Blow up solutions for Sinh-Gordon equation with residual mass

Abstract

We are concerned with the Sinh-Gordon equation in bounded domains. We construct blow up solutions with residual mass exhibiting either partial or asymmetric blow up, i.e. where both the positive and negative part of the solution blow up. This is the first result concerning residual mass for the Sinh-Gordon equation showing in particular that the concentration-compactness theory with vanishing residuals of Brezis-Merle can not be extended to this class of problems.

1 Introduction

We are concerned with the following Sinh-Gordon equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\rho ^+\dfrac{e^u}{\int _{\Omega }e^udx}-\rho ^-\dfrac{e^{-u}}{\int _{\Omega }e^{-u}dx}=0\quad &{} \text{ in } \Omega \\ u=0 &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega \subset {{\mathbb {R}}}^2$ is smooth and bounded and $\rho ^+, \rho ^-$ are two positive parameters. The latter problem arises as a mean field equation in the study of the equilibrium turbulence [21, 24]. Moreover, it is also related to constant mean curvature surfaces [20, 29]. Observe that for $\rho ^-=0$, (1.1) reduces to the standard Liouville equation which has been extensively studied in the literature. Therefore, many efforts have been done to study existence [3, 12,13,14,15] and blow up phenomena [1, 10, 16, 17, 20, 23, 25, 26, 28] for this class of problems.

In the present paper we further explore the blow up phenomenon of (1.1). Let $u_n$ be a sequence of solutions to (1.1) corresponding to $\rho ^{\pm }=\rho _n^{\pm }\le C$. Define the positive and negative blow up set as

$$\begin{aligned} S_{\pm }:=\left\{ x\in {\bar{\Omega }} : \ \exists x_n\rightarrow x \ s.t. \ \pm u_n(x_n)-\log \int _{\Omega }e^{\pm u_n}dx + \log \rho _n^{\pm } \rightarrow +\infty \ as \ n\rightarrow \infty \right\} . \end{aligned}$$

We have $S_{\pm }\cap \partial \Omega =\emptyset$ by [1], and $S_{\pm }\subset \Omega$ is finite by the argument of [5]. For $p\in S_{\pm }$ the local mass is defined by

$$\begin{aligned} m_{\pm } (p)=\lim _{r\rightarrow 0}\lim _{n\rightarrow \infty }\dfrac{\rho _n^{\pm }\int _{B_r(p)}e^{\pm u_n}dx}{\int _{\Omega } e^{\pm u_n}dx}. \end{aligned}$$

By [16, 20] we know that $m_{\pm }(p)$ satisfy a quantization property, i.e. $m_{\pm }(p)\in 8\pi {\mathbb {N}}$. Moreover, in view of the relation

$$\begin{aligned} (m_+(p)-m_-(p))^2=8\pi (m_+(p)+m_-(p)), \end{aligned}$$

see for example [23], the couple $(m_+, m_-)$, up to the order, takes the value in the set

$$\begin{aligned} \Sigma :=\Big \{8\pi \left( \frac{k(k-1)}{2}, \frac{k(k+1)}{2}\right) , \ k\in {\mathbb {N}}\setminus \{0\} \Big \}, \end{aligned}$$

(1.2)

see [16, 20]. Finally, by standard analysis [23], one has, for $n\rightarrow +\infty$,

$$\begin{aligned} \rho _n^{\pm }\dfrac{e^{\pm u_n}}{\int _{\Omega }e^{\pm u_n}dx} \rightharpoonup \sum _{p\in S_{\pm }} m_{\pm }(p)\delta _{p} + r_{\pm }, \end{aligned}$$

in the sense of measures, where $r_{\pm }\in L^1(\Omega )$ are residual terms. From the above convergence, $\rho ^{\pm }$ will be called global masses of the blow up solutions. Observe that both the local masses and the residual terms affect the global masses. In striking contrast with the concentration-compactness theory of Brezis-Merle [5], the latter residuals may not be zero a priori. This fact has important effects in the blow up analysis, variational analysis and Leray-Schauder degree theory of (1.1). One of the goals of the present paper is to provide the first explicit example of blow up solutions exhibiting residual terms, thus confirming that the concentration-compactness theory can not be extended to this class of problems.

1.1 Partial blow up

We start here with a related problem, that is partial blow up with prescribed global mass. More precisely, we look for blowing up solutions $-u_n$ with $\rho ^-_n\rightarrow 8\pi k, k\in {\mathbb {N}}$, such that $u_n$ have prescribed global mass $\rho ^+_n=\rho ^+\in (0, 8\pi )$. To this end we introduce

$$\begin{aligned} {\mathcal {F}}_k\Omega :=\biggr \{{{\varvec{\xi }}}:=(\xi _1,\cdots ,\xi _k)\in \Omega ^k: \ \xi _i\ne \xi _j \text{ for } i \ne j\biggr \} \end{aligned}$$

(1.3)

and consider the following singular (at $\xi _i\in \Omega$) mean field equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta z(x,{{\varvec{\xi }}})+\rho ^+\dfrac{h(x, {{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}}{\int _{\Omega }h(x, {{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}dx}=0 \quad &{}\text{ in }~ \Omega ,\\ z(x,{{\varvec{\xi }}})=0 &{} \text{ on } \partial \Omega \end{array}\right. } \end{aligned}$$

(1.4)

where ${{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega$ and $h(x,{{\varvec{\xi }}})=e^{-8\pi \sum _{i=1}^kG(x,\xi _i)}$. Here G(x, y) is the Green function of the Laplace operator in $\Omega$ with Dirichlet boundary condition and we denote its regular part by H(x, y). Equation (1.4) is the Euler-Lagrange equation of the functional

$$\begin{aligned} I_{{\varvec{\xi }}}(z):= \frac{1}{2} \int _{\Omega }|\nabla z|^2dx-\rho ^+\log \left( \int _{\Omega } h(x,{\varvec{\xi }})e^z dx \right) . \end{aligned}$$

To the latter functional and (a combination of) the Green functions we associate the following map:

$$\begin{aligned} \Lambda ({{\varvec{\xi }}}):=\frac{1}{2}I_{{\varvec{\xi }}}(z(\cdot ,{{\varvec{\xi }}}))-32\pi ^2\Big ( \sum _{i=1}^kH(\xi _i,\xi _i) +\sum _{j\ne i}G(\xi _i,\xi _j)\Big ). \end{aligned}$$

(1.5)

It is known by [2] that if $\Omega$ is simply connected and $\rho ^+\in (0, 8\pi )$, then for any ${{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega$ there exists a unique solution to (1.4) and the solution is non-degenerate, in the sense that the linearized problem admits only the trivial solution. Then, by making use of the implicit function theorem it is not difficult to show that the function $\Lambda$ is smooth, see for example [8]. Finally, as in [22], a compact set ${\mathcal {K}}\subset {\mathcal {F}}_k\Omega$ of critical points of $\Lambda$ is said to be $C^1$-stable if, fixed a neighborhood ${\mathcal {U}}$ of ${\mathcal {K}}$, any map $\Phi : {\mathcal {U}}\rightarrow {{\mathbb {R}}}$ sufficiently close to $\Lambda$ in $C^1$-sense has a critical point in ${\mathcal {U}}$.

The first result of this paper is the following.

Theorem 1.1

Let $\Omega$ be simply connected, $\rho ^+\in (0, 8\pi )$ and let ${\mathcal {K}}\subset {\mathcal {F}}_k\Omega$, $k\in {\mathbb {N}}$, be a $C^1$-stable set of critical points of $\Lambda$. Then, there exists $\lambda _0>0$ such that for any $\lambda \in (0,\lambda _0)$ there exists a solution $u_\lambda$ of (1.1) with $\rho ^{\pm }=\rho _\lambda ^{\pm }$ such that the following two properties hold:

• 1. $\rho ^+_\lambda =\rho ^+ , \ \rho ^-_\lambda \rightarrow 8k\pi$ as $\lambda \rightarrow 0$.

• 2. There exist ${{\varvec{\xi }}}(\lambda )\in {\mathcal {F}}_k\Omega$ and $\delta _i(\lambda )>0$ such that $d({{\varvec{\xi }}}, {\mathcal {K}})\rightarrow 0, \ \delta _i\rightarrow 0$ and

  $$\begin{aligned} u_\lambda (x)\rightarrow z(x,{{\varvec{\xi }}})-\sum _{i=1}^k\Big (\log \frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}+8\pi H(x, \xi _i) \Big ) \quad \text{ in } H^1_0(\Omega ), \end{aligned}$$

  as $\lambda \rightarrow 0$, where z solves (1.4) .

Some comments are in order. The assumptions that $\Omega$ is simply connected and $\rho ^+\in (0, 8\pi )$ guarantee the existence of a unique non-degenerate solution to (1.4): in general, the above result holds true whenever such solution exists. For example, one can drop the condition on $\Omega$ by assuming $\rho ^+$ to be sufficiently small, see for example [8].

On the other hand, if $\Omega$ is simply connected and $\rho ^+\in (0, 8\pi )$ it is not difficult to show that for $k=1$ the minimum of $\Lambda$ is a $C^1$-stable set of critical points of $\Lambda$, see for example [8]. Moreover, for non-simply connected domains the function $\Lambda$ always admits a $C^1$-stable set of critical points [7].

Therefore, the conclusion of Theorem 1.1 holds true if either $\Omega$ is simply connected, $\rho ^+\in (0, 8\pi )$ and $k=1$, or $\Omega$ is multiply connected, $\rho ^+$ sufficiently small and $k\ge 1$. Finally, the location of the blow up set can be determined by using the following expression, which can be derived similarly as in [8]:

$$\begin{aligned} \partial _{\xi _j}\Lambda ({{\varvec{\xi }}})=8\pi \frac{\partial z}{\partial x}(\xi _j,{{\varvec{\xi }}})-32\pi ^2\Big ( \frac{\partial H}{\partial x}(\xi _j,\xi _j)+\sum _{i\ne j}\frac{\partial G}{\partial x}(\xi _i,\xi _j)\Big ). \end{aligned}$$

(1.6)

1.2 Asymmetric blow up

We next construct blow up solutions with residual mass exhibiting the asymmetric blow up, i.e. where both the positive and negative part of the solution blow up. Since the local masses $(m_+,m_-)$ belong to the set $\Sigma$ defined in (1.2), for $k \ge 2$ we look for blowing up solution $u_n$ with $\rho _n^-\rightarrow 4\pi k(k+1)$ and $\rho _n^+= \rho ^+=4\pi k(k-1)+\rho _0$, where $\rho _0\in (0, 8\pi )$ is a fixed residual mass. For simplicity of presentation we assume that k is odd, the case of k even being similar. We consider here l-symmetric domains $\Omega$ with $l\ge 2$ even, i.e. if $x\in \Omega$ then ${\mathcal {R}}_l\cdot x\in \Omega$, where

$$\begin{aligned} {\mathcal {R}}_l:=\left( \begin{array}{cc} \cos \frac{2\pi }{l}&{}\sin \frac{2\pi }{l}\\ \\ -\sin \frac{2\pi }{l}&{}\cos \frac{2\pi }{l} \end{array} \right) , \quad l\ge 2 \text{ even }. \end{aligned}$$

(1.7)

Consider then the following singular (at $x=0$) mean field equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta z(x)+\rho _0\dfrac{e^{z(x)-8k\pi G(x,0)}}{\int _{\Omega }e^{z(x)-8k\pi G(x,0)}dx}=0 \quad &{}\text{ in } ~\Omega ,\\ z(x)=0 &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(1.8)

Again by [2] we know that if $\Omega$ is simply connected and $\rho ^0\in (0, 8\pi )$, then there exists a unique non-degenerate solution to (1.8).

The second result of this paper is the following.

Theorem 1.2

Let $\Omega$ be a simply connected l-symmetric domain according to (1.7) and $\rho ^+=4\pi k(k-1)+\rho _0$ with $k\in {\mathbb {N}}$ odd, $l\ge 2$ even and $\rho _0\in (0, 8\pi )$. Then, there exists $\lambda _0>0$ such that for any $\lambda \in (0,\lambda _0)$, there exists a solution $u_\lambda$ of (1.1) with $\rho ^{\pm }=\rho _\lambda ^{\pm }$ such that the following two properties hold:

• 1. $\rho ^+_\lambda =\rho ^+ , \ \rho ^-_\lambda \rightarrow 4\pi k(k+1)$ as $\lambda \rightarrow 0$.

• 2. There exists $\delta _i(\lambda )\rightarrow 0$ (defined in (4.2)) such that

  $$\begin{aligned} u_\lambda (x)\rightarrow z(x)+\sum _{i=1}^k(-1)^i\Big (\log \frac{1}{(\delta _i(\lambda )^{\alpha _i} +|x|^{\alpha _i})^2}+4\pi \alpha _iH(x, 0) \Big ) \quad \text{ in } H^1_0(\Omega ), \quad \alpha _i=4i-2, \end{aligned}$$

  as $\lambda \rightarrow 0$, where z solves (1.8) .

Observe that the assumption that $\Omega$ is simply connected and $\rho _0\in (0, 8\pi )$ is used only to ensure the existence of a non-degenerate solution to (1.8): in general, the above result holds true whenever such solution exists. On the other hand, the symmetry condition of the domain is imposed to rule out the degeneracy of the singular Liouville equation.

The argument follows the strategy introduced in [8, 9] for the Toda system, that is a system of Liouville-type equations, and it is based on the perturbation method starting from an approximate solution and studying the invertibility of the linearized problem. The main difficulty is due to the coupling of the local and global nature of the problem since we are prescribing both the local and global masses. In particular, blow up solutions of (1.1) with local masses $(4\pi k(k-1), 4\pi k(k+1))$ have been constructed in [11] by superposing k different bubbles with alternating sign. Gluing the solution of (1.8) to the latter blow up solutions we are able to construct blow up solutions with residual mass, that is with $\rho ^+_n=\rho ^+=4\pi k(k-1)+\rho _0$ and $\rho ^-_n\rightarrow 4\pi k(k+1)$ for any $k\ge 2$. In this generality the latter construction is quite delicate and technically more difficult compared to the one in [9, 11], since we have more sign-changing singular bubbles, and since one need to consider all the interactions of different bubbles and also the interaction with the global solution z(x) which makes the linear theory more complicated, see the linear theory in Sect. 4.3. We remark that the same strategy can be carried out for more general asymmetric Sinh-Gordon equations, for example for the Tzitzéica equation [18].

The paper is organized as follows. Sect. 2 contains some notation and preliminary results which will be used in the paper. Sect. 3 is devoted to the proof of Theorem 1.1 while the proof of Theorem 1.2 is derived in Sect. 4.

2 Preliminaries

In this section we collect some notation and useful information that we will use in this paper. We shall write

$$\begin{aligned} \Vert u\Vert =\Big ( \int _{\Omega }|\nabla u|^2dx\Big )^{\frac{1}{2}} \quad \text{ and } \quad \Vert u\Vert _p=\Big ( \int _{\Omega }u^pdx\Big )^{\frac{1}{p}} \end{aligned}$$

to denote the norm in $H_0^1(\Omega )$ and in $L^p(\Omega )$, respectively, for $1\le p\le +\infty$. For $\alpha \ge 2$, let us define the Hilbert spaces:

$$\begin{aligned} L_{\alpha }({{\mathbb {R}}}^2):= & {} L^2\Big ( {{\mathbb {R}}}^2, \dfrac{|y|^{\alpha -2}}{(1+|y|^\alpha )^2}dy \Big ), \\ H_{\alpha }({{\mathbb {R}}}^2):= & {} \{u\in W_{loc}^{1,2}({{\mathbb {R}}}^2)\cap L_\alpha ({{\mathbb {R}}}^2): \Vert \nabla u\Vert _{L^2({{\mathbb {R}}}^2)}<\infty \}, \end{aligned}$$

with $\Vert u\Vert _{L_{\alpha }}$ and $\Vert u\Vert _{H_{\alpha }}:=(\Vert \nabla u\Vert _{L^2({{\mathbb {R}}}^2)}^2+\Vert u\Vert _{L_{\alpha }}^2)^{\frac{1}{2}}$ denoting their norms, respectively. For simplicity, we will denote $L_2$ and $H_2$ by L and H, respectively. Let us recall that the embedding $H_\alpha ({{\mathbb {R}}}^2)\rightarrow L_\alpha ({{\mathbb {R}}}^2)$ is compact [11]. For any $p>1$, let $i_p^*: L^p(\Omega )\rightarrow H_0^1(\Omega )$ be the adjoint operator of the embedding $i_p: H_0^1(\Omega )\rightarrow L^{\frac{p}{p-1}}(\Omega )$, i.e. for $v\in L^{p}(\Omega )$ , $u=i_p^*(v)$ if and only if in the weak sense

$$\begin{aligned} -\Delta u=v \text{ in } \Omega , \qquad u=0 \text{ on } \partial \Omega . \end{aligned}$$

Then one has $\Vert i_p^*(v)\Vert _{H_0^1(\Omega )}\le c_p \Vert v\Vert _p$ for some constant $c_p>0$ depending only on $\Omega$ and $p>1$.

The symbol $B_r(p)$ will stand for the open metric ball of radius r and center p. To simplify the notation we will write $B_r$ for balls which are centered at 0. Throughout the whole paper c, C will stand for constants which are allowed to vary among different formulas or even within the same line.

3 Partial blow up

3.1 Approximate solutions

In order to prove Theorem 1.1 we introduce the associated equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\rho ^+\dfrac{e^u}{\int _\Omega e^u}-\lambda e^{-u}=0 \quad \text{ in } \Omega , \\ u=0 \quad \text{ on } \partial \Omega \end{array}\right. } \end{aligned}$$

(3.1)

where $\lambda >0$ will be suitably chosen small. By the definition of $i_p^*$, problem (3.1) is equivalent to the following:

$$\begin{aligned} u=i_p^*(F(u)), \, u\in H_0^1(\Omega ) \end{aligned}$$

(3.2)

where $F(u)=\rho ^+g(u)-\lambda f(u)$ and

$$\begin{aligned} g(u)=\frac{e^u}{\int _\Omega e^u dx}, \, f(u)=e^{-u}. \end{aligned}$$

(3.3)

First let us introduce the approximate solutions we will use. Recall that solutions of the following regular Liouville equation [6]:

$$\begin{aligned} \Delta w+e^w=0 \quad \text{ in } {{\mathbb {R}}}^2, \qquad \int _{{{\mathbb {R}}}^2}e^w dx<\infty , \end{aligned}$$

are given by

$$\begin{aligned} w_{\delta , \xi }(x)=\log \frac{8\delta ^2}{(\delta ^2+|x-\xi |^2)^2} \end{aligned}$$

for $\delta >0, \ \xi \in {{\mathbb {R}}}^2$ and we set

$$\begin{aligned} w(x)=\log \frac{8}{(1+|x|^2)^2}. \end{aligned}$$

Since we are considering Dirichlet boundary condition, let us introduce the projection:

$$\begin{aligned} \Delta Pu=\Delta u \quad \text{ in } \Omega , \qquad Pu=0 \quad \text{ on } \partial \Omega . \end{aligned}$$

By the maximum principle,

$$\begin{aligned} Pw_{\delta , \xi }(x)=w_{\delta , \xi }(x)-\log 8\delta ^2+8\pi H(x, \xi )+O(\delta ^2) \quad \text{ in } C^1\text{-sense }, \end{aligned}$$

(3.4)

where H(x, y) is the regular part of the Green’s function of the Dirichlet Laplacian in $\Omega$, $G(x,y)=\frac{1}{2\pi }\log \frac{1}{|x-y|}+H(x,y)$, see [11] for the derivation of (3.4)

Let $k\ge 1$, fix ${{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega$ and consider $z(x, {{\varvec{\xi }}})$ which is the unique solution to (1.4). The approximate solutions we will use are given by

$$\begin{aligned} W=z(x, {{\varvec{\xi }}})-\sum _{i=1}^k Pw_i(x), \quad w_i(x)=w_{\delta _i,\xi _i}(x), \end{aligned}$$

(3.5)

where the parameters $\delta _i$ are suitably chosen such that

$$\begin{aligned} 8\delta _i^2=\lambda d_i({{\varvec{\xi }}}), \quad d_i({{\varvec{\xi }}})=\exp \Big [8\pi (H(\xi _i,\xi _i)+\sum _{j\ne i} G(\xi _i,\xi _j))-z(\xi _i,{{\varvec{\xi }}}) \Big ]. \end{aligned}$$

(3.6)

Our aim is to find a solution u to (3.1) of the form $u=W+\phi$ where $\phi$ is small in some sense. Before we go further, let us first collect some useful well-known facts.

As it is shown in [4], any solution $\psi \in H$ of

$$\begin{aligned} \Delta \psi +e^{w_{\delta , \xi }}\psi =0\quad \mathrm { in }\quad {{\mathbb {R}}}^2, \end{aligned}$$

can be expressed as a linear combination of

$$\begin{aligned} Z_{\delta , \xi }^0(x)=\frac{\delta ^2-|x-\xi |^2}{\delta ^2+|x-\xi |^2}, \quad Z_{\delta ,\xi }^i(x)=\frac{x_i-\xi _i}{\delta ^2+|x-\xi |^2}, \ i=1,2. \end{aligned}$$

Moreover, the projections of $Z_{\delta , \xi }^i$ have the following expansion:

$$\begin{aligned} PZ_{\delta ,\xi }^0(x)= & {} Z_{\delta , \xi }^0(x)+1+O(\delta ^2),\quad \nonumber \\ PZ_{\delta , \xi }^i(x)= & {} Z_{\delta , \xi }^i(x)+O(1), \ i=1,2 \quad \text{ in } C^1\text{-sense }. \end{aligned}$$

(3.7)

Finally, by straightforward computations and taking into account the choice of $\lambda$ in (3.6) the following estimates hold true [8]:

Lemma 3.1

For any ${\mathcal {C}}\subset {\mathcal {F}}_k\Omega$ compact and ${{\varvec{\xi }}}\in {\mathcal {C}}$, one has

$$\begin{aligned} \Vert Pw_i\Vert= & {} O(|\log \lambda |^{\frac{1}{2}}),\quad \Vert \nabla _{{\varvec{\xi }}}Pw_i\Vert =O(\lambda ^{-\frac{1}{2}}), \\ \Vert W\Vert= & {} O(|\log \lambda |^{\frac{1}{2}}),\quad \Vert \nabla _{{\varvec{\xi }}}W\Vert =O(\lambda ^{-\frac{1}{2}}), \end{aligned}$$

and there exists some $a>0$ such that for any $i=1,\cdots ,k$ and $j=1,2$, it holds that

$$\begin{aligned} \Vert PZ_i^j\Vert =a\lambda ^{-\frac{1}{2}}(1+o(1)) ,\quad \Vert \nabla _{{\varvec{\xi }}}PZ_i^j\Vert =O\left( \frac{1}{\lambda }\right) , \end{aligned}$$

(3.8)

and

$$\begin{aligned} \langle PZ_i^j, PZ_l^k\rangle =o\left( \frac{1}{\lambda }\right) \quad \text{ if } \quad i\ne l \text{ or } j\ne k, \end{aligned}$$

(3.9)

where $Z_i^j=Z_{\delta _i,\xi _i}^j$ and $\langle u,v\rangle =\int _{\Omega }\nabla u\cdot \nabla v\, dx$.

In the section, we set

$$\begin{aligned} K={\text {Span}}\{PZ_i^j, i=1,\cdots ,k \, j=1,2\} \end{aligned}$$

(3.10)

and

$$\begin{aligned} K^\perp =\{\phi \in H_0^1(\Omega ), \int _{\Omega }\nabla \phi \cdot \nabla PZ_i^j dx=0, \, i=1,\cdots , k, \, j=1,2\}. \end{aligned}$$

(3.11)

Denote by

$$\begin{aligned} \Pi : H_0^1(\Omega )\rightarrow K, \, \Pi ^\perp : H_0^1(\Omega )\rightarrow K^\perp \end{aligned}$$

be the corresponding projections. To solve (3.1), it is equivalent to solve the following system:

$$\begin{aligned} \Pi (u-i_p^*(F(u)))=0, \, \Pi ^\perp (u-i_p^*(F(u)))=0. \end{aligned}$$

(3.12)

3.2 Estimate of the error

We next estimate the error of the approximate solution:

$$\begin{aligned} R=\Delta W+\rho ^+ \frac{e^W}{\int _\Omega e^W}-\lambda e^{-W} \text{ and } {\mathcal {R}}=i_p^*(R). \end{aligned}$$

Lemma 3.2

For any $p\ge 1$ we have, for ${{\varvec{\xi }}}\in {\mathcal {C}}\subset {\mathcal {F}}_k\Omega$, ${\mathcal {C}}$ compact,

$$\begin{aligned} \Vert R\Vert _p=O(\lambda ^{\frac{2-p}{2p}}), \quad \Vert \partial _{{\varvec{\xi }}}R\Vert _p=O(\lambda ^{\frac{1-p}{p}}). \end{aligned}$$

Moreover, $\Vert {\mathcal {R}}\Vert \le c_p\Vert R\Vert _p$ for some $c_p>0$ depending on $p, \Omega$.

Proof

By the definition of W,

$$\begin{aligned} \begin{aligned} R&=\Delta W+\rho ^+ \frac{e^W}{\int _\Omega e^W}-\lambda e^{-W}\\&=\Delta (z(x,{{\varvec{\xi }}})-\sum _i Pw_i)+\rho ^+\frac{e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}{\int _\Omega e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}-\lambda e^{\sum _i Pw_i-z(x,{{\varvec{\xi }}})}\\&=\Big (\sum _i e^{w_i}-\lambda e^{\sum _i Pw_i-z(x,{{\varvec{\xi }}})}\Big )+\Big (\Delta z(x,{{\varvec{\xi }}})+\rho ^+\frac{e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}{\int _\Omega e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}\Big )\\&:=E_1(x)+E_2(x). \end{aligned} \end{aligned}$$

Estimate of $E_1=\Big (\sum _i e^{w_i}-\lambda e^{\sum _i Pw_i-z(x,{{\varvec{\xi }}})}\Big )$. Take $\eta >0$ such that $|\xi _i-\xi _j|\ge 2\eta$ and $d(\xi _i,\partial \Omega )\ge 2\eta$. First, using (3.4), we have

$$\begin{aligned} \begin{aligned} W&=z(x,{{\varvec{\xi }}})-\sum _i Pw_i=z(x,{{\varvec{\xi }}})-\sum _i \Big [ \log \frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}+8\pi H(x, \xi _i)\Big ]+O(\lambda ). \end{aligned} \end{aligned}$$

Hence, on $B_{\eta }(\xi _i)$, writing $x=\xi _i+\delta _i y$, one has

$$\begin{aligned} \begin{aligned} e^{-W(x)}&=e^{\sum _{i=1}^k \Big [ \log \frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}+8\pi H(x, \xi _i)\Big ]-z(x, {{\varvec{\xi }}})}(1+O(\lambda ))\\&=e^{w(y)}\cdot \exp \Big (8\pi H(\xi _i,\xi _i)+\sum _{j\ne i}8\pi G(\xi _i,\xi _j)\\&\quad -4\log \delta _i-\log 8-z(\xi _i,{{\varvec{\xi }}}) \Big )(1+O(\lambda )+O(\delta _i|y|))\\&=\frac{d_i({{\varvec{\xi }}})}{8\delta _i^4}e^{w(y)} (1+O(\lambda )+O(\delta _i|y|)). \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} e^{w_i}-\lambda e^{-W(x)}&=\frac{8}{\delta _i^2(1+|y|^2)^2}\left[ 1-\frac{\lambda }{8\delta _i^2}d_i({{\varvec{\xi }}})+O(\lambda )+O(\delta _i|y|)\right] \\&=O\left( \frac{1}{(1+|y|^2)^2}\right) +O\left( \frac{|y|}{\lambda ^{\frac{1}{2}}(1+|y|^2)^2}\right) . \end{aligned} \end{aligned}$$

(3.13)

It follows that

$$\begin{aligned} \Vert e^{w_i}-\lambda e^{-W(x)}\Vert _{L^p(B(\xi _i,\eta ))}=O(\lambda ^{\frac{2-p}{2p}})~\text { for any }p\ge 1. \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert e^{w_j}\Vert _{L^\infty (B(\xi _i,\eta ))}=O(\lambda ) \text{ for } j\ne i \ \text{ and } \ \Vert e^{w_i}-\lambda e^{-W(x)}\Vert _{L^\infty (\Omega \setminus \cup _i B(\xi _i,\eta ))}=O(\lambda ). \end{aligned}$$

Combining the above estimates,

$$\begin{aligned} \Vert E_1\Vert _{p}=O(\lambda ^{\frac{2-p}{2p}}) \text{ for } p\ge 1. \end{aligned}$$

(3.14)

Estimate of $E_2=\Big (\Delta z(x,{{\varvec{\xi }}})+\rho ^+\frac{e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}{\int _\Omega e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}\Big )$. First of all,

$$\begin{aligned} \begin{aligned} W&=z(x,{{\varvec{\xi }}})-\sum _i Pw_i\\&=z(x,{{\varvec{\xi }}})+2\sum _i \log (\delta _i^2+|x-\xi _i|^2)-8\pi \sum _i H(x,\xi _i)+O(\lambda )\\&=\log h(x, {{\varvec{\xi }}})+z(x,{{\varvec{\xi }}})+2\sum _i\log \frac{\delta _i^2+|x-\xi _i|^2}{|x-\xi _i|^2}+O(\lambda ), \end{aligned} \end{aligned}$$

(3.15)

where

$$\begin{aligned} h(x, {{\varvec{\xi }}})=\prod _{i=1}^k |x-\xi _i|^4\exp [-8\pi H(x, \xi _i)] =\prod _{i=1}^k\exp \Big (-8\pi G(x,\xi _i)\Big ). \end{aligned}$$

So

$$\begin{aligned} e^W=h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}+O(\lambda ). \end{aligned}$$

(3.16)

One has

$$\begin{aligned} E_2=\Delta z(x,{{\varvec{\xi }}})+\rho ^+\frac{e^W}{\int _{\Omega } e^W}=\Delta z(x, {{\varvec{\xi }}})+\rho ^+\frac{h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}}{\int _{\Omega }h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}}+O(\lambda ) =O(\lambda ), \end{aligned}$$

since $z(x,{{\varvec{\xi }}})$ is a solution of (1.4). Thus

$$\begin{aligned} \Vert E_2\Vert _{\infty }=O(\lambda ). \end{aligned}$$

(3.17)

Estimate of $\partial _{{\varvec{\xi }}}E_1$. Next we consider the derivatives. By straightforward computations we get

$$\begin{aligned} \begin{aligned} \partial _{\xi _i^j}E_1&=\sum _\ell e^{w_\ell }\partial _{\xi _i^j}w_{\ell }+\lambda e^{-W}\partial _{\xi _i^j}W\\&=\lambda e^{-W}\partial _{\xi _i^j} z(x, {{\varvec{\xi }}})+\left( \sum _i e^{w_i}-\lambda e^{-W}\right) \sum _{\ell =1}^k\partial _{\xi _i^j}Pw_\ell \\&\quad -\sum _{\ell }e^{w_\ell }\partial _{\xi _i^j}(Pw_\ell -w_\ell )-\sum _{\ell \ne i}e^{w^i}\partial _{\xi _i^j}Pw_\ell \\&:=I_1+I_2+I_3+I_4. \end{aligned} \end{aligned}$$

It is then not difficult to show that

$$\begin{aligned} \begin{aligned} \Vert I_1\Vert _p&\le \Vert E_1\Vert _p+\sum _i\Vert e^{w_i}\Vert _p=O(\lambda ^{\frac{1-p}{p}}),\\ \Vert I_2\Vert _p&\le \Vert E_1\Vert _p\Vert \partial _{{\varvec{\xi }}}Pw_j\Vert _\infty =O(\lambda ^{\frac{1-p}{p}}),\\ \Vert I_3\Vert _p&\le \Vert e^{w_i}\Vert _p \Vert \partial _{{\varvec{\xi }}}(Pw_j-w_j)\Vert _\infty =O(\lambda ^{\frac{1-p}{p}}),\\ \Vert I_4\Vert _p&=0. \end{aligned} \end{aligned}$$

Combining all the above estimates,

$$\begin{aligned} \Vert \partial _{{\varvec{\xi }}}E_1\Vert _p=O(\lambda ^{\frac{1-p}{p}}). \end{aligned}$$

(3.18)

Estimate of $\partial _{{\varvec{\xi }}}E_2$. The estimate of the derivative of $E_2$ is analogous. Using the equation satisfied by $z(x,{{\varvec{\xi }}})$ in (1.4) and (3.15),

$$\begin{aligned} \begin{aligned} \frac{1}{\rho ^+}\partial _{\xi _i^j}E_2&=-\frac{(\partial _{\xi _i^j}z(x, {{\varvec{\xi }}})h+\partial _{\xi _i^j}h)e^{z(x,{{\varvec{\xi }}})}}{\int _{\Omega }he^{z(x, {{\varvec{\xi }}})}dx}+\frac{he^{z(x, {{\varvec{\xi }}})}\int _{\Omega }(\partial _{\xi _i^j}z(x, {{\varvec{\xi }}})h+\partial _{\xi _i^j}h)e^{z(x,{{\varvec{\xi }}})}}{(\int _{\Omega }he^{z(x, {{\varvec{\xi }}})}dx)^2}\\&\quad +\frac{e^W\partial _{\xi _i^j}W}{\int _{\Omega }e^W}-\frac{e^W\int _{\Omega }e^W\partial _{\xi _i^j}Wdx}{(\int _\Omega e^W)^2}\\&=O(\lambda ). \end{aligned} \end{aligned}$$

Thus we have

$$\begin{aligned} \Vert \partial _{{\varvec{\xi }}}E_2\Vert _\infty =O(\lambda ). \end{aligned}$$

(3.19)

Finally, combining the estimates for $E_1$ and $E_2$, we have

$$\begin{aligned} \Vert R\Vert _p=O(\lambda ^{\frac{2-p}{2p}}), \quad \Vert \partial _{{\varvec{\xi }}}R\Vert _p=O(\lambda ^{\frac{1-p}{p}}). \end{aligned}$$

Once we get the estimate for R, the estimate for ${\mathcal {R}}$ follows directly. $\square$

3.3 The linear operator

In this subsection, we consider the following problem: given $h\in H_0^1(\Omega )$ we look for a function $\phi \in H_0^1(\Omega )$ and $c_{ij}$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta \phi +\rho ^+\left( \dfrac{e^W\phi }{\int _\Omega e^Wdx}-\dfrac{e^W\int _\Omega e^W\phi dx}{(\int _\Omega e^Wdx)^2}\right) +\sum _{i=1}^k e^{w_i}\phi =\Delta h+\sum _{i,j}c_{ij}e^{w_i}Z_i^j,\\ \\ \int _{\Omega }\nabla \phi \nabla PZ_i^j dx=0,~j=1,2,~i=1,\cdots ,k. \end{array}\right. } \end{aligned}$$

(3.20)

It is equivalent to

$$\begin{aligned} \phi -i_p^*(M(W)[\phi ])=h-\sum _{ij}c_{ij}PZ_i^j, \, \phi \in K^\perp \end{aligned}$$

(3.21)

where

$$\begin{aligned} M(W)[\phi ]=\rho ^+\left( \dfrac{e^W\phi }{\int _\Omega e^Wdx}-\dfrac{e^W\int _\Omega e^W\phi dx}{(\int _\Omega e^Wdx)^2}\right) +\sum _{i=1}^k e^{w_i}\phi . \end{aligned}$$

Let $L: K^\perp \rightarrow K^\perp$ be the linear operator defined by

$$\begin{aligned} L(\phi )=\phi -\Pi ^\perp (i_p^*(M(W)[\phi ])), \end{aligned}$$

then the problem is equivalent to first solving $\phi$ for

$$\begin{aligned} L(\phi )=\Pi ^\perp (h) \end{aligned}$$

(3.22)

and then finding $c_{ij}$ for

$$\begin{aligned} \Pi (i_p^*(M(W)[\phi ]))=\Pi (h)-\sum _{ij}c_{ij}PZ_i^j. \end{aligned}$$

(3.23)

First we have the following apriori estimate:

Lemma 3.3

Let ${\mathcal {C}}\subset {\mathcal {F}}_k\Omega$ be a fixed compact set. Then, there exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0,\lambda _0)$, ${{\varvec{\xi }}}\in {\mathcal {C}}$ and $h\in H_0^1(\Omega )$, any solution $\phi \in H_0^1(\Omega )$ of

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta \phi +\rho ^+\left( \dfrac{e^W\phi }{\int _\Omega e^Wdx}-\dfrac{e^W\int _\Omega e^W\phi dx}{(\int _\Omega e^Wdx)^2} \right) +\sum _{i=1}^k e^{w_i}\phi =\Delta h,\\ \\ \int _{\Omega }\nabla \phi \cdot \nabla PZ_i^j dx=0,~j=1,2,~i=1,\cdots ,k, \end{array}\right. } \end{aligned}$$

(3.24)

satisfies

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda | \Vert h\Vert . \end{aligned}$$

Proof

We prove it by contradiction. Assume there exist $\lambda _n\rightarrow 0$, ${{\varvec{\xi }}}_n\rightarrow {{\varvec{\xi }}}^*\in {\mathcal {F}}_k \Omega$, $h_n\in H_0^1(\Omega )$ and $\phi _n \in H_0^1(\Omega )$ which solves (3.24) with

$$\begin{aligned} \Vert \phi _n\Vert =1, \ |\log \lambda _n|\Vert h_n\Vert \rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

For $i=1, \cdots , k$, define ${\tilde{\phi }}_i(y)$ as

$$\begin{aligned} \begin{aligned} {\tilde{\phi }}_i(y)= {\left\{ \begin{array}{ll} \phi _i(\delta _i y+\xi _i), \quad &{}y\in {\tilde{\Omega }}_i=\frac{\Omega -\xi _i}{\delta _i},\\ 0, &{}y\in {{\mathbb {R}}}^2\setminus {\tilde{\Omega }}_i. \end{array}\right. } \end{aligned} \end{aligned}$$

In the following, we omit the index n for simplicity.

Step 1. We claim that

$$\begin{aligned} {\tilde{\phi }}_i(y)\rightarrow \gamma _i\frac{1-|y|^2}{1+|y|^2} \text { weakly in }H({{\mathbb {R}}}^2)\text { and strongly in }L({{\mathbb {R}}}^2), \end{aligned}$$

(3.25)

and

$$\begin{aligned} \phi \rightarrow 0 \text { weakly in }H_0^1(\Omega )\text { and strongly in }L^q(\Omega )\text { for }q\ge 2. \end{aligned}$$

(3.26)

Let $\psi \in C_0^\infty (\Omega \setminus \{\xi ^*_1, \cdots ,\xi _k^*\})$, multiply equation (3.24) by $\psi$ and integrate, then

$$\begin{aligned} \begin{aligned}&-\int _{\Omega }\nabla \psi \cdot \nabla \phi +\sum _{i=1}^k\int _{\Omega }e^{w_i}\phi \psi dx +\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi \psi dx}{\int _{\Omega }e^Wdx}-\dfrac{\int _{\Omega }e^W\phi dx \int _{\Omega } e^W\psi dx}{(\int _{\Omega }e^Wdx)^2}\right) \\&\quad =\int _{\Omega }\Delta h \psi dx. \end{aligned} \end{aligned}$$

By the assumption on $\phi$, using the fact that in $\Omega \setminus \{\xi _1^*,\cdots , \xi _k^*\}, \ e^{w_i}=O(\lambda )$ and $e^W=h(x, {{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}+O(\lambda )$, one has

$$\begin{aligned} \phi \rightarrow \phi ^* \text { weakly in }H_0^1(\Omega )\text { and strongly in }L^q(\Omega )\text { for }q\ge 2, \end{aligned}$$

which gives

$$\begin{aligned} -\int _{\Omega }\nabla \phi ^* \cdot \nabla \psi dx+\rho ^+\left( \dfrac{\int _{\Omega }he^z\phi ^*\psi dx}{\int _{\Omega } he^zdx}-\dfrac{\int _{\Omega }he^z\psi dx\int _{\Omega }he^z\phi ^*dx}{(\int _{\Omega }he^zdx)^2}\right) =0. \end{aligned}$$

So $\Vert \phi ^*\Vert _{H_0^1(\Omega )}\le 1$ and it solves

$$\begin{aligned} \Delta \phi ^*+\rho ^+\left( \dfrac{he^z\phi ^*}{\int _{\Omega }he^zdx} -\dfrac{he^z\int _{\Omega }he^z\phi ^*dx}{(\int _{\Omega } he^zdx)^2}\right) =0. \end{aligned}$$

By the non-degeneracy of $z(x, {{\varvec{\xi }}})$, we can get that $\phi ^*=0$. Thus (3.26) is proved.

Now let us prove (3.25). Multiplying (3.24) again by $\phi$ and integrating,

$$\begin{aligned} \int _{\Omega }|\nabla \phi |^2dx-\sum _{i=1}^k\int _{\Omega }e^{w_i}\phi ^2dx-\rho ^ +\left( \dfrac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx} -\dfrac{(\int _{\Omega }e^W\phi dx)^2}{(\int _{\Omega }e^Wdx)^2} \right) =\int _{\Omega }\nabla h\cdot \nabla \phi dx. \end{aligned}$$

From the above equation, one can get that

$$\begin{aligned} \begin{aligned} \int _{{\tilde{\Omega }}_i}e^w {\tilde{\phi }}_i^2dx&=\int _{\Omega }e^{w_i}\phi ^2dx\\&\le \int _{\Omega }|\nabla \phi |^2dx-\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx} -\dfrac{(\int _{\Omega }e^W\phi dx)^2}{(\int _{\Omega }e^Wdx)^2}\right) -\int _{\Omega }\nabla h\cdot \nabla \phi dx\\&\le 1+o(1)+\Vert h\Vert =O(1) \end{aligned} \end{aligned}$$

where we used (3.26). So we get that ${\tilde{\phi }}_i$ is bounded in $H({{\mathbb {R}}}^2)$. There exists ${\tilde{\phi }}_0$ such that

$$\begin{aligned} {\tilde{\phi }}_i \rightarrow {\tilde{\phi }}_0 \text { weakly in }H({{\mathbb {R}}}^2)\text { and strongly in }L({{\mathbb {R}}}^2). \end{aligned}$$

Let ${\tilde{\psi }}\in C_0^\infty ({{\mathbb {R}}}^2)$ and define $\psi _i={\tilde{\psi }}(\frac{x-\xi _i}{\delta _i})$. Multiplying (3.24) by $\psi _i$ and integrating over $\Omega$,

$$\begin{aligned}&\int _{\Omega } \nabla \phi \cdot \nabla \psi _idx -\sum _{j}\int _{\Omega }e^{w_j}\phi \psi _idx -\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi \psi _idx}{\int _{\Omega }e^Wdx}-\dfrac{\int _{\Omega } e^W\phi dx \int _{\Omega } e^W\psi _idx}{(\int _{\Omega } e^Wdx)^2} \right) \nonumber \\&\quad =\int _{\Omega } \nabla h\cdot \nabla \psi _idx.\nonumber \\ \end{aligned}$$

(3.27)

Since $\psi _i(x)=0$ if $|x-\xi _i|\ge R\delta _i$ for some $R>0$, we have

$$\begin{aligned} \int _{\Omega }e^{w_j}\phi \psi _idx=O(\delta _j^2) \quad \text { for }j\ne i. \end{aligned}$$

Passing to the limit in (3.27), we have

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}\nabla {\tilde{\phi }}_0\cdot \nabla {\tilde{\psi }}dx-\int _{{{\mathbb {R}}}^2}e^w {\tilde{\phi }}_0{\tilde{\psi }}dx=0. \end{aligned}$$

Moreover, by the orthogonality condition in (3.24), we have

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}{\tilde{\phi }}_0e^w \frac{y_j}{1+|y|^2}dy=0, \ j=1,2. \end{aligned}$$

So we deduce that

$$\begin{aligned} {\tilde{\phi }}_0=\gamma _i\frac{1-|y|^2}{1+|y|^2}. \end{aligned}$$

Step 2. We claim that $\gamma _i=0$ for $i=1,\cdots ,k$. Multiplying equation (3.24) by $PZ_i^0$ and integrate over $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\nabla \phi \cdot \nabla PZ_i^0dx-\sum _{j}\int _{\Omega }e^{w_j}\phi PZ_i^0dx -\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi PZ_i^0dx}{\int _{\Omega }e^Wdx} -\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega }e^W PZ_i^0dx}{(\int e^Wdx)^2} \right) \\&\quad =\int _{\Omega }\nabla h\cdot \nabla PZ_i^0dx. \end{aligned}\nonumber \\ \end{aligned}$$

(3.28)

Since

$$\begin{aligned} \int _{\Omega }\nabla \phi \cdot \nabla PZ_i^0dx=\int _{\Omega }e^{w_i}\phi Z_i^0dx=\int _{{\tilde{\Omega }}_i}e^w Z^0 {\tilde{\phi }}_idy \end{aligned}$$

where $Z^0=\frac{1-|y|^2}{1+|y|^2}$ and by (3.7),

$$\begin{aligned} \begin{aligned} \sum _j \int _{\Omega }e^{w_j}\phi PZ_i^0dx&=\int _{\Omega }e^{w_i}\phi PZ_i^0dx +\sum _{j\ne i}\int _{\Omega }e^{w_j}\phi PZ_i^0dx\\&=\int _{{\tilde{\Omega }}_i}e^w{\tilde{\phi }}_i (1+Z^0(y)+O(\delta _i^2))dy +\sum _{j\ne i}\int _{\Omega }e^{w_j}\phi PZ_i^0dx\\&=\int _{{\tilde{\Omega }}_i}e^w {\tilde{\phi }}_i (1+Z^0(y))dy+O(\lambda ^{\frac{1}{p}}), \end{aligned} \end{aligned}$$

for some $p>1$, by Hölder inequality. Moreover, by (3.26), (3.7) and (3.15), one has

$$\begin{aligned} \rho ^+\left( \dfrac{\int _{\Omega }e^W\phi PZ_i^0dx}{\int _{\Omega }e^Wdx} -\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega }e^W PZ_i^0dx}{(\int _{\Omega }e^Wdx)^2} \right) =O(\lambda ). \end{aligned}$$

From (3.28) and the above estimates, one has

$$\begin{aligned} \lim _{\lambda \rightarrow 0}|\log \lambda |\int _{{\tilde{\Omega }}_i}e^w{\tilde{\phi }}_i dy=0. \end{aligned}$$

(3.29)

Next we multiply equation (3.24) by $Pw_i$ and integrate over $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\nabla \phi \cdot \nabla Pw_idx-\sum _j \int _{\Omega }e^{w_j}\phi Pw_idx-\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi Pw_idx}{\int _{\Omega }e^Wdx} -\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega }e^WPw_idx}{(\int _{\Omega }e^Wdx)^2} \right) \\&\quad =\int _{\Omega }\nabla h\cdot \nabla Pw_idx. \end{aligned} \end{aligned}$$

Now we estimate the above equation term by term.

$$\begin{aligned} \int _{\Omega }\nabla \phi \cdot \nabla Pw_i dx=\int _{\Omega }e^{w_i}\phi dx =\int _{{\tilde{\Omega }}_i}e^w{\tilde{\phi }}_idy=o(1) \end{aligned}$$

by (3.25) and the fact that

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}e^w\frac{1-|y|^2}{1+|y|^2}dy=0. \end{aligned}$$

By the expansion of $Pw_i$,

$$\begin{aligned} \begin{aligned}&\sum _j\int _{\Omega }e^{w_j}\phi Pw_idx=\int _{\Omega }e^{w_i}\phi Pw_idx+\sum _{j\ne i}\int _{\Omega }e^{w_j}\phi Pw_idx\\&\quad =\int _{{\tilde{\Omega }}_i}e^w{\tilde{\phi }}_i\Big (-4\log \delta _i-2\log (1+|y|^2)+8\pi H(\xi _i,\xi _i)+O(\delta _i|y|+\delta _i^2)\Big )dy\\&\qquad +\sum _{j\ne i}\int _{{\tilde{\Omega }}_j}e^w {\tilde{\phi }}_j(8\pi G(\xi _i,\xi _j)+O(\delta _j|y|+\delta _j^2))dy\\&\quad =\gamma _i\int _{{{\mathbb {R}}}^2}e^w\frac{1-|y|^2}{1+|y|^2}[-2\log (1+|y|^2)]dy+o(1). \end{aligned} \end{aligned}$$

Moreover,

$$\begin{aligned} \rho ^+\left( \dfrac{\int _{\Omega }e^W\phi Pw_idx}{\int _{\Omega }e^Wdx} -\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega }e^WPw_idx}{(\int _{\Omega }e^Wdx)^2}\right) =o(1) \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega }\nabla h\cdot \nabla Pw_idx=O(\Vert h\Vert _p\Vert Pw_i\Vert )=O(\log \lambda )^{\frac{1}{2}}\Vert h\Vert =o(1). \end{aligned}$$

Combining all the above estimates, we have

$$\begin{aligned} \gamma _i\int _{{{\mathbb {R}}}^2}e^w\frac{1-|y|^2}{1+|y|^2}[-2\log (1+|y|^2)]dy=0, \end{aligned}$$

which implies that $\gamma _i=0$ since

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}e^w\frac{1-|y|^2}{1+|y|^2}[-2\log (1+|y|^2)]dy\ne 0. \end{aligned}$$

Step 3. Finally, we derive a contradiction.

Multiply equation (3.24 ) by $\phi$ and integrate:

$$\begin{aligned} \begin{aligned} \int _{\Omega }|\nabla \phi |^2dx-\sum _i\int _{\Omega }e^{w_i}\phi ^2dx-\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx}-\dfrac{(\int _{\Omega } e^W\phi dx)^2}{(\int _{\Omega }e^Wdx)^2}\right) =\int _{\Omega }\nabla h\cdot \nabla \phi dx. \end{aligned} \end{aligned}$$

From the estimates in step 1-2 and the assumptions on $\phi$ and h, it is not difficult to show that the left hand side of the above equation tends to 1, while the right hand side has limit 0. This is a contradiction which concludes the proof. $\square$

Now we can derive a priori estimates for problem (3.20).

Proposition 3.4

Let ${\mathcal {C}}\subset {\mathcal {F}}_k\Omega$ be a compact set. Then, there exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0, \lambda _0)$, ${{\varvec{\xi }}}\in {\mathcal {C}}$ and $h\in H_0^1(\Omega )$, if $(\phi , c_{ij})$ is a solution of (3.20), we have

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda |\Vert h\Vert . \end{aligned}$$

Proof

By Lemma 3.3 and (3.8) , we know that

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda |\left( \Vert h\Vert +\sum _{ij}|c_{ij}|\Vert PZ_i^j\Vert \right) \le C|\log \lambda |\left( \Vert h\Vert +\sum _{ij}\frac{1}{\sqrt{\lambda }}|c_{ij}|\right) . \end{aligned}$$

In order to estimate $c_{ij}$, multiply the equation (3.20) by $PZ_i^j$ and integrating over $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\phi e^{w_i}(PZ_i^j-Z_i^j)dx+\sum _{\ell \ne i}\int _{\Omega }e^{w_\ell }\phi PZ_i^jdx +O\left( \int _{\Omega }|\phi ||PZ_i^j|dx+\int _{\Omega }|\phi |\int _{\Omega }|PZ_i^j|dx\right) \\&\quad =\int _{\Omega }\nabla h\cdot \nabla PZ_i^j+c_{ij}\int _{\Omega } e^{w_i}Z_i^jPZ_i^j dx+\sum _{k\ne i, \ell \ne j}o\left( \frac{|c_{k\ell }|}{\lambda }\right) , \end{aligned} \end{aligned}$$

where in the last line we use (3.9). Since for any $q\ge 1$,

$$\begin{aligned} \begin{aligned} \int _{\Omega }\phi e^{w_i}(PZ_i^j-Z_i^j)dx+\sum _{\ell \ne i}\int _{\Omega }e^{w_\ell }\phi PZ_i^jdx&=O(\Vert \phi \Vert (\Vert e^{w_i}\Vert _q+\Vert e^{w_\ell }PZ_i^j\Vert _q))=O\left( \lambda ^{\frac{1-q}{q}}\Vert \phi \Vert \right) ,\\ O\left( \int _{\Omega }|\phi ||PZ_i^j|+\int _{\Omega }|\phi |\int _{\Omega }|PZ_i^j|\right)&=O(\Vert PZ_i^j\Vert _2\Vert \phi \Vert )=O\left( |\log \lambda |^{\frac{1}{2}}\Vert \phi \Vert \right) ,\\ \int _{\Omega }\nabla h\cdot \nabla PZ_i^j&=O(\Vert h\Vert \Vert PZ_i^j\Vert )=O\left( \frac{1}{\sqrt{\lambda }}\Vert h\Vert \right) ,\\ \end{aligned} \end{aligned}$$

we have

$$\begin{aligned} |c_{ij}|+o\left( \sum _{k\ne j,\ell \ne i}|c_{k,\ell }|\right) =O\left( \lambda ^{\frac{1}{q}}\Vert \phi \Vert +\lambda |\log \lambda |^{\frac{1}{2}}\Vert \phi \Vert +\lambda ^{\frac{1}{2}}\Vert h\Vert \right) . \end{aligned}$$

Summing all $|c_{ij}|$ up and choosing suitable $q\in (1, 2)$, we can get that

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda |\Vert h\Vert . \end{aligned}$$

$\square$

From the above a priori estimate and the Fredholm alternative it is then standard to derive the following existence result.

Proposition 3.5

Let ${\mathcal {C}}\subset {\mathcal {F}}_k\Omega$ be a compact set. Then, there exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0, \lambda _0)$, ${{\varvec{\xi }}}\in {\mathcal {C}}$ and $h\in H^1_0(\Omega )$, there exists a unique solution $(\phi , c_{ij})$ of (3.20), which satisfies

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda |\Vert h\Vert . \end{aligned}$$

Proof

For the first equation (3.22), since $\phi \rightarrow \Pi ^\perp (i_p^*(M(W)[\phi ]))$ is the compact operator in $K^\perp$, the existence and uniqueness of $\phi$ follows from the Fredholm alternative and the above a priori estimate, and then $c_{ij}$ are determined by (3.23). Once existence of $(\phi , c_{ij})$ is obtained, the estimate follows from Proposition 3.4. $\square$

3.4 Nonlinear problem

The aim of this subsection is to find $(\phi , \{c_{ij}\})$ such that $u=W_{\xi }+\phi _{\xi }$ solves

$$\begin{aligned} \left\{ \begin{array}{l} \Delta u+\rho ^+\dfrac{e^u}{\int _{\Omega }e^udx}-\lambda e^{-u}=\sum _{ij}c_{ij}e^{w_i}Z_i^j, \\ \\ \int _{\Omega }\nabla \phi \nabla PZ_i^jdx=0, \quad j=1,2,~i=1,\cdots ,k. \end{array} \right. \end{aligned}$$

For this purpose, we shall find a solution $\phi$ of

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\Delta \phi +\rho ^+\Big (\dfrac{e^W\phi }{\int _\Omega e^Wdx}-\dfrac{e^W\int _\Omega e^W\phi dx}{(\int _\Omega e^Wdx)^2} \Big )+\sum _{i=1}^k e^{w_i}\phi =-\Big (R+{S}(\phi )+{N}(\phi )\Big )\\ &{}\quad +\sum _{ij}c_{ij}e^{w_i}Z_i^j, \\ \\ &{}\int _{\Omega }\nabla \phi \cdot \nabla PZ_i^jdx=0, \quad j=1,2,~i=1,\cdots ,k. \end{array}\right. } \end{aligned}$$

(3.30)

where R is the error term defined in Sect. 3.2,

$$\begin{aligned} \begin{aligned} {N}(\phi )&=-\lambda \Big (f(W+\phi )-f(W)-f'(W)\phi \Big )+\rho ^+\Big ( g(W+\phi )-g(W)-g'(W)\phi \Big ),\\ {S}(\phi )&=-\left( \sum _{i=1}^k e^{w_i}+\lambda f'(W)\right) \phi , \\ f(W)&=e^{-W}, \, g(W)=\frac{e^W}{\int _\Omega e^W dx}. \end{aligned} \end{aligned}$$

From the above linear theory, the existence of a solution to the nonlinear problem (3.30) follows a standard strategy using contraction mapping.

Proposition 3.6

Let ${\mathcal {C}}\subset {\mathcal {F}}_k\Omega$ be compact set. For any $\epsilon >0$ sufficiently small, there exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0, \lambda _0)$ and ${{\varvec{\xi }}}\in {\mathcal {C}}$, there exists a unique $(\phi , c_{ij})$ solution of (3.30) satisfying the estimates:

$$\begin{aligned} \Vert \phi \Vert \le C\lambda ^{\frac{1}{2}-\epsilon }, \ \Vert \partial _{\xi _i^j}\phi \Vert \le C\lambda ^{-\epsilon }, \ |c_{ij}|\le C\lambda . \end{aligned}$$

(3.31)

Proof

Denote the solution to (3.20) by $\phi :=T(h)$. Then (3.30) is equivalent to

$$\begin{aligned} \phi =T(i_p^*(R+N(\phi )+S(\phi )))=:{\mathcal {T}}(\phi ). \end{aligned}$$

The solution $\phi$ can be obtained through contraction mapping. Define

$$\begin{aligned} {\mathcal {B}}=\{\phi \in K^\perp , \Vert \phi \Vert \le \Lambda |\log \lambda |\lambda ^{\frac{2-p}{2p}}\} \end{aligned}$$

for $\Lambda$ large and $\lambda$ small and p close to 1.

From Proposition 3.5 and the error estimate for R, for $\phi , \phi _1, \phi _2\in {\mathcal {B}}$, similarly to the estimate in Proposition 4.10 in [8], one has

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {T}}\phi \Vert&\le C_p|\log \lambda |\Vert i_p^*(R+N(\phi )+S(\phi ))\Vert \\&\le C_p|\log \lambda |(\Vert R\Vert _p+\Vert N(\phi )\Vert _p+\Vert S(\phi )\Vert _p)\\&\le \Lambda |\log \lambda |\lambda ^{\frac{2-p}{2p}}, \\&\text{ and } \\ \Vert {\mathcal {T}}(\phi _1)-{\mathcal {T}}(\phi _2)\Vert&\le C_p(\Vert N(\phi _1)-N(\phi _2)\Vert _p+\Vert S(\phi _1)-S(\phi _2)\Vert _p)\\&\le \frac{1}{2}\Vert \phi _1-\phi _2\Vert . \end{aligned} \end{aligned}$$

So ${\mathcal {T}}$ maps ${\mathcal {B}}$ into itself and it is a contraction mapping. For $\epsilon$ small, we can choose p sufficiently close to 1, such that ${\mathcal {B}}\subset \{\phi , \, \Vert \phi \Vert \le C\lambda ^{\frac{1}{2}-\epsilon }\}\}$. Since ${\mathcal {T}}$ is a contraction mapping in ${\mathcal {B}}$, we can also get that the fixed point in ${\mathcal {B}}$ is unique, i.e. the solution $\phi$ is unique in ${\mathcal {B}}$. The estimate for $\phi$ follows from the above estimates. The estimates for $\partial _{\xi _i}\phi$ are obtained similarly to Proposition 4.10 in [8] and the estimate for $c_{ij}$ follows from Proposition 3.4. $\square$

3.5 The reduced problem

We introduce here the finite-dimensional reduction. In the previous subsection we have found a solution $u=W+\phi$ to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\rho ^+\dfrac{e^u}{\int _{\Omega }e^udx}-\lambda e^{-u}=\sum _{ij}c_{ij} e^{w_i}Z_i^j\\ \\ \int _{\Omega }\nabla \phi \cdot \nabla PZ_i^jdx=0,~j=1,2,~i=1,\cdots ,k. \end{array}\right. } \end{aligned}$$

Consider now the associated energy functional:

$$\begin{aligned} J(u)=\frac{1}{2}\int _{\Omega }|\nabla u|^2dx-\rho ^+\log \int _{\Omega }e^udx-\lambda \int _{\Omega }e^{-u}dx \end{aligned}$$

(3.32)

and let ${\tilde{J}}({{\varvec{\xi }}})=J(W_{{\varvec{\xi }}}+\phi _{{\varvec{\xi }}})$.

Lemma 3.7

Let ${{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega$ be a critical point of ${\tilde{J}}$, then for $\lambda$ small, $u=W_{{{\varvec{\xi }}}}+\phi _{{{\varvec{\xi }}}}$ is a solution of (3.1).

Proof

If ${{\varvec{\xi }}}$ is a critical point of ${\tilde{J}}({\varvec{\xi }})$, then one has

$$\begin{aligned} \langle J'(u), \partial _{{\varvec{\xi }}}(W_{{\varvec{\xi }}}+\phi _{{\varvec{\xi }}})\rangle =0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \langle \sum _{ij}c_{ij}e^{w_i}Z_i^j, \partial _{\xi _\ell ^s}(W_{{\varvec{\xi }}}+\phi _{{\varvec{\xi }}})\rangle =0 \quad \text{ for } \ell =1,\cdots ,k, \ s=1,2. \end{aligned}$$

(3.33)

Let us fix $q>1$. Since

$$\begin{aligned} \Vert e^{w_i}Z_i^j\Vert _q=O(\lambda ^{\frac{2-3q}{2q}}), \end{aligned}$$

(3.34)

combining the estimate (3.31), one has

$$\begin{aligned} \int _{\Omega }e^{w_i}Z_i^j \partial _{\xi _\ell ^s}\phi _{{\varvec{\xi }}}dx= & {} O(\Vert e^{w_i}Z_i^j\Vert _q\Vert \partial _{\xi }\phi \Vert )=O\left( \lambda ^{\frac{2-3q}{2q}-\epsilon }\right) =o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

(3.35)

$$\begin{aligned} \int _{\Omega }e^{w_i}Z_i^j \partial _{\xi _\ell ^s}W_{{\varvec{\xi }}}dx= & {} -\int _{\Omega }PZ_\ell ^s e^{w_i}Z_i^j dx+O\left( \frac{1}{\sqrt{\lambda }}\right) =\frac{a}{\lambda }\delta _{i\ell }\delta _{j s}+o\left( \frac{1}{\lambda }\right) ,\nonumber \\ \end{aligned}$$

(3.36)

Combining the estimates (3.35) and (3.36), we conclude that

$$\begin{aligned} c_{ij}+o(1)\sum _{\ell \ne i, s\ne j}c_{\ell s}=0, \end{aligned}$$

which implies that all $c_{ij}$ are zero. So the corresponding u is a solution of (3.1) as desired. $\square$

Recall the definition of $\Lambda$ in (1.5). We next consider the expansion of the energy.

Proposition 3.8

It holds

$$\begin{aligned} J(W)=\Lambda ({{\varvec{\xi }}})-8\pi k\log \lambda -(16\pi -24\pi \log 2)k+o(1), \end{aligned}$$

${\mathcal {C}}^1$ uniformly in ${{\varvec{\xi }}}$ in compact sets of $\Omega$.

Proof

By the definition of J(W) and W, one has

$$\begin{aligned} \begin{aligned} J(W)&=\frac{1}{2}\int _{\Omega }\left( |\nabla z|^2+\sum _{i=1}^k |\nabla Pw_i|^2-2\sum _{i=1}^k \nabla Pw_i\cdot \nabla z+2\sum _{i\ne j}\nabla Pw_i\cdot \nabla Pw_j\right) dx\\&\quad -\rho ^+\log \int _{\Omega }e^Wdx-\lambda \int _{\Omega }e^{-W}dx. \end{aligned} \end{aligned}$$

Using (3.16),

$$\begin{aligned} \begin{aligned} \frac{1}{2}\int _{\Omega }|\nabla z|^2dx-\rho ^+\log \int _{\Omega }e^Wdx =\frac{1}{2}\int _{\Omega }|\nabla z|^2dx -\rho ^+\log \int _{\Omega } h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}dx+O(\lambda ). \end{aligned} \end{aligned}$$

While using (3.13) and the estimate for $E_1$,

$$\begin{aligned} \begin{aligned} \lambda \int _{\Omega }e^{-W}dx&=\sum _{i=1}^k\int _{\Omega }e^{w_i}dx+o(1)=8k\pi +o(1),\\ \int _{\Omega }\nabla Pw_i\cdot \nabla z dx&=\int _{\Omega } e^{w_i}z(x,{{\varvec{\xi }}})dx =\int _{{\tilde{\Omega }}_i}\frac{8}{(1+|y|^2)^2}z(\delta _i y+\xi _i, {{\varvec{\xi }}})dy=8\pi z(\xi _i,{{\varvec{\xi }}})+o(1), \end{aligned} \end{aligned}$$

where ${\tilde{\Omega }}_i=(\Omega -\xi _i)/\delta _i$. Moreover, using the expansion (3.4)

$$\begin{aligned} \begin{aligned} \int _{\Omega }|\nabla Pw_i|^2dx&=\int _{\Omega }e^{w_i}Pw_idx\\&=\int _{\Omega }e^{w_i}\Big (\log \frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}+8\pi H(x,\xi _i)+O(\lambda ) \Big )dx\\&=64\pi ^2 H(\xi _i,\xi _i)-2\int _{{\tilde{\Omega }}_i}e^{w_i}(\log \delta _i^2+\log (1+|y|^2))dy+o(1)\\&=64\pi ^2 H(\xi _i,\xi _i)-16\pi \log \delta _i^2-16\pi +o(1)\\&=64\pi ^2 H(\xi _i,\xi _i)-16\pi \log \frac{\lambda d_i({{\varvec{\xi }}})}{8}-16\pi +o(1)\\&=-64\pi ^2 H(\xi _i,\xi _i)-128\pi ^2\sum _{j\ne i}G(\xi _i,\xi _j)+16\pi z(\xi _i,{{\varvec{\xi }}})\\&\quad -16\pi \log \lambda -16\pi +48\pi \log 2+o(1), \end{aligned} \end{aligned}$$

and for $i\ne j$,

$$\begin{aligned} \begin{aligned} \int _{\Omega }\nabla Pw_i \cdot \nabla Pw_jdx&=\int _{\Omega }e^{w_i}\Big (\log \frac{1}{(\delta _j^2+|x-\xi _j|^2)^2}+8\pi H(x,\xi _j)+O(\lambda ) \Big )dx\\&=64\pi ^2 G(\xi _i,\xi _j)+o(1). \end{aligned} \end{aligned}$$

Combining all the above estimates, we have

$$\begin{aligned} \begin{aligned} J(W)&=\frac{1}{2}\int _{\Omega }|\nabla z|^2dx-\rho ^+\log \int _{\Omega } h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}dx-8\pi k\log \lambda \\&\quad -32\pi ^2\sum _{i=1}^k\left( H(\xi _i,\xi _i)+\sum _{j\ne i}G(\xi _i,\xi _j)\right) -(16\pi -24\pi \log 2)k+o(1)\\&=\Lambda ({{\varvec{\xi }}})-8\pi k\log \lambda -(16\pi -24\pi \log 2)k+o(1). \end{aligned} \end{aligned}$$

Next, we consider the derivative of J(W).

$$\begin{aligned} \begin{aligned} \partial _{\xi _i^j} J(W)&=\int _{\Omega }\left( -\Delta W-\rho ^+\dfrac{e^W}{\int _{\Omega } e^Wdx}+\lambda e^{-W}\right) \partial _{\xi _i^j}Wdx=-\int _{\Omega }(E_1(x)+E_2(x))\partial _{\xi _i^j}Wdx \\&=4\int _{\Omega }E_1(z)Z_i^j dx+o(1) =4\int _{\Omega }\left( \sum _\ell e^{w_\ell } -\lambda e^{-W}\right) Z_i^jdx, \end{aligned} \end{aligned}$$

where $E_1, E_2$ were introduced in Lemma 3.2 and where we used

$$\begin{aligned} \partial _{\xi _i^j}W=-4PZ_i^j+O(1). \end{aligned}$$

Using the definition of $w_i$ and $Z_i^j$, for $\ell \ne i$

$$\begin{aligned} \begin{aligned} \int _{\Omega } e^{w_\ell }Z_i^jdx&=\int _{\Omega }\frac{8\delta _\ell ^2}{(\delta _\ell ^2+|x-\xi _\ell |^2)^2}\frac{x_j-\xi _i^j}{\delta _i^2+|x-\xi _i|^2}dx=8\pi \frac{\xi _\ell ^j-\xi _i^j}{|\xi _\ell -\xi _i|^2}+o(1). \end{aligned} \end{aligned}$$

Moreover, taking $\eta >0$ such that $|\xi _i-\xi _j|\ge 2\eta$ and $d(\xi _i,\partial \Omega )\ge 2\eta$, we have

$$\begin{aligned} \begin{aligned} \int _{B(\xi _\ell , \eta )}\lambda e^{-W}Z_i^jdx&=\lambda \int _{B(\xi _\ell ,\eta )}\exp \Big [8\pi \sum _i H(x,\xi _i)\\&\quad -z(x,{{\varvec{\xi }}})+O(\lambda )\Big ]\frac{x_j-\xi _i^j}{\delta _i^2+|x-\xi _i|^2} \prod _{i=1}^k\frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}dx\\&=\frac{\lambda }{\delta _\ell ^2}\int _{{\tilde{\Omega }}_\ell }\exp \Big [8\pi H(\xi _\ell ,\xi _\ell )\\&\quad +8\pi \sum _{j\ne \ell }G(\xi _\ell ,\xi _j)-z(\xi _\ell , {{\varvec{\xi }}})\Big ]\frac{1}{(1+|y|^2)^2}\frac{\xi _\ell ^j-\xi _i^j}{|\xi _\ell -\xi _i|^2}dx+o(1)\\&=8\pi \frac{\xi _\ell ^j-\xi _i^j}{|\xi _\ell -\xi _i|^2}+o(1). \end{aligned} \end{aligned}$$

Let

$$\begin{aligned} \gamma (x, {{\varvec{\xi }}})=8\pi H(x, \xi _i)+8\pi \sum _{j\ne i}G(x, \xi _j)-z(x,{{\varvec{\xi }}}). \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned} \int _{B(\xi _i, \eta )}\lambda e^{-W}Z_i^jdx&=\lambda \int _{B(\xi _i,\eta )}\exp \Big [8\pi \sum _i H(x,\xi _i)-z(x,{{\varvec{\xi }}})+O(\lambda )\Big ]\\&\quad \frac{x_j-\xi _i^j}{\delta _i^2+|x-\xi _i|^2} \prod _{i=1}^k\frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}dx\\&=\frac{\lambda }{\delta _i^3}\int _{{\tilde{\Omega }}_i}\frac{1}{(1+|y|^2)^2}\frac{y_j}{1+|y|^2}\\&\quad \exp \Big [8\pi H(\xi _i+\delta _i y, \xi _i)+8\pi \sum _{j\ne i}G(\xi _i+\delta _i y,\xi _j)\\&\qquad -z(\xi _i+\delta _i y, {{\varvec{\xi }}})\Big ]dy+o(1)\\&=\frac{8}{\delta _i}\int _{B(0, \frac{\eta }{\delta _i})}\frac{y_j}{(1+|y|^2)^3}exp[\gamma (\xi _i+\delta _i y, {{\varvec{\xi }}})-\gamma (\xi _i,{{\varvec{\xi }}})]dy+o(1)\\&=\frac{8}{\delta _i}\int _{{{\mathbb {R}}}^2}\frac{y_j}{(1+|y|^2)^3}\frac{\partial \gamma }{\partial x}(\xi _i, {{\varvec{\xi }}})\cdot \delta _i y \,dy +o(1)\\&=2\pi \frac{\partial \gamma }{\partial x}(\xi _i, {{\varvec{\xi }}})+o(1). \end{aligned} \end{aligned}$$

Finally,

$$\begin{aligned} \begin{aligned} |\int _{\Omega \setminus \bigcup _i B(\xi _i, \eta )}\lambda e^{-W}Z_i^jdx|\le C\lambda \int _{\Omega \setminus \bigcup _i B(\xi _i, \eta )}e^{\sum _{\ell }Pw_\ell }|Z_i^j|dx \le C\lambda =o(1). \end{aligned} \end{aligned}$$

Combining the above estimates, we have

$$\begin{aligned} \begin{aligned} \partial _{\xi _i^j}J(W)=-8\pi \frac{\partial \gamma }{\partial x}(\xi _i,{{\varvec{\xi }}})+o(1)=\partial _{\xi _i^j}\Lambda ({{\varvec{\xi }}})+o(1), \end{aligned} \end{aligned}$$

as desired, where we used (1.6). $\square$

Finally, we have the following expansion of the reduced energy.

Proposition 3.9

It holds

$$\begin{aligned} {\tilde{J}}({{\varvec{\xi }}}):=J(W_{{\varvec{\xi }}}+\phi _{{\varvec{\xi }}})=J(W_{{\varvec{\xi }}})+o(1), \end{aligned}$$

${\mathcal {C}}^1$ uniformly in ${{\varvec{\xi }}}$ in compact sets of ${\mathcal {F}}_k\Omega$.

Proof

To simplify the notation, we shall drop the sub-index ${{\varvec{\xi }}}$ in the proof. It is not difficult to show that

$$\begin{aligned} \begin{aligned} J(W+\phi )-J(W)&=\frac{1}{2}\int _{\Omega }|\nabla \phi |^2dx+\int _{\Omega }\nabla W\cdot \nabla \phi dx+\lambda \int _{\Omega }e^{-W}(1-e^{-\phi })dx\\&\quad +\rho ^+\Big (\log \int _{\Omega }e^Wdx-\log \int _{\Omega }e^{W+\phi }\Big )dx\\&=-\int _{\Omega }\Delta z(x,{{\varvec{\xi }}})\phi dx-\rho ^+\int _{\Omega } \dfrac{h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}\phi }{\int _{\Omega }h(x, {{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}dx}dx\\&\quad +\int _{\Omega }\sum _ie^{w_i}\phi dx-\lambda \int _{\Omega }e^{-W}\phi dx\\&\quad +\rho ^+\Big ( \log \int _{\Omega }e^Wdx-\log \int _{\Omega }e^{W+\phi }dx\\&\quad + \int _{\Omega }\dfrac{h(x,{{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}\phi }{\int _{\Omega }h(x, {{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}dx}dx\Big )\\&\quad +\lambda \int _{\Omega }e^{-W}(1-e^{-\phi }+\phi )dx+\Vert \phi \Vert ^2=o(1). \end{aligned} \end{aligned}$$

Next we consider the derivatives.

$$\begin{aligned} \begin{aligned} \partial _{\xi _i^j}[J(W+\phi )-J(W)]&=-\int _{\Omega }\left( \Delta (W+\phi )+\rho ^+\dfrac{e^{W+\phi }}{\int _{\Omega }e^{W+\phi }dx} -\lambda e^{-(W+\phi )}\right) \partial _{\xi _i^j}\phi dx\\&\quad -\int _{\Omega }\left[ \Delta \phi +\rho ^+\left( \frac{e^{W+\phi }}{\int _{\Omega } e^{W+\phi }dx} -\frac{e^W}{\int _{\Omega }e^Wdx}\right) \right. \\&\left. \quad -\lambda (e^{-(W+\phi )}-e^{-W}) \right] \partial _{\xi _i^j}Wdx\\&=\sum _{i,j}\int _{\Omega }c_{ij}e^{w_i}Z_i^j \partial _{\xi _i^j}\phi dx -\int _{\Omega }\Delta \phi \partial _{\xi _i^j}W dx -\int _{\Omega }\lambda e^{-W}\phi \partial _{\xi _i^j}W dx\\&\quad +\int _{\Omega }\lambda (e^{-(W+\phi )}-e^{-W}+e^{-W}\phi )\partial _{\xi _i^j}W dx\\&\quad +\rho ^+\int _{\Omega }\left( \frac{e^{W+\phi }}{\int e^{W+\phi }}-\frac{e^W}{\int e^W}\right) \partial _{\xi _i^j}W dx. \end{aligned} \end{aligned}$$

Using the estimate for $c_{ij}$ in Proposition 3.6 and (3.34), we have

$$\begin{aligned} \begin{aligned} \sum _{i,j}\int _{\Omega }c_{ij}e^{w_i}Z_i^j \partial _{\xi _i^j}\phi dx&=O\left( \sum _{i,j} |c_{ij}|\Vert \partial _{\xi _i^j}\phi \Vert \cdot \Vert e^{w_i}Z_i^j\Vert _q\right) =O(\lambda ^{\frac{2-3q}{2q}+1-\epsilon })=o(1), \end{aligned} \end{aligned}$$

provided q is sufficiently close to 1. Recalling the definitions of f, g in (3.3) we exploit now the estimates in [8, Lemma 4.7]. For some $\theta \in (0,1)$ and p sufficiently close to 1 we have

$$\begin{aligned} \begin{aligned} \int _{\Omega }\lambda (e^{-(W+\phi )}-e^{-W}+e^{-W}\phi )\partial _{\xi _i^j}Wdx&=\int _{\Omega }\lambda f''(W+\theta \phi )\phi ^2 \partial _{\xi _i^j}Wdx\\&=O(\Vert \lambda f''(W+\theta \phi )\phi ^2\Vert _p\Vert \partial _{\xi _i^j}W\Vert _q) \\&=O(\lambda ^{\frac{1-pq}{pq}-\frac{1}{2}+1-2\epsilon })=o(1). \end{aligned} \end{aligned}$$

Moreover, for some ${{\tilde{\theta }}}\in (0,1)$ and suitable p, q

$$\begin{aligned} \begin{aligned} \rho ^+\int _{\Omega }\left( \frac{e^{W+\phi }}{\int e^{W+\phi }}-\frac{e^W}{\int e^W}\right) \partial _{\xi _i^j}W\,dx&=\rho ^+\int _{\Omega }g'(W+{{\tilde{\theta }}} \phi )\phi \partial _{\xi _i^j}W\,dx \\&=O(\Vert g'(W+{{\tilde{\theta }}} \phi )\phi \Vert _p \Vert \partial _{\xi _i^j}W\Vert _q) \\&=O(\lambda ^{\frac{1}{2}-\epsilon })=o(1). \end{aligned} \end{aligned}$$

Recall that

$$\begin{aligned} \lambda e^{-W}=\sum _{i=1}^k e^{w_i}+O(\lambda )\quad \text { and }\quad \partial _{\xi _i^j}W=-4PZ_i^j+O(1), \end{aligned}$$

for ${{\varvec{\xi }}}$ in compact sets of ${\mathcal {F}}_k\Omega$. Then

$$\begin{aligned} \begin{aligned} \lambda \int _{\Omega }e^{-W}\phi \partial _{\xi _i^j}Wdx&=-4\sum _{\ell =1}^k \int _{\Omega }e^{w_\ell }\phi PZ_{i}^j dx+o(1)\\&=-4\int _{\Omega }e^{w_i}Z_i^j\phi dx-4\sum _{\ell \ne i}\int _{\Omega }e^{w_\ell }\phi Z_i^jdx+o(1)\\&=-4\int _{\Omega }\nabla \phi \cdot \nabla PZ_i^j dx+o(1)=o(1) \end{aligned} \end{aligned}$$

by the orthogonality condition satisfied by $\phi$. Moreover, again by the orthogonality condition we have

$$\begin{aligned} \begin{aligned} \int _{\Omega }\Delta \phi \partial _{\xi _i^j}Wdx&=-\int _{\Omega }\nabla \phi \cdot \nabla \partial _{\xi _i^j}Wdx =-4\int _{\Omega }\nabla \phi \cdot (\nabla PZ_i^j+O(1))dx \\&=O(1)\int _{\Omega }|\nabla \phi |dx=o(1). \end{aligned} \end{aligned}$$

Combining the above estimates, we have

$$\begin{aligned} \partial _{\xi _i^j}{\tilde{J}}({{\varvec{\xi }}})=\partial _{\xi _i^j}J(W)+o(1), \end{aligned}$$

(3.37)

as desired. $\square$

Proof of Theorem 1.1

Let ${\mathcal {K}}\subset {\mathcal {F}}_k\Omega$ be a $C^1$-stable set of critical points of $\Lambda$. Then, by Propositions 3.8-3.9, for $\lambda >0$ small, there exists ${{\varvec{\xi }}}_\lambda$ critical point of ${\tilde{J}}$ and $d({{\varvec{\xi }}}_\lambda , {\mathcal {K}})\rightarrow 0$ as $\lambda \rightarrow 0$. By Lemma 3.7, $u_\lambda =W_{\xi _\lambda }+\phi _{\xi _\lambda }$ is a solution of (3.1). It follows that $u_\lambda$ solves the original problem (1.1) with $\rho ^+_\lambda =\rho ^+$ and

$$\begin{aligned} \rho _\lambda ^-=\lambda \int _{\Omega }e^{-u_\lambda }dx=\lambda \int _{\Omega }e^{-W_{\xi _\lambda }}dx+o(1)=8k\pi +o(1). \end{aligned}$$

Moreover, from the definition of $u_\lambda$, and using (3.4), (3.5) and (3.31), we can derive the second property in Theorem 1.1. $\square$

4 Asymmetric blow up

4.1 Approximate solutions

In this section we will derive the proof of Theorem 1.2. To this end we will always assume that $\Omega$ is $l-$symmetric for $l\ge 2$ even according to (4.1) below. Therefore, we will consider symmetric functions such that

$$\begin{aligned} u(x)=u ({\mathcal {R}}_l\cdot x), \end{aligned}$$

(4.1)

recall (1.7), and define

$$\begin{aligned} {\mathcal {H}}_l:=\left\{ u\in H_0^1(\Omega ), \ u \text{ satisfies } (4.1)\right\} . \end{aligned}$$

Consider problem (3.1) and let $k\ge 2$ be an odd integer. In order to construct blow up solutions with local masses $(4\pi k(k-1), 4\pi k(k+1))$, we need to consider the following singular Liouville equation. Let $\alpha \ge 2$. It is known [27] that

$$\begin{aligned} w_\delta ^\alpha (x)=\log \frac{2\alpha ^2\delta ^\alpha }{(\delta ^\alpha +|x|^\alpha )^2}, \quad \delta >0, \end{aligned}$$

solves the problem

$$\begin{aligned} \Delta w+|x|^{\alpha -2}e^w=0~\mathrm {in}~{{\mathbb {R}}}^2, \quad \int _{{{\mathbb {R}}}^2}|x|^{\alpha -2}e^wdx<\infty , \end{aligned}$$

and

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}|x|^{\alpha -2}e^wdx=4\pi \alpha . \end{aligned}$$

Similarly to the previous section, let Pu be the projection of the function u into $H_0^1(\Omega )$. We look here for a sign changing solution of the form

$$\begin{aligned} u=W+\phi (x), \quad W(x)=z(x)+\sum _{i=1}^k(-1)^i Pw_i(x), \end{aligned}$$

where $\phi$ is a small error term, z(x) is the unique solution of (1.8) and $Pw_i=Pw_{\delta _i}^{\alpha _i}$ with

$$\begin{aligned} \alpha _i=4i-2, \quad \delta _i=d_i\lambda ^{\frac{k-i+1}{4i-2}}, \, d_i>0, \quad i=1,\cdots ,k. \end{aligned}$$

(4.2)

The latter parameters are chosen such that the interaction of different bubbles is small. More precisely, the following functions will play an important role in the interaction estimate:

$$\begin{aligned} \Theta _i(y)= & {} Pw_i(\delta _i y)-w_i(\delta _i y)-(\alpha _i-2)\log |\delta _iy|\nonumber \\&+\sum _{j\ne i}(-1)^{j-i}Pw_j-z(\delta _i y)+\log \lambda , \quad \text{ i } \text{ odd }, \end{aligned}$$

(4.3)

$$\begin{aligned} T_i(y)= & {} Pw_i(\delta _i y)-w_i(\delta _i y)-(\alpha _i-2)\log |\delta _iy|\nonumber \\&+\sum _{j\ne i}(-1)^{j-i}Pw_j+z(\delta _i y)-\log Q, \quad \text{ i } \text{ even }, \end{aligned}$$

(4.4)

where

$$\begin{aligned} Q=\rho _0^{-1}\int _{\Omega }e^{z-8k\pi G(x,0)}dx. \end{aligned}$$

(4.5)

As we will see in the sequel, in order to make these two functions small, we will need to choose $\delta _i$ and $\alpha _i$ such that

$$\begin{aligned} (\alpha _i-2)+\sum _{j<i}(-1)^{j-i}2\alpha _j=0, \quad i=1,\cdots ,k, \end{aligned}$$

(4.6)

and

$$\begin{aligned}&-\alpha _i\log \delta _i-\log (2\alpha _i^2) -2\sum _{j>i}(-1)^{j-i}\alpha _j\log \delta _j-z(0)\nonumber \\&\quad +\sum _{j=1}^k(-1)^{j-i}h_j(0)+\log \lambda =0, \quad \text{ i } \text{ odd }, \end{aligned}$$

(4.7)

$$\begin{aligned}&-\alpha _i\log \delta _i-\log (2\alpha _i^2) -2\sum _{j>i}(-1)^{j-i}\alpha _j\log \delta _j+z(0)\nonumber \\&+\sum _{j=1}^k(-1)^{j-i}h_j(0)-\log Q=0, \quad \text{ i } \text{ even }, \end{aligned}$$

(4.8)

where $h_i(x)=4\pi \alpha _iH(x,0).$ From (4.6) we deduce that $\alpha _1=2$ and $\alpha _i=\alpha _{i-1}+4$ for $i\ge 2$ which implies the choice of $\alpha _i$ in (4.2). On the other hand, from (4.7) and (4.8) one easily deduces that

$$\begin{aligned} \delta _k^{\alpha _k}=\lambda e^{\sum _j(-1)^{j-k}h_j(0)-z(0)-\log (2\alpha _k^2)} =\lambda e^{8k\pi H(0,0)-z(0)-\log (2\alpha _k^2)}, \end{aligned}$$

and

$$\begin{aligned} \delta _{i-1}^{\alpha _{i-1}}=\dfrac{\delta _i^{\alpha _i}}{4\alpha _i^2\alpha _{i-1}^2Q}\lambda . \end{aligned}$$

From the above identities, one can get that

$$\begin{aligned} \delta _i=d_i\lambda ^{\frac{k-i+1}{4i-2}}, \end{aligned}$$

for some $d_i>0$, which implies (4.2).

We estimate now $\Theta _i$ and $T_i$. First, using the maximum principle it is not difficult to see that

$$\begin{aligned} \begin{aligned} Pw_i(x)&=w_i(x)-\log (2\alpha _i^2\delta _i^{\alpha _i})+h_i(x)+O(\delta _i^{\alpha _i})\\&=-2\log (\delta _i^{\alpha _i}+|x|^{\alpha _i})+h_i(x)+O(\delta _i^{\alpha _i}) \end{aligned} \end{aligned}$$

(4.9)

and for $i,j=1,\cdots ,k$,

$$\begin{aligned} Pw_i(\delta _j y)= {\left\{ \begin{array}{ll} -2\alpha _i\log (\delta _j|y|)+h_i(0) +O\Big (\frac{1}{|y|^{\alpha _i}}\Big (\frac{\delta _i}{\delta _j}\Big )^{\alpha _i}\Big )\\ +O(\delta _j|y|)+O(\delta _i^{\alpha _i})\quad \text{ if } ~i<j\\ \\ -2\alpha _i\log \delta _i-2\log (1+|y|^{\alpha _i})+h_i(0)\\ +O(\delta _i(y))+O(\delta _i^{\alpha _i})\quad \text{ if } ~i=j\\ \\ -2\alpha _i\log \delta _i+h_i(0) +O\Big (|y|^{\alpha _i}\Big (\frac{\delta _j}{\delta _i}\Big )^{\alpha _i}\Big )\\ +O(\delta _j|y|)+O(\delta _i^{\alpha _i}) \quad \text{ if } ~i>j. \end{array}\right. } \end{aligned}$$

(4.10)

where $h_i(x)=4\pi \alpha _iH(x,0)$.

Remark 4.1

From the above expansion, one can get that for $|x|\ge \delta _0$ for $\delta _0>0$ small, the following expansion holds:

$$\begin{aligned} \begin{aligned} \sum _{i=1}^k(-1)^iPw_i(x)=4\pi \sum _i(-1)^i\alpha _iH(x,0)-2\sum _i(-1)^i\alpha _i\log |x|+O(\delta _k^{\alpha _k}) \end{aligned} \end{aligned}$$

From the definition of $\alpha _i$ we have $\sum _{i=1}^k(-1)^i\alpha _i=(-1)^k2k$ and hence, for k odd it holds

$$\begin{aligned} \sum _{i=1}^k(-1)^i Pw_i(x)=-8kG(x,0)+O(\lambda ). \end{aligned}$$

We next introduce the following shrinking annulus

$$\begin{aligned} A_j=\left\{ x\in \Omega , \sqrt{\delta _{j-1}\delta _j}\le |x|\le \sqrt{\delta _j\delta _{j+1}}\right\} , \quad j=1,\cdots ,k, \end{aligned}$$

(4.11)

where $\delta _0:=0$ and $\delta _{k+1}:=+\infty$.

Lemma 4.2

For any $y\in \frac{A_i}{\delta _i}$, the following estimates hold:

$$\begin{aligned}&\Theta _i(y)=O(\delta _i|y|+\lambda ), \quad \text{ i } \text{ odd }, \end{aligned}$$

(4.12)

$$\begin{aligned}&T_i(y)=O(\delta _i|y|+\lambda ), \quad \text{ i } \text{ even }. \end{aligned}$$

(4.13)

In particular,

$$\begin{aligned} \sup _{y\in \frac{A_i}{\delta _i}}|\Theta _i(y)|+\sup _{y\in \frac{A_i}{\delta _i}}|T_i(y)|=O(1). \end{aligned}$$

(4.14)

Proof

Consider $y\in \frac{A_i}{\delta _i}$. From (4.10), and using (4.6) and (4.7), for i odd,

$$\begin{aligned} \begin{aligned} \Theta _i(y)&=-\alpha _i\log \delta _i-\log (2\alpha _i^2)+h_i(0)-(\alpha _i-2)\log |\delta _iy|+O(\delta _i|y|+\delta _i^{\alpha _i})\\&\quad +\sum _{j<i}(-1)^{i-j}\Big [-2\alpha _j\log (\delta _i|y|)+h_j(0)+O\Big (\frac{1}{|y|^{\alpha _j}}\Big (\frac{\delta _j}{\delta _i} \Big )^{\alpha _j}\Big )+O(\delta _i|y|+\delta _j^{\alpha _j})\Big ]\\&\quad +\sum _{j>i}(-1)^{j-i}\Big [-2\alpha _j\log \delta _j+h_j(0)+O\Big (|y|^{\alpha _j}\Big (\frac{\delta _i}{\delta _j} \Big )^{\alpha _j}\Big )+O(\delta _i|y|+\delta _j^{\alpha _j})\Big ]\\&\quad -z(0)+\log \lambda +O(\delta _i|y|)\\&=\Big [\sum _{j=1}^k(-1)^{j-i}h_j(0)-\alpha _i\log \delta _i-\log (2\alpha _i^2)-2\sum _{j>i}(-1)^{j-i}\alpha _j\log \delta _j-z(0)+\log \lambda \Big ]\\&\quad (=0 \text { because of }(4.7))\\&\quad -\log |\delta _i|y||\Big [(\alpha _i-2)+\sum _{j<i}(-1)^{i-j}2\alpha _j\Big ]\\&\quad (=0 \text { because of }(4.6))\\&\quad +O(\delta _i|y|)+\sum _j\delta _j^{\alpha _j}+\sum _{j>i}O\Big (|y|^{\alpha _j}\Big (\frac{\delta _i}{\delta _j} \Big )^{\alpha _j}\Big )+\sum _{j<i}O\Big (\frac{1}{|y|^{\alpha _j}}\Big (\frac{\delta _j}{\delta _i} \Big )^{\alpha _j}\Big )\\&=O(\delta _i|y|)+\sum _j\delta _j^{\alpha _j}+\sum _{j>i}O\Big (|y|^{\alpha _j}\Big (\frac{\delta _i}{\delta _j} \Big )^{\alpha _j}\Big )+\sum _{j<i}O\Big (\frac{1}{|y|^{\alpha _j}}\Big (\frac{\delta _j}{\delta _i} \Big )^{\alpha _j}\Big )\\&=O(\delta _i|y|+\lambda ). \end{aligned} \end{aligned}$$

Similarly, for i even,

$$\begin{aligned} \begin{aligned} T_i(y)&=\Big [\sum _{j=1}^k(-1)^{j-i}h_j(0)-\alpha _i\log \delta _i-\log (2\alpha _i^2)-2\sum _{j>i}(-1)^{j-i}\alpha _j\log \delta _j+z(0)-\log Q\Big ]\\&\quad (=0 \text { because of }(4.8))\\&\quad -\log |\delta _i|y||\Big [(\alpha _i-2)+\sum _{j<i}(-1)^{i-j}2\alpha _j\Big ]\\&\quad (=0 \text { because of }(4.6))\\&\quad +O(\delta _i|y|)+\sum _j\delta _j^{\alpha _j}+\sum _{j>i}O\Big (|y|^{\alpha _j}\Big (\frac{\delta _i}{\delta _j} \Big )^{\alpha _j}\Big )+\sum _{j<i}O\Big (\frac{1}{|y|^{\alpha _j}}\Big (\frac{\delta _j}{\delta _i} \Big )^{\alpha _j}\Big )\\&=O(\delta _i|y|)+\sum _j\delta _j^{\alpha _j}+\sum _{j>i}O\Big (|y|^{\alpha _j}\Big (\frac{\delta _i}{\delta _j} \Big )^{\alpha _j}\Big )+\sum _{j<i}O\Big (\frac{1}{|y|^{\alpha _j}}\Big (\frac{\delta _j}{\delta _i} \Big )^{\alpha _j}\Big )\\&=O(\delta _i|y|+\lambda ). \end{aligned} \end{aligned}$$

Finally, (4.14) follows from the above two estimates since $\delta _i|y|=O(1)$ when $y\in \frac{A_i}{\delta _i}$. $\square$

Finally, we will need the following non-degeneracy result for entire singular Liouville equations which was derived in [11, Theorem 6.1] for $l=2$ and which can be extended to any $l\ge 2$ even.

Proposition 4.3

Assume $\phi : {{\mathbb {R}}}^2\rightarrow {{\mathbb {R}}}$ satisfying (4.1) is a solutions of

$$\begin{aligned} \Delta \phi +2\alpha ^2\frac{|y|^{\alpha -2}}{(1+|y|^\alpha )^2}\phi =0 \quad \text { in }\quad {{\mathbb {R}}}^2, ~\quad \int _{{{\mathbb {R}}}^2}|\nabla \phi |^2dy<\infty , \end{aligned}$$

with $\alpha \ge 2$ and $\frac{\alpha }{2}$ odd. Then,

$$\begin{aligned} \phi (y)=\gamma \frac{1-|y|^\alpha }{1+|y|^\alpha }, \quad \text{ for } \text{ some } \gamma \in {{\mathbb {R}}}. \end{aligned}$$

4.2 Estimate of the error term

In this subsection we estimate the error of the approximate solution. To this end, set

$$\begin{aligned} \begin{aligned} E_1&=\rho ^+\dfrac{e^W}{\int _{\Omega }e^W dx}-\sum _{i\ even}|x|^{\alpha _i-2}e^{w_i}-\rho _0\dfrac{e^{z-8k\pi G(x,0)}}{\int _{\Omega }e^{z-8k\pi G(x,0)} dx},\\ E_2&=\lambda e^{-W}-\sum _{i \ odd}|x|^{\alpha _i-2}e^{w_i}. \end{aligned} \end{aligned}$$

Lemma 4.4

For any $q\ge 1$ sufficiently close to 1, the following holds:

$$\begin{aligned} \Vert E_1\Vert _{q}=O\Big (\lambda ^{\frac{2-q}{2q(2k-1)}}\Big ), \quad \Vert E_2\Vert _q=O\Big (\lambda ^{\frac{2-q}{2q(2k-1)}}\Big ). \end{aligned}$$

Proof

First we consider $E_2$. Recall the definition of the annulus $A_i$ in (4.11).

$$\begin{aligned} \begin{aligned} \int _{\Omega }E_2^qdx&=\sum _{i=1}^k \int _{A_i}E_2^qdx =\sum _{i \ odd}\int _{A_i}E_2^qdx+\sum _{i \ even}\int _{A_i}E_2^qdx=I_1+I_2. \end{aligned} \end{aligned}$$

One has

$$\begin{aligned} \begin{aligned} I_1&=\sum _{i \ odd}\int _{A_i}E_2^qdx=\sum _{i \ odd}\int _{A_i}|\lambda e^{\sum _{l\ odd}Pw_l-\sum _{l \ even}Pw_l-z}-\sum _{j \ odd}|x|^{\alpha _j-2}e^{w_j}|^qdx\\&\le C\sum _{i \ odd}\int _{A_i}||x|^{\alpha _i-2}e^{w_i}-\lambda e^{\sum _{l \ odd}Pw_l-\sum _{l \ even}Pw_l-z}|^qdx\\&\quad +C\sum _{i, j\ odd,\ i\ne j}\int _{A_i}||x|^{\alpha _j-2}e^{w_j}|^qdx\\&=I_{11}+I_{12}. \end{aligned} \end{aligned}$$

Let us estimate $I_{11}$. For fixed i odd,

$$\begin{aligned} \begin{aligned}&\int _{A_i}||x|^{\alpha _i-2}e^{w_i}-\lambda e^{\sum _{l \ odd}Pw_l-\sum _{l \ even}Pw_l-z}|^qdx\\&\quad =\int _{A_i}|x|^{q(\alpha _i-2)}e^{qw_i}|1-e^{Pw_i-w_i-(\alpha _i-2)\log |x|+\sum _{j\ne i \ odd}Pw_j-\sum _{l \ even}Pw_l-z+\log \lambda }|^qdx\\&\quad =C\delta _i^{2-2q}\int _{\frac{A_i}{\delta _i}}\frac{|y|^{q(\alpha _i-2) }}{(1+|y|^{\alpha _i})^{2q}}|1-e^{\Theta _i(y)}|^qdy=C\delta _i^{2-2q}\int _{\frac{A_i}{\delta _i}}\frac{|y|^{q(\alpha _i-2) }}{(1+|y|^{\alpha _i})^{2q}}|\Theta _i(y)|^qdy \\&\quad ( \text{ using } (4.12))\\&\quad =O\Big (\delta _i^{2-2q}\int _{\frac{A_i}{\delta _i}}\frac{|y|^{q(\alpha _i-2) }}{(1+|y|^{\alpha _i})^{2q}}|\delta _i |y|+\lambda |^qdy \Big )=O(\delta _i^{2-2q}\lambda ^q+\delta _i^{2-q})=O(\delta _1^{2-2q}\lambda ^q+\delta _k^{2-q})\\&\quad =O(\lambda ^{q+k(1-q)}+\lambda ^{\frac{2-q}{2(2k-1)}}) =O(\lambda ^{\frac{2-q}{2(2k-1)}}), \end{aligned} \end{aligned}$$

provided that q is close to 1. Therefore, we get $I_{11}=O(\lambda ^{\frac{2-q}{2(2k-1)}}).$

For $I_{12}$, fix $j\ne i$ odd,

$$\begin{aligned} \begin{aligned}&\int _{A_i}||x|^{\alpha _j-2}e^{w_j}|^qdx=C\int _{A_i}\left( \frac{|x|^{\alpha _j-2}\delta _j^{\alpha _j}}{(\delta _j^{\alpha _j} +|x|^{\alpha _j})^2}\right) ^qdx\\&\quad =C\delta _j^{2-2q}\int _{\frac{\sqrt{\delta _{i-1}\delta _i}}{\delta _j}\le |y|\le \frac{\sqrt{\delta _i\delta _{i+1}}}{\delta _j}}\frac{|y|^{q(\alpha _j-2) }}{(1+|y|^{\alpha _j})^{2q}}dy \\&\quad ={\left\{ \begin{array}{ll} O\left( \delta _j^{2-2q}\Big (\frac{\sqrt{\delta _i\delta _{i+1}}}{\delta _j}\Big ) ^{(\alpha _j-2)q+2}\right) \quad &{} \text{ for } ~j>i\\ \\ O\left( \delta _j^{2-2q}\Big (\frac{\sqrt{\delta _i\delta _{i-1}}}{\delta _j}\Big ) ^{-(\alpha _j+2)q+2}\right) \quad &{} \text{ for } ~j<i \end{array}\right. }\\&\quad ={\left\{ \begin{array}{ll} O\Big (\delta _3^{2-2q}(\frac{\delta _{k-1}}{\delta _k})^{(\alpha _k-2)q+2}\Big ) =O\left( \lambda ^{\frac{(k-2)(1-q)}{5} +\frac{(2k+1)(2(k-1)q+1)}{4(k-1)^2-1}}\right) \\ \\ O\Big (\delta _1^{2-2q}(\frac{\delta _{k-2}}{\delta _{k-1}})^{q(2+\alpha _{k-2})-2}\Big ) =O\left( \lambda ^{k(1-q)+\frac{(2k+1)(2(k-2)q-1)}{4(k-2)^2-1}}\right) \end{array}\right. }\\&\quad =O\Big (\lambda ^{\frac{2-q}{2(2k-1)}}\Big ). \end{aligned} \end{aligned}$$

(4.15)

provided that q is close to 1. Therefore, $\Vert I_1\Vert _q=O\Big (\lambda ^{\frac{2-q}{2q(2k-1)}}\Big )$.

Next, let us estimate $I_2$. For l even fixed,

$$\begin{aligned} \begin{aligned}&\int _{A_l}E_2^qdx\le C\int _{A_l}|\lambda e^{-W}|^qdx+C\sum _{i \ odd}\int _{A_l}||x|^{\alpha _i-2}e^{w_i}|^qdx=I_{21}+I_{22}. \end{aligned} \end{aligned}$$

We have,

$$\begin{aligned} \begin{aligned} I_{21}&=C\int _{A_l}|\lambda e^{-Pw_l-\sum _{j\ne l \ even}Pw_j-z+\sum _{i \ odd}Pw_i}|^qdx\\&\quad =C\lambda ^q \delta _l^2\int _{\frac{A_l}{\delta _l}}|e^{-w_l(\delta _l y)-(\alpha _l-2)\log |\delta _l y|-T_l(y)-\log Q}|^qdy\\&\quad ( \text{ using } (4.13))\\&=O\Big (\delta _l^{2+2q}\lambda ^q \int _{{\sqrt{\frac{\delta _{l-1}}{\delta _l}}\le |y|\le \sqrt{\frac{\delta _{l+1}}{\delta _l}}} } \frac{(1+|y|^{\alpha _l})^{2q}}{|y|^{(\alpha _l-2)q}}(1+\delta _l|y|+\lambda )^qdy \Big )\\&=O\Big (\delta _l^{2+2q}\lambda ^q \Big [\Big (\frac{\delta _{l+1}}{\delta _l}\Big )^{\frac{(\alpha _l+2)q}{2}+1} +\Big (\frac{\delta _{l}}{\delta _{l-1}}\Big )^{\frac{(\alpha _l-2)q}{2}-1} \Big ]\Big )\\&\quad =O\Big (\delta _{2}^{2+2q}\lambda ^q\Big [\Big (\frac{\delta _{3}}{\delta _2}\Big )^{\frac{(\alpha _2+2)q}{2}+1} +\Big (\frac{\delta _{2}}{\delta _{1}}\Big )^{\frac{(\alpha _2-2)q}{2}-1} \Big ] \Big )\\&=O\Big (\lambda ^{q+\frac{(k-1)(1+q)}{3}-\frac{(2k+1)(2q-1)}{6}}\Big )=O\Big (\lambda ^{\frac{2-q}{2(2k-1)}}\Big ), \end{aligned} \end{aligned}$$

(4.16)

if q is close to 1. Moreover, similarly to the estimate of $I_{12}$, one can also get that $I_{22}=O\Big (\lambda ^{\frac{2-q}{2(2k-1)}}\Big ).$

Combining all the above estimates, one has

$$\begin{aligned} \int _{\Omega }E_2^qdx=O(\lambda ^{\frac{2-q}{2(2k-1)}}). \end{aligned}$$

(4.17)

Next we consider $E_1$. First we need to estimate $\int _{\Omega }e^{W}dx$. For i even fixed,

$$\begin{aligned} \begin{aligned} \int _{A_i}e^Wdx&=\int _{A_i}e^{Pw_i-w_i+z+\sum _{j\ne i}(-1)^{j-i}Pw_j-(\alpha _i-2)\log |x|}|x|^{\alpha _i-2}e^{w_i}dx\\&=\int _{\frac{A_i}{\delta _i}}e^{T_i(y)+\log Q}|\delta _i y|^{\alpha _i-2}e^{w_i(\delta _i y)}\delta _i^2dy\\&=\int _{\frac{A_i}{\delta _i}}e^{\log Q+O(\delta _i |y|+\lambda )}|\delta _i y|^{\alpha _i-2}e^{w_i(\delta _i y)}\delta _i^2dy=4\pi \alpha _iQ+O(\lambda ^{\frac{1}{2(2k-1)}}), \end{aligned} \end{aligned}$$

where we have used Lemma 4.2 for the estimate of $T_i(y)$ and the fact that

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2} dy=4\pi \alpha _i. \end{aligned}$$

For $i<k$ odd and fixed, reasoning as in (4.16) with $q=1$, one has

$$\begin{aligned} \begin{aligned} \int _{A_i}e^Wdx=\int _{A_i}e^{-Pw_i-\sum _{j\ne i}(-1)^{j-i}Pw_j+z}dx =O(\lambda ^{\frac{2k-5}{6}}). \end{aligned} \end{aligned}$$

Finally for $i=k$ which is odd, using Remark 4.1,

$$\begin{aligned} \begin{aligned} \int _{A_k}e^Wdx&=\int _{A_k}e^z e^{-Pw_k-\sum _{j\ne k}(-1)^{j-k}Pw_j}dx\\&=\int _{|x|>\sqrt{\delta _{k-1}\delta _k}}e^{z-8k\pi G(x,0)}dx +O(\delta _k^{\alpha _k})+O(\lambda ^{\frac{1}{2(2k-1)}})\\&=\int _{\Omega }e^{z-8k\pi G(x,0)}dx+O(\lambda ^{\frac{1}{2(2k-1)}}). \end{aligned} \end{aligned}$$

In conclusion, one has

$$\begin{aligned} \begin{aligned} \int _{\Omega }e^Wdx&=\int _{\Omega }e^{z-8k\pi G(x,0)}dx+\sum _{i \ even}4\pi \alpha _iQ+O(\lambda ^{\frac{1}{2(2k-1)}})\\&=\frac{\rho ^+}{\rho _0}\int _{\Omega }e^{z-8k\pi G(x,0)}dx+O(\lambda ^{\frac{1}{2(2k-1)}}), \end{aligned} \end{aligned}$$

(4.18)

where we used the definition of Q in (4.5) and the fact that

$$\begin{aligned} \sum _{i \ even}4\pi \alpha _iQ=\frac{\rho ^+-\rho _0}{\rho _0}\int _{\Omega }e^{z-8k\pi G(x,0)}dx, \end{aligned}$$

since $\sum _{i\ even}4\pi \alpha _i=4\pi k(k-1)=\rho ^+-\rho _0$.

With the estimate for $\int _{\Omega }e^Wdx$ in hand, we now consider $E_1$.

$$\begin{aligned} \begin{aligned} \int _{\Omega }E_1^qdx&=\sum _{i\ even}\int _{A_i}E_1^qdx+\sum _{l\ odd}\int _{A_l}E_1^qdx=J_1+J_2. \end{aligned} \end{aligned}$$

First for i even fixed,

$$\begin{aligned} \begin{aligned} \int _{A_i}E_1^qdx&=\int _{A_i}|\rho ^+\frac{e^W}{\int _{\Omega }e^Wex}-\rho _0\frac{e^{z-8k\pi G(x,0)}}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\\&\quad - |x|^{\alpha _i-2}e^{w_i}-\sum _{j\ne i \ even}|x|^{\alpha _j-2}e^{w_j}|^qdx\\&\le C\int _{A_i}|\rho ^+\frac{e^W}{\int _{\Omega }e^Wex}-|x|^{\alpha _i-2}e^{w_i}|^qdx+C\int _{A_i}|\rho _0\frac{e^{z-8k\pi G(x,0)}}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}|^qdx\\&\quad +C\sum _{j\ne i \ even}\int _{A_i}||x|^{\alpha _j-2}e^{w_j}|^qdx\\&=C\int _{A_i}|\rho ^+\frac{e^W}{\int _{\Omega }e^Wdx}-|x|^{\alpha _i-2}e^{w_i}|^qdx +O(\lambda ^{\frac{2-q}{2(2k-1)}})+O(\delta _{i+1}^{4kq+2})\\&=C\delta _i^{2-2q}\int _{\frac{A_i}{\delta _i}}\frac{|y|^{(\alpha _i-2)q}}{(1+|y|^{\alpha _i})^{2q}}\\&\quad \left| 1-e^{Pw_i(\delta _i y)-w_i(\delta _i y)-(\alpha _i-2)\log |\delta _i y|+\sum _{j\ne i}(-1)^{j-i}Pw_j+z+\log \frac{\rho ^+}{\int _{\Omega }e^Wdx}}\right| ^qdx\\&\quad ( \text{ by } (4.13))\\&=C\delta _i^{2-2q}\int _{\frac{A_i}{\delta _i}}\frac{|y|^{(\alpha _i-2)q}}{(1+|y|^{\alpha _i})^{2q}}\\&\quad \left| 1-e^{T_i(y)+\log Q +\log \frac{\rho _0}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+O(\lambda ^{\frac{1}{2(2k-1)}})}\right| ^qdx\\&=C\delta _i^{2-2q}\int _{\frac{A_i}{\delta _i}}\frac{|y|^{(\alpha _i-2)q}}{(1+|y|^{\alpha _i})^{2q}} |\delta _i|y|+O(\lambda ^{\frac{1}{2(2k-1)}})|^qdy=O(\lambda ^{\frac{2-q}{2(2k-1)}}). \end{aligned} \end{aligned}$$

So we have

$$\begin{aligned} J_1=O\Big (\lambda ^{\frac{2-q}{2(2k-1)}}\Big ). \end{aligned}$$

(4.19)

Next, consider $J_2$. For $l<k$ odd and fixed, similarly to the estimates in (4.16), (4.15) and using (4.18)

$$\begin{aligned} \begin{aligned} \int _{A_l}|E_1|^qdx&=O(1)\Big (\int _{A_l}|e^{-Pw_l-\sum _{j\ne l}(-1)^{j-l}Pw_j+z}|^qdx+\int _{A_l}\left| \frac{e^{z-8k\pi G(x,0)}}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\right| ^qdx \\&\quad +\sum _{j\ even}\int _{A_l}||x|^{\alpha _j-2}e^{w_j}|^qdx\Big ) =O(\lambda ^{\frac{2-q}{2(2k-1)}}). \end{aligned} \end{aligned}$$

Finally, we consider the case $l=k$ which is odd: using (4.18) and (4.15)

$$\begin{aligned}\begin{aligned} \int _{A_k}E_1^qdx&\le C\int _{A_k}\left| \rho ^+\dfrac{e^{z+\sum _{i}(-1)^iPw_i}}{\int _{\Omega } e^Wdx} -\rho _0\dfrac{e^{z-8k\pi G(x,0)}}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\right| ^qdx\\&\quad +C\sum _{i\ even}\int _{A_k}|x|^{(\alpha _i-2)q}e^{qw_i}dx\\&=C\int _{A_k}\left| \rho _0\frac{e^{z+\sum _{i}(-1)^iPw_i}}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}-\rho _0\frac{e^{z-8k\pi G(x,0)}}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\right| ^qdx +O(\lambda ^{\frac{2-q}{2(2k-1)}})\\&=O(\delta _k^{2q})+O(\lambda ^{\frac{2-q}{2(2k-1)}}) =O(\lambda ^{\frac{2-q}{2(2k-1)}}). \end{aligned} \end{aligned}$$

In conclusion, one has

$$\begin{aligned} \Vert E_1\Vert _q=O\Big (\lambda ^{\frac{2-q}{2q(2k-1)}}\Big ). \end{aligned}$$

$\square$

4.3 The linear theory

In this subsection, we consider the linear problem: given $h\in {\mathcal {H}}_l$, we look for $\phi \in {\mathcal {H}}_l$ such that

$$\begin{aligned} \Delta \phi +\rho ^+\left( \dfrac{e^W\phi }{\int _\Omega e^Wdx}-\dfrac{e^W\int _\Omega e^W\phi dx}{(\int _\Omega e^Wdx)^2}\right) +\lambda e^{-W}\phi =\Delta h\quad \text{ in }~\Omega . \end{aligned}$$

(4.20)

First we have the following apriori estimate:

Lemma 4.5

There exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0,\lambda _0)$, $h\in {\mathcal {H}}_l$ and $\phi \in {\mathcal {H}}_l$ solution of (4.20) we have

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda | \Vert h\Vert . \end{aligned}$$

We start by listing some straightforward integrals which will be useful in the proof of Lemma 4.5.

Lemma 4.6

The following hold:

$$\begin{aligned}&\int _{{{\mathbb {R}}}^2}\frac{|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}\frac{1-|y|^{\alpha _i}}{1+|y|^{\alpha _i}}dy=0, \end{aligned}$$

(4.21)

$$\begin{aligned}&\int _{{{\mathbb {R}}}^2}2\alpha _i^2\frac{|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}\frac{1-|y|^{\alpha _i}}{1+|y|^{\alpha _i}}\log (1+|y|^{\alpha _i})^2dy=-4\pi \alpha _i, \end{aligned}$$

(4.22)

$$\begin{aligned}&\int _{{{\mathbb {R}}}^2}2\alpha _i^2\frac{|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}\frac{1-|y|^{\alpha _i}}{1+|y|^{\alpha _i}}\log |y|dy=-4\pi . \end{aligned}$$

(4.23)

Proof of Lemma 4.5

We prove it by contradiction. Assume there exist $\lambda _n\rightarrow 0$, $h_n\in {\mathcal {H}}_l$ and $\phi _n \in {\mathcal {H}}_l$ which solves (4.20) such that

$$\begin{aligned} \Vert \phi _n\Vert =1, \quad \ |\log \lambda _n|\Vert h_n\Vert \rightarrow 0 \quad \text{ as } n\rightarrow \infty . \end{aligned}$$

In the following, we omit the index n for simplicity. For $i=1, \cdots , k$, define ${\tilde{\phi }}_i(y)$ as

$$\begin{aligned} \begin{aligned} {\tilde{\phi }}_i(y)={\left\{ \begin{array}{ll} \phi _i(\delta _i y), \quad &{}y\in {\tilde{\Omega }}_i=\frac{\Omega }{\delta _i},\\ 0, &{}y\in {{\mathbb {R}}}^2\setminus {\tilde{\Omega }}_i. \end{array}\right. } \end{aligned} \end{aligned}$$

Step 1. We claim that

$$\begin{aligned} \phi \rightarrow 0~\text { weakly in }H_0^1(\Omega )\text { and strongly in } L^q(\Omega )\text { for }q\ge 2. \end{aligned}$$

(4.24)

and

$$\begin{aligned} {\tilde{\phi }}_i \text { is bounded in }H_{\alpha _i}({{\mathbb {R}}}^2) \end{aligned}$$

Letting $\psi \in C_0^\infty (\Omega \setminus \{0\})$ and multiplying equation (4.20) by $\psi$ and integrating, one has

$$\begin{aligned} \begin{aligned}&-\int _{\Omega }\nabla \psi \cdot \nabla \phi dx+\int _{\Omega }\lambda e^{-W}\phi \psi dx\\&\quad +\rho ^ +\left( \dfrac{\int _{\Omega }e^W\phi \psi dx}{\int _{\Omega }e^Wdx}-\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega } e^W\psi dx}{(\int _{\Omega }e^Wdx)^2} \right) =\int _{\Omega }\Delta h \psi dx. \end{aligned} \end{aligned}$$

(4.25)

By the assumption on $\phi$, using the fact that in compact sets of $\Omega \setminus \{0\}$,

$$\begin{aligned}e^W=e^{z(x)-8k\pi G(x,0)}+O(\lambda )\quad \text{ and } \quad \lambda e^{-W}=O(\lambda ), \end{aligned}$$

one has

$$\begin{aligned} \phi \rightarrow \phi ^* \text { weakly in }H_0^1(\Omega )\text { and strongly in }L^q(\Omega )\text { for }q\ge 2 \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&-\int _{\Omega }\nabla \phi ^* \cdot \nabla \psi dx+\rho ^+\dfrac{\int _{\Omega } e^{z-8k\pi G(x,0)}\phi ^*\psi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}}\\&\quad -\rho ^+\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\psi dx\int _{\Omega }e^{z-8k\pi G(x,0)}\phi ^*dx}{(\int _{\Omega }e^{z-8k\pi G(x,0)}dx)^2}=0. \end{aligned} \end{aligned}$$

So $\Vert \phi ^*\Vert _{H_0^1(\Omega )}\le 1$ and it solves

$$\begin{aligned} \Delta \phi ^*+\rho ^+\left( \dfrac{e^{z-8k\pi G(x,0)}\phi ^*}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}-\dfrac{e^{z-8k\pi G(x,0)}\int _{\Omega }e^{z-8k\pi G(x,0)}\phi ^*dx}{(\int _{\Omega }e^{z-8k\pi G(x,0)}dx)^2}\right) =0. \end{aligned}$$

By the non-degeneracy of z(x) we get $\phi ^*=0$. Thus (4.24) is proved.

Now we prove that ${\tilde{\phi }}_i$ is bounded in $H_{\alpha _i}({{\mathbb {R}}}^2)$. First it is easy to check that

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}|\nabla {\tilde{\phi }}_i|^2dy=\int _{\Omega }|\nabla \phi _i|^2dx\le 1 \quad \text{ for } i=1,\cdots ,k. \end{aligned}$$

(4.26)

We multiply (4.20) again by $\phi$ and integrate,

$$\begin{aligned}&\int _{\Omega }|\nabla \phi |^2dx-\int _{\Omega }\lambda e^{-W}\phi ^2dx\nonumber \\&\quad -\rho ^+\Big (\frac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx} -\frac{(\int _{\Omega }e^W\phi dx)^2}{(\int _{\Omega }e^Wdx)^2} \Big )=\int _{\Omega }\nabla h\cdot \nabla \phi dx. \end{aligned}$$

(4.27)

From the above equation, one can get that,

$$\begin{aligned} \begin{aligned} \int _{\Omega }\lambda e^{-W} \phi _i^2dx&\le \int _{\Omega }|\nabla \phi |^2dx-\rho ^+\Big (\frac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx} -\frac{(\int _{\Omega }e^W\phi dx)^2}{(\int _{\Omega }e^Wdx)^2} \Big )-\int _{\Omega }\nabla h\cdot \nabla \phi dx\\&\le 1+o(1)+\Vert h\Vert =O(1) \end{aligned} \end{aligned}$$

where we used (4.24). Let i be odd. Lemma 4.4 gives

$$\begin{aligned} \int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi ^2dx\le C, \end{aligned}$$

or equivalently

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}\frac{|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^2dy\le C. \end{aligned}$$

Combined with (4.26), we deduce that ${\tilde{\phi }}_i$ is bounded in $H_{\alpha _i}({{\mathbb {R}}}^2)$ when i is odd.

We consider now the case for i even. From (4.18), $e^W=e^{z-8k\pi G(x,0)}+O(\lambda )$ uniformly on compact sets of $\Omega \setminus \{0\}$ and recalling (4.24), we get that

$$\begin{aligned} \int _{\Omega }e^W\phi dx=O(1). \end{aligned}$$

(4.28)

Moreover, by (4.27) one can get that

$$\begin{aligned} \rho ^+\left( \dfrac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx} -\dfrac{(\int _{\Omega }e^W\phi dx)^2}{(\int _{\Omega }e^Wdx)^2} \right) =O(1). \end{aligned}$$

(4.29)

Combining (4.28) and (4.29), we have

$$\begin{aligned} \int _{\Omega }e^W\phi ^2dx=O(1). \end{aligned}$$

(4.30)

By Lemma 4.4, (4.24) and (4.30), $\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi ^2dx=O(1)$ for i even, which implies that

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}\frac{|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^2dy=O(1). \end{aligned}$$

So we get that also for i even, ${\tilde{\phi }}_i$ is bounded in $H_{\alpha _i}({{\mathbb {R}}}^2)$.

Step 2. We claim that

$$\begin{aligned} {\tilde{\phi }}_i(y)\rightarrow \gamma _i\frac{1-|y|^2}{1+|y|^2} \text { weakly in } H_{\alpha _i}({{\mathbb {R}}}^2)\text { and strongly in } L_{\alpha _i}({{\mathbb {R}}}^2), \gamma _i\in {\mathbb {R}}. \end{aligned}$$

(4.31)

From Step 1, we know that ${\tilde{\phi }}_i\rightarrow {\tilde{\phi }}_i^*$ weakly in $H_{\alpha _i}({{\mathbb {R}}}^2)$ and strongly in $L_{\alpha _i}({{\mathbb {R}}}^2)$. Consider ${\tilde{\psi }}\in C_0^\infty ({{\mathbb {R}}}^2\setminus \{0\})$ and let ${\mathcal {K}}$ be its support. For n large, one has

$$\begin{aligned} {\mathcal {K}}\subset \frac{A_i}{\delta _i}=\left\{ y\in {\tilde{\Omega }}_i, \sqrt{\frac{\delta _{i-1}}{\delta _i}}\le |y|\le \sqrt{\frac{\delta _{i+1}}{\delta _i}}\right\} . \end{aligned}$$

Define $\psi _i={\tilde{\psi }}(\frac{x}{\delta _i})$. Multiplying (4.20) by $\psi _i$ and integrating over $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\nabla \phi \cdot \nabla \psi _idx -\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi \psi _idx}{\int _{\Omega }e^Wdx} -\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega }e^W\psi _idx}{(\int _{\Omega }e^Wdx)^2} \right) \\&\quad -\int _{\Omega }\lambda e^{-W}\phi \psi _idx =\int _{\Omega }\nabla h\cdot \nabla \psi _idx. \end{aligned} \end{aligned}$$

(4.32)

Consider first i even. According to Lemma 4.4, one has

$$\begin{aligned} \begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^W\phi dx}{\int _{\Omega }e^Wdx}&=\sum _{j\ even}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}\phi dx+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o(1)\\&=\sum _{j\ even}\int _{{{\mathbb {R}}}^2}\frac{2\alpha _j^2|y|^{\alpha _j-2}{\tilde{\phi }}_j}{(1+|y|^{\alpha _i})^2}dy+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o(1)\\&=\sum _{j\ even}\int _{{{\mathbb {R}}}^2}\frac{2\alpha _j^2|y|^{\alpha _j-2}{\tilde{\phi }}_j^*}{(1+|y|^{\alpha _i})^2}dy+o(1) \end{aligned} \end{aligned}$$

where in the last line we used (4.24). Similarly, one has

$$\begin{aligned} \begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^W\phi \psi _idx}{\int _{\Omega }e^Wdx}&=\int _{{{\mathbb {R}}}^2} \dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\psi }}{\tilde{\phi }}_i^*dy+o(1),\\ \rho ^+\dfrac{\int _{\Omega }e^W\psi _idx}{\int _{\Omega }e^Wdx}&=\int _{{{\mathbb {R}}}^2} \frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\psi }}dy+o(1),\\ \lambda \int _{\Omega }e^{-W}\phi \psi _idx&=\sum _{j\ odd}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}\phi \psi _idx+o(1)=o(1). \end{aligned} \end{aligned}$$

Thus, ${\tilde{\phi }}_i^*$ satisfies

$$\begin{aligned} \begin{aligned}&\int _{{{\mathbb {R}}}^2}\nabla {\tilde{\phi }}_i^*\cdot \nabla {\tilde{\psi }}_idy-\int _{{{\mathbb {R}}}^2}\dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*{\tilde{\psi }}dy\\&\quad =-\dfrac{1}{\rho ^+}\Big (\int _{{{\mathbb {R}}}^2}\dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\psi }}dy \Big )\Big (\int _{{{\mathbb {R}}}^2} \dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy\Big ). \end{aligned} \end{aligned}$$

From this we deduce that the function

$$\begin{aligned} {\tilde{\phi }}_i^*-\frac{1}{\rho ^+}\int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy\in H_{\alpha _i}({{\mathbb {R}}}^2) \end{aligned}$$

is a solution of

$$\begin{aligned} \Delta \phi +\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}\phi =0\quad \text{ in } \quad {{\mathbb {R}}}^2\setminus \{0\}. \end{aligned}$$

(4.33)

Since $\int |\nabla {\tilde{\phi }}_i^*|^2dy\le 1$, ${\tilde{\phi }}_i^*$ is a solution in the whole space ${{\mathbb {R}}}^2$. By Proposition 4.3, we get that ${\tilde{\phi }}_i^*-\frac{1}{\rho ^+}\int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy =\gamma _i \frac{1-|y|^{\alpha _i}}{1+|y|^{\alpha _i}}$ for some $\gamma _i$. By(4.21) one has

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}\dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy =\dfrac{1}{\rho ^+}\int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}dy\int _{{{\mathbb {R}}}^2}\dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy \end{aligned}$$

which implies that

$$\begin{aligned} \left( \frac{4\pi \alpha _i}{\rho ^+}-1\right) \int _{{{\mathbb {R}}}^2}\dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy=0. \end{aligned}$$

Since $\rho ^+\ne 4\pi \alpha _i$ we deduce that

$$\begin{aligned} {\tilde{\phi }}_i^*=\gamma _i\frac{1-|y|^{\alpha _i}}{1+|y|^{\alpha _i}}. \end{aligned}$$

Hence, (4.31) is proved for i even.

We next turn to i odd. In this case, we consider (4.32) with i odd and estimate each term separately,

$$\begin{aligned} \begin{aligned} \int _{\Omega }e^W\psi _idx=o(1), \quad \int _{\Omega }e^W\phi \psi _idx=o(1), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \lambda \int _{\Omega } e^{-W}\phi \psi _idx=\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi \psi _idx+o(1)=\int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2 }}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i{\tilde{\psi }}dy +o(1). \end{aligned} \end{aligned}$$

Hence, ${\tilde{\phi }}_i^*$ satisfies

$$\begin{aligned} \int _{{{\mathbb {R}}}^2}\nabla {\tilde{\phi }}_i^*\cdot \nabla {\tilde{\psi }}dy-\int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}^*_i{\tilde{\psi }}dy =0, \end{aligned}$$

namely ${\tilde{\phi }}_i^*$ is a solution of

$$\begin{aligned} \Delta \phi +\dfrac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}\phi =0\quad \text{ in } \quad {{\mathbb {R}}}^2\setminus \{0\}, \end{aligned}$$

and again we conclude by using Proposition 4.3.

Step 3. In this step, we will prove some estimates on the speed of convergence. We set

$$\begin{aligned} \sigma _i(\lambda ):=|\log \lambda |\int _{{{\mathbb {R}}}^2}2\alpha _i^2 \frac{|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_idy. \end{aligned}$$

(4.34)

We will show that

$$\begin{aligned} {\left\{ \begin{array}{ll} \sigma _i(\lambda )=o(1)&{} \text{ for } \text{ i } \text{ odd }\\ \\ \sigma _i(\lambda )-\dfrac{4\pi \alpha _i}{\rho ^+}\Big (\sum _{j\ even}\sigma _j(\lambda )+|\log \lambda |\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\Big )=o(1)\quad &{}\text { for i even}. \end{array}\right. } \end{aligned}$$

Set $Z_i^0=\frac{\delta _i^{\alpha _i}-|x|^{\alpha _i}}{\delta _i^{\alpha _i}+|x|^{\alpha _i}}$, we know that $Z_i^0$ is a solution of

$$\begin{aligned} \Delta Z+|x|^{\alpha _i-2}e^{w_i}Z=0\quad \text{ in } \quad {{\mathbb {R}}}^2. \end{aligned}$$

Let $PZ_i^0$ be its the projection onto $H_0^1(\Omega )$, that is

$$\begin{aligned} \Delta PZ_i^0+|x|^{\alpha _i-2}e^{w_i}Z_i^0=0 \ \text{ in } \ \Omega , \quad PZ_i^0=0 \ \text{ on } \ \partial \Omega . \end{aligned}$$

By maximum principle one can show

$$\begin{aligned} PZ_i^0=Z_i+1+O(\delta _i^{\alpha _i})=\frac{2\delta _i^{\alpha _i}}{\delta _i^{\alpha _i}+|x|^{\alpha _i}} +O(\delta _i^{\alpha _i}), \end{aligned}$$

(4.35)

which implies

$$\begin{aligned} PZ_i^0(\delta _jy)= {\left\{ \begin{array}{ll} O\left( \frac{1}{|y|^{\alpha _i}}(\frac{\delta _i}{\delta _j})^{\alpha _i}\right) +O(\delta _i^{\alpha _i})&{} \text{ for } i<j,\\ \\ \frac{2}{1+|y|^{\alpha _i}}+O(\delta _i^{\alpha _i}),\quad &{} \text{ for } i=j,\\ \\ 2+O\left( |y|^{\alpha _i}(\frac{\delta _j}{\delta _i})^{\alpha _i}\right) +O(\delta _i^{\alpha _i})&{} \text{ for } i>j, \end{array}\right. } \end{aligned}$$

(4.36)

and

$$\begin{aligned} \Vert PZ_i^0\Vert _q^q=O(\delta _i^2), \ q>1. \end{aligned}$$

(4.37)

First we consider i even. Multiply (4.20) by $PZ_i^0$ and integrate over $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\nabla \phi \cdot \nabla PZ_i^0dx -\rho ^+\left( \dfrac{\int _{\Omega }e^W\phi PZ_i^0dx}{\int _{\Omega }e^Wdx} -\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega } e^W PZ_i^0dx}{(\int _{\Omega }e^Wdx)^2} \right) \\&\quad -\int _{\Omega }\lambda e^{-W}\phi PZ_i^0dx=-\int _{\Omega }\nabla h\cdot \nabla PZ_i^0dx. \end{aligned} \end{aligned}$$

(4.38)

For the first term,

$$\begin{aligned} \begin{aligned} \int _{\Omega }\nabla \phi \cdot \nabla PZ_i^0dx=-\int _{\Omega }\phi \Delta PZ_i^0dx =\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi Z_i^0 dx. \end{aligned} \end{aligned}$$

(4.39)

By Lemma 4.4, (4.24), (4.35) and (4.37),

$$\begin{aligned} \begin{aligned} \dfrac{\int _{\Omega }e^W\phi PZ_i^0dx}{\int _{\Omega }e^Wdx}&=\sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi PZ_i^0dx\\&\quad +\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}PZ_i^0\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\\&=\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi dx+\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi Z_i^0 dx\\&\quad +\sum _{j\ne i \ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi PZ_i^0 dx +o\Bigr (\frac{1}{|\log \lambda |}\Bigr ) \end{aligned} \end{aligned}$$

(4.40)

For $j\ne i$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi PZ_i^0dx= \int _{{\tilde{\Omega }}_j} \frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j PZ_i^0(\delta _j y)dy\\&\quad ={\left\{ \begin{array}{ll} \int _{{{\mathbb {R}}}^2}\frac{4\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_jdy +O\Big ( \int _{{\tilde{\Omega }}_j}\Big (|y|^{\alpha _i} (\frac{\delta _j}{\delta _i})^{\alpha _i} +\delta _i^{\alpha _i}\Big ) \frac{|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2} {\tilde{\phi }}_j \Big )dy,~ &{} \text{ for } i>j,\\ \\ O\Big ( \int _{{\tilde{\Omega }}_j}\Big ( \frac{1}{|y|^{\alpha _i}}(\frac{\delta _i}{\delta _j})^{\alpha _i} +\delta _i^{\alpha _i}\Big ) \frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2} {\tilde{\phi }}_j\Big )dy,&{} \text{ for } i<j, \end{array}\right. } \\&\quad ={\left\{ \begin{array}{ll} \frac{2\sigma _j(\lambda )}{|\log \lambda |}+o\bigr (\frac{1}{|\log \lambda |}\bigr ),\quad &{} \text{ for } i>j,\\ \\ o\bigr (\frac{1}{|\log \lambda |}\bigr ), &{} \text{ for } i<j, \end{array}\right. } \end{aligned}\nonumber \\ \end{aligned}$$

(4.41)

where we used (4.36).

Replace $\phi$ by 1 in the estimate of (4.41) , one has for $j\ne i$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}PZ_i^0dx= \int _{{\tilde{\Omega }}_j} \frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}PZ_i^0(\delta _j y)dy\\&\quad ={\left\{ \begin{array}{ll} \int _{{{\mathbb {R}}}^2}\frac{4\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}dy +O\Big ( \int _{{\tilde{\Omega }}_j}\Big (|y|^{\alpha _i} (\frac{\delta _j}{\delta _i})^{\alpha _i} +\delta _i^{\alpha _i}\Big ) \frac{|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2} \Big )dy,~ &{} \text{ for } i>j,\\ \\ O\Big ( \int _{{\tilde{\Omega }}_j}\Big ( \frac{1}{|y|^{\alpha _i}}(\frac{\delta _i}{\delta _j})^{\alpha _i} +\delta _i^{\alpha _i}\Big ) \frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}\Big )dy,&{} \text{ for } i<j, \end{array}\right. } \\&\quad ={\left\{ \begin{array}{ll} 8\pi \alpha _j+o\bigr (\frac{1}{|\log \lambda |}\bigr ),\quad &{} \text{ for } i>j,\\ \\ o\bigr (\frac{1}{|\log \lambda |}\bigr ), &{} \text{ for } i<j, \end{array}\right. } \end{aligned}\nonumber \\ \end{aligned}$$

(4.42)

Using Lemma 4.4 , (4.42) and (4.37),

$$\begin{aligned} \begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^WPZ_i^0dx}{\int _{\Omega }e^Wdx}&=\sum _{j\ even}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}PZ_i^0dx+ \rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}PZ_i^0dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\\&=\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}PZ_i^0 dx+\sum _{j\ne i \ even }\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}PZ_i^0dx+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\\&=\int _{\Omega }|x|^{\alpha _i-2}e^{w_i} dx+\int _{\Omega }|x|^{\alpha _i-2}e^{w_i} Z_i^0 dx\\&\quad +\sum _{j\ne i \ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j} PZ_i^0 dx +o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\\&(\text{ using } \, (4.21))\\&=4\pi \alpha _i+\sum _{j<i\ even}8\pi \alpha _j+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ). \end{aligned}\nonumber \\ \end{aligned}$$

(4.43)

Moreover,

$$\begin{aligned} \begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^W\phi dx}{\int _{\Omega }e^Wdx}&=\sum _{i \ even}\int _{\Omega } |x|^{\alpha _i-2}e^{w_i}\phi dx +\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ) \end{aligned}\nonumber \\ \end{aligned}$$

(4.44)

and again by Lemma 4.4 and (4.41)

$$\begin{aligned} \begin{aligned} \lambda \int _{\Omega }e^{-W}\phi PZ_i^0dx&=\sum _{j\ odd}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}\phi PZ_i^0dx+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\\&=\sum _{j<i\ odd}\frac{2\sigma _j(\lambda )}{|\log \lambda |}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ). \end{aligned} \end{aligned}$$

(4.45)

Finally, for the last term,

$$\begin{aligned} \int _{\Omega }\nabla h\cdot \nabla PZ_i^0dx=O(\Vert h\Vert \Vert PZ_i^0\Vert )=o\Bigr (\frac{1}{|\log \lambda |}\Bigr ). \end{aligned}$$

(4.46)

Combining (4.38), (4.41), (4.39), (4.40), (4.43), (4.44), (4.45) and (4.46), we deduce that for i even,

$$\begin{aligned} \begin{aligned}&\frac{4\pi (\alpha _i+\sum _{j<i\ even}2\alpha _j)}{\rho ^+}\Big (\sum _{j\ even}\frac{\sigma _j(\lambda )}{|\log \lambda |}+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\Big )\\&\quad -\frac{1}{|\log \lambda |}(\sigma _i(\lambda )+\sum _{j<i}2\sigma _j(\lambda )) =o\Bigr (\frac{1}{|\log \lambda |}\Bigr ). \end{aligned} \end{aligned}$$

(4.47)

Next we consider (4.38) for i odd. In this case, again we estimate (4.38) term by term. Similarly to the estimate for i even, first by Lemma 4.4, (4.37) and (4.41), one has

$$\begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^W\phi PZ_i^0dx}{\int _{\Omega }e^Wdx}= & {} \sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi PZ_i^0dx\nonumber \\&+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi PZ_i^0dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\nonumber \\= & {} \sum _{j<i\ even }\frac{2\sigma _j(\lambda )}{|\log \lambda |}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ), \nonumber \\ \rho ^+\dfrac{\int _{\Omega }e^W PZ_i^0dx}{\int _{\Omega }e^Wdx}= & {} \sum _{j\ even}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j} PZ_i^0dx\nonumber \\&+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)} PZ_i^0dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\nonumber \\= & {} \sum _{j<i\ even }8\pi \alpha _j+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ), \end{aligned}$$

(4.48)

$$\begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^W \phi dx}{\int _\Omega e^W dx}= & {} \sum _{j\ even}\int |x|^{\alpha _j-2}e^{w_j} \phi dx\nonumber \\&+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\nonumber \\= & {} \sum _{j\ even}\frac{\sigma _j(\lambda )}{|\log \lambda |}+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ),\nonumber \\ \end{aligned}$$

(4.49)

and

$$\begin{aligned} \begin{aligned} \lambda \int _{\Omega }e^{-W}\phi PZ_i^0dx&=\sum _{j\ odd}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi PZ_i^0dx+o\Bigr (\frac{1}{|\log \lambda |}\Bigr )\\&=\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi Z_i^0dx+\frac{\sigma _i(\lambda )}{|\log \lambda |}+\sum _{j<i\ odd}\frac{2\sigma _j(\lambda )}{|\log \lambda |}+o\Bigr (\frac{1}{|\log \lambda |}\Bigr ). \end{aligned} \end{aligned}$$

Combining all these terms, one can get that for i odd,

$$\begin{aligned} \begin{aligned}&\frac{8\pi \sum _{j<i\ even}2\alpha _j}{\rho ^+}\Big (\sum _{j\ even}\frac{\sigma _j(\lambda )}{|\log \lambda |}+\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\Big )\\&\quad -\frac{1}{|\log \lambda |}(\sigma _i(\lambda )+\sum _{j<i}2\sigma _j(\lambda )) =o\Bigr (\frac{1}{|\log \lambda |}\Bigr ). \end{aligned} \end{aligned}$$

(4.50)

By considering the difference of (4.47) and (4.50), one has the following:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{4\pi \alpha _{i+1}}{\rho ^+}\Big (\sum \limits _{j\ even}\sigma _j(\lambda )+|\log \lambda |\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\Big ) -\sigma _{i+1}-\sigma _i=o(1)\quad ~\text { for i odd},\\ \\ \frac{4\pi \alpha _{i}}{\rho ^+}\Big (\sum \limits _{j\ even}\sigma _j(\lambda )+|\log \lambda |\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\Big ) -\sigma _{i+1}-\sigma _i=o(1)\quad ~\text { for i even}. \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.51)

From (4.50), we first have $\sigma _1(\lambda )=o(1)$. From (4.51), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \sigma _i(\lambda )=o(1)&{}\text { for i odd}\\ \\ \sigma _i(\lambda )-\frac{4\pi \alpha _i}{\rho ^+}\Big (\sum _{j\ even}\sigma _j(\lambda )+|\log \lambda |\rho _0\dfrac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}\Big )=o(1)~&{}\text { for i even}. \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.52)

Step 4. We claim that $\gamma _i=0$ for $i=1,\cdots ,k$.

When i is even, multiplying equation (4.20) by $Pw_i$ and integrating over $\Omega$,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\nabla \phi \cdot \nabla Pw_idx-\rho ^+\Big (\dfrac{\int _{\Omega }e^W\phi Pw_idx}{\int _{\Omega }e^Wdx}-\dfrac{\int _{\Omega }e^W\phi dx\int _{\Omega }e^WPw_idx}{(\int _{\Omega }e^Wdx)^2} \Big )\\&\quad -\lambda \int _{\Omega }e^{-W}\phi Pw_idx=\int _{\Omega }\nabla h\cdot \nabla Pw_idx. \end{aligned} \end{aligned}$$

(4.53)

Now we estimate the above equation term by term. For the first term, we have

$$\begin{aligned} \int _{\Omega } \nabla \phi \cdot \nabla Pw_idx=\int _{\Omega } |x|^{\alpha _i-2}e^{w_i}\phi dx=\int _{{{\mathbb {R}}}^2}|y|^{\alpha _i-2}e^{w_i(\delta _i y)}{\tilde{\phi }}_idy=o(1) \end{aligned}$$

(4.54)

by (4.31) and (4.21).

To estimate the second term, by Lemma 4.4 and (4.24), we have

$$\begin{aligned} \begin{aligned} \rho ^+\frac{\int _{\Omega }e^W \phi Pw_i dx}{\int _{\Omega }e^Wdx}&=\sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j} \phi Pw_idx +\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi Pw_i}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Big (\frac{1}{|\log \lambda |}\Big )\\&=\sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j} \phi Pw_idx+o(1). \end{aligned}\nonumber \\ \end{aligned}$$

(4.55)

By (4.10) and (4.2), we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi Pw_idx =\int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_jPw_i(\delta _j y)dy\\&\\&\quad ={\left\{ \begin{array}{ll} \int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j(-2\alpha _i\log \delta _i+h_i(0))dy\\ +O\Big (\int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j (|y|^{\alpha _j}(\frac{\delta _j}{\delta _i})^{\alpha _i}+\delta _j|y|+\delta _i^{\alpha _i} )dy\Big )\quad &{} \text{ for } j<i\\ \\ \int _{{{\tilde{\Omega }}}_i}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i (-2\alpha _i\log \delta _i-2\log (1+|y|^{\alpha _i})+h_i(0))dy\\ +O\Big (\int _{{{\tilde{\Omega }}}_i}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i (\delta _i|y|+\delta _i^{\alpha _i})dy \Big ) &{} \text{ for } j=i\\ \\ \int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j (-2\alpha _i\log (\delta _j|y|)+h_i(0))dy\\ +O\Big (\int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j \Big (\frac{1}{|y|^{\alpha _i}}(\frac{\delta _i}{\delta _j})^{\alpha _i}+\delta _j|y|+\delta _i^{\alpha _i} \Big )dy \Big ) &{} \text{ for } j>i \end{array}\right. }\\&\\&\quad ={\left\{ \begin{array}{ll} \int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j[-2\alpha _i\log d_i-2(k-i+1)\log \lambda +h_i(0)]dy+o(1) &{} \text{ for } j<i\\ \\ \int _{{{\tilde{\Omega }}}_i}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i [-2\alpha _i\log d_i-2(k-i+1)\log \lambda -2\log (1+|y|^{\alpha _i})+h_i(0)]dy+o(1) &{} \text{ for } j=i\\ \\ \int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j [-2\alpha _i\log d_j-2(k-j+1)\frac{2i-1}{2j-1}\log \lambda -2\alpha _i\log |y|+h_i(0)]dy+o(1)~&{} \text{ for } j>i. \end{array}\right. } \end{aligned}\nonumber \\ \end{aligned}$$

(4.56)

Based on (4.56), by the definition of $\sigma _j(\lambda )$ (see (4.34)), Lemma 4.6 and (4.31), we get

$$\begin{aligned} \begin{aligned}&\int _{\Omega }|x|^{\alpha _j-2}e^{w_j}\phi Pw_idx\\&\\&={\left\{ \begin{array}{ll} 2(k-i+1)\sigma _j(\lambda )+o(1) &{} \text{ for } j<i\\ \\ 2(k-i+1)\sigma _i(\lambda )+\int _{{{\tilde{\Omega }}}_i}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i [-2\log (1+|y|^{\alpha _i})]dy+o(1) &{} \text{ for } j=i\\ \\ 2(k-j+1)\frac{2i-1}{2j-1}\sigma _j(\lambda )+\int _{{{\tilde{\Omega }}}_j}\frac{2\alpha _j^2|y|^{\alpha _j-2}}{(1+|y|^{\alpha _j})^2}{\tilde{\phi }}_j [-2\alpha _i\log |y|]dy+o(1)\quad &{} \text{ for } j>i \end{array}\right. }\\&\\&\quad ={\left\{ \begin{array}{ll} 2(k-i+1)\sigma _j(\lambda )+o(1) &{} \text{ for } j<i\\ \\ 2(k-i+1)\sigma _i(\lambda )+4\pi \alpha _i\gamma _i+o(1) &{} \text{ for } j=i\\ \\ 2(k-j+1)\frac{2i-1}{2j-1}\sigma _j(\lambda )+8\pi \alpha _i\gamma _j+o(1) &{} \text{ for } j>i, \end{array}\right. } \end{aligned}\nonumber \\ \end{aligned}$$

(4.57)

where we used [11, (4.18)-(4.20) ].

Then by (4.57) and (4.55), one has

$$\begin{aligned} \begin{aligned} \rho ^+\frac{\int _{\Omega }e^W \phi Pw_i dx}{\int _{\Omega }e^Wdx}&=\sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j} \phi Pw_idx +\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi Pw_i}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\Big (\frac{1}{|\log \lambda |}\Big )\\&=\sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j} \phi Pw_idx+o(1)\\&=4\pi \alpha _i\Bigr (\gamma _i+\sum _{j>i\ even}2\gamma _j\Bigr )+2(k-i+1)\Bigr (\sigma _i(\lambda )+\sum _{j<i\ even}\sigma _j(\lambda )\Bigr )\\&\quad +\sum _{j>i\ even}2(k-j+1)\frac{2i-1}{2j-1}\sigma _j(\lambda )+o(1). \end{aligned}\nonumber \\ \end{aligned}$$

(4.58)

Similarly, by replacing $\phi$ by 1 in (4.57), one can deduce that

$$\begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^WPw_idx}{\int _{\Omega }e^Wdx}= & {} 8\pi |\log \lambda |\left( \sum _{j\le i\ even}(k-i+1)\alpha _j\right. \nonumber \\&\left. +\sum _{j>i\ even}(k-j+1)\frac{2i-1}{2j-1}\alpha _j\right) +O(1), \end{aligned}$$

(4.59)

and the estimate for $\rho ^+\frac{\int _{\Omega }e^W\phi dx}{\int _{\Omega }e^W dx}$ has been obtained in (4.48).

Moreover,

$$\begin{aligned} \begin{aligned}&\lambda \int _{\Omega }e^{-W}\phi Pw_idx=\sum _{j\ odd}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}\phi Pw_idx+o(1)\\&\quad =8\pi \alpha _i\sum _{j>i\ odd}\gamma _j+\sum _{j<i\ odd}2(k-i+1)\sigma _j(\lambda )\\&\qquad +\sum _{j>i\ odd}2(k-j+1)\frac{2i-1}{2j-1}\sigma _j(\lambda )+o(1), \end{aligned} \end{aligned}$$

(4.60)

and

$$\begin{aligned} \int _{\Omega }\nabla h\cdot \nabla Pw_idx=O(\Vert h\Vert _p\Vert Pw_i\Vert )=O(\log \lambda )^{\frac{1}{2}}\Vert h\Vert =o(1). \end{aligned}$$

(4.61)

Putting all the estimates in (4.54), (4.58), (4.59), (4.48), (4.60) and (4.61) into (4.53), we get that for i even,

$$\begin{aligned} \begin{aligned}&4\pi \alpha _i(\gamma _i+\sum _{j>i}2\gamma _j)+\sum _{j\le i}2(k-i+1)\sigma _j+\sum _{j>i}2(k-j+1)\frac{2i-1}{2j-1}\sigma _j\\&\quad -\frac{8\pi }{\rho ^+}\sum _{j\le i\ even}(k-i+1)\alpha _j\Big (\sum _{l \ even}\sigma _l(\lambda )+|\log \lambda |\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx} \Big )\\&\quad -\frac{8\pi }{\rho ^+}\sum _{j>i \ even}(k-j+1)\frac{2i-1}{2j-1}\alpha _j \Big (\sum _{l \ even}\sigma _l(\lambda ) +|\log \lambda |\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx} \Big )=o(1). \end{aligned}\nonumber \\ \end{aligned}$$

(4.62)

Next we consider i odd. Similarly to the previous estimates, one has

$$\begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^W \phi Pw_i dx}{\int _{\Omega }e^Wdx}= & {} \sum _{j\ even}\int _{\Omega }|x|^{\alpha _j-2}e^{w_j} \phi Pw_idx +\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi Pw_i}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx}+o\left( \frac{1}{|\log \lambda |}\right) \nonumber \\= & {} 8\pi \alpha _i\sum _{j>i\ even}\gamma _j+\sum _{j<i\ even}2(k-i+1)\sigma _j(\lambda )\nonumber \\&+\sum _{j>i\ even}2(k-j+1)\frac{2i-1}{2j-1}\sigma _j(\lambda )+o(1) \end{aligned}$$

(4.63)

$$\begin{aligned} \rho ^+\dfrac{\int _{\Omega }e^WPw_idx}{\int _{\Omega } e^Wdx}= & {} 8\pi |\log \lambda |\Big (\sum _{j<i\ even}(k-i+1)\alpha _j \nonumber \\&+\sum _{j>i\ even}(k-j+1)\frac{2i-1}{2j-1}\alpha _j\Big )+O(1), \end{aligned}$$

(4.64)

and

$$\begin{aligned} \begin{aligned}&\lambda \int _{\Omega } e^{-W}\phi Pw_idx=\sum _{j\ odd}\int _{\Omega } |x|^{\alpha _j-2}e^{w_j}\phi Pw_idx+o(1)\\&\quad =4\pi \alpha _i\gamma _i+8\pi \alpha _i\sum _{j>i\ odd}\gamma _j +\sum _{j\le i\ odd}2(k-i+1)\sigma _j(\lambda )\\&\qquad +\sum _{j>i\ odd}2(k-j+1)\frac{2i-1}{2j-1}\sigma _j(\lambda )+o(1). \end{aligned} \end{aligned}$$

(4.65)

Putting all the estimates in (4.54), (4.63), (4.64), (4.48), (4.65) and (4.61) into (4.53), we have for i odd,

$$\begin{aligned} \begin{aligned}&4\pi \alpha _i(\gamma _i+\sum _{j>i}2\gamma _j)+\sum _{j\le i}2(k-i+1)\sigma _j+\sum _{j>i}2(k-j+1)\frac{2i-1}{2j-1}\sigma _j\\&\quad -\frac{8\pi }{\rho ^+}\sum _{j< i\ even}(k-i+1)\alpha _j\Big (\sum _{l \ even}\sigma _l(\lambda )+|\log \lambda |\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx} \Big )\\&\quad -\frac{8\pi }{\rho ^+}\sum _{j>i \ even}(k-j+1)\frac{2i-1}{2j-1}\alpha _j \Big (\sum _{l \ even}\sigma _l(\lambda ) +|\log \lambda |\rho _0\frac{\int _{\Omega }e^{z-8k\pi G(x,0)}\phi dx}{\int _{\Omega }e^{z-8k\pi G(x,0)}dx} \Big )=o(1). \end{aligned}\nonumber \\ \end{aligned}$$

(4.66)

Combining (4.52), (4.62) and (4.66), we have

$$\begin{aligned} 4\pi \alpha _i\Bigr (\gamma _i+\sum _{j>i}2\gamma _j\Bigr )=o(1), \end{aligned}$$

from which we deduce that $\gamma _i=0$ for $i=1,\cdots ,k$.

Step 5. Finally, we derive a contradiction.

Multiplying equation (4.20) by $\phi$ and integrating, we get

$$\begin{aligned} \begin{aligned} \int _{\Omega }|\nabla \phi |^2dx-\lambda \int _{\Omega }e^{-W}\phi ^2dx-\rho ^+\Big ( \dfrac{\int _{\Omega }e^W\phi ^2dx}{\int _{\Omega }e^Wdx}-\dfrac{(\int _{\Omega } e^W\phi dx)^2}{(\int _{\Omega } e^Wdx)^2}\Big ) =\int _{\Omega } \nabla h\cdot \nabla \phi dx. \end{aligned} \end{aligned}$$

From (4.24) and the assumptions on $\phi$ and h, we have that the left hand side of the above equation tends to 1 while the right hand side is of order o(1). This yields a contradiction. $\square$

Using the a priori estimates in Lemma 4.5 and the Fredholm alternative, we have the following existence result similarly to the proof of Proposition 3.5:

Proposition 4.7

There exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0,\lambda _0)$, $h\in {\mathcal {H}}_l$ there exists a unique solution $\phi \in {\mathcal {H}}_l$ solution of (4.20) satisfying

$$\begin{aligned} \Vert \phi \Vert \le C|\log \lambda | \Vert h\Vert . \end{aligned}$$

4.4 Conclusion

By exploiting the linear theory developed in the previous subsection it is then standard to derive an existence result for the nonlinear problem (4.67) based on the contraction mapping, similarly to Proposition 3.6. We here give the sketch proof.

Proposition 4.8

For any $\epsilon >0$ sufficiently small, there exist $\lambda _0>0$ and $C>0$ such that for any $\lambda \in (0, \lambda _0)$, there exists a unique $\phi \in {\mathcal {H}}_l$ solution of

$$\begin{aligned} \Delta (W+\phi )+\rho ^+\frac{e^{W+\phi }}{\int _{\Omega }e^{W+\phi }dx}-\lambda e^{-W-\phi }=0\quad \text{ in } \quad \Omega \end{aligned}$$

(4.67)

satisfying

$$\begin{aligned} \Vert \phi \Vert \le C\lambda ^{\frac{1}{2(2k-1)}-\epsilon }. \end{aligned}$$

(4.68)

Proof

The proof is similar to Proposition 3.6 in the last section. The nonlinear problem (4.67) is equivalent to

$$\begin{aligned} \Delta \phi +\rho ^+\left( \dfrac{e^W\phi }{\int _\Omega e^Wdx}-\dfrac{e^W\int _\Omega e^W\phi dx}{(\int _\Omega e^Wdx)^2}\right) +\lambda e^{-W}\phi =-({\bar{E}}+{\bar{N}}(\phi )) \end{aligned}$$

(4.69)

where

$$\begin{aligned} {\bar{E}}= & {} \Delta W+\rho ^+\frac{e^W}{\int _\Omega e^W dx}-\lambda e^{-W}, \\ {\bar{N}}(\phi )= & {} \rho ^+[g(W+\phi )-g(W)-g'(W)\phi ]-\lambda [f(W+\phi )-f(W)-f'(W)\phi ], \\ f(W)= & {} e^{-W} \text{ and } g(W)=\frac{e^W}{\int _\Omega e^W dx}. \end{aligned}$$

Denote the solution to (4.20) by $\phi :={\bar{T}}(h)$, then (4.69) is equivalent to

$$\begin{aligned} \phi ={\bar{T}}(i_p^*({\bar{E}}+{\bar{N}}(\phi )))=:\bar{{\mathcal {T}}}(\phi ). \end{aligned}$$

Define

$$\begin{aligned} \bar{{\mathcal {B}}}=\{\phi \in {\mathcal {H}}_l, \Vert \phi \Vert \le \Lambda |\log \lambda |\lambda ^{\frac{2-p}{2p(2k-1)}}\} \end{aligned}$$

for $\Lambda$ large and $\lambda$ small. If we choose p sufficiently close to 1, one can see that $\bar{{\mathcal {B}}}\subset \{\phi , \, \Vert \phi \Vert \le C\lambda ^{\frac{1}{2(2k-1)}-\epsilon } \}\}$.

From Proposition 4.7 and the error estimate for ${\bar{E}}$, for $\phi , \phi _1, \phi _2\in \bar{{\mathcal {B}}}$, similarly to the estimate in Proposition 5.4 in [9], one has

$$\begin{aligned} \begin{aligned} \Vert \bar{{\mathcal {T}}}\phi \Vert&\le C_p|\log \lambda |\Vert i_p^*({\bar{E}}+{\bar{N}}(\phi ))\Vert \\&\le C_p|\log \lambda |(\Vert {\bar{E}}\Vert _p+\Vert {\bar{N}}(\phi )\Vert _p\\&\le \Lambda |\log \lambda |\lambda ^{\frac{2-p}{2p(2k-1)}}, \\&\text{ and } \\ \Vert \bar{{\mathcal {T}}}(\phi _1)-\bar{{\mathcal {T}}}(\phi _2)\Vert&\le C_p\Vert {\bar{N}}(\phi _1)-{\bar{N}}(\phi _2)\Vert _p\\&\le \frac{1}{2}\Vert \phi _1-\phi _2\Vert . \end{aligned} \end{aligned}$$

So $\bar{{\mathcal {T}}}$ maps $\bar{{\mathcal {B}}}$ into itself and it is a contraction mapping, we can get that the solution $\phi$ is unique in $\bar{{\mathcal {B}}}$. The estimate for $\phi$ follows from the above estimate. $\square$

Proof of Theorem 1.2

By Proposition 4.8, $u_\lambda =W_{\lambda }+\phi _\lambda$ is a solution to the original problem (1.1) with $\rho ^+_\lambda =\rho ^+=4\pi k(k-1)+\rho _0$ and $\rho ^-_{\lambda }=\lambda \int _{\Omega }e^{-u_\lambda }dx$. Then by Lemma 4.4 and (4.68)

$$\begin{aligned} \begin{aligned} \rho ^-_\lambda&=\lambda \int _{\Omega }e^{-u_\lambda }dx=\lambda \int _{\Omega }e^{-W_\lambda }dx+o(1) =\sum _{i\ odd}\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}dx+o(1)\\&=\sum _{i\ odd}4\pi \alpha _i+o(1)=4\pi k(k+1)+o(1). \end{aligned} \end{aligned}$$

Moreover, from the definition of $u_\lambda$ and properties (3.37), (4.68) and (4.9), the second property in Theorem 1.2 can be derived easily. $\square$