Uniqueness of bubbling solutions of mean field equations with non-quantized singularities

TL;DR

This paper proves the uniqueness of bubbling solutions in singular mean field equations on compact Riemann surfaces, especially when blowup points coincide with bubbling sources and under specific non-degeneracy conditions.

Contribution

It extends previous results by establishing uniqueness of bubbling solutions with non-quantized singularities and non-degenerate assumptions.

Findings

Uniqueness of bubbling solutions when blowup points coincide with bubbling sources.

Uniqueness under non-degeneracy when source strength is not multiple of 4π.

Extension of prior work by Bartolucci et al.

Abstract

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with bubbling sources. If the strength of the bubbling sources at blowup points are not multiple of $4\pi$ we prove that bubbling solutions are unique under non-degeneracy assumptions. This work extends a previous work of Bartolucci, et, al \cite{bart-4}.

Equations

$$\Delta_{g}v+\rho\bigg{(}\frac{he^{v}}{\int_{M}he^{v}{\rm d}\mu}-\frac{1}{vol_{g}(M)}\bigg{)}=\sum_{j=1}^{N}4\pi\alpha_{j}(\delta_{q_{j}}-\frac{1}{vol_{g}(M)})\quad{\rm in}\ \;M,$$

$$\left\{\begin{array}[]{ll}-\Delta_{g}G(x,p)=\delta_{p}-1\quad{\rm in}\ \;M\\ \int_{M}G(x,p){\rm d}\mu=0,\end{array}\right.$$

$$G(x,p)=-\frac{1}{2\pi }\log dist(x,p)+R(x,p)$$

$$\Delta_{g}w+\rho\bigg{(}\frac{He^{w}}{\int_{M}He^{w}{\rm d}\mu}-1\bigg{)}=0\quad{\rm in}\ \;M,$$

$$w(x)=v(x)+4\pi \sum _{j=1}^N{\alpha }_{j}G(x,q_j),$$

$$H(x)=h(x)\prod _{j=1}^N{e}^{-4\pi {\alpha }_{j}G(x,q_j)}.$$

$$H(x)=h_j(x)|x-q_j{|}^{2{\alpha }_j},|x-q_j|\ll 1,1\le j\le N,$$

$$\left\{\begin{array}[]{ll}p_{j}=q_{j}&{\rm if}\ 1\leq j\leq\tau,\\ p_{j}\notin\{q_{1},\cdots,q_{N}\}&{\rm if}\ \tau+1\leq j\leq m,\end{array}\right.$$

$${\alpha }_1=\cdots ={\alpha }_t>{\alpha }_l,l\ge t+1,\text{mbox}where1\le t\le \tau .$$

$$I_{\rho }(w)=\frac{1}{2}{\int }_M|\nabla w{|}^2+\rho {\int }_{M}w-\rho \log {\int }_{M}H{e}^w,$$

$$G_j^{\ast }(x)=8\pi (1+{\alpha }_j)R(x,p_j)+8\pi \sum _{l\ne j}^{1,\cdots ,m}(1+{\alpha }_l)G(x,p_l),$$

$$\displaystyle L(\mathbf{p})=\sum_{j=1}^{t}\big{[}\Delta\log h(p_{j})+\rho_{*}-N^{*}-2K(p_{j})\big{]}(h_{j}(p_{j}))^{\frac{1}{1+\alpha_{1}}}e^{\frac{G_{j}^{*}(p_{j})}{1+\alpha_{1}}},$$

$$D(\mathbf{p})=\left(\begin{array}{c}\nabla (\log h_1+G_1^{\ast })(p_1)\\ \cdots \\ \nabla (\log h_t+G_t^{\ast })(p_t)\end{array}\right),$$

$${\rho }_{\ast }=8\pi \sum _{j=1}^m(1+{\alpha }_j),N^{\ast }=4\pi \sum _{j=1}^N{\alpha }_j$$

$$f^{*}(x_{\tau+1},\cdots,x_{m})=\sum_{j=\tau+1}^{m}\big{[}\log h(x_{j})+4\pi R(x_{j},x_{j})\big{]}+4\pi\sum_{l\neq j}^{\tau+1,\cdots,m}G(x_{l},x_{j}).$$

$$\left\{\begin{array}[]{lll}\Delta v+\rho\frac{he^{v}}{\int_{\Omega}he^{v}{\rm d}x}=\sum_{j=1}^{N}4\pi\alpha_{j}\delta_{q_{j}}&{\rm in}\;\ \Omega,\\ v=0&{\rm on}\;\ \partial\Omega,\end{array}\right.$$

$$v_k\mathrm{blows}\mathrm{up}\mathrm{at}{p}_j\in \Omega ,1\le j\le m,$$

$$\left\{\begin{array}[]{lll}-\Delta G_{\Omega}(x,p)=\delta_{p}&{\rm in}\;\ \Omega,\\ G_{\Omega}(x,p)=0&{\rm on}\;\ \partial\Omega,\end{array}\right.$$

$$G_{j,\Omega }^{\ast }(x)=8\pi (1+{\alpha }_j)R_{\Omega }(x,p_j)+8\pi \sum _{l\ne j}^{1,\cdots ,m}(1+{\alpha }_l)G_{\Omega }(x,p_l),$$

$$\displaystyle L_{\Omega}(\mathbf{p})=\sum_{j=1}^{t}\big{[}\Delta\log h(p_{j})-N^{*}\big{]}(h_{j}(p_{j}))^{\frac{1}{1+\alpha_{1}}}e^{\frac{G_{j}^{*}(p_{j})}{1+\alpha_{1}}},$$

$$D_{\Omega }(\mathbf{p})=\left(\begin{array}{c}\nabla (\log h_1+G_1^{\ast })(p_1)\\ \cdots \\ \nabla (\log h_t+G_t^{\ast })(p_t)\end{array}\right).$$

$$f_{\Omega}^{*}(x_{\tau+1},\cdots,x_{m})=\sum_{j=\tau+1}^{m}\big{[}\log h(x_{j})+4\pi R(x_{j},x_{j})\big{]}+4\pi\sum_{l\neq j}^{\tau+1,\cdots,m}G(x_{l},x_{j}),$$

$$u_{k}=w_{k}-\log\bigg{(}\int_{M}He^{w_{k}}{\rm d}\mu\bigg{)}$$

$${\Delta }_g{u}_k+{\rho }_k(H{e}^{u_k}-1)=0\mathrm{in}M.$$

$${\int }_{M}H{e}^{u_k}\mathrm{d}\mu =1.$$

$$u_k-{\bar{u}}_k\to \sum _{j=1}^{m}8\pi (1+{\alpha }_j)G(x,p_j)\mathrm{in}{\mathrm{C}}_{loc}^2(M\backslash \{p_1,\cdots ,p_m\})$$

$${\bar{u}}_k={\int }_M{u}_k\mathrm{d}\mu .$$

$$M=\bigcup _{j=1}^m{\overline{M}}_j;M_j\cap M_l=\varnothing ,\mathrm{if}j\ne l;B(p_j,3r_0)\subset M_j,j=1,\cdots ,m.$$

$$\lambda_{k,j}=\left\{\begin{array}[]{lcl}u_{k}(p_{j})&&{\rm if}\ \,\alpha_{j}\neq 0,\\ u_{k}(p_{k,j})\mathrel{\mathop{:}}=\max_{B(p_{j},r_{0})}u_{k}&&{\rm if}\ \,\alpha_{j}=0.\end{array}\right.$$

$$\Delta U_{k,j}+{\rho }_k{h}_j(p_{k,j})|x-p_{k,j}{|}^{2{\alpha }_j}{e}^{U_{k,j}}=0\mathrm{in}{\mathbb{R}}^2$$

$$U_{k,j}(x)=\lambda_{k,j}-2\log\Big{(}1+\frac{\rho_{k}h_{j}(p_{k,j})}{8(1+\alpha_{j})^{2}}e^{\lambda_{k,j}}|x-p_{k,j}|^{2(1+\alpha_{j})}\Big{)}.$$

$$\big{|}u_{k}(x)-U_{k,j}(x)\big{|}\leq C,\quad x\in B(p_{j},r_{0}).$$

Full text

1. Introduction

The main goal of this article is to study the uniqueness property of the following mean field equations with singularities:

[TABLE]

where $(M,g)$ be a Riemann surface with the metric $g$, $\Delta_{g}$ is the Laplace-Beltrami operator ($-\Delta_{g}\geq 0$), $h$ is a positive smooth function on $M$, $q_{1},\cdots,q_{N}$ are distinct points on $M$, $\rho>0,\alpha_{j}>-1$ are constants, $\delta_{q_{j}}$ is the Dirac measure at $q_{j}\in M$. Equation (1.1) is one of the most extensively studied elliptic PDE in the past few decades, partly due to its immense and profound connections with many branches of mathematics and Physics. In conformal geometry, (1.1) represents a metric on M with conic singularity (see [21, 36, 37]). Also it is derived from the mean field limit of point vortices in the Euler flow [9, 10] and serves as a model equation in the Chern-Simons-Higgs theory [31, 33, 40] and in the electroweak theory [1], etc. The literature for the study of various form of (1.1) is just too numerous to be listed in any reasonable way.

Recently it was found by Lin-Yan [26] that the uniqueness property is particularly important for equations with concentration phenomenon. In their work [26] they proved the first uniqueness property for bubbling solutions of Chern-Simon-Higgs equation and computed the exact number of solutions in certain special cases. In an important work [3] Bartolucci, et. al, extended Lin-Yan’s result for mean field equation (1.1) if the blowup points are not singular sources. Our goal in this article is to further extend the uniqueness property to the case that some singular sources coincide with blowup points.

To write the main equation in an equivalent form, we invoke the standard Green’s function$G(x,p)$:

[TABLE]

where the volume of $M$ is assumed to be $1$ for convenience. Then it is well known that in a neighborhood of $p$, $G(x,p)$ can be written as

[TABLE]

where $dist(x,p)$ is the geodesic distance from $p$ to $x$ for $x$ close to $p$.

Using $G(x,p)$ we write (1.1) as

[TABLE]

where

[TABLE]

and

[TABLE]

Note that in a local coordinate near $q_{j}$,

[TABLE]

for some $h_{j}(x)>0$.

We say that $\{v_{k}\}$ is a sequence of bubbling solutions of (1.1) if the corresponding $w_{k}$ defined by (1.4) tends to infinity as $k$ goes to infinity. The places that $w_{k}$ tends to infinity are called blowup points of $v_{k}$ or $w_{k}$. In this article we use $p_{1},...,p_{m}$ to denote blowup points. Let $q_{1},...,q_{N}$ be the location of singular sources. If none of $p_{1},...p_{m}$ is a singular source, Bartolucci, et. al have obtained the uniqueness of the blow up solution in [3]. Thus in this article we consider two cases: either all blowup points are singular sources or part of blowup points coincide with singular sources. In more precise terms let

[TABLE]

for some $1\leq\tau\leq m$. Thus if $\tau=m$ all blowup points are singular sources, if $1\leq\tau<m$, some blowup points are singular sources and some are not. Let $4\pi\alpha_{j}$ be the strength of the singular source at $p_{j}$, so we have $\alpha_{j}=0$ if $j>\tau$. Since the largest $\alpha_{j}$ matters the most we require the first $t$ of them to have this strength:

[TABLE]

It is well known that equation (1.3) is the Euler-Lagrange equation of the variational form:

[TABLE]

for $w\in H^{1}(M)$. Since adding a constant to any solution of (1.3) certainly gives to another solution, the space of solutions for (1.3) is the set of all $H^{1}(M)$ function with average equal to [math]. The discussion on the variational structure of (1.3) can be found in [28].

To state the main results we use the following notations:

[TABLE]

where $h_{j}$ is defined in (1.6), and

[TABLE]

Our first result is when all blowup points are singular sources:

Theorem 1.1**.**

Let $v_{k}^{(1)}$ and $v_{k}^{(2)}$ be two sequences of bubbling solutions of (1.1) with $\rho_{k}^{(1)}=\rho_{k}=\rho_{k}^{(1)}$ and $\alpha_{j}\in\mathbb{R}^{+}\setminus\mathbb{N}\,(1\leq j\leq m)$. If $L(\mathbf{p})\neq 0$ and $D(\mathbf{p})=0$, then $v_{k}^{(1)}=v_{k}^{(2)}$ for $k$ large enough.

Note that we use $\mathbb{N}$ to denote the set of positive integers. The assumption that $\alpha_{j}\in\mathbb{R}^{+}\setminus\mathbb{N}$ implies that all blowup points are singular sources. It is also very essential to require $\alpha_{j}$ to be non-integer, since quantized singular sources ( if the strength is $4\pi N$) exhibit non-simple blowup phenomenon [24][38] that has to be studied in a separate work in the future.

The assumption of $D({\bf p})$ is also very interesting. It is well known that if $p$ is not a singular source, the vanishing rate of $D({\bf p})$ is very fast for a regular blowup point ( [22],[13]).

Our second main result is about the uniqueness of bubbling solutions when some blowup points are non-quantized singular sources and some are regular points. So in this case we require $1\leq\tau<m$ and for $(x_{\tau+1},\cdots,x_{m})\in M\times\cdots\times M$, we define

[TABLE]

It is well known that $(p_{\tau+1},\cdots,p_{m})$ is a critical point of $f^{*}$.

Theorem 1.2**.**

Let $v_{k}^{(1)}$ and $v_{k}^{(2)}$ be two sequences of bubbling solutions of (1.1) with $\rho_{k}^{(1)}=\rho_{k}=\rho_{k}^{(1)}$ and $0\leq\alpha_{j}<1(1\leq j\leq m)$. Suppose $1\leq\tau<m$, $L(\mathbf{p})\neq 0$, $D(\mathbf{p})=0$ and \det\big{(}D^{2}f^{*}(p_{\tau+1},\cdots,p_{m})\big{)}\neq 0, then $v_{k}^{(1)}=v_{k}^{(2)}$ for $k$ large enough.

The notation $D^{2}f^{*}$ in Theorem 1.2 stands for the Hessian tensor field on $M$. Theorems 1.1 and Theorem 1.2 are clearly extensions of the main theorem in [3], where the uniqueness of bubbling solutions around regular blowup points is established. Here in our work, the assumptions of $L(\mathbf{p})$ and $D(\mathbf{p})$ are only placed on singular sources with the strongest strength.

In addition to the importance of application, the proof of the main theorems requires extremely delicate local analysis, just like the argument in [3]. Our argument relies heavily on the result of the second author in [42], Chen-Lin’s refined estimates in [16, 17] and the argument used by Lin-Yan [26] and Bartolucci-et-al [3]. Even though the outline of our paper is similar to those used in [26, 3] we have to establish accurate estimates for certain terms in an iterative manner.

The proof of Theorem 1.1 and Theorem 1.2 can also be applied to solve the following locally defined Dirichlet boundary problem: Let $\Omega$ be an open and bounded domain in $\mathbb{R}^{2}$ with regular boundary $\partial\Omega\in C^{2}$, $v$ be a solution of

[TABLE]

where $h>0$ is a $C^{1}$ funation in $\Omega$, $q_{1},\cdots,q_{N}$ are distinct points in $\Omega$, $\rho>0$, $\alpha_{j}>0$ are constants.

Let $\{v_{k}\}$ be a sequence of solutions to (1.13) with $\rho=\rho_{k}$. We say

[TABLE]

if $\rho\frac{he^{v}}{\int_{\Omega}he^{v}{\rm d}x}\rightharpoonup 8\pi\sum_{j=1}^{N}(1+\alpha_{j})\delta_{p_{j}}$ in $\Omega$ in the sense of measure, where $\alpha_{j}=0$ if $p_{j}\notin\{q_{1}\cdots,q_{N}\}$. Similar to notations for the first part, we assume there exist $1\leq t\leq\tau\leq m$ such that $\alpha_{1}=\cdots\alpha_{t}>\alpha_{i}$, $i\geq t+1$ and $\alpha_{\tau+1}=\cdots\alpha_{m}$.

Let $G_{\Omega}$ be the Green’s function defined by

[TABLE]

and $R_{\Omega}(x,p)=G_{\Omega}(x,p)+\frac{1}{2\pi}\log|x-p|$ be the regular part of $G_{\Omega}(x,p)$. In order to state the uniqueness results of (1.13) we denote $N^{*}=4\pi\sum_{j=1}^{m}\alpha_{j}$ and

[TABLE]

Then we have the following result similar to Theorem 1.1.

Theorem 1.3**.**

Let $v_{k}^{(1)}$ and $v_{k}^{(2)}$ be two sequences of solutions of (1.13) (1.14) with $\rho_{k}^{(1)}=\rho_{k}=\rho_{k}^{(1)}$ and $\alpha_{j}\in\mathbb{R}^{+}\setminus\mathbb{N}\,(1\leq j\leq m)$. If $L_{\Omega}(\mathbf{p})\neq 0$ and $D_{\Omega}(\mathbf{p})=0$, then $v_{k}^{(1)}=v_{k}^{(2)}$ for $k$ large enough.

If the set of blowup points is a mixture of non-quantized singular sources and regular points, we also have a uniqueness result. Let

[TABLE]

and $D^{2}f_{\Omega}^{*}$ be the Hessian tensor field on $M$. In this case, $(p_{\tau+1},\cdots,p_{m})$ is a critical point of $f_{\Omega}^{*}$. Then, we obtain the following result.

Theorem 1.4**.**

Let $v_{k}^{(1)}$ and $v_{k}^{(2)}$ be two sequences of solutions of (1.13) (1.14) with $\rho_{k}^{(1)}=\rho_{k}=\rho_{k}^{(1)}$ and $0\leq\alpha_{j}<1(1\leq j\leq m)$. If $L_{\Omega}(\mathbf{p})\neq 0$, $D_{\Omega}(\mathbf{p})=0$ and \det\big{(}D^{2}f_{\Omega}^{*}(p_{\tau+1},\cdots,p_{m})\big{)}\neq 0, then $v_{k}^{(1)}=v_{k}^{(2)}$ for $k$ large enough.

When we were in the final stage of writing this article, we found that Bartolucci, et, al [4] posted an article on arxiv.org about the same topic. Their theorem is a special case of our results and both works were carried out independently.

The organization of this paper is as follows. Section 2 is dedicated to notations and preliminary sharp estimates for bubbling solutions of equation (1.1). In section 3 we consider the differences between two bubbling sequences and establish many estimates near each blowup point and away from all blowup points. In section 4 we derive some Pohozaev-type identities and evaluate each term carefully. These Pohozaev identities play a key role in the proof of the main theorems. Finally the proof of Theorem 1.1 is placed in section 5 and that of Theorem 1.2 can be found in section 6. At the end of section 6, we list the brief sketch of the proof of Theorems 1.3 and 1.4 based on well known facts [27].

2. Preliminary Estimates

Since the proof of the main theorems requires very delicate analysis, in this section we list some established estimates in [13, 16, 41, 42].

Let $w_{k}$ be a sequence of solutions of (1.3) with $\rho=\rho_{k}$. Suppose that $w_{k}$ blows up at $m$ points $\{p_{1}\cdots,p_{m}\}$ as we have stated in section one. To describe the bubbling profile of $w_{k}$ near $p_{j}$, we set

[TABLE]

and write the equation for $u_{k}$ as

[TABLE]

It is easy to observe from the definition of $u_{k}$ that

[TABLE]

From previous works of Liouville equations ( for example [13] ),

[TABLE]

where $\bar{u}_{k}$ is the average of $u_{k}$ on $M$:

[TABLE]

For the convenience later we fix $r_{0}>0$ small and $M_{j}\subset M,1\leq j\leq m$ such that

[TABLE]

According to this definition $M_{1}=M$, if $m=1$.

Then we use $\lambda_{k,j}$ to denote

[TABLE]

and let $U_{k,j}$ be a global solution of

[TABLE]

with the expression ($U_{k,j}$ is called a standard bubble):

[TABLE]

It is well-known [25, 5] that $u_{k}$ can be approximated by the standard bubbles $U_{k,j}$ near $p_{j}$ with $O(1)$ error:

[TABLE]

As a consequence,

[TABLE]

for some $C$ independent of $k$. Furthermore, it is established in [2] that $\rho_{*}=\lim_{k\to+\infty}\rho_{k}$.

Later, sharper estimates were obtained in [42, 16] for $1\leq j\leq\tau$ and in [13, 41, 22] for $\tau+1\leq j\leq m$. In order to apply those estimates, we might consider the equation in terms of the flat metric and introduce the following notations.

In $B(p_{j},r_{0})$, the flat metric is {\rm d}s^{2}=e^{\phi_{j}}\big{(}({\rm d}x_{1})^{2}+({\rm d}x_{2})^{2}\big{)} with $\phi_{j}$ satisfying

[TABLE]

where [math] is the coordinate of $p_{j}$, $\Delta=\sum_{i=1}^{2}\frac{\partial^{2}}{\partial x_{i}^{2}}$. In this local coordinate, equation (2.2) is equivalent to

[TABLE]

If we denote $\tilde{h}_{j}=h_{j}e^{\phi_{j}}$, then (2.11) can be written as follows:

[TABLE]

To state the more refined asymptotic analysis we introduce the following notations:

[TABLE]

[TABLE]

[TABLE]

where $R(x,p_{k,j})$ is the regular part of $G(x,p_{k,j})$. Finally for $x\in B(p_{k,j},r_{0})$, set

[TABLE]

[TABLE]

2.1. Sharper estimates

If $\alpha_{j}\in\mathbb{R}^{+}\setminus\mathbb{N}$, in order to obtain the refined estimates of the bubbling solutions, the second author considered the harmonic function $\psi_{k,j}$ in [42], which satisfies

[TABLE]

With the help of $\psi_{k,j}$, Zhang and Chen-Lin proved the following sharp estimate in [42].

Theorem 1**.**

[42, 16]**

For $x\in B(p_{k,j},r_{0})$, it holds that

[TABLE]

where $a=\nabla(\log h_{j}+G_{k,j})(p_{k,j})\in\mathbb{R}^{2}$ and

[TABLE]

In [16],the following estimates for $\psi_{k,j}$ and $\sigma_{k}$ are established:

Theorem 2**.**

[16]**

[TABLE]

[TABLE]

Then, by Theorem 1 and Theorem 2, we have

[TABLE]

For the case $\tau+1\leq j\leq m$, the estimate for $\eta_{k,j}$, established in [13][41][22], is

[TABLE]

Moreover, according to the proof of Theorem 3.5 in [16], the following estimate holds:

[TABLE]

As a consequence, we have

[TABLE]

For the difference between $\rho_{k}$ and $\rho_{*}$, $\rho_{k}$ and $8\pi(1+\alpha_{j})$, the following estimates also have been proved in [16, 13].

Theorem 3**.**

[16, 13]**

[TABLE]

with fixed $\gamma\in(0,\min({\alpha_{1},\frac{1}{2}}))$ small and

[TABLE]

If $\tau<m$, as in [13] and [3], non-degeneracy condition \det\big{(}D^{2}f^{*}(p_{\tau+1},\cdot,p_{m})\big{)}\neq 0 leads to

[TABLE]

Futhermore, in [16], the authors showed that

[TABLE]

2.2. The kernel of the linearized equations

In the proof of the uniqueness, we need some facts about the linearized equation after the appropriate rescale.

For $\alpha\in\mathbb{R}^{+}\setminus\mathbb{N}$, Chen-Lin proved the following lemma in [16].

Lemma 1**.**

Suppose $\alpha>0$ is not an integer, $\varphi$ is a $C^{2}$-function that satisfies

[TABLE]

where $U_{\alpha}(x)=\log\frac{8(1+\alpha)^{2}}{(1+|x|^{2(1+\alpha)})^{2}}$ and $\kappa\in(0,1)$. Then there exists some constant $b_{0}$ such that

[TABLE]

For $\alpha=0$, Chen-Lin proved the following lemma in [13].

Lemma 2**.**

Let $\varphi$ be a $C^{2}$-function of

[TABLE]

where $U(x)=\log\frac{8}{(1+|x|^{2})^{2}}$ and $\kappa\in[0,1)$. Then there exist constants $b_{0}$, $b_{1}$, $b_{2}$ such that

[TABLE]

where

[TABLE]

3. The difference between $u_{k}^{(1)}$ and $u_{k}^{(2)}$

The way we prove the main theorems is by contradiction. So we assume that $u_{k}^{(1)}$ and $u_{k}^{(2)}$ are two different sequences of solutions to (1.3) with $\rho_{k}^{(1)}=\rho_{k}=\rho_{k}^{(2)}$, and common blowup points located at $p_{1},\cdots,p_{m}$. For $i=1,2$, we use the following notations

[TABLE]

with obvious interpretations in the context. Finally the following three functions are defined by the difference of $u_{1}^{k}$ and $u_{2}^{k}$:

[TABLE]

Clearly $\varsigma_{k}$ satisfies

[TABLE]

As the first step of our proof, we give an initial estimate of $\parallel u_{k}^{(1)}-u_{k}^{(2)}\parallel_{L^{\infty}(M)}$ using $L({\bf p})\neq 0$:

Lemma 3.1**.**

Under the assumption of $L(\mathbf{p})\neq 0$, we have

[TABLE]

Proof.

Step 1. For $x\in B(p_{k,j},r_{0}),1\leq j\leq m$, by (2.16) (2.21) (2.13) (2.15) and Theorem 3, we have

[TABLE]

Theorem 3 and $L(\mathbf{p})\neq 0$ give rise to

[TABLE]

which immediately implies

[TABLE]

Then by (3.6) and (2.24), what holds for one point is also true at other blowup points:

[TABLE]

On the other hand, using (2.28) in direct computation, we have,

[TABLE]

Thus $u_{k}^{(1)}$ and $u_{k}^{(2)}$ are close in the interior of the ball $B(p_{k,j}^{(1)},r_{0})$:

[TABLE]

Step 2. For $x\in M\backslash\bigcup_{j=1}^{m}B(p_{k,j}^{(1)},r_{0})$, we first use the Green’s representation formula to write u_{k}^{(1)}-u_{k}^{(2)}-\big{(}\bar{u}_{k}^{(1)}-\bar{u}_{k}^{(2)}\big{)} in three parts:

[TABLE]

Before we evaluate each one of them we recall a few facts: First

[TABLE]

Next for $x\in M\backslash\bigcup_{j=1}^{m}B(p_{k,j}^{(1)},r_{0})$, $y\in B(p_{k,j}^{(1)},\frac{r_{0}}{2})$,

[TABLE]

Then using symmetry, scaling, and the closeness between $u_{k}^{(i)}$ with standard bubbles, we have

[TABLE]

The closeness between $\rho_{k,j}^{(1)}$ and $\rho_{k,j}^{(2)}$ leads to the smallness of $I_{2}$ (see (2.13) (2.25) and (2.26)):

[TABLE]

For $I_{3}$, the magnitude of $u_{k}^{(i)}$ outside the bubbling area determines the smallness of $I_{3}$:

[TABLE]

Therefore

[TABLE]

To eliminate the averages in (3.10) we take advantage of (2.23) and (3.6):

[TABLE]

Using (3.11) in (3.10) we arrive at

[TABLE]

for all $x\in M\backslash\bigcup_{j=1}^{m}B(p_{k,j}^{(1)},r_{0})$. Lemma 3.1 is established.

∎

As an immediate application, Lemma 3.1 gives ( see (3.3) )

[TABLE]

To simply the notations, we set

[TABLE]

and

[TABLE]

which satisfies

[TABLE]

for $|z|<r_{0}\epsilon_{k,j}^{-1}$.

The following lemma determines the limit of $\varsigma_{k,j}$ in both situations:

Lemma 3.2**.**

( The limit of* $\varsigma_{k,j}$ )*

(i)* For $1\leq j\leq\tau$,*

[TABLE]

where $\varphi_{j,0}$ is

[TABLE]

(ii)* If $\tau<m$ and $\tau+1\leq j\leq m$, there exist constants $b_{j,0}$, $b_{j,1}$ and $b_{j,2}$ such that*

[TABLE]

where $\varphi_{j,i}$ are

[TABLE]

Proof.

(i) For $1\leq j\leq\tau$, it is easy to use (3.13) (2.16) (2.21) and (2.22) to obtain

[TABLE]

Therefore, $\varsigma_{k,j}\rightarrow\varsigma_{j}$ in $C_{loc}(\mathbb{R}^{2})$ and $\varsigma_{j}$ satisfies

[TABLE]

Since it is obvious to have $|\varsigma_{j}|\leq 1$ from $|\varsigma_{k,j}|\leq 1$, we apply Lemma 1 to have $\varsigma_{j}=b_{j,0}\varphi_{j,0}$ for some constant $b_{j,0}$ and

[TABLE]

That is $\varsigma_{k,j}\rightarrow b_{j,0}\varphi_{j,0}$ in $C_{loc}(\mathbb{R}^{2})$.

(ii) For $\tau+1\leq j\leq m$, by (3.13) (2.16) and (2.22), we have $\varsigma_{k,j}\rightarrow\varsigma_{j}$ in $C_{loc}(\mathbb{R}^{2})$, where

[TABLE]

In this case we use Lemma 2 to conclude that

[TABLE]

for some constants $b_{j,0}$, $b_{j,1}$ and $b_{j,2}$. Lemma 3.2 is established.

∎

Our next goal is to prove that all $b_{j,0}$ are the same, and equal to the limit of $\varsigma_{k}$ away from the bubbling area. Our approach is similar to the corresponding parts in [26] for the Chern-Simons-Higgs equation and in [3] for regular mean field equations.

Lemma 3.3**.**

There exists a constant $b_{0}$ such that

[TABLE]

Moreover, $b_{j,0}=b_{0}$ for all $1\leq j\leq m$.

Proof.

Starting from the equation for $\varsigma_{k}$:

[TABLE]

we observe from (3.13) (2.14) and (2.20) that $c_{k}\rightarrow 0$ in $C_{loc}(M\backslash\{p_{1},\cdots,p_{m}\})$. Since $\|\varsigma_{k}\|_{L^{\infty}(M)}\leq 1$, $\varsigma_{k}\rightarrow\varsigma_{0}$ in $C_{loc}(M\backslash\{p_{1},\cdots,p_{m}\})$, where $\varsigma_{0}$ satisfies

[TABLE]

The bound for $\varsigma$: $\|\varsigma_{0}\|_{L^{\infty}(M)}\leq 1$, which comes from $\|\varsigma_{k}\|_{L^{\infty}(M)}\leq 1$, yields the smoothness of $\varsigma_{0}$ on the whole manifold. Thus $\varsigma_{0}\equiv-b_{0}$ in $M$ for some constant $b_{0}$. In particular,

[TABLE]

For $1\leq j\leq m$, let

[TABLE]

be a sequence of solutions of

[TABLE]

Recall that

[TABLE]

Using (2.28) (2.16) (2.21), (2.22) and integration by parts, we find, for $d\in(0,r_{0})$, that

[TABLE]

By scaling $x=\epsilon_{k,j}z+p_{k,j}^{(1)}$, (2.21), (2.22) and the estimate of $\parallel u_{k}^{(1)}-u_{k}^{(2)}\parallel_{L^{\infty}(M)}$, it is not hard to obtain

[TABLE]

Let $\varsigma_{k,j}(r)$ be the spherical average of $\varsigma_{k}$:

[TABLE]

where $r=|x-p_{k,j}^{(1)}|$. Then (3.20) yields

[TABLE]

For any $r\in(R\epsilon_{k,j},r_{0})$, we also notice that

[TABLE]

Then we conclude

[TABLE]

Integrating (3.21) from $R\epsilon_{k,j}$ to $r$, we get for all $r\in(R\epsilon_{k,j},r_{0})$

[TABLE]

The first term of (3.22) is almost a constant ( Lemma 3.2 ):

[TABLE]

where $\lim_{R\to+\infty}o_{R}(1)=0$. Then it is easy to see from (3.22) and (3.19) that $b_{j,0}=b_{0}$ for all $1\leq j\leq m$. ∎

Next we introduce a few quantities to be used later. For $1\leq j\leq m$, let

[TABLE]

[TABLE]

It is easy to see that in $M\setminus\{p_{k,1}^{(1)},\cdots,p_{k,m}^{(1)}\}$,

[TABLE]

Set

[TABLE]

and

[TABLE]

then we estimate $\nabla v_{k,j}^{(i)}$ away from the bubbling area:

Lemma 3.4**.**

For any $\theta\in(0,r_{0})$ small enough and $x\in B(p_{k,j}^{(1)},2r_{0})\setminus B(p_{k,j}^{(1)},\theta)$, the gradient of $v_{k,j}^{(i)}$ is very close to that of a harmonic function:

[TABLE]

Proof.

By (2.14) (2.20) and Theorem 3, we have

[TABLE]

for $i=1,2$, $1\leq j\leq m$. Consequently using (3.23)$\sim$(3.25) we have

[TABLE]

∎

Next we estimate $\varsigma_{k}$ and its derivatives away from blowup points.

Lemma 3.5**.**

Given $\theta\in(0,r_{0})$ small enough, we have

[TABLE]

Proof.

From the Green’s representation formula for $u_{k}^{(i)}$ and the definition of $\varsigma_{k}$, we have the following expression of $\varsigma_{k}$:

[TABLE]

Then for $x\in M\setminus\bigcup_{j=1}^{m}B(p_{k,j}^{(1)},\theta)$ we evaluate the integral in three parts:

[TABLE]

Note that $J_{3}=\emptyset$ if $\tau=m$. Then it follows from the definition of $\eta_{k,j}^{(1)}$, (2.21) and (2.22) that

[TABLE]

For $1\leq j\leq\tau$, using $y=\epsilon_{k,j}z+p_{k,j}^{(1)}$ in the evaluation of the identity above, we have

[TABLE]

If $\tau<m$, let us recall that $\alpha_{j}=0$ for $\tau+1\leq j\leq m$. Similarly, by the standard scaling, Lemma 3.2 and symmetry, we have

[TABLE]

where

[TABLE]

For the second order terms in the expansion of $G$, we have

[TABLE]

Consequently for $J_{2}$ and $J_{3}$ we have

[TABLE]

[TABLE]

Observing (3.30) (3.32) and (3.33), we conclude that (3.29) holds. Then by standard estimates we also have

[TABLE]

∎

4. Estimates associated with Pohozaev identities

In this section, we establish some sharp estimates for certain terms crucial for evaluation of Pohozaev identities.

The first important quantity is $A_{k,j}$, defined in (3.29) and the study of which is through the following Pohozaev identity:

Lemma 4.1**.**

For $1\leq j\leq m$ and any $r\in(0,r_{0})$, it holds that

[TABLE]

This Pohozaev identity has been used in [3] and [26], we include the proof for the convenience of the readers.

Proof.

First we observe that for any two smooth functions $u$ and $v$,

[TABLE]

Replacing $u$, $v$ by $v_{k,j}^{(1)}-v_{k,j}^{(2)}$ and $v_{k,j}^{(1)}+v_{k,j}^{(2)}$ respectively in (4.2), we have

[TABLE]

By the definition of $v_{k,j}^{(i)}$, we see that, for $x\in B(p_{k,j}^{(1)},r_{0})$,

[TABLE]

Using (4.4) and

[TABLE]

the right hand side (RHS) of (4.3) can be written as:

[TABLE]

Since $\varsigma_{k}=\frac{v_{k,j}^{(1)}-v_{k,j}^{(2)}}{\parallel v_{k,j}^{(1)}-v_{k,j}^{(2)}\parallel_{L^{\infty}(M)}}$ and $\nu=\frac{x-p_{k,j}^{(1)}}{r}$, we have

[TABLE]

On the other hand,

[TABLE]

Then (4.1) follows from (4.3), (4.5) and (4.6).

∎

Remark 4.1**.**

It is easy to see Pohozaev-type identity (4.1) also holds for $\alpha_{j}>-1$.

Lemma 4.2**.**

For all $1\leq j\leq m$,

[TABLE]

Proof.

From (3.28) and (3.34), we find that

[TABLE]

For $x\in\partial B(p_{k,j}^{(1)},r)$ we use the Green’s representation formula to estimate $\varsigma_{k}(x)$:

[TABLE]

where

[TABLE]

It is easy to see that the last term is rather minor:

[TABLE]

Setting

[TABLE]

we now have

[TABLE]

Thus

[TABLE]

Now we take the global cancellation property into consideration: for any fixed $\theta\in(0,r)$,

[TABLE]

Using (3.25) (4.9) and (4.2), we have

[TABLE]

where $B(p_{k,j}^{(1)},r)$ and $B(p_{k,j}^{(1)},\theta)$ are replaced by $B_{r}$, $B_{\theta}$ respectively for simplicity. Therefore

[TABLE]

Further direct computation yields

[TABLE]

where $\lim_{\theta\to 0}o_{\theta}(1)=0$, and we have used the fact that all the terms related to $l\neq j$ are minor. Let us observe that

[TABLE]

Obviously, from (4.10)$\sim$(4.12) we can see that

[TABLE]

which together with (4.8), concludes the proof of Lemma 4.2.

∎

Lemma 4.3**.**

[TABLE]

[TABLE]

Proof.

We use $K_{1},K_{2},K_{3}$ to denote the three terms on the right hand of (4.1). The first two terms are quite easy to estimate:

[TABLE]

[TABLE]

More work is needed for

[TABLE]

First we use $\nabla\phi_{j}(p_{k,j}^{(1)})=0$ to write $\nabla(\log\tilde{h}_{j}+\phi_{k,j})(x)$ as

[TABLE]

Then we evaluate $K_{3}$ in three cases:

$\mathbf{Case\ 1:}$ $1\leq j\leq t$ $(\alpha_{j}=\alpha_{1})$. The assumption $D(\mathbf{p})=0$ and (3.23) (2.27) imply

[TABLE]

Thus, after the scaling $x=\epsilon_{k,j}z+p_{k,j}^{(1)}$, the first order term can be estimated as follows:

[TABLE]

For the second order term that contains $D^{2}(\log\tilde{h}_{j}+\phi_{k,j})(p_{k,j}^{(1)})$, we have

[TABLE]

The expression above can be greatly simplified by this beautiful identity:

[TABLE]

Consequently, Lemma 3.2 and the two identities above lead to

[TABLE]

Also elementary estimate gives

[TABLE]

Therefore, we complete the proof of (4.13) by using (4.16) (4.17) and (4.19)$\sim$(4.21). Note that the leading term in the second order term is ignored at this stage, since the requirement of error in the current step is still crude.

$\mathbf{Case\ 2:}$ $t+1\leq j\leq\tau$ $(0<\alpha_{j}<\alpha_{1})$. For the first term it is easy to see that

[TABLE]

For the second order term we have

[TABLE]

where we used the scaling $x=\epsilon_{k,j}z+p_{k,j}^{(1)}$ and $\alpha_{j}<\alpha_{1}$. Similar to (4.21), we know

[TABLE]

Therefore (4.14) follows from (4.16), (4.17) and (4.22)$\sim$(4.24).

$\mathbf{Case\ 3:}$ $\tau+1\leq j\leq m$ $(\alpha_{j}=0)$. In view of (2.29), we get

[TABLE]

The first order term is rather small:

[TABLE]

For the second order term we have

[TABLE]

where we have used Lemma 3.2 and symmetry. It is easy to see

[TABLE]

and

[TABLE]

Finally, by scaling we immediately observe that

[TABLE]

Therefore, $K_{3}$ is small in this case as well.

[TABLE]

where $\alpha_{1}>0$ is used. Lemma 4.3 is established.

∎

Since $|A_{k,j}|=O(1)$, (4.1) along with Lemma 4.2 and Lemma 4.3 implies the initial estimate for $A_{k,j}$.

Corollary 4.1**.**

[TABLE]

∎

Based on (4.29), we can improve the estimates in (4.7) and (3.29):

[TABLE]

[TABLE]

The identity (4.31), which is the refined $C^{1}$-estimate of $\varsigma_{k}$ away from the blowup points, will help to improve the estimate of RHS of Pohozaev-type identity (4.1) and the estimate of $D\varsigma_{k}$. The later one will play a part in section 6. In order to achieve this goal, we analyse the projections of $\varsigma_{k,j}$ in more detail.

For $1\leq j\leq\tau$, we recall the equation of $\varsigma_{k}$ in $B(p_{k,j}^{(1)},r_{0})$:

[TABLE]

and set the following quantities for convenience:

[TABLE]

Then the equation for $\varsigma_{k}$ becomes

[TABLE]

where

[TABLE]

After scaling $x=\epsilon_{k,j}z+p_{k,j}^{(1)}$, we have

[TABLE]

where

[TABLE]

For each integer $l\geq 0$ we define the projections of frequency $l$ as

[TABLE]

Obviously the study of $\xi_{l}$ is representative enough. (4.32) shows that $\xi_{l}$ satisfies

[TABLE]

where

[TABLE]

and $a_{k,j}^{1}$ is the first component of $a_{k,j}$. Moreover, from (4.31) we obtain that $\xi_{l}(0)=o(1)$ for all $l\geq 1$ and

[TABLE]

From the equation of $\xi_{l}$ and the maximum principle, we only need to consider the finite $l$. Without loss of generality, we consider $1\leq l\leq l_{0}$ in the following analysis. Let us denote $\delta_{l,j}=\frac{l}{1+\alpha_{j}}$ and consider the homogeneous ordinary differential equation

[TABLE]

By direct computation, we can verify that the following two functions are two fundamental solutions of (4.34)

[TABLE]

Using $|\xi_{l}|\leq 1$ we have $C_{l,2}=0$, that is

[TABLE]

where $C_{l,1}$ is a constant, and

[TABLE]

for

[TABLE]

It is easy to see that $w^{{}^{\prime}}=(\xi_{l,1}\xi_{l,2}^{{}^{\prime}}-\xi_{l,1}^{{}^{\prime}}\xi_{l,2})^{{}^{\prime}}=-\frac{1}{r}w$, which means $w(r)\sim\frac{1}{r}$. Next, let us estimate $\xi_{l}$ in $(0,\frac{r_{0}}{\epsilon_{k,j}})$ for $l\geq 1$.

For $1\leq j\leq t$, the assumption $D(\mathbf{p})=0$ implies $a_{k,j}=O(\epsilon_{k,1}^{2})$. Furthermore, for $t+1\leq j\leq\tau$, it is easy to see that $\epsilon_{k,j}=o(\epsilon_{k,1})$. Therefore, for all $1\leq j\leq\tau$, we estimate $\tilde{E}_{l}$ as follows

[TABLE]

Roughly,

[TABLE]

By using the above estimates (4.37) for $\tilde{E}_{l}$ and (4.35), we have

[TABLE]

Direct computation shows, for $0<r<1$

[TABLE]

and for $1<r<\frac{r_{0}}{\epsilon_{k,j}}$

[TABLE]

Consequently

[TABLE]

Combining (4.36) and (4.38), we rewrite $\tilde{E}_{l}$ as

[TABLE]

Then repeating the above argument $n_{0}$ times, we obtain

[TABLE]

Then, from (4.33), we have

[TABLE]

and

[TABLE]

Similarly,

[TABLE]

In other words, all projections of high frequency of $\varsigma_{k,j}$ $(1\leq j\leq\tau)$ are $o(\epsilon_{k,1})$.

Using (4.41) and (4.42), we now obtain an important sharper estimate of the right hand side of (4.1).

Lemma 4.4**.**

[TABLE]

Proof.

In view of (4.16)$\sim$(4.18) and (4.22)$\sim$(4.24), we only need to improve the estimate in (4.22). In other words, it is enough to prove the following estimate

[TABLE]

In fact, by the change of variable $x=\epsilon_{k,j}z+p_{k,j}^{(1)}$, we have

[TABLE]

where we used the fact $\alpha_{j}<\alpha_{1}$ and the definition of $n_{0}$. Then from symmetry and the estimates of high frequency of $\varsigma_{k,j}$, which are (4.41) and (4.42), we have the following estimate

[TABLE]

Therefore, (4.44) holds. Finally, combining (4.44) with the proof of Lemma 4.3, we obtain the esstimate (4.43).

∎

Based on the Pohozaev-type identity (4.1) and its refined estimates, which are (4.30) (4.14) (4.15) and (4.43), we can improve the estimate for $A_{k,j}$ and prove $b_{0}=0$.

Corollary 4.2**.**

[TABLE]

∎

Proposition 4.1**.**

$b_{0}=0$. In particular, $b_{j,0}=0$, for $1\leq j\leq m$.

Proof.

Now the global cancellation property of $f_{k}$ plays a crucial role:

[TABLE]

From (4.1) (4.30) (4.14) (4.15) and (4.43), we can see

[TABLE]

On the other hand, from (2.24), it holds

[TABLE]

As a consequence, we obtain

[TABLE]

which together with the assumption $L(\mathbf{p})\neq 0$ implies $b_{0}=0$. In particular, $b_{j,0}=0$ for $1\leq j\leq m$.

∎

5. Proof of Theorem 1.1

Proof of Theorem 1.1 .

Let $p_{k}^{*}$ be a maximum point of $\varsigma_{k}$, which says

[TABLE]

In view of Lemma 3.3 and Proposition 4.1, we obtain the fact

[TABLE]

Therefore,

[TABLE]

for some $p_{j}\in\{p_{1},\cdots,p_{m}\}$. Moreover, denoting $s_{k}=|p_{k}^{*}-p_{k,j}^{(1)}|$, by Lemma 3.2 and Proposition 4.1, it holds

[TABLE]

Thus,

[TABLE]

Setting $\tilde{\varsigma_{k}}(x)=\varsigma_{k}(s_{k}x+p_{k,j}^{(1)})$, $|x|<s_{k}^{-1}r$, where $r>0$ small enough, then $\tilde{\varsigma_{k}}$ satisfies

[TABLE]

On the other hand, by (5.1), we also have

[TABLE]

In view of (5.3) and $|\tilde{\varsigma_{k}}|\leq 1$, we see that $\tilde{\varsigma_{k}}\rightarrow\tilde{\varsigma_{0}}$ in $C_{loc}(\mathbb{R}^{2}\backslash\{0\})$, where $\tilde{\varsigma_{0}}$ satisfies $\Delta\tilde{\varsigma_{0}}=0$ in $\mathbb{R}^{2}\backslash\{0\}$. Since $|\tilde{\varsigma_{0}}|\leq 1$, we have $\Delta\tilde{\varsigma_{0}}=0$ in $\mathbb{R}^{2}$. Hence $\tilde{\varsigma_{0}}$ is a constant.

Recalling that $\frac{|p_{k}^{*}-p_{k,j}^{(1)}|}{s_{k}}=1$ and (5.4), we find that $\tilde{\varsigma_{0}}\equiv 1$ or $\tilde{\varsigma_{0}}\equiv-1$. Therefore, we obtain that for $k$ large enough

[TABLE]

By using Lemma 3.3, we have

[TABLE]

for fixed $d>0$ small enough and arbitrary $R>0$ large enough.

However, by (5.3), $\epsilon_{k,j}\ll s_{k}$. Thus, $|\varsigma_{k}(s_{k})|<\frac{1}{4}$ for $k$ large enough, which contradicts with (5.5). Theorem 1.1 is established.

∎

6. Proof of Theorem 1.2

In this section, we will analyse the behavior of $u_{k}^{(1)}$ and $u_{k}^{(2)}$ whose common blowup points include singular source(s) and regular point(s). So in this section $\tau<m$, $0<\alpha_{j}<1$ for $1\leq j\leq\tau$ and $\alpha_{j}=0$ for $\tau+1\leq j\leq m$. Our argument is similar to the approach in [3] where all blowup points are regular points. The fact that $\varsigma_{k,j}\rightarrow 0$ in $C_{loc}(\mathbb{R}^{2})$ for all $1\leq j\leq m$ plays a vital role.

In [26], Lin-Yan obtained the following Pohozaev-type identity:

Lemma 3**.**

[3, 26]**

For $\tau+1\leq j\leq m$, it holds

[TABLE]

By Lemma 4.6 in [3] and Appendix D in [26], we have:

[TABLE]

for $i=1,2$ and $\tau+1\leq j\leq m$. The detail of this proof can be found in [3].

The LHS of (6.1) boils down to sharp estimates of $\nabla v_{k,j}^{(i)}$ and $\nabla\varsigma_{k}$ on $\partial B(p_{k,j}^{(1)},r)$. The estimate for $\nabla v_{k,j}^{(i)}$ is established in Lemma 3.4, and the following lemma provides the estimates for $\nabla\varsigma_{k}$ (see (3.34) for comparison).

Lemma 6.1**.**

For any $\theta\in(0,r)$ small enough, it holds

[TABLE]

Proof.

Using the same notations in (3.30) and (3.33), now we only need to show

[TABLE]

Indeed, from (4.45) and the assumption $0<\alpha_{1}<1$, we have

[TABLE]

Recall that

[TABLE]

Based on (4.41) and (4.42), by the method similar to the proof of (4.44) in Lemma 4.4, we have

[TABLE]

Therefore,

[TABLE]

Consequently (6.3) holds in C^{1}\big{(}M\setminus\bigcup_{j=1}^{m}B(p_{k,j}^{(1)},\theta)\big{)} and the gradient estimate is

[TABLE]

∎

By the improved estimates of $\nabla v_{k,j}^{(i)}$ and $\nabla\varsigma_{k}$ in (3.28) and (6.6), we can estimate the left hand of (6.1) just like Lemma 4.7 in [3] or Appendix D in [26] and the result is:

[TABLE]

Finally we prove $b_{j,1}=b_{j,2}=0$ for all $j$.

Proposition 6.1**.**

$b_{j,1}=b_{j,2}=0$, for all $j=\tau+1,\cdots,m$. In particular,

[TABLE]

Proof.

Obviously, (6.1) together with (6.2) and (6.7) implies, for all $i=1,2$, and $j=\tau+1,\cdots,m$,

[TABLE]

Set $\vec{b}=(\tilde{b}_{\tau+1,1}B_{\tau+1},\tilde{b}_{\tau+1,2}B_{\tau+1},\cdots,\tilde{b}_{m,1}B_{m},\tilde{b}_{m,2}B_{m})$, where

[TABLE]

Then by (2.28) and letting $k\to+\infty$, we obtain that

[TABLE]

By using the non-degeneracy assumption \det\big{(}D^{2}f^{*}(p_{\tau+1},\cdots,p_{m})\big{)}\neq 0, we conclude that

[TABLE]

Proposition 6.1 is established.

∎

Proof of Theorem 1.2 .

From Lemma 3.3 and Proposition 4.1 $\varsigma_{k}$ tends to [math] in $C_{loc}(M\backslash\{p_{1},\cdots,p_{m}\})$. By Lemma 3.2 and Proposition 6.1, we have

[TABLE]

Theorem 1.2 follows just like the last step of the proof of Theorem 1.1.

∎

Finally, we finish to prove Theorem 1.3 and Theorem 1.4 about Dirichlet problems.

Proof of Theorems 1.3, 1.4 .

For the blowup solutions to (1.13), the corresponding estimates as in section 2 have been also obtained in [16][42] for $\alpha_{j}\in\mathbb{R}^{+}\setminus\mathbb{N}$ and in [13][41][22] for $\alpha_{j}=0$. Those preliminary estimates have almost the same form except for $\phi_{j}=0$ and $K\equiv 0$, where $\phi_{j}$ are the conformal factor at $p_{j}$ and $K$ is the Gaussian curvature of $M$.

Then, under the assumption of regularity about $\partial\Omega$ and $q_{j}\in\Omega$ $(1\leq j\leq N)$, [27] has showed that the blowup points of (1.13) are far away from $\partial\Omega$ via the moving plane method and the Pohozaev identities. Consequently, the terms coming from the boundary of domain are included in the error term. In other words, those boundary terms do not affect our argument.

On the other hand, the vital part of estimates obtained in section 3, 4 and 6 only come from local analysis, Therefore, such results still work for the Dirichlet problem (1.13).

Thus, Theorem 1.3 and Theorem 1.4 can be proved as Theorem 1.1 and Theorem 1.4, respectively.

∎