Second order classification for singular Liouville equations with a coefficient function

1. Problem Statement

The authors investigate the existence and classification of blow-up solutions for the singular Liouville equation on the unit ball $B_1 \subset \mathbb{R}^2$:
$$\begin{cases} -\Delta v = \lambda |x|^{2\alpha} V(x) e^v & \text{in } B_1, \\ v = 0 & \text{on } \partial B_1, \end{cases}$$
where:

• $\lambda > 0$ is a small parameter ($\lambda \to 0^+$).
• $V(x)$ is a positive, smooth potential function.
• The singularity is located at the origin ($x=0$).

The paper specifically focuses on the case where the singularity strength is an integer, $\alpha = N \in \mathbb{N}$. In this regime, the behavior of blow-up solutions is complex. Solutions can exhibit:

1. Simple blow-up: The solution behaves like a single "bubble" centered at the origin.
2. Non-simple blow-up: The solution develops multiple local maxima (bubbles) near the origin, often distributed symmetrically, violating the spherical Harnack inequality.

The central question addressed is: Under what conditions on the coefficient function $V(x)$ does a sequence of solutions blow up at the origin, and does this blow-up occur in a simple or non-simple manner?

2. Methodology

The authors employ a combination of integral identities (Pohozaev type) for the necessary condition and the Lyapunov-Schmidt reduction method for the sufficient condition.

A. Necessary Condition (Pohozaev Identities)

To determine the constraints on $V(x)$ required for blow-up to occur, the authors derive specific Pohozaev-type identities.

• Test Functions: They utilize test functions involving singular terms like $\frac{x_i}{|x|^2}$ and combinations of derivatives to exploit the invariance of the limiting problem under inversion ($x \mapsto x/|x|^2$).
• Domain Analysis: The identities are derived on the punctured domain $B_1 \setminus B_\epsilon$ and limits are taken as $\epsilon \to 0$.
• Asymptotic Analysis: They use refined estimates for the blow-up profile (derived from previous works [3], [39]) to evaluate boundary terms on the inner circle $|x|=\epsilon$.
• Key Insight: By analyzing the leading order terms of these identities, they relate the Hessian matrix of $V$ at the origin to the geometry of the blow-up profile.

B. Sufficient Condition (Lyapunov-Schmidt Reduction)

To prove the existence of solutions under specific conditions, they construct solutions via perturbation theory.

• Approximate Solution: They construct an approximate solution $PW_{\lambda, b}$ by projecting a "standard bubble" solution of the limiting Liouville equation onto the space $H^1_0(B_1)$. The bubble is centered at a parameter $b$ (which scales with $\sqrt{\lambda}$).
• Linearization: They analyze the linearized operator around this approximate solution. Crucially, they rely on the non-degeneracy of the limiting bubble solutions (established in [17], [21]), ensuring the kernel of the linearized operator is spanned only by variations in the parameters of the bubble family.
• Reduction: The problem is reduced to finding critical points of a finite-dimensional reduced functional. This involves solving a nonlinear equation for the remainder term $\phi$ and then finding the parameter $b$ that eliminates the Lagrange multipliers (orthogonality conditions).

3. Key Assumptions and Setup

The main results are established for the case $N=1$ (i.e., $\alpha=1$) on the unit ball.

• Potential $V$: $V \in C^3(B_1)$, $V > 0$, $V(0)=1$, $\nabla V(0)=0$, and $0$ is a non-degenerate critical point.
• Expansion: Near the origin, $V(x)$ admits the expansion:
  $$V(x) = 1 + \gamma_1 x_1^2 + \gamma_2 x_2^2 + O(|x|^3)$$
  (after a suitable rotation of coordinates).

4. Main Results

Theorem 1.2: Second-Order Classification

The paper provides a necessary and sufficient condition for the existence of a blow-up sequence $v_k$ at the origin for $N=1$.

Condition:
$$\det D^2 V(0) = \gamma_1 \gamma_2 > 0$$
This means the eigenvalues of the Hessian matrix of $V$ at the origin must have the same sign (i.e., $V$ must have a local minimum or a local maximum at the origin, but not a saddle point).

Consequences:

1. Existence: If $\gamma_1 \gamma_2 > 0$, there exists a sequence of solutions blowing up at the origin.
2. Nature of Blow-up: The blow-up is simple. The solution behaves asymptotically like a single bubble.
3. Location of the Bubble: The center of the bubble, denoted by $b_1$ (in the complex plane representation), is determined by the relative magnitudes of $\gamma_1$ and $\gamma_2$:
   • If $|\gamma_1| < |\gamma_2|$, then $b_1 > 0$.
   • If $|\gamma_1| > |\gamma_2|$, then $b_1 < 0$.
   • If $\gamma_1 = \gamma_2$ (radial symmetry), then $b_1 = 0$ (the bubble is centered exactly at the origin).
4. Exclusion of Non-Simple Blow-up: If $0$ is a non-degenerate critical point, non-simple blow-up is impossible. This resolves a gap in previous literature where non-simple blow-up was known to exist for specific symmetric potentials but its general exclusion was unclear.

5. Technical Significance and Contributions

• Resolution of the $N \in \mathbb{N}$ Case: While the case $\alpha \notin \mathbb{N}$ is well-understood (always simple blow-up), the integer case $N \in \mathbb{N}$ allows for non-simple blow-up. This paper rigorously classifies when simple blow-up occurs and proves that non-simple blow-up cannot occur if the potential has a non-degenerate critical point at the singularity.
• Second-Order Classification: Previous results (e.g., [13], [41]) established that $\Delta V(0) = 0$ is necessary for non-simple blow-up. This paper goes further by analyzing the second derivatives (Hessian) to provide a complete classification for $N=1$.
• Methodological Innovation:
  • The derivation of the necessary condition relies on a delicate Pohozaev identity involving singular test functions, which captures information about the Hessian matrix that standard identities miss.
  • The authors note that their method fails for $N \geq 2$ because the Pohozaev identities do not provide enough information to control the second derivatives of $V$ in higher-order singular cases (Remark 2.8).
• Geometric Interpretation: The result links the geometric properties of the potential $V$ (specifically the curvature/eigenvalues of the Hessian) directly to the topological structure of the blow-up (single vs. multiple bubbles).

6. Conclusion

The paper establishes that for the singular Liouville equation with integer singularity strength $N=1$, the existence of blow-up solutions at the origin is strictly governed by the definiteness of the Hessian of the potential $V$ at that point. If the Hessian is definite ($\det D^2 V(0) > 0$), simple blow-up occurs, and the bubble shifts slightly away from the origin depending on the anisotropy of $V$. If the Hessian is indefinite (saddle point), no such blow-up sequence exists. This provides a complete "second-order" classification for the $N=1$ case, significantly advancing the understanding of singular Liouville equations in mathematical physics and conformal geometry.