On $SU(3)$ Toda system with multiple singular sources

TL;DR

This paper investigates the existence and non-existence of solutions for a singular $SU(3)$ Toda system with multiple sources in the plane, extending previous results and exploring higher-dimensional analogs.

Contribution

It generalizes Luo-Tian's results for a singular Liouville equation to the $SU(3)$ Toda system with multiple singular sources and studies higher order equations in $ eal^n$.

Findings

Established conditions for existence of solutions.

Proved non-existence under certain parameter constraints.

Extended analysis to higher order Liouville equations.

Abstract

We consider the singular $SU(3)$ Toda system with multiple singular sources \begin{align*} \left\{\begin{array}{ll}-\Delta w_1=2e^{2w_1}-e^{w_2}+2\pi\sum_{\ell=1}^m\beta_{1,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\ \rule{0cm}{.5cm} -\Delta w_2=2e^{2w_2}-e^{w_1}+2\pi\sum_{\ell=1}^m\beta_{2,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^2 \\ w_i(x)=-2\log|x|+O(1)\quad\text{as }|x|\to\infty,\, i=1,2, \end{array}\right.\end{align*} with $m\geq 3$ and $\beta_{i,\ell}\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on $\beta_{i,\ell}$. This generalizes Luo-Tian's \cite{Luo-Tian} result for a singular Liouville equation in $\mathbb{R}^2$. We also study existence results for a higher order singular Liouville equation in $\mathbb{R}^n$.

Equations

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta w_{1}=2e^{2w_{1}}-e^{w_{2}}+2\pi\sum_{\ell=1}^{m}\beta_{1,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^{2}\\ \rule{0.0pt}{14.22636pt}-\Delta w_{2}=2e^{2w_{2}}-e^{w_{1}}+2\pi\sum_{\ell=1}^{m}\beta_{2,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^{2}\\ w_{i}(x)=-2\log|x|+O(1)\quad\text{as }|x|\to\infty,\,i=1,2,\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta w_{1}=2e^{2w_{1}}-e^{w_{2}}+2\pi\sum_{\ell=1}^{m}\beta_{1,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^{2}\\ \rule{0.0pt}{14.22636pt}-\Delta w_{2}=2e^{2w_{2}}-e^{w_{1}}+2\pi\sum_{\ell=1}^{m}\beta_{2,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^{2},\end{array}\right.$$

$$-\Delta w=e^{2w}+2\pi \sum _{\ell =1}^m{\beta }_{\ell }{\delta }_{P_{\ell }}\text{in }{\mathbb{R}}^2.$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta w=e^{2w}&\quad\text{in }\mathbb{R}^{2}\setminus\{P_{1},P_{2},\dots,P_{m}\}\\ w(x)=-\beta_{\ell}\log|x-P_{\ell}|+h_{\ell}(x)&\quad\text{around each }P_{\ell}\\ \rule{0.0pt}{14.22636pt}w(x)=-2\log|x|+h_{m+1}(x)&\quad\text{as }|x|\to\infty\\ \beta_{\ell}\in(0,1)&\quad\ell=1,\dots,m\end{array}\right.$$

$$\sum _{\ell =1}^m{\beta }_i<2\text{and }\sum _{\ell \ne j}{\beta }_{\ell }>{\beta }_j\text{for every }j=1,2,\dots ,m.$$

$$\displaystyle\left\{\begin{array}[]{ll}w_{i}(x)=-\beta_{i,\ell}\log|x-P_{\ell}|+h_{i,\ell}&\quad\text{around each point }P_{\ell}\\ \rule{0.0pt}{14.22636pt}w_{i}(x)=-2\log|x|+h_{i,m+1}&\quad\text{as }|x|\to\infty\\ h_{i,\ell}\text{ is continuous in a neighborhood of }P_{\ell},\end{array}\right.$$

$$u_i(x)=w_i(x)+\sum _{\ell =1}^m{\beta }_{i,\ell }\log |x-P_{\ell }|,i=1,2.$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta u_{1}=2K_{1}e^{2u_{1}}-K_{2}e^{2u_{2}}\quad\text{in }\mathbb{R}^{2}\\ -\Delta u_{2}=2K_{2}e^{2u_{2}}-K_{1}e^{2u_{1}}\quad\text{in }\mathbb{R}^{2}\\ \rule{0.0pt}{14.22636pt}K_{i}(x):=\prod_{\ell=1}^{m}\frac{1}{|x-P_{\ell}|^{2\beta_{i,\ell}}}\quad i=1,2.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}u_{i}(x)=-{\beta_{i}}\log|x|+\text{ a bounded continuous function}&\quad\text{on }B_{1}^{c}\\ \rule{0.0pt}{14.22636pt}\beta_{i}:=2-\sum_{\ell=1}^{m}\beta_{i,\ell},&\quad i=1,2,\end{array}\right.$$

$$3(1+{\beta }_{i,j})<2\sum _{\ell =1}^m{\beta }_{i,\ell }+\sum _{\ell =1}^m{\beta }_{3-i,\ell },\sum _{\ell =1}^m{\beta }_{i,\ell }<2,\text{for }j=1,2,\dots ,m,i=1,2.$$

$$\sum _{\ell =1}^m{\beta }_{1,\ell }>1+{\beta }_{i,j}\text{for every }i=1,2,j=1,\dots ,m,$$

$$\sum _{\ell =1,\ell \ne j}^m{\beta }_{i,\ell }>\max \{{\beta }_{1,j},{\beta }_{2,j}\}\text{for every }j=1,\dots ,m,i=1,2,$$

$$(-\Delta {)}^{\frac{n}{2}}w=e^{nw}+{\gamma }_n\sum _{\ell =1}^m{\beta }_{\ell }{\delta }_{P_{\ell }}\text{in }{\mathbb{R}}^n$$

$$w(x)=-2\log |x|+O(1)\text{as }|x|\to \infty .$$

$$\frac{1}{{\gamma }_n}(-\Delta {)}^{\frac{n}{2}}\log \frac{1}{|x|}={\delta }_0.$$

$$u_i(x):=\frac{1}{2\pi }\sum _{j=1}^2{a}_{i,j}{\int }_{{\mathbb{R}}^2}\log \left(\frac{1}{|x-y|}\right)K_j(y)e^{2(u_j(y))}dy+c_i,i=1,2,$$

$$\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right).$$

$$\sum _{j=1}^2{a}_{i,j}{\int }_{{\mathbb{R}}^2}{K}_j{e}^{2u_j}dx=2\pi {\beta }_i,i=1,2,$$

$${\int }_{{\mathbb{R}}^2}{K}_i{e}^{2u_i}dx=2\pi {\bar{\beta }}_i,{\bar{\beta }}_i:=\frac{1}{3}(2{\beta }_i+{\beta }_{3-i}),i=1,2.$$

$${\bar{\beta }}_i>0,{\bar{\beta }}_i<1-{\beta }_{i,\ell }\text{for every }i=1,2,\ell =1,\dots ,m.$$

$$X:=C_0({\mathbb{R}}^2)\times C_0({\mathbb{R}}^2),\Vert \mathbf{v}\Vert :=\max \{\Vert v_1{\Vert }_{L^{\infty }({\mathbb{R}}^2)},\Vert v_2{\Vert }_{L^{\infty }({\mathbb{R}}^2)}\}\text{for }\mathbf{v}=(v_1,v_2)\in X,$$

$$u_0(x)=-\log |x|\text{ on }{B}_1^c.$$

$${\int }_{{\mathbb{R}}^2}\overline{K_i}{e}^{2(v+c_{i,v})}dx=2\pi {\bar{\beta }}_i,{\bar{K}}_i:=K_i{e}^{2{\beta }_i{u}_0},i=1,2,$$

$${\bar{v}}_i(x):=\frac{1}{2\pi }\sum _{j=1}^2{a}_{i,j}{\int }_{{\mathbb{R}}^2}\log \left(\frac{1}{|x-y|}\right){\bar{K}}_j(y)e^{2(v_j(y)+c_{j,v_j})}dy-{\beta }_i{u}_0(x),i=1,2.$$

$${\bar{v}}_i(x):=\frac{1}{2\pi }\sum _{j=1}^2{a}_{i,j}{\int }_{{\mathbb{R}}^2}\log \left(\frac{|x|}{|x-y|}\right){\bar{K}}_j(y)e^{2(v_j(y)+c_{j,v_j})}dy,i=1,2.$$

$$\Vert \mathbf{v}{\Vert }_X\le C\text{for every }(\mathbf{v},t)\in X\times [0,1]\text{ satisfying }\mathbf{v}=tT(\mathbf{v}).$$

$${\psi }_i^k:=v_i^k+c_i^k,c_i^k:=c_{i,v_i^k}+\frac{1}{2}\log t^k.$$

$$\displaystyle\left\{\begin{array}[]{ll}\psi_{1}^{k}(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log\left(\frac{1}{|x-y|}\right)\left(2\bar{K}_{1}(y)e^{2\psi_{1}^{k}(y)}-\bar{K}_{2}(y)e^{2\psi_{2}^{k}(y)}\right)dy-t^{k}\beta_{1}u_{0}(x)+c_{1}^{k}\\ \rule{0.0pt}{22.76228pt}\psi_{2}^{k}(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log\left(\frac{1}{|x-y|}\right)\left(2\bar{K}_{2}(y)e^{2\psi_{2}^{k}(y)}-\bar{K}_{1}(y)e^{2\psi_{1}^{k}(y)}\right)dy-t^{k}\beta_{2}u_{0}(x)+c_{2}^{k}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}\psi_{1}^{k}(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log\left(\frac{|x|}{|x-y|}\right)\left(2\bar{K}_{1}(y)e^{2\psi_{1}^{k}(y)}-\bar{K}_{2}(y)e^{2\psi_{2}^{k}(y)}\right)dy+c_{1}^{k}\\ \rule{0.0pt}{22.76228pt}\psi_{2}^{k}(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log\left(\frac{|x|}{|x-y|}\right)\left(2\bar{K}_{2}(y)e^{2\psi_{2}^{k}(y)}-\bar{K}_{1}(y)e^{2\psi_{1}^{k}(y)}\right)dy+c_{2}^{k}.\end{array}\right.$$

$$\max \{\sup {\psi }_1^k,\sup {\psi }_2^k\}\to \infty .$$

Full text

[TABLE]

with $m\geq 3$ and $\beta_{i,\ell}\in[0,1)$. We prove the existence and non-existence results under suitable assumptions on $\beta_{i,\ell}$. This generalizes Luo-Tian’s [31] result for a singular Liouville equation in $\mathbb{R}^{2}$. We also study existence results for a higher order singular Liouville equation in $\mathbb{R}^{n}$.

The first author is supported by the SNSF Grant No. P2BSP2-172064

The third author is partially supported by NSERC

1. Introduction

We consider the following singular $SU(3)$ Toda system with multiple singular sources

[TABLE]

where $P_{1},\dots,P_{m}$ are distinct points in $\mathbb{R}^{2}$, $\beta_{i,\ell}\in[0,1)$ and $\delta_{P}$ denotes the Dirac measure at $P$ (notice that source terms are written with a plus sign). When $w_{1}=w_{2},\beta_{1,l}=\beta_{2,l}=\beta_{l}$, the above system reduces to the singular Liouville equation

[TABLE]

The Toda system (1.3) and the Liouville equation (1.4) have been widely studied in the literature due to its important role in geometry and mathematical physics. For instance, Eq. (1.4) is related to the problem of prescribing Gaussian curvature on surfaces with conical singularity, and abelian gauge in Chern-Simons theory [4, 7, 37, 38]. The Toda system (1.3) appears in the description of holomorphic curves in $\mathbb{CP}^{3}$ [10, 12, 16, 18], and in the non-abelian Chern-Simmon theory [19, 36, 42]. For classification and blow-up analysis to the (singular) Liouville equation and the $SU(n)$ Toda system we refer the reader to [5, 7, 8, 9, 11, 14, 13, 21, 22, 23, 24, 27, 25, 30, 33, 35, 28, 29, 26] and the references therein.

Luo-Tian [31] gave a necessary and sufficient condition for the existence of singular metric with three or more conical singularities on the $2$-sphere, whose equivalent statement on $\mathbb{R}^{2}$ is the following theorem:

Theorem A** ([31]).**

Let $m\geq 3$. Let $P_{1},\dots,P_{m}$ be $m$ distinct points in $\mathbb{R}^{2}$. Then there exist continuous functions $h_{\ell}$ around $P_{\ell}$ for $\ell=1,\dots,m$, a bounded continuous function $h_{m+1}$ outside a compact set, and a solution $w$ to

[TABLE]

if and only if

[TABLE]

Moreover, the solution is unique.

Troyanov [39] studied singular metrics with $2$ singulirities (i.e., $m=2$) and constant curvature $1$ on the $2$-sphere, and showed that the order of both singularities are equal (i.e., $\beta_{1}=\beta_{2}<1$). A necessary and sufficient condition on $\{\beta_{1},\beta_{2},\beta_{3}\}\subset(-\infty,1)$ for the existence of singular metric on the $2$-sphere has been given in [20, 40]. See also [6, 5, 32] and the references therein for various existence results on compact surfaces.

In this paper we study Problem (1.9) in the context of $SU(3)$ Toda system. More precisely, we prove existence and non-existence of solutions $(w_{1},w_{2})$ to (1.3) satisfying

[TABLE]

for $i=1,2$ and $\ell=1,\dots,m$, and $h_{i,m+1}$ is bounded outside a compact set. We write

[TABLE]

Then $w_{i}$ solves (1.3) if and only if $u_{i}$ solves

[TABLE]

The condition (1.14) in terms of $u_{i}$ is

[TABLE]

provided $u_{i}$ is continuous.

For Toda system with singular sources, the only complete result is [28] in which the case of single source, i.e., $m=1$ is completely solved by PDE and integrable system theory. In [26], some special cases of $m=2$ are classified using higher order hypergeometric equations. The following theorem gives the first existence result when $m\geq 3$:

Theorem 1.1**.**

Let $m\geq 3$. Let $\{\beta_{i,\ell}:i=1,2,\,\ell=1,2,\dots m\}\subset[0,1)$ be such that

[TABLE]

Then given $m$ distinct points $\{P_{\ell}\}_{\ell=1}^{m}\subset\mathbb{R}^{2}$ there exists continuous solution $(u_{1},u_{2})$ to (1.18) such that (1.21) holds.

Note that if $\sum_{\ell}{\beta_{1,\ell}}=\sum_{\ell}\beta_{2,\ell}$, then the first condition of (1.22) reduces to

[TABLE]

which is stronger than (1.10). We shall show that an equivalent condition of (1.10) for the Toda system, namely a condition of the form

[TABLE]

is not sufficient for the existence of solutions to (1.18) satisfying the asymptotic behavior (1.21). See Lemma 3.2.

In [31], the existence of a solution to (1.4) is proved by a variational argument. In this paper we propose a new proof on the existence via fixed point theory. The crucial step in which we need condition (1.22) is Proposition 2.1 below, a compactness result which follows from the blow-up analysis of sequences of solutions (see Lemma 5.2). This compactness is used to prove the a priori bounds necessary to run the fixed point argument of [3, 22, 41]. Let us point out that condition (1.23) is sufficient to rule-out a “full blow-up” phenomena (that is, after a suitable rescaling, the limiting profile is a $SU(3)$ Toda system in $\mathbb{R}^{2}$) for a sequence of solutions to (1.18)-(1.21) (for “half blow-up” and “full blow-up” phenomena see e.g., [2, 17, 34]). In particular, condition (1.23) is sufficient to prove the a priori estimate when $\beta_{1,\ell}=\beta_{2,\ell}=\beta_{\ell}$ and $u_{1}=u_{2}$, that is, a priori estimate for the singular Liouville problem (1.9) . Moreover, the same method also works for a higher order generalization of it.

Theorem 1.2**.**

Let $m\geq 3$ and $n\geq 2$. For $\ell=1,2,\dots,m$ let $\beta_{\ell}\in(0,1)$ be such that (1.10) holds. Then given $m$ distinct points $\{P_{\ell}\}_{\ell=1}^{m}\subset\mathbb{R}^{n}$ there exists a solution $w\in C^{0}(\mathbb{R}^{n}\setminus\{P_{1},\dots,P_{m}\})$ to

[TABLE]

satisfying the asymptotic behavior

[TABLE]

Here $\gamma_{n}:=\frac{(n-1)!}{2}|S^{n}|$ is such that

[TABLE]

2. Proof of Theorem 1.1

It is well-know that if $(u_{1},u_{2})$ is a solution to (1.18) with $\beta_{i,\ell}<1$ and $u_{i}$, $K_{i}e^{2u_{i}}\in L^{1}_{loc}(\mathbb{R}^{2})$, then $u_{i}$ is continuous. On the other hand, if $(u_{1},u_{2})$ is a continuous solution to (1.18)-(1.21) with $\beta_{i,\ell}<1$, then $K_{i}e^{2u_{i}}=O(|x|^{-4})$ as $|x|\to\infty$. In particular, $\log|\cdot|K_{i}e^{2u_{i}}\in L^{1}(\mathbb{R}^{2})$, and $u_{i}$ satisfies the integral equation

[TABLE]

for some $c_{i}\in\mathbb{R}$, where $(a_{i,j})$ is the $SU(3)$ Cartan matrix

[TABLE]

Moreover, the asymptotic behavior (1.21) implies that

[TABLE]

that is

[TABLE]

Thus, Theorem 1.1 is equivalent to the existence of solution $(u_{1},u_{2})$ to (2.1)-(2.2). Moreover, (1.22) in terms of $\bar{\beta}_{i}$ is

[TABLE]

In order to prove existence of solution to (2.1)-(2.2) we use a fixed point argument on the space

[TABLE]

where $C_{0}(\mathbb{R}^{2})$ denotes the space of continuous functions vanishing at infinity. We fix $u_{0}\in C^{\infty}(\mathbb{R}^{2})$ such that

[TABLE]

For $v\in C_{0}(\mathbb{R}^{2})$ let $c_{i,v}\in\mathbb{R}$ be the unique number so that

[TABLE]

where $\bar{\beta}_{i}$ is as in (2.2). Now we define $T:X\to X$, $(v_{1},v_{2})\mapsto(\bar{v}_{1},\bar{v}_{2})$, where we have set

[TABLE]

As $\beta_{i}=2\bar{\beta}_{i}-\bar{\beta}_{3-i}$, for $x\in B_{1}^{c}$ this can be written as

[TABLE]

Using that $\bar{K}_{i}=O(|x|^{-4})$ for $|x|$ large, one can show that $(\bar{v}_{1},\bar{v}_{2})\in X$. Moreover, the operator $T$ is compact (see e.g. the proof of [22, Lemma 4.1]).

The following proposition is crucial in proving existence of fixed point of $T$.

Proposition 2.1**.**

There exists $C>0$ such that

[TABLE]

Proof.

We assume by contradiction that the proposition is false. Then there exists ${\bf v}^{k}=(v_{1}^{k},v_{2}^{k})$ and $t^{k}\in(0,1]$ with ${\bf v}^{k}=t^{k}T({\bf v}^{k})$ such that $\|{\bf v}^{k}\|\to\infty$. We set

[TABLE]

Then we have

[TABLE]

For $|x|\geq 1$ this is equivalent to

[TABLE]

Since $\|{\bf v}^{k}\|\to\infty$, we necessarily have

[TABLE]

Without any loss of generality we assume that $\sup\psi^{k}_{1}\geq\sup\psi_{2}^{k}$. We fix $x^{k}\in\mathbb{R}^{2}$ such that

[TABLE]

If $x^{k}$ is bounded then, up to a subsequence, $x^{k}\to x^{\infty}$.

We consider the following three cases.

Case 1 $x^{\infty}\in\mathbb{R}^{2}\setminus\{P_{\ell}:\ell=1,2,\dots,m\}$.

By Lemma 5.2 (see also [23, 30]) we have

[TABLE]

where the blow-up value at a point $P$ is defined by

[TABLE]

This contradicts (2.3) as $\sigma_{i}(x^{\infty})\leq\bar{\beta}_{i}<1.$

Case 2 $x^{\infty}\in\{P_{\ell}:\ell=1,2,\dots,m\}$.

Without loss of generality we assume that $x^{\infty}=P_{1}$. Notice that

[TABLE]

for some positive continuous functions $f_{1}$ and $f_{2}$ in a small neighborhood of the point $P_{1}$. In particular, the functions $w_{i}^{k}(x):=\psi_{i}^{k}(x-P_{1})$ satisfies the conditions of Lemma 5.2 for some $R>0$, and we get

[TABLE]

a contradiction to (2.3).

Case 3 $|x^{k}|\to\infty$.

We set

[TABLE]

and extend them continuously at the origin. Then $\tilde{\psi}_{i}^{k}$ satisfies

[TABLE]

for $x\in B_{1}$. Since $\tilde{K}_{i}(0)>0$ for $i=1,2$, and

[TABLE]

one obtains a contradiction as in Case 1.

We conclude the proposition.

∎

Proof of Theorem 1.1 It follows from Proposition 2.1 and Schauder fixed point theorem that the operator $T$ has a fixed point, say $(v_{1},v_{2})$. Then setting

[TABLE]

one sees that $(u_{1},u_{2})$ is a solution to (1.18)-(1.21).

3. Non-existence results

We show that Theorem 1.1 is not true if the assumption (1.22) is replaced by (1.23). Let us fix $\beta_{1},\dots,\beta_{7}\in(0,1)$ such that the assumptions $\mathcal{A}1)$ to $\mathcal{A}5)$ hold:

• $\mathcal{A}1)$

$\beta_{4}+\sum_{\ell=1}^{4}\beta_{\ell}=2$

• $\mathcal{A}2)$

$\beta_{2}+\beta_{3}<\beta_{1}$

• $\mathcal{A}3)$

$\beta_{4}<\frac{1}{3}$

• $\mathcal{A}4)$

$\beta_{4}+\sum_{\ell=5}^{7}\beta_{\ell}=2$

• $\mathcal{A}5)$

$\beta_{4}+\beta_{5}<1$.

It is easy to see that $\mathcal{A}1)$ and $\mathcal{A}2)$ implies that

• $\mathcal{A}6)$

$\beta_{4}+\beta_{1}>1$ and $\beta_{4}+\beta_{\ell}<1$ for $\ell=2,3.$

We shall show an non-existence result to the Toda system (1.3) satisfying (1.14) for the following choice of $\{\beta_{i,\ell}\}$:

[TABLE]

Let us point out that we can choose $\{\beta_{\ell}\}$ satisfying $\mathcal{A}1)$ to $\mathcal{A}5)$ in such a way that $\{\beta_{i,\ell}\}$ satisfy (1.23) with $m=7$, $i=1,2$. For instance, one can simply take

[TABLE]

For these $\beta_{\ell}$’s one has

[TABLE]

and hence $\{\beta_{i,\ell}\}$ does not satisfy (1.22).

We begin with the following non-existence result for a singular Liouville equation.

Lemma 3.1**.**

Let $\beta_{\ell}\in(0,1)$ with $\ell=1,2,3,4$ be such that $\mathcal{A}1)$ to $\mathcal{A}3)$ hold. Let $P_{1},P_{2},P_{3}$ be fixed three distinct points in $\mathbb{R}^{2}$. Then, for $|P_{4}|$ large enough, there exists no continuous solution to

[TABLE]

Proof.

Assume by contradiction that there exists a sequence of solutions $(u^{k})$ to (3.6) with

[TABLE]

Notice that the assymptotic behavior

[TABLE]

is equivalent to

[TABLE]

Step 1 We have

[TABLE]

To prove this we use Kelvin transform. Up to a small translation, we can assume that none of $P_{1},P_{2},P_{3}$ is the origin. We set

[TABLE]

for some $c^{k}\in\mathbb{R}$. Then setting $Q_{\ell}:=\frac{P_{\ell}}{|P_{\ell}|^{2}}$ for $\ell=1,2,3,4$ we see that

[TABLE]

Using that $|x||y||\frac{x}{|x|^{2}}-\frac{y}{|y|^{2}}|=|x-y|$, $\mathcal{A}1)$, and for suitably chosen $c^{k}$, we obtain

[TABLE]

In fact, as $\tilde{u}^{k}=O_{k}(1)$ in $B_{1}$, it satisfies the above equation at the origin as well, that is,

[TABLE]

As $|P_{4}|\to\infty$, we have that $Q_{4}\to 0$. By $\mathcal{A}3)$ one gets

[TABLE]

for some $\varepsilon>0$. Hence, by Lemma 5.1 we obtain

[TABLE]

Step 1 follows immediately from the relation

[TABLE]

Step 2 No blow-up occurs on bounded domains, that is, for every $R>0$,

[TABLE]

Writing $\bar{u}^{k}=u^{k}-\beta_{4}\log|P_{4}|$ we see that

[TABLE]

where

[TABLE]

It follows that $K_{1}\to 1$ in $C^{0}_{loc}(\mathbb{R}^{2})$ as $k\to\infty$, and $K_{0}$ does not depend on $k$.

Assume by contradiction that $\bar{u}^{k}$ is not locally uniformly bounded from above. Then, as blow-up points are discrete, there exists $\delta>0$ such that

[TABLE]

for some $x_{0}\in\mathbb{R}^{2}$. If $x_{0}\not\in\{P_{2},P_{3},P_{4}\}$, then one can show that

[TABLE]

a contradiction as $\beta_{4}<1$. Thus, $x_{0}=P_{\ell_{0}}$ for some $\ell_{0}\in\{1,2,3\}$, and in fact, the set of all blow-up points is a subset of $\{P_{1},P_{2},P_{3}\}$. We fix $R>0$ such that $\bar{B}_{2R}(x_{0})\cap\{P_{1},P_{2},P_{3}\}=\{x_{0}\}$. Then $\bar{u}^{k}$ is uniformly bounded from above in $B_{2R}(x_{0})\setminus B_{\frac{R}{2}}(x_{0})$. Using this, and as $\bar{u}^{k}$ satisfies the integral equation

[TABLE]

for some $C^{k}\in\mathbb{R}$, we get that

[TABLE]

Hence, by the remark after Lemma 5.2 we have (this can be shown easily by a local Pohozaev type identity to the above integral equation satisfied by $\bar{u}_{k}$)

[TABLE]

Thus $2\beta_{4}\geq\sigma(x_{0})=2(1-\beta_{\ell_{0}})$. This and $\mathcal{A}6)$ imply that $\ell_{0}=1$, that is, $P_{1}$ is the only blow-up point. In particular, $\bar{u}^{k}\to-\infty$ locally uniformly outside $P_{1}$. Therefore, by Step 1 and (3.7) we get

[TABLE]

a contradiction to $\mathcal{A}6)$. This finishes Step 2.

Since $\bar{u}^{k}$ is locally uniformly bounded from above, up to a subsequence, either $\bar{u}^{k}\to\infty$ locally uniformly, or $\bar{u}^{k}\to\bar{u}$ in $C_{loc}^{0}(\mathbb{R}^{2})$. In the first case we get a contradiction to

[TABLE]

thanks to Step 1. Therefore, only the later case can occur, and the limit function $\bar{u}$ satisfies

[TABLE]

Again by Step 1, we have that

[TABLE]

which is equivalent to

[TABLE]

Thus,

[TABLE]

satisfies (1.9) with $m=3$, where $\beta_{1},\beta_{2},\beta_{3}$ satisfy $\mathcal{A}2)$. This contradicts the necessary condition (1.10) in Theorem A. ∎

Remark 1**.**

Problem (3.6) is super critical under the assumptions $\mathcal{A}1)$ and $\mathcal{A}2)$. To be more precise, if one uses fixed point arguments (as described in Section 4) to prove the lemma, then one would not be able to rule-out a blow-up phenomena around the point $P_{1}$. This is due to the fact that the energy of a singular bubble at $P_{1}$ is $4\pi(1-\beta_{1})$, which is smaller than the total energy $4\pi\beta_{4}$.

The super criticality of the Problem (3.6) under $\mathcal{A}1)$ and $\mathcal{A}2)$ can also be seen from the point of view of singular Moser-Trudinegr inequality, see e.g. [1, 9, 32, 15, 38] and the references therein.

Now we are in a position to prove non-existence of solution to the Toda system (1.3)-(1.14) for the choice of $\{\beta_{i,\ell}\}$ as in (3.5). More precisely, we have:

Lemma 3.2**.**

Let $\beta_{\ell}\in(0,1)$ with $\ell=1,\dots,7$ be such that $\mathcal{A}1)$ to $\mathcal{A}5)$ hold. Let $\{\beta_{i,\ell}:i=1,2,\,\ell=1,\dots,7\}$ be as in (3.5). Let $P_{1},\dots,P_{4}$ be such that Problem (3.6) has no solution. Let $P_{5}$ be a fixed point (different from $P_{1},\dots,P_{4}$). Then for $|P_{6}|,|P_{7}|$ large ($P_{6}\neq P_{7}$) there exists no solution to (1.18) with $m=7$ such that

[TABLE]

Proof.

We assume by contradiction that there is a sequence of solutions $(u_{i}^{k})$ with

[TABLE]

that is, $u_{i}^{k}$ satisfies

[TABLE]

where

[TABLE]

Notice that $K_{1}$ does not depend on $k$, $K_{1}\in L^{1}(\mathbb{R}^{2})$, thanks to the assumption $\beta_{4}<1$, and

[TABLE]

We claim that $u_{1}^{k}\to u$ locally uniformly in $\mathbb{R}^{2}$, where $u$ satisfies

[TABLE]

Then one can show that $u(x)=-2\beta_{4}\log|x|+O(1)$ as $|x|\to\infty$. In particular, $\bar{u}(x)=u(x)+\frac{1}{2}\log 2$ is a solution to the Problem (3.6), a contradiction to our assumption on $P_{1},\dots,P_{4}$ that the Problem (3.6) has no solution.

We prove the claim in few steps.

Step 1 We have

[TABLE]

The proof is very similar to that of Step 1 in Lemma 3.1. Here we give a sketch of it.

We set

[TABLE]

so that $\tilde{u}^{k}_{1}$ satisfies ($\tilde{K}$ does not depend on $k$)

[TABLE]

[TABLE]

Now we can apply Lemma 5.1 with $\beta=0$, thanks to the assumption $\mathcal{A}3)$, to get that $\tilde{u}^{k}_{1}\leq C$ in a neighborhood of the origin. Step 1 follows.

Setting

[TABLE]

we shall show that $S_{1}\cup S_{2}=\emptyset$. We start with:

Step 2 $S_{1}\subseteq\{P_{1},\dots,P_{4}\}$ and $S_{2}\subseteq\{P_{5}\}$.

For $x_{0}\in S_{1}\cup S_{2}$ we can write

[TABLE]

where $\alpha_{1}\in\{0,\beta_{1},\dots,\beta_{4}\}$, $\alpha_{2}\in\{0,\beta_{5}\}$ and $\alpha_{1}\alpha_{2}=0$. By Lemma 5.1 and $\mathcal{A}3)$ one gets $S_{1}\subseteq\{P_{1},\dots,P_{4}\}$ and $S_{2}\subseteq\{P_{5}\}$.

Step 3 $S_{1}\cup S_{2}=\emptyset$.

It is well-known that $u^{k}_{i}$ satisfies the integral equation

[TABLE]

For $x_{0}\in S_{1}\cup S_{2}$ let $R>0$ be such that $\bar{B}_{R}(x_{0})\cap(S_{1}\cup S_{2})=\{x_{0}\}$, and $x_{0}$ is the only singularity for $K_{1},K_{2}$ on $\bar{B}_{R}(x_{0})$. Then, from the above integral representation, one can show that

[TABLE]

In particular, $u^{k}_{i}$ and $K_{i}$ satisfy all the assumptions in Lemma 5.2. Therefore, if $S_{2}=\{P_{5}\}$, then as $\sigma_{1}(P_{5})=0$, we must have $\sigma_{2}(P_{5})=1-\beta_{5}$. This implies that

[TABLE]

a contradiction to $\mathcal{A}5)$. Hence, $S_{2}=\emptyset$.

Now we assume that $\beta_{\ell_{0}}\in S_{1}$ for some $\ell_{0}\in\{1,\dots,4\}$. Then, in a similar way we get that $\beta_{4}\geq 1-\beta_{\ell_{0}}$. In fact, by $\mathcal{A}6)$, a strict inequality holds, that is, $\beta_{4}>1-\beta_{\ell_{0}}$. Since

[TABLE]

we must have that the cardinality of $S_{1}$ is at least $2$, thanks to Step 1. Taking $P_{\ell_{1}}\in S_{1}$ with $\ell_{1}\in\{1,\dots,4\}\setminus\{\ell_{0}\}$, and again using that $\sigma({P_{\ell_{1}}})=1-\beta_{\ell_{1}}$, we obtain

[TABLE]

a contradiction to $\mathcal{A}1)$.

We conclude Step 3.

Step 4 $u_{1}^{k}\to\bar{u}_{1}$ in $C^{0}_{loc}(\mathbb{R}^{2})$ where $\bar{u}_{1}$ satisfies (3.13).

Since $S_{1}\cup S_{2}=\emptyset$, up to a subsequence, one of the following holds:

• i)

$u_{i}^{k}\to\bar{u}_{i}$ in $C^{0}_{loc}(\mathbb{R}^{2})$ for $i=1,2$

• ii)

$u_{1}^{k}\to\bar{u}_{1}$ in $C^{0}_{loc}(\mathbb{R}^{2})$ and $u_{2}^{k}\to-\infty$ locally uniformly in $\mathbb{R}^{2}$

• iii)

$u_{2}^{k}\to\bar{u}_{2}$ in $C^{0}_{loc}(\mathbb{R}^{2})$ and $u_{1}^{k}\to-\infty$ locally uniformly in $\mathbb{R}^{2}$

• iv)

$u_{i}^{k}\to-\infty$ locally uniformly in $\mathbb{R}^{2}$ for $i=1,2$.

It follows from Step 1, and the integral condition $\int_{\mathbb{R}^{2}}K_{1}e^{2u_{1}}dx=2\pi\beta_{4}$ that either $i)$ or $ii)$ holds, and $\bar{u}_{1}$ satisfies the integral condition

[TABLE]

Now we assume by contradiction that $i)$ holds. Then the limit functions $(\bar{u}_{1},\bar{u}_{2})$ satisfy the system

[TABLE]

where $\bar{K}_{2}(x):=|x-P_{5}|^{-2\beta_{5}}$ is the limit of $K_{2}$ as $k\to\infty$. Then one has

[TABLE]

and together with $\bar{K}_{2}e^{2\bar{u}_{2}}\in L^{1}(\mathbb{R}^{2})$ we have $\beta_{5}+2\gamma-\beta_{4}>1$. Hence, $\beta_{4}+\beta_{5}>1$, a contradiction to $\mathcal{A}5)$.

Thus, $ii)$ holds, and (3.17) reduces to a single equation (3.13).

We conclude the lemma. ∎

4. Higher order singular Liouville equation

The proof of Theorem 1.2 is very similar to that of Theorem 1.1 (see also [22]). Here we give a sketch of it.

Writing

[TABLE]

Theorem 1.2 is equivalent to prove the existence of solution $u\in C^{0}(\mathbb{R}^{n})$ to

[TABLE]

satisfying the asymptotic behavior

[TABLE]

As before we fix $u_{0}\in C^{\infty}(\mathbb{R}^{n})$ such that $u_{0}(x)=-\log|x|$ for $|x|\geq 1$, and we look for a solution $u$ to (4.1) of the form

[TABLE]

where $c$ is a normalizing constant and $v\in X$, where

[TABLE]

Then $u$ satisfies (4.1) if and only if $v=u-\beta u_{0}-c$ satisfies

[TABLE]

The function $\bar{K}$ satisfies

[TABLE]

For $v\in X$, we fix $c_{v}\in\mathbb{R}$ so that

[TABLE]

We define a compact operator

[TABLE]

[TABLE]

It follows that $\bar{v}\in C^{0}(\mathbb{R}^{n})$ (in fact, Hölder continuous), and by (4.5)

[TABLE]

We claim that there exists $C>0$ such that

[TABLE]

Then by Schauder fixed point theorem the operator $T$ has a fixed point $v$ in $X$, and consequently we get a continuous solution to (4.1) satisfying (4.2).

To prove (4.7) we assume by contradiction that there exists $(v^{k},t^{k})\in X\times[0,1]$ such that $\|v^{k}\|_{X}\to\infty$ and $v^{k}=t^{k}T(v^{k})$, that is

[TABLE]

Then we can choose $x^{k}\in\mathbb{R}^{n}$ so that

[TABLE]

The crucial ingredients to obtain a contradiction are Lemma 5.3, and the relation

[TABLE]

which follows from the second condition in (1.10). Up to a subsequence, we distinguish the following two cases:

Case 1 $x^{k}\to x^{\infty}\in\mathbb{R}^{n}\setminus\{P_{\ell}:\ell=1,2,\dots.m\}.$

In a small neighborhood of $x^{\infty}$ we have for some $c_{0}>0$

[TABLE]

where $\alpha\in\{0,\beta_{1},\dots,\beta_{m}\}$. Using (4.8)-(4.9) one gets a contradiction as in [22], see also [3, 41].

Case 2 $|x^{k}|\to\infty$.

Setting

[TABLE]

we obtain $\tilde{\psi}_{k}(\tilde{x}^{k})\to\infty$, and $\tilde{\psi}_{k}$ satisfies

[TABLE]

where

[TABLE]

Note that $\tilde{K}$ is smooth around the origin and

[TABLE]

one can proceed as in Case 1. Thus, $\psi^{k}\leq C$ on $\mathbb{R}^{n}$, and we have (4.7).

5. Some useful lemmas

The following lemma is a generalizations of Brezis-Merle [11] type results, compare [8, Theorem 5].

Lemma 5.1**.**

Let $(u^{k})$ be a sequence of solutions to

[TABLE]

for some $\varepsilon>0$ and $\alpha\in[0,1)$. Assume that $g^{k}\geq 0$, $\|g^{k}\|_{L^{1}(B_{1})}\leq C$, $0\leq f^{k}\leq C$ and $\inf_{B_{1}\setminus B_{\delta}}f^{k}\geq C_{\delta}^{-1}$ for some $0<\delta<\frac{1}{3}$. Then $u^{k}$ is locally uniformly bounded from above in $B_{1}$.

Proof.

We write $u^{k}=v^{k}+h^{k}$, where $h^{k}$ is harmonic in $B_{1}$ and

[TABLE]

Since $g_{k}\geq 0$, by Jensen’s inequality one gets that

[TABLE]

Notice that

[TABLE]

Since $\delta<\frac{1}{3}$, fixing $\delta+\frac{1}{3}<r_{1}<r_{2}<1-\delta$ we see that

[TABLE]

Therefore, by mean value theorem,

[TABLE]

Thus, $\int_{B_{1}}(h^{k})^{+}dx\leq C$. If $\rho^{k}:=\int_{B_{\frac{1}{2}}}|h^{k}|dx\leq C$ then we have

[TABLE]

In particular, $(h^{k})$ is bounded in $C^{0}_{loc}(B_{1})$. If $\rho^{k}\to\infty$, then

[TABLE]

This shows that $(h^{k})$ is locally uniformly bounded from above in $B_{1}$. This leads to

[TABLE]

Using this uniform bound, and Hölder inequality with $p=\frac{1}{1-\alpha-\varepsilon/2}$, one gets $v^{k}\leq C$ in $B_{r}$ for $0<r<1$, and the lemma follows. ∎

A strong version (precise quantization value of $\sigma_{1},\sigma_{2}$) of the following lemma is proven in [27, 29]. See [30] for a Pohozaev type identity for regular $SU(3)$ Toda system.

Lemma 5.2** ([27, 29]).**

Let $(u_{1}^{k},u_{2}^{k})$ be a sequence of solutions to

[TABLE]

for some $\alpha_{1},\alpha_{2}<1$, and $B_{1}$ is the unit ball in $\mathbb{R}^{2}$. Assume that [math] is the only blow-up point, that is,

[TABLE]

Then setting

[TABLE]

we have

[TABLE]

In particular, if $(\sigma_{1},\sigma_{2})\neq(0,0)$ then

[TABLE]

Remark 2**.**

If $\alpha_{1}=\alpha_{2}=\alpha$, $K_{1}^{k}=K_{2}^{k}$ and $u_{1}^{k}=u_{2}^{k}$ in the above lemma, then $\sigma_{1}=\sigma_{2}=2(1-\alpha)$.

Theorem 5.3** ([22, 35]).**

Let $u$ be a normal solution to

[TABLE]

for some $\alpha>-1$ and $n\geq 2$, that is, $u$ satisfies the integral equation

[TABLE]

for some $C\in\mathbb{R}.$ Then $\Lambda=\Lambda_{1}(1+\alpha)$, $\Lambda_{1}:=2\gamma_{n}$.