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

Abstract: 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 $γ$-homogeneous with $γ> 1$. 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.

• The paper introduces a variational framework that overcomes noncompactness in mass-supercritical regimes via a two-stage linearization method.
• It establishes the existence of mountain-pass (saddle) solutions for ergodic MFG systems, highlighting criticality thresholds and optimal inequality attainment.
• The analysis unifies mass-subcritical, critical, and supercritical regimes, offering new insights into solution multiplicity and bifurcation phenomena.

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

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Introduction and Problem Setting

This paper addresses the existence and structure of mountain-pass solutions for second-order stationary (ergodic) mean-field game (MFG) systems on $\mathbb{R}^n$ under mass-supercritical coupling and in the absence of a confining potential. The focus is on MFGs governed by highly nonlinear, homogeneous Hamiltonians and strongly aggregating, nonlinear local couplings. Formally, the system investigated is: $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$ where $u$ is the value function, $m$ is the population density, $H$ is a smooth strictly convex Hamiltonian, $f(m) = -m^\alpha$ with supercritical exponent $\alpha$, and $V(x)\equiv 0$ (potential-free case).

The mass-supercritical regime ($\alpha > \alpha^* := \gamma'/n$, with $\gamma'$ the conjugate exponent to the Hamiltonian's homogeneity) is especially challenging: the associated variational structure yields an action functional that is unbounded below, so standard minimization approaches and classical mountain-pass theorems do not apply directly.

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Variational Framework and Analytical Obstacles

The authors leverage a constrained variational approach, as stationary MFGs with local couplings can be formulated as constrained optimization problems of the form: $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$0 where

$$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$1

and $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$2 is the Legendre transform of $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$3.

In the mass-supercritical case, $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$4 is unbounded from below, and the lack of global compactness and regularity due to the lack of a potential further undermine direct minimization. Additionally, the nonsmooth structure of the admissible set and constraint functional precludes straightforward use of deformation-type arguments to obtain mountain-pass solutions.

One key technical novelty is the adoption of an $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$5 normalization constraint (as opposed to $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$6 mass constraint), inspired by analogies with mass-constrained nonlinear Schrödinger PDEs, which better interacts with the critical exponents involved in the Pohozaev identity and yields better regularity and compactness.

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Existence of Solutions: Two-Stage Linearization and Vanishing Potential Method

To overcome variational nonsmoothness and compactness difficulties, the paper develops a two-step regularization/linearization scheme:

1. Coercive Approximation: First, problems are analyzed in the presence of a confining potential $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$7 (with mild regularity/growth), which provides good compactness and allows the construction of $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$8-normalized minimizers.
2. Two-Stage Linearization: The nonlinear normalization constraint is addressed via two successive linearizations: first by mollifying the density, then by linearizing the constraint functional in the minimization process, allowing the existence of a regular minimizer together with a value function $$\left\{ \begin{array}{ll} -\Delta u + H(\nabla u) + \lambda = f(m) + V(x), & x \in \mathbb{R}^n \ \Delta m + \nabla\cdot (m \nabla H(\nabla u)) = 0, & x \in \mathbb{R}^n \ \int_{\mathbb{R}^n} m \, dx = M > 0 & \end{array} \right.$$9 solving the Hamilton–Jacobi component.
3. Limit Passage: Through uniform regularity estimates (maximal regularity for the Hamilton–Jacobi equation, exponential decay for the densities, and compactness from the potential), the vanishing-potential limit is taken. As $u$0, this scheme yields a solution to the original potential-free system in the whole space.

The functional $u$1 is shown to possess a mountain-pass geometry in the scaling-invariant regime, and the associated critical level is obtained by a minimax characterization. The solution is not a minimizer but a genuine saddle, associated to an optimal constant in a MFG-adapted Gagliardo–Nirenberg type inequality.

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Sharp Inequalities and Structure of Solutions

A fundamental byproduct is the derivation and attainment of an optimal Gagliardo–Nirenberg-type inequality: $u$2 for $u$3 in the Sobolev-subcritical range. The optimal densities are proved to be exactly the (scaled) mountain-pass critical points of the MFG variational structure, showing that the mountain-pass solutions are not only critical points but actually extremizers of a sharp functional inequality.

The paper delivers:

• Existence of classical (regular) solutions $u$4 to the original ergodic MFG system with aggregating nonlinearity and no potential, which correspond to mountain-pass (not minimizing) critical points.
• Explicit scaling/categorization of the solution regime in terms of $u$5 relative to critical exponents. In the supercritical regime, minimizers do not exist, but mountain-pass solutions do.
• A unification of minimization (mass-subcritical), critical (minimal-mass-critical), and saddle (supercritical) solution structures, clarifying their variational context.

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Technical Contributions

Several analytic advances underpin these results:

• Local and global maximal regularity for viscous Hamilton–Jacobi equations with superlinear homogeneous Hamiltonians, crucial for handling degenerate coefficients and unbounded domains.
• A compactness framework for sequences of densities and fluxes in the whole Euclidean space, leveraging exponential decay (via Pohozaev-type arguments) and regularity.
• Two-stage linearization to handle critical point theory for nonsmooth constraints.
• Sharp Pohozaev-type identities linking variational integrals and functional critical points, essential for connecting existence results to extremal inequalities.

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Implications and Outlook

This work resolves a longstanding open question about the existence of mountain-pass solutions in focusing, stationary second-order MFGs in the mass-supercritical regime, for general $u$6 strictly convex homogeneous Hamiltonians and local power-type coupling. The solution theory developed here provides new insights into the energy landscape and solution multiplicity of MFG systems, showing that nonuniqueness and saddle-type solutions arise naturally in supercritical interaction regimes.

From a mathematical standpoint, the results illuminate deep connections with critical point theory for nonlinear PDEs, particularly in the way scaling, noncompactness, and optimal inequalities interact. The extension and adaptation of techniques from mass-constrained nonlinear Schrödinger models to MFG systems is particularly noteworthy.

On the theoretical side, these findings point to new avenues for:

• Classification of solution multiplicity and concentration phenomena in large-population interacting agent systems.
• Development of a more complete bifurcation theory, analogously to nonlinear Schrödinger and chemotaxis models, for ergodic MFGs.
• Investigation of stability and dynamical behavior of mountain-pass (and other non-minimizing) solutions.

Practically, a precise grasp of solution multiplicity, lack of uniqueness, and the corresponding energy structures in MFGs is essential for understanding the emergence of patterns, singular behaviors, and regime transitions in large interacting systems, with applications in economics, statistical physics, and collective dynamics.

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Conclusion

The paper establishes the existence and variational characterization of mountain-pass solutions in mass-supercritical, aggregation-driven viscous ergodic mean-field games on $u$7 with homogeneous convex Hamiltonians. The construction utilizes constrained minimization, a novel two-stage linearization scheme, rigorous passage to the potential-free limit, and variational identities. Optimal functional inequalities are established, and solution regimes are classified according to the criticality of the coupling exponent. This advances the mathematical understanding of stationary MFG systems and opens up new directions for bifurcation theory and qualitative analysis in high-dimensional mean-field games (2604.01662).