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On the Classification of blow-up solutions of a singular Liouville equation on the disk

Zhijie Chen, Houwang Li, Tuoxin Li, Juncheng Wei

Abstract

We study the blow-up behavior of solutions to the singular Liouville equation \[ Δ\tilde u+λe^{\tilde u}=4παδ_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $α>0$, $λ>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $α>0$ and $λ\in(0,λ_α)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

On the Classification of blow-up solutions of a singular Liouville equation on the disk

Abstract

We study the blow-up behavior of solutions to the singular Liouville equation where , and is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed and , the singular Liouville equation has exactly solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each , a unique -peak sequence whose blow-up points are the vertices of a regular -gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

Paper Structure

This paper contains 11 sections, 18 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Locating the blow-up set
  3. Local uniqueness results
  4. Local uniqueness of $m$-peak bubble solutions
  5. Local uniqueness of non-quantized singular bubble solution
  6. Local uniqueness of quantized singular bubble solutions
  7. Proof of Theorem \ref{['thm main_class']}
  8. Non-degeneracy results
  9. Non-degeneracy of $m$-peak bubble solutions
  10. Non-degeneracy of radial solutions
  11. Analysis of the Matrix

Key Result

Theorem 1.1

Let $m<1+\alpha$ be a positive integer and Then $(z_1,\cdots, z_m)\in (B^*)^m\setminus \Theta_m$ is a critical point of $\Phi_m$ if and only if $z_1,\cdots, z_m$ are the vertices of a regular $m$-polygon satisfying Moreover, $\Phi_{m'}$ has no critical points in $(B^*)^{m'}\setminus \Theta_{m'}$ for any integer $m'\ge1+\alpha$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 25 more