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Asymptotic Analysis and Uniqueness of blowup solutions of non-quantized singular mean field equations

Daniele Bartolucci, Wen Yang, Lei Zhang

TL;DR

The paper tackles the problem of uniqueness for bubbling solutions of a singular mean field equation on compact Riemann surfaces, allowing blowup at regular points or non-quantized singular sources. It develops a robust local–global blowup analysis, deriving fourth-order asymptotic expansions around each blowup point and introducing global nondegeneracy quantities $L(\mathbf{p})$ and $D(\mathbf{p})$ to govern uniqueness. A key advance is removing the prior restriction that positive singular blowup sources must be critical points of Kirchhoff–Routh type functionals, replacing it with refined estimates and a comprehensive concentration theory for Liouville-type equations. The results yield uniqueness of bubbling configurations for fixed blowup data and pave the way for nondegeneracy results and topological degree computations, with implications for mean field models in geometric PDEs and related vortex theories.

Abstract

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions as far as blowup points are either regular points or non-quantized singular sources. In particular the uniqueness result covers the most general case extending or improving all previous works of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4,bart-4-2} and Wu-Zhang \cite{wu-zhang-ccm}. For example, unlike previous results, we drop the assumption of singular sources being critical points of a suitably defined Kirchoff-Routh type functional. Our argument is based on refined estimates, robust and flexible enough to be applied to a wide range of problems requiring a delicate blowup analysis. In particular we come up with several new estimates of independent interest about the concentration phenomenon for Liouville-type equations.

Asymptotic Analysis and Uniqueness of blowup solutions of non-quantized singular mean field equations

TL;DR

The paper tackles the problem of uniqueness for bubbling solutions of a singular mean field equation on compact Riemann surfaces, allowing blowup at regular points or non-quantized singular sources. It develops a robust local–global blowup analysis, deriving fourth-order asymptotic expansions around each blowup point and introducing global nondegeneracy quantities and to govern uniqueness. A key advance is removing the prior restriction that positive singular blowup sources must be critical points of Kirchhoff–Routh type functionals, replacing it with refined estimates and a comprehensive concentration theory for Liouville-type equations. The results yield uniqueness of bubbling configurations for fixed blowup data and pave the way for nondegeneracy results and topological degree computations, with implications for mean field models in geometric PDEs and related vortex theories.

Abstract

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions as far as blowup points are either regular points or non-quantized singular sources. In particular the uniqueness result covers the most general case extending or improving all previous works of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4,bart-4-2} and Wu-Zhang \cite{wu-zhang-ccm}. For example, unlike previous results, we drop the assumption of singular sources being critical points of a suitably defined Kirchoff-Routh type functional. Our argument is based on refined estimates, robust and flexible enough to be applied to a wide range of problems requiring a delicate blowup analysis. In particular we come up with several new estimates of independent interest about the concentration phenomenon for Liouville-type equations.
Paper Structure (13 sections, 25 theorems, 584 equations)

This paper contains 13 sections, 25 theorems, 584 equations.

Table of Contents

  1. Introduction
  2. Preliminary Estimates
  3. A uniqueness lemma
  4. Local asymptotic analysis near a blowup point
  5. Asymptotic analysis for $\alpha>0$
  6. Asymptotic analysis around a negative pole
  7. Asymptotic analysis around a regular blowup point
  8. Proof of Theorem \ref{['main-theorem-2']}
  9. First estimates about $\sigma_k$.
  10. Proof of $b_0=0$: improved estimates about $\xi_k$.
  11. Proof of $b_1=b_2=0$
  12. Proof of Other Theorems
  13. Appendix

Key Result

Theorem 1.1

Let $\nu_k^{(1)}$ and $\nu_k^{(2)}$ 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 blowup set: $\{p_1,\cdots,p_m\}$. Suppose $(\alpha_1,\cdots,\alpha_N)$ satisfies (largest-s), $\alpha_M>0$, $L(\mathbf{p})\neq 0$ and, as far

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem1']}.
  • Lemma 2.2
  • ...and 53 more