Skew-Symmetric Singular Liouville System

• The skew-symmetric singular Liouville system is a coupled nonlinear elliptic PDE model arising in non-relativistic Chern-Simons theory with purely mutual interaction and prescribed vortex strengths.
• It employs a variational approach with an indefinite mixed gradient term and a variable transformation to reduce complexity and ensure coercivity for solution analysis.
• Critical thresholds, aided by improved Moser–Trudinger inequalities and bubble-test functions, underpin uniform compactness and multiplicity results via Morse theory.

The skew-symmetric singular Liouville system is a class of coupled nonlinear elliptic partial differential equations that arises as the self-dual equations of the non-relativistic $[U(1)]^2$ Chern-Simons model with purely mutual interaction. The system is characterized by an off-diagonal coupling matrix $K_{ij}$ with $K_{ij}=0$ for $i=j$ and $K_{12}=K_{21}=1$, and it incorporates singularities associated with vortex points having prescribed strengths. This system embodies a mean-field reformulation relevant to models in gauge theory and statistical mechanics, particularly where mutual interaction dominates and self-interaction is absent (Jevnikar et al., 11 Jan 2026).

1. Mathematical Formulation

Let $(M, g)$ denote a closed Riemannian surface with total area normalized to $|M|=1$. Vortex singularities are located at points $p_1,\dots, p_N \in M$, each associated with a strength parameter $\alpha_j \geq 0$. The Green's function $G_{p}$ solves $-\Delta_g G_{p} = \delta_p - 1$, subject to $\int_M G_p \, dV_g = 0$.

Upon implementing the standard desingularization procedure, namely

$$u_i \mapsto u_i + 4\pi\sum_{j=1}^N \alpha_j G_{p_j}, \qquad h_i \mapsto \widetilde h_i(x) = h_i(x) e^{-4\pi\sum_j \alpha_j G_{p_j}(x)},$$

the system is reformulated for zero-average variables $u_i \in H^1(M)$, $\int_M u_i = 0$: $$\begin{cases} -\Delta_g u_1 = \rho_2 \left( \frac{\widetilde h_2 e^{u_2}}{\int_M \widetilde h_2 e^{u_2}} - 1 \right),\ -\Delta_g u_2 = \rho_1 \left( \frac{\widetilde h_1 e^{u_1}}{\int_M \widetilde h_1 e^{u_1}} - 1 \right), \end{cases} \qquad \int_M u_i\, dV_g = 0,$$ where $\rho_1, \rho_2 > 0$ are coupling constants, and each $\widetilde h_i > 0$ behaves locally as $d(x, p_j)^{2\alpha_j}$ near $p_j$.

The underlying mean-field system generalizes to

$$-\Delta_g u_i = \lambda \sum_{j=1}^2 K_{ij} e^{u_j} - 4\pi \sum_j \alpha_j \delta_{p_j}, \qquad K = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}.$$

2. Variational Structure and Indefinite Functional

System $(\ast)$ is the Euler-Lagrange equation for the action functional

$$\mathcal J_\rho(u_1, u_2) = \int_M \nabla u_1 \cdot \nabla u_2 \, dV_g - \rho_2 \log \int_M \widetilde h_2 e^{u_2}\, dV_g - \rho_1 \log \int_M \widetilde h_1 e^{u_1}\, dV_g,$$

posed on the zero-mean Hilbert product space $H^1_0(M) \times H^1_0(M)$. The presence of the mixed gradient term $\int_M \nabla u_1 \cdot \nabla u_2$ renders $\mathcal J_\rho$ strongly indefinite, invalidating direct application of classical minimax methods or mountain-pass arguments typically used in variational analysis.

3. Coercivity via Reduction to One Variable

To address indefiniteness, one employs the variable transformation: $u_1 = F - G,\qquad u_2 = F + G,$ with $F, G \in H^1_0(M)$. This yields

$$\mathcal J_\rho(F-G, F+G) = \int_M |\nabla F|^2 - |\nabla G|^2\, dV_g - \rho_2 \log \int_M \widetilde h_2 e^{F+G} - \rho_1 \log \int_M \widetilde h_1 e^{F-G}.$$

For fixed $F$, the auxiliary functional

$$I_\rho(F,G) = \int_M |\nabla G|^2 \, dV_g + \rho_2 \log \int_M \widetilde h_2 e^{F+G} + \rho_1 \log \int_M \widetilde h_1 e^{F-G}$$

is strictly convex and coercive in $G$, guaranteeing a unique minimizer $\widetilde G(F) \in H^1_0(M)$ for each $F$.

The reduced functional

$$\widetilde J_\rho(F) = \int_M |\nabla F|^2\, dV_g - I_\rho(F, \widetilde G(F)),$$

is of class $C^1$ and remains indefinite, but its critical points yield solutions to the original system.

4. Compactness, Blow-Up, and Critical Thresholds

Blow-up analysis, notably by Gu–Lin–Zhang, identifies possible failure points for compactness corresponding to the ‘critical set’: $$\Lambda = \left\{ (\rho_1, \rho_2) :\, \frac{\rho_1 \rho_2}{\rho_1 + \rho_2} = 4\pi n,\, n \in \mathbb N \right\}.$$ If $(\rho_1, \rho_2) \notin \Lambda$, all $C^2$ bounds are obtained for solutions, and there are no Palais–Smale failures at finite level for $\widetilde J_\rho$. This ensures the validity of variational and Morse-theoretic arguments in these parameter regimes.

5. Topology of Sublevels and Barycenter Homology

For intervals $[a, b]$ excluding critical values, sublevel inclusions $$\widetilde J_\rho^a \hookrightarrow \widetilde J_\rho^b$$ are deformation retracts, rendering high sublevels contractible. For

$$\rho_1, \rho_2 > 8k\pi,\qquad \frac{\rho_1 + \rho_2}{2} < 8(k+1)\pi,$$

low sublevels $\widetilde J_\rho^{-L}$ possess the homology of the $k$-th formal barycenter set: $$M_k = \left\{ \sum_{i=1}^k t_i \delta_{x_i} : \sum t_i = 1,\, t_i \geq 0,\, x_i \in M \right\}.$$ Continuous maps $\Psi : \widetilde J_\rho^{-L} \to M_k$ and $\Phi : M_k \to \widetilde J_\rho^{-L}$ demonstrate $\Psi \circ \Phi$ is homotopic to the identity on $M_k$, confirming that $\widetilde J_\rho^{-L}$ is not contractible, with an injective map at the homology level: $$H_*(M_k) \hookrightarrow H_*(\widetilde J_\rho^{-L}).$$

6. Morse Theory and Solution Counting

With generic choices for $(g, h_1, h_2)$, all critical points of $\widetilde J_\rho$ are shown to be nondegenerate. By employing Morse inequalities for Hilbert-manifold $C^1$-functionals and leveraging deformation-retraction results in lieu of Palais–Smale, one obtains lower bounds for the number of critical points in level intervals $[-L, L]$ by the Betti numbers of $(\widetilde J_\rho^L, \widetilde J_\rho^{-L})$: $$\#\{\text{critical points}\} \geq \sum_q \operatorname{rank} H_q(\widetilde J_\rho^{-L}) \geq \sum_q \operatorname{rank} H_q(M_k).$$ For surfaces of positive genus $g>0$, and under the above parameter regime, the multiplicity result is given by

$$\#\{\text{solutions}\} \geq \binom{k + g - 1}{g - 1}.$$

7. Analytical Foundations and Main Theorems

Improved Moser–Trudinger inequalities underpin concentration-compactness techniques; if a normalized measure is supported across $\ell$ well-separated regions,

$$2\log\!\int_M\!h\,e^F \leq \frac{1+\varepsilon}{8\pi \ell} \int_M |\nabla F|^2 + C.$$

Explicit bubble-test functions on $M_k$,

$$\varphi_{\lambda, \sigma}(y) = \log \sum_{i=1}^k t_i \left( \frac{\lambda}{1 + \lambda^2 d(y, x_i)^2} \right)^2 - \log \pi,$$

demonstrate energy concentration as $\lambda \to \infty$.

The main results can be summarized as follows:

• Theorem A (Uniform Compactness): If $(\rho_1, \rho_2)$ is outside $\Lambda$, all solutions are $C^2$-bounded on $M$.
• Theorem B (Existence on $S^2$ without Singularities): For $M=S^2$, $\alpha_j=0$ and adequate parameter constraints, at least one solution exists.
• Theorem C (Multiplicity on High-Genus Surfaces): For genus $g>0$, with generic background data and under suitable conditions, the solution count satisfies $$\#\{\text{solutions}\} \geq \binom{k + g - 1}{g - 1}$$.

These theorems collectively establish a comprehensive existence and multiplicity framework for the skew-symmetric singular Liouville system within the context of non-relativistic Chern-Simons models with purely mutual interaction (Jevnikar et al., 11 Jan 2026).