Sharp Estimate for the Critical Parameters of SU(3) Toda System with Arbitrary Singularities, I

Abstract

To obtain the a priori estimate of Toda system, the first step is to determine all the possible local masses of blow up solutions. In this paper we study this problem and improve the main results in (Anal. PDE 8, 807–837, 2015). Our method is based on a recent work by Eremenko–Gabrielov–Tarasov (Illinois J. Math. 58, 739–745, 2014).

Appendix

In this section, we shall study the theorem of [13, Theorem 4.1] and interpret it in terms of the language which could be used in this paper.

Let ${\mathscr{M}}$ be a compact Riemann surface, and $\{p_{1},\dots ,p_{n}\}$ is a finite set of points of ${\mathscr{M}}$. In a local coordinate system $x\in {\Omega }\subset \mathbb {C}$, a corresponding metric g₀ is called conformal with conical singularity p_i of angle 2π(β_i + 1), β_i > − 1 if there exists local coordinates $x\in {\Omega }\subset \mathbb {C}$ and $u\in C^{2}({\Omega }\setminus \{p_{0},\dots ,p_{n+1}\})$ such that g₀ = e^udx² and $g_{0}\sim |x|^{2{\upbeta }_{i}}$, for the local coordinate x which is equal to 0 at p_i. Here u is the solution of the following equation

$$ {\Delta} u+2e^{u}=4\pi\sum\limits_{i=0}^{n+1}{\upbeta}_{i}\delta_{p_{i}}\quad\text{in }~\mathcal{M},\qquad {\int}_{\mathcal{M}}e^{u}dx<+\infty. $$

(A.1)

In [13] the authors consider the problem: let $p_{0},\dots ,p_{n+1}$ be distinct points on Riemann sphere $\mathbb {S}^{2}$ and numbers ${\upbeta }_{1},\dots ,{\upbeta }_{n}$ are positive inters, while β₀ and β_n+ 1 are not integers. They have the following result:

Theorem A.1

Suppose $p_{0},\dots ,p_{n+1}$ are n + 2 distinct points on the Riemann sphere and numbers β₀, β_n+ 1 > − 1 are not integers and β_i, 1 ≤ i ≤ n are positive integers. Then the necessary and sufficient conditions for the existence of (A.1) on ${\mathscr{M}}=\mathbb {S}^{2}$ are the following:

• (a) If $[{\upbeta }_{0}]+[{\upbeta }_{n+1}]+{\sum }_{i=1}^{n}{\upbeta }_{i}$ is even, then β₀ −β_n+ 1 is an integer, and

  $$ |[{\upbeta}_{0}]-[{\upbeta}_{n+1}]|\leq\sum\limits_{i=1}^{n}{\upbeta}_{i}. $$

• (b) If $[{\upbeta }_{0}]+[{\upbeta }_{n+1}]+{\sum }_{i=1}^{n}{\upbeta }_{i}$ is odd, then β₀ + β_n+ 1 is an integer, and

  $$ [{\upbeta}_{0}]+[{\upbeta}_{n+1}]+3\leq\sum\limits_{i=1}^{n}{\upbeta}_{i}. $$

Let us interpret Theorem A.1 for the following equation on $\mathbb {R}^{2}$:

$$ {\Delta} u+2e^{u}=4\pi\sum\limits_{i=0}^{n}\alpha_{i}\delta_{p_{i}}\quad\text{in }~\mathbb{R}^{2},\qquad {\int}_{\mathbb{R}^{2}}e^{u}dx<+\infty, $$

(A.2)

where $p_{0},\dots ,p_{n}$ are distinct points in $\mathbb {R}^{2}$, α₀ > − 1 and α_i, 1 ≤ i ≤ n are positive integers.

Theorem A.2

Let $p_{0},\dots ,p_{n}$ be n + 1 distinct points in $\mathbb {R}^{2}$ and u be a solution of (A.2). Suppose α_i, 1 ≤ i ≤ n are positive integers and α₀ > − 1 is not an integer, then any solution u of (A.2) satisfies

$$ \frac{1}{2\pi}{\int}_{\mathbb{R}^{2}}e^{u}dx=\sum\limits_{i=0}^{n+1}\alpha_{i}+2 $$

for some α_n+ 1 > − 1 such that either α₀ − α_n+ 1 or α₀ + α_n+ 1 is an integer. Moreover, we have:

• (i) If α₀ − α_n+ 1 is an integer, then $|\alpha _{0}-\alpha _{n+1}|\leq {\sum }_{i=1}^{n}\alpha _{i}$.

• (ii) If α₀ + α_n+ 1 is an integer, then $\alpha _{0}+\alpha _{n+1}+2\leq {\sum }_{i=1}^{n}\alpha _{i}$.

Proof

Notice that in terms of notations of Theorem A.1, we have

$$ \alpha_i={\upbeta}_i,\quad 0\leq i\leq n+1. $$

Thus, Theorem A.2 follows from Theorem A.1. □

We denote

$$ M_{u}:=\frac{1}{2\pi}{\int}_{\mathbb{R}^{2}}e^{u}dx. $$

(A.3)

A direct consequence of Theorem A.2 is the following,

Proposition A.1

Let M_u be defined in (A.3). Suppose the assumption of Theorem A.2 holds. Then we have

$$ M_{u}= \left\{\begin{array}{ll} 2(\alpha_{0}+1)+2\ell_{1}\quad &\text{if }~\alpha_{0}\notin\mathbb{N},~\alpha_{0}-\alpha_{n+1}\in\mathbb{Z},\\ 2+2\ell_{2}\quad &\text{if }~\alpha_{0}\notin\mathbb{N},~\alpha_{0}+\alpha_{n+1}\in\mathbb{Z},\\ 2+2\ell_{3}\quad &\text{if }~\alpha_{0}\in\mathbb{N}, \end{array}\right. $$

where $\ell _{1},\ell _{2},\ell _{3}\in \mathbb {N}\cup \{0\}$ and

$$ 0\leq \ell_{1},\ell_{2}\leq \sum\limits_{i=1}^{n}\alpha_{i}\quad \text{ and }\quad \ell_{3}>\frac12\left( \sum\limits_{i=0}^{n}\alpha_{i}-1\right). $$

Proof

If $\alpha _{0}\notin \mathbb {N}$ and $\alpha _{0}-\alpha _{n+1}\in \mathbb {Z}$, then using Theorem A.2 we get

$$ M_{u}=2\alpha_{0}+2+\left( \alpha_{n+1}-\alpha_{0}+\sum\limits_{i=1}^{n}\alpha_{i}\right) = 2(\alpha_{0}+1)+2\ell_{1}, $$

with $\ell _{1}\in \mathbb {N}\cup \{0\}$ and $0\leq \ell _{1}\leq {\sum }_{i=1}^{n}\alpha _{i}$. Similarly, we could obtain the conclusion for $\alpha _{0}\notin \mathbb {N}$ and $\alpha _{0}+\alpha _{n+1}\in \mathbb {Z}$. If $\alpha _{0}\in \mathbb {N}$, by [24, Theorem 2.1] we have M_u is multiple of 4. Together with the fact that M_u is strictly positive, we get M_u = 2 + 2ℓ₃ with $\ell _{3}\in \mathbb {N}$. On the other hand, by the standard potential analysis we get

$$ u(x)=-2\left( M_u-\sum\limits_{i=0}^n\alpha_i\right)\log|x|+O(1)\quad \text{at }~\infty. $$

It together with ${\int \limits }_{\mathbb {R}^{2}}e^{u}dx<+\infty$ implies that

$$ 2+2\ell_3>1+\sum\limits_{i=0}^n\alpha_i, $$

which is equivalent to

$$ \ell_3>\frac12\left( \sum\limits_{i=0}^n\alpha_i-1\right). $$

Hence, we finish the proof. □