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A simple proof of the Uniqueness of blow-up solutions of mean field equations

Lina Wu, Wenming Zou

TL;DR

This work establishes a simple proof of the uniqueness of blow-up solutions for mean field equations on compact Riemann surfaces, including singular cases with negative poles ($\alpha_M\le 0$). It leverages a Green's function reformulation and refined local asymptotics to extend prior results and to derive a concise contradiction-argument that forces all relevant vanishing parameters to be zero, hence uniqueness. The analysis combines improved regular-point expansions, negative-source bubbling, and careful energy balance near multiple blow-up points, offering a more accessible route than previous technical proofs. The results have implications for related Dirichlet problems and broaden the scope of uniqueness in mean field-type equations with singular inputs.

Abstract

For a regular mean field equation defined on a compact Riemann surface, an important work of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4} proved a uniqueness theorem for blow-up solutions under non-degeneracy assumptions. However, the proof is highly nontrivial and challenging to read. In this article, we not only provide a simple proof for the regular equation but also extend our proof to the case of singular equations with negative singular poles. Our proof supplements what is not written in a recent outstanding work by Bartolucci-Yang-Zhang \cite{byz-1}.

A simple proof of the Uniqueness of blow-up solutions of mean field equations

TL;DR

This work establishes a simple proof of the uniqueness of blow-up solutions for mean field equations on compact Riemann surfaces, including singular cases with negative poles (). It leverages a Green's function reformulation and refined local asymptotics to extend prior results and to derive a concise contradiction-argument that forces all relevant vanishing parameters to be zero, hence uniqueness. The analysis combines improved regular-point expansions, negative-source bubbling, and careful energy balance near multiple blow-up points, offering a more accessible route than previous technical proofs. The results have implications for related Dirichlet problems and broaden the scope of uniqueness in mean field-type equations with singular inputs.

Abstract

For a regular mean field equation defined on a compact Riemann surface, an important work of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4} proved a uniqueness theorem for blow-up solutions under non-degeneracy assumptions. However, the proof is highly nontrivial and challenging to read. In this article, we not only provide a simple proof for the regular equation but also extend our proof to the case of singular equations with negative singular poles. Our proof supplements what is not written in a recent outstanding work by Bartolucci-Yang-Zhang \cite{byz-1}.
Paper Structure (7 sections, 8 theorems, 164 equations)

This paper contains 7 sections, 8 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminary Estimates
  3. Local asymptotic analysis near a blow-up point
  4. Asymptotic analysis around a negative source
  5. Asymptotic analysis around a regular blow-up point
  6. Proof of the Main Theorem
  7. First estimates about $\sigma_k$.

Key Result

Theorem A

($\alpha_M>0$) Let $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of bubbling solutions of (m-equ) with the same $\rho_k$: $\rho_k^{(1)}=\rho_k=\rho_k^{(2)}$ and the same blow-up set: $\{p_1,\cdots,p_m\}$. Suppose $(\alpha_1,\cdots,\alpha_N)$ satisfies (largest-s), $\alpha_M>0$, $L(\mathbf{p})\neq 0$

Theorems & Definitions (12)

  • Theorem A
  • Theorem 1.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 2 more