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).

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Correspondence to Wen Yang.

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Dedicated to Professor Jürgen Jost on the occasion of his 65th birthday.

Appendix

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 pi of angle 2πi + 1), βi > − 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 = eudx2 and \(g_{0}\sim |x|^{2{\upbeta }_{i}}\), for the local coordinate x which is equal to 0 at pi. 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 βn+ 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, βn+ 1 > − 1 are not integers and βi, 1 ≤ in are positive integers. Then the necessary and sufficient conditions for the existence of (A.1) on \({\mathscr{M}}=\mathbb {S}^{2}\) are the following:

  1. (a)

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

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

    If \([{\upbeta }_{0}]+[{\upbeta }_{n+1}]+{\sum }_{i=1}^{n}{\upbeta }_{i}\) is odd, then β0 + βn+ 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 αi, 1 ≤ in 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 αi, 1 ≤ in 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 αn+ 1 > − 1 such that either α0αn+ 1 or α0 + αn+ 1 is an integer. Moreover, we have:

  1. (i)

    If α0αn+ 1 is an integer, then \(|\alpha _{0}-\alpha _{n+1}|\leq {\sum }_{i=1}^{n}\alpha _{i}\).

  2. (ii)

    If α0 + αn+ 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 Mu 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 Mu is multiple of 4. Together with the fact that Mu is strictly positive, we get Mu = 2 + 23 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. □

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Lin, CS., Yang, W. Sharp Estimate for the Critical Parameters of SU(3) Toda System with Arbitrary Singularities, I. Vietnam J. Math. (2020). https://doi.org/10.1007/s10013-020-00450-y

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Keywords

  • SU(3)-Toda system
  • Conical singularity
  • Critical parameter
  • A priori estimate
  • Blowup solutions

Mathematics Subject Classification (2010)

  • 35J60
  • 35J55