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Local uniqueness and non-degeneracy of blowup solutions for regular Liouville systems

Zetao Cheng, Haoyu Li, Lei Zhang

TL;DR

This work addresses local uniqueness and non-degeneracy of blowup solutions for regular Liouville systems on compact surfaces. It develops extremely precise higher-order expansions around bubbling points, refines coordinate systems to control boundary oscillations, and builds a Fredholm framework to interpret the Pohozaev identity within systems. By comparing two blowup solutions and propagating refined estimates through all orders, the authors prove that blowup profiles are locally unique and the linearized operator is non-degenerate under natural assumptions. The methods extend single-equation results to Liouville systems with continuum energy distributions, tackling the challenges of multiple interacting bubbles and the limited information provided by Pohozaev identities. The results rely on refined inner/outer expansions, a detailed expansion of the energy functional $\Lambda_{I,N}(\rho^k)$, and a robust invertibility/Fredholm theory to establish the vanishing of all frequency components, yielding the desired uniqueness and non-degeneracy.

Abstract

We study the following Liouville system defined on a compact Riemann surface $M$, \begin{equation} -Δu_i=\sum_{j=1}^n a_{ij}ρ_j\Big(\frac{h_j e^{u_j}}{\int_Ωh_j e^{u_j}}-1\Big)\mbox{ in }M\mbox{ for }i=1,\cdots,n,\nonumber \end{equation} where the coefficient matrix $A=(a_{ij})_{n\times n}$ is nonnegative, $h_1, \ldots, h_n$ are positive smooth functions, and $ρ_1, \ldots, ρ_n$ are positive constants. For the blowup solutions, we establish their uniqueness and non-degeneracy based on natural assumptions. The main results significantly generalize corresponding results for single Liouville equations \cite{BartJevLeeYang2019,BartYangZhang20241,BartYangZhang20242}. To overcome several substantial difficulties, we develop certain tools and extend them into a more general framework applicable to similar situations. Notably, to address the considerable challenge of a continuum of standard bubbles, we refine the techniques from Huang-Zhang \cite{HuangZhang2022} and Zhang \cite{Zhang2006,Zhang2009} to achieve extremely precise pointwise estimates. Additionally, to address the limited information provided by the Pohozaev identity, we develop a useful Fredholm theory to discern the exact role that the Pohozaev identity plays for systems. The considerable difference between systems and a single equation is also reflected in the location of blowup points, where the uncertainty of the energy type of the blowup point makes it difficult to determine the sufficiency of pointwise estimates. In this regard, we extend our highly precise pointwise estimates to any finite order. This aspect is drastically distinct from analyses of single equations.

Local uniqueness and non-degeneracy of blowup solutions for regular Liouville systems

TL;DR

This work addresses local uniqueness and non-degeneracy of blowup solutions for regular Liouville systems on compact surfaces. It develops extremely precise higher-order expansions around bubbling points, refines coordinate systems to control boundary oscillations, and builds a Fredholm framework to interpret the Pohozaev identity within systems. By comparing two blowup solutions and propagating refined estimates through all orders, the authors prove that blowup profiles are locally unique and the linearized operator is non-degenerate under natural assumptions. The methods extend single-equation results to Liouville systems with continuum energy distributions, tackling the challenges of multiple interacting bubbles and the limited information provided by Pohozaev identities. The results rely on refined inner/outer expansions, a detailed expansion of the energy functional , and a robust invertibility/Fredholm theory to establish the vanishing of all frequency components, yielding the desired uniqueness and non-degeneracy.

Abstract

We study the following Liouville system defined on a compact Riemann surface , \begin{equation} -Δu_i=\sum_{j=1}^n a_{ij}ρ_j\Big(\frac{h_j e^{u_j}}{\int_Ωh_j e^{u_j}}-1\Big)\mbox{ in }M\mbox{ for }i=1,\cdots,n,\nonumber \end{equation} where the coefficient matrix is nonnegative, are positive smooth functions, and are positive constants. For the blowup solutions, we establish their uniqueness and non-degeneracy based on natural assumptions. The main results significantly generalize corresponding results for single Liouville equations \cite{BartJevLeeYang2019,BartYangZhang20241,BartYangZhang20242}. To overcome several substantial difficulties, we develop certain tools and extend them into a more general framework applicable to similar situations. Notably, to address the considerable challenge of a continuum of standard bubbles, we refine the techniques from Huang-Zhang \cite{HuangZhang2022} and Zhang \cite{Zhang2006,Zhang2009} to achieve extremely precise pointwise estimates. Additionally, to address the limited information provided by the Pohozaev identity, we develop a useful Fredholm theory to discern the exact role that the Pohozaev identity plays for systems. The considerable difference between systems and a single equation is also reflected in the location of blowup points, where the uncertainty of the energy type of the blowup point makes it difficult to determine the sufficiency of pointwise estimates. In this regard, we extend our highly precise pointwise estimates to any finite order. This aspect is drastically distinct from analyses of single equations.

Paper Structure

This paper contains 23 sections, 26 theorems, 351 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Preliminary Results
  3. A higher order expansion around a blowup point
  4. An initial coordinate and classical estimates
  5. Choosing a refined coordinate
  6. A refined expansion inside the bubbling disks
  7. A refined estimate outside the bubbling disks
  8. A refined expansion of $\Lambda_{I,N}(\rho^k)$
  9. A refined Pohozaev identity
  10. Preliminary estimates on the difference between solutions
  11. Comparison between $\varepsilon_t^{(1),k}$ and $\varepsilon_t^{(2),k}$
  12. Comparison between $p_t^{(1),k}$ and $p_t^{(2),k}$
  13. Comparison between $u^{(1),k}$ and $u^{(2),k}$
  14. Proof of Theorem \ref{['t:MAIN']} and Theorem \ref{['non-deg']}
  15. A preliminary expansion of the difference between $u^{(1),k}$ and $u^{(2),k}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $u^{(1),k}=(u_1^{(1),k},..,u_n^{(1),k})$ and $u^{(2),k}=(u_1^{(2),k},...,u_n^{(2),k})$ be any two blowup solutions of (e:00k) with the same parameters $\rho^k=(\rho_1^k,..,\rho_n^k)$ and $q^*=(q^*_1,\cdots,q^*_N)$. Suppose $(\mathcal{H}_1)$, $(\mathcal{H}_2)$ and $(G)$ hold and one of the three

Figures (1)

  • Figure 1: The effect of the symmetric difference of the domain

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Proposition 2.1
  • Claim 3.1
  • Remark 3.1
  • Lemma 3.1
  • Proposition 3.1
  • Theorem 3.1
  • ...and 38 more