Existence of mountain-pass solutions for stationary mean-field game systems in whole space and bounded domains

Establish the existence of mountain-pass type solutions for the viscous stationary mean-field game system (MFG-SS), consisting of the coupled Hamilton–Jacobi and Fokker–Planck equations with mass constraint, in both the whole space R^n and in bounded domains.

Background

The paper studies the stationary viscous mean-field game (MFG) system comprising a Hamilton–Jacobi equation coupled to a Fokker–Planck equation with a mass constraint. In the mass-supercritical regime, direct variational approaches are difficult because the natural functional is unbounded below, and standard tools such as the classical deformation lemma do not apply due to non-smoothness.

Prior work established local minimizers in bounded domains (under certain regimes), but local minimizers are believed not to exist in the whole space. The authors note that, before their contribution, the existence of mountain-pass solutions—critical points characterized by a min–max principle—"remains largely open" both in the whole space and in bounded domains. Their results address a potential-free, mass-supercritical setting in the whole space; the broader problem as stated remains a key theme in the literature.