}).

Background

The paper proves the existence of classical mountain-pass solutions for a potential-free stationary MFG system in the mass-supercritical regime via an L^{1+\alpha}-constrained variational formulation, a two-stage linearization argument, and a vanishing-potential limit. These solutions correspond to critical points at the mountain-pass level and also optimize a Gagliardo–Nirenberg-type inequality.

While existence and variational characterization are established, the authors explicitly note that the stability of these mountain-pass solutions is not addressed in the paper and remains an open problem, particularly in contrast to local minimizers whose stability has been studied in related settings.