Nonexistence of Extremals for a Trudinger–Moser Inequality on a Riemann Surface with Boundary

Abstract

It is known that the Adimurthi–Druet inequality admits extremal function, when the perturbation parameter $\alpha$ is small. However, the problem that bothered us is when extremal function of the Adimurthi–Druet inequality does not exist. Recently, Mancini–Thizy (J. Differential Equations) first solved this problem by the method of energy estimate. After that, Yang (Sci. China Math.) extended the result to a closed Riemann surface. In this paper, on a compact Riemann surface with boundary, we consider the nonexistence of extremal function for a Trudinger–Moser inequality with the Neumann boundary condition. Moreover, this result complements our work in (Math. Inequal. Appl.).

1 Introduction

Let $\Omega \subseteq {\mathbb {R}}^N (N\ge 2)$ be a smooth bounded domain, $C_0^{\infty }(\Omega )$ is the space of smooth functions with compact support in $\Omega$ and $W_0^{1,p}(\Omega )\ (p \ge 1)$ be the completion of $C_0^{\infty }(\Omega )$ under the Sobolev norm

$$\begin{aligned} \Vert u\Vert _{W_0^{1,p}(\Omega )}=\Vert \nabla _{{\mathbb {R}}^N}u\Vert _p, \end{aligned}$$

where $\nabla _{{\mathbb {R}}^N}$ is the gradient operator on ${\mathbb {R}}^N$ and $\Vert \cdot \Vert _p$ denotes the standard $L^p$-norm. As a limit case of the Sobolev embedding, the classical Trudinger–Moser inequality was proved by Yudovich [38], Pohožaev [28], Peetre [27], Trudinger [31] and Moser [25], namely

$$\begin{aligned} J_{\beta }(\Omega )=\sup _{u\in W_0^{1,N}(\Omega ),\Vert \nabla _{{\mathbb {R}}^N} u\Vert _N\le 1}\int _\Omega e^{\beta |u|^\frac{N}{N-1} }{\mathrm {d}}x<+\infty , \ \ \forall \ \beta \le \alpha _N, \end{aligned}$$

(1)

where $\alpha _N=N\omega _{N-1}^{1/(N-1)}$ and $\omega _{N-1}$ represents the area of the unit sphere in ${{\mathbb {R}}^N}$. When $\beta > \alpha _N$, all integrals in (1) are still finite, but the supremum is infinite. In this sense, $\alpha _N$ is the best constant for this inequality. It is interesting to know whether or not the supremum in (1) can be attained. In the case that $\Omega$ is a unit ball ${\mathbb {B}}\subset {\mathbb {R}}^N$, Carleson–Chang [4] first obtained an extremal function for the supremum $J_{\alpha _N}({\mathbb {B}})$. This result was extended by Struwe [29] to the case that $\Omega$ is close to a unit ball in the sense of measure, by Flucher [12] to a general domain $\Omega \subset {\mathbb {R}}^2$ and by Lin [17] to an arbitrary domain $\Omega \subset {\mathbb {R}}^N$.

Trudinger–Moser inequalities were studied on Riemannian manifolds by Aubin [3], Cherrier [5] and Fontana [13]. In particular, let $\Sigma$ be an N-dimensional closed Riemannian manifold, $W^ { 1 ,N } ( \Sigma )$ be the usual Sobolev space. Then there holds

$$\begin{aligned} \sup _{u \in W ^ { 1,N} ( \Sigma ), \int _\Sigma u {\mathrm {d}}v_g = 0, \Vert \nabla _g u \Vert _ N \le 1 } \int _\Sigma e^{\beta |u|^{\frac{N}{N-1}} }{\mathrm {d}}v_g<+\infty ,\ \ \forall \ \beta \le \alpha _N, \end{aligned}$$

(2)

where $\nabla _g$ stands for the gradient operator on $\Sigma$, ${\mathrm {d}}v_g$ stands for the volume element and $\alpha _N$ has the same definition in (1). Based on the works of Ding–Jost–Li–Wang [8] and Adimurthi–Struwe [2], Li [15, 16] proved the existence of extremal function for the supremum in (2) by the method of blow-up analysis. When $\Sigma$ is a two-dimensional compact Riemann surface with smooth boundary, Yang [33] obtained the same inequality as (2), namely

$$\begin{aligned} \sup _ {u \in W^{1,2}(\Sigma ), \int _\Sigma u {\mathrm {d}}v_g = 0, \Vert \nabla _g u \Vert _ { 2 } \le 1 }\int _ { \Sigma } e^{\ \beta u^ 2 } {\mathrm {d}}v_g<+\infty , \ \forall \ \beta \le 2\pi . \end{aligned}$$

(3)

This inequality is sharp in the sense that if $\beta > 2\pi$, all integrals in (3) are still finite, but the supremum is infinite. Furthermore, the supremum in (3) can be attained. Later various versions of (1) were considered by many authors. In this paper, we are interested in Adimurthi–Druet [1] inequality, which is an improvement of the standard Trudinger–Moser inequality (1) by adding a $L^2$-type perturbation, which is

$$\begin{aligned} \sup _ { u \in W _ { 0 } ^ { 1,2 } ( \Omega ) , \Vert \nabla _{{\mathbb {R}}^2} u \Vert _ { 2 } \le 1 } \int _ { \Omega } e^{4 \pi u ^ { 2 } \left( 1 + \alpha \Vert u \Vert _ { 2 } ^ { 2 } \right) } {\mathrm {d}}x<+\infty , \ \forall \ 0 \le \alpha < \lambda _ { 1 } ( \Omega ), \end{aligned}$$

(4)

where $\lambda _ { 1 } ( \Omega )$ is the first eigenvalue of the Laplacian with Dirichlet boundary condition in $\Omega .$ This inequality is sharp in the sense that if $\alpha \ge \lambda _ { 1 } ( \Omega )$, all integrals in (4) are still finite, but the supremum is infinite. Obviously, (4) is reduced to (1) when $\alpha =0$. Later Yang [34] obtained the same inequality as (4) on a closed Riemann surface, namely

$$\begin{aligned} \sup _ {u \in W^{1,2}(\Sigma ), \int _\Sigma u {\mathrm {d}}v_g = 0, \Vert \nabla _g u \Vert _ 2 \le 1 }\int _ { \Sigma } e^{4\pi u^ 2(1+\alpha \Vert u\Vert _2^2) } {\mathrm {d}}v_g<+\infty , \ \forall \ 0 \le \alpha < \lambda ^*_ { 1 } ( \Sigma ), \end{aligned}$$

(5)

where $\lambda ^*_ { 1 } ( \Sigma )$ is the first eigenvalue of the Laplace–Beltrami operator with respect to the metric g. When $\alpha \ge \lambda ^* _ { 1 } ( \Sigma )$, the above supremum is infinite. Furthermore, one important result he got was that extremal function of the supremum in (5) exist only for sufficiently small $\alpha > 0$.

Lu–Yang [18] considered the mean value zero case. They extended (3) and (4) to a version, which is

$$\begin{aligned} \sup _ {u \in W^{1,2}(\Omega ), \int _{\Omega } u {\mathrm {d}}x=0, \Vert \nabla _{{\mathbb {R}}^2} u \Vert _ { 2 } \le 1 } \int _ { \Omega } e^{ 2 \pi u ^ 2 \left( 1 + \alpha \Vert u \Vert _ 2 ^ 2 \right) } {\mathrm {d}}x< + \infty , \ \forall \ 0 \le \alpha < \lambda ^* _ 1 ( \Omega ), \end{aligned}$$

(6)

where $\lambda ^* _ 1( \Omega )$ denotes the first nonzero Neumann eigenvalue of the Laplacian operator. This inequality is sharp in the sense that all integrals in (6) are still finite when $\alpha \ge \lambda ^* _ 1 ( \Omega )$, but the supremum is infinite. Moreover, they also obtained that the supremum is attained only for sufficiently small $\alpha > 0$. After that, the author strengthened (6) on a compact Riemann surface with smooth boundary in [39]. Precisely, let $\Sigma$ be a compact Riemann surface with smooth boundary and

$$\begin{aligned} \lambda _1(\Sigma )= \inf _ { u \in W^{1,2}(\Sigma ) , \int _ { \Sigma } u {\mathrm {d}}v_g = 0 , u \not \equiv 0 } \frac{ \Vert \nabla _g u\Vert ^2_2}{\Vert u\Vert _2^2} \end{aligned}$$

(7)

denotes the first eigenvalue of the Laplace–Beltrami operator with respect to the zero mean value condition. Then there holds

$$\begin{aligned} \sup _{ u \in W^{1,2}(\Sigma ), \int _{\Sigma }u {\mathrm {d}}v_g = 0, \Vert \nabla _g u\Vert _2^2 \le 1 } \int _{\Sigma } e^{ 2\pi u^{2}\left( 1+\alpha \Vert u\Vert _2^2\right) }{\mathrm {d}}v_g<+\infty , \ \forall \ 0 \le \alpha <\lambda _1(\Sigma ), \end{aligned}$$

(8)

When $\alpha \ge \lambda _1(\Sigma )$, the above supremum is infinite. In addition, various extensions of the inequality (4) were obtained by Lu–Yang [19] to a version involving $L^{p}$-norms for any $p>1$ and by Zhu [40] to the result in high dimensions with the $L^{p}$-versions with $1<p \le N$. Later various stronger versions of (4) were obtained by Tintarev [30], de Souza–do Ó [6, 9] and Nguyen [26].

For a long time, we had a question about whether or not extremal function of Adimurthi–Druet’s inequality (4) exists for $\alpha$ sufficiently close to $\lambda _1(\Omega )$. Based on the technique of energy estimate introduced by Malchiodi–Martinazzi [20], Mancini–Martinazzi [21] once again proved that the supremum $J_{4\pi }({\mathbb {B}})$ admits extremal function, which was first obtained by Carleson–Chang [4]. Using this method of energy estimate, Mancini–Thizy [22] first proved that the supremum in (4) has no extremal function, when $\alpha$ is sufficiently close to $\lambda _1(\Omega )$. Recently, Yang [36] extended the result to a closed Riemann surface, namely the supremum in (5) has no extremal function when $\alpha$ is sufficiently close to $\lambda _1(\Omega )$. In addition, Wang [32] generalized Mancini–Thizy’s result to a version involving $L^{p}$-norms for any $p>1$.

In this paper, we extend the result of Yang [36] from a closed Riemann surface to a compact Riemann surface with smooth boundary. In other words, our aim is to prove the nonexistence of extremal function for (8) when $\alpha$ is sufficiently close to $\lambda _1(\Sigma )$. Our main result reads

Theorem 1

Let $\Sigma$ be a compact Riemann surface with smooth boundary $\partial \Sigma$ and $\lambda _1(\Sigma )$ be defined by (7). We denote

$$\begin{aligned} \mathcal { S } = \left\{ u \in W^{1,2}(\Sigma ) : \Vert \nabla _g u\Vert _2^2 \le 1 \ \, \mathrm {and}\, \int _{\Sigma }u {\mathrm {d}}v_g = 0 \right\} \end{aligned}$$

(9)

and

$$\begin{aligned} F_{\alpha }(u)= \int _{\Sigma } e^{ 2\pi u^2\left( 1+\alpha \Vert u\Vert _2^2\right) }{\mathrm {d}}v_g. \end{aligned}$$

(10)

Then there exists some $\alpha ^{*}\in \left( 0,\lambda _{1}(\Sigma )\right)$, such that for any $\alpha \in (\alpha ^{*}, \lambda _{1}(\Sigma ))$, the supremum $\sup _{u \in \mathcal {S}}F_{\alpha }(u)$ has no extremal function.

We should point out that the blow-up occurs on the boundary $\partial \Sigma$ in our case, which is more difficult to handle than the cases of Mancini–Thizy [22] and Yang [36]. To be able to deal with this situation, we use the key ingredient in the proof of our theorem is isothermal coordinate system on $\partial \Sigma$, which has just been provided by Yang–Zhou [37] via Riemann mapping theorems involving the boundary.

The paper is organized as follows. In Sect. 2, we will prove Theorem 1 by the method of reduction to absurdity. This proof relies on the key energy estimates of Proposition 1, whose proof is given in Sect. 3. Without loss of generality, we do not distinguish sequence and subsequence in the following, and we denote various constants by the same C.

2 Proof of Theorem 1

Suppose Theorem 1 does not hold. Then for any $\alpha \in \left( 0,\ \lambda _{1}(\Sigma )\right)$, the supremum $\sup _{u \in \mathcal {S}}F_{\alpha }(u)$ can be attained. That is to say, we can take a sequence of numbers $\{\alpha _k\}_{k=1}^{+\infty }$ increasingly tending to $\lambda _1(\Sigma )$. Moreover, for any fixed k, there exists a function $u_k\in \mathcal {S}$ such that

$$\begin{aligned} \sup _{u \in \mathcal {S}}F_{\alpha _k}(u)= F_{\alpha _k}(u_k), \end{aligned}$$

(11)

where $\mathcal {S}$ is defined by (9). In view of (8), we obtain

$$\begin{aligned} \lim _{k \rightarrow +\infty } \left( \sup _{u \in \mathcal {S}}F_{\alpha _k}(u)\right) =\sup _{u \in \mathcal {S}}F_{\lambda _1(\Sigma )}(u)=+\infty . \end{aligned}$$

(12)

It follows from Lemma 1 in [39] that

$$\begin{aligned} \Vert \nabla _g u_k\Vert _2^2=1. \end{aligned}$$

(13)

From a direct calculation, we derive that $u_k$ satisfies the Euler–Lagrange equation

$$\begin{aligned} \left\{ \begin{aligned}&\Delta _g u_k=\frac{1}{\lambda _k}{\beta _k}u_ke^{\sigma _ku_k^2}+\gamma _k u_k-\frac{\mu _k}{\lambda _k} \ \ \mathrm { in } \ \ \Sigma ,\\&\frac{\partial u_k}{\partial \mathbf {n}}=0 \ \ \mathrm { on } \ \ \partial \Sigma , \end{aligned} \right. \end{aligned}$$

(14)

where $\mathbf {n}$ denotes the outward normal vector on $\partial \Sigma$,

$$\begin{aligned} \left\{ \begin{aligned}&\sigma _k= 2\pi \left( 1+\alpha _k\Vert u_k\Vert _2^2 \right) ,\ \beta _k=\frac{1+\alpha _k \Vert u_k\Vert _2^2}{1+2\alpha _k \Vert u_k\Vert _2^2},\ \gamma _k=\frac{\alpha _k}{1+2\alpha _k \Vert u_k\Vert _2^2},\\&\lambda _k=\int _{\Sigma }{u}_k^2e^{\sigma _ku_k^2 }{\mathrm {d}}v_g,\ \mu _k=\frac{\beta _k}{|\Sigma |}\int _{\Sigma }{u}_ke^{\sigma _ku_k^2}{\mathrm {d}}v_g, \end{aligned} \right. \end{aligned}$$

(15)

and $|\Sigma |$ is the area of $\Sigma$. Denote $c_k=|u_k\left( x_k\right) |=\max _{\overline{\Sigma }}|u_k|$. Without loss of generality, we assume

$$\begin{aligned} c_k=u_k\left( x_k\right) \rightarrow +\infty \ \ \mathrm {as}\ \ k\rightarrow +\infty \end{aligned}$$

(16)

and $x_k \rightarrow x_{0}$ as $k \rightarrow +\infty$. Applying maximum principle to (14), we have $x_0\in \partial \Sigma .$ Using the fact of $1+t e^{t }\ge e^{t }$ for any $t \ge 0$, we get

$$\begin{aligned} \lambda _k \ge \frac{1}{\sigma _k}\int _{\Sigma }\left( e^{\sigma _k u_k^2 }-1\right) {\mathrm {d}}v_g, \end{aligned}$$

which along with (11) and (12) leads to

$$\begin{aligned} \lambda _k \rightarrow +\infty \ \ \mathrm { as } \ \ k \rightarrow +\infty . \end{aligned}$$

(17)

Following Lemmas 3.2 and 3.4 in [36], we obtain

$$\begin{aligned}&\displaystyle \frac{c_k^2}{\lambda _k}=o_k(1), \end{aligned}$$

(18)

$$\begin{aligned}&\displaystyle \lim _{k \rightarrow +\infty }c_k\left\| u_k\right\| _{2q} =+ \infty ,\ \ \forall \ q \ge 1,\ \end{aligned}$$

(19)

$$\begin{aligned}&\displaystyle \int _{\Sigma } \frac{1}{\lambda _k} c_k {\beta _k} u_k e^{\sigma _k u_k^2} {\mathrm {d}}v_g=1+o_k(1), \end{aligned}$$

(20)

where $o_k(1)\rightarrow 0$ as $k \rightarrow +\infty$. Furthermore, using the same argument as the proof of Lemma 2 in [39], we get

$$\begin{aligned} \left\{ \begin{aligned}&u_k\rightharpoonup 0\, \, \mathrm {weakly}\, \, \mathrm {in}\, \, W^{1,2}(\Sigma ),\\&u_k\rightarrow 0\, \, \mathrm {strongly}\, \, \mathrm {in}\, \, L^p(\Sigma ) , \forall p>1,\\&u_k \rightarrow 0\, \, \mathrm {in} \, \, C_{loc}^{1}\left( \Sigma \backslash \left\{ x_{0}\right\} \right) . \end{aligned} \right. \end{aligned}$$

(21)

Moreover, there hold $\sigma _k \rightarrow 2 \pi$, $\beta _k \rightarrow 1$ and $\gamma _k \rightarrow \lambda _{1}(\Sigma )$ as $k \rightarrow +\infty$. To proceed, we state the key energy estimates.

Proposition 1

Let a sequence of function $\{u_k\}^{+\infty }_{k=1}$ satisfy (11), (13) and (14). Then there holds

$$\begin{aligned} \int _{\Sigma }\left| \nabla _g u_k\right| ^2 {\mathrm {d}}v_g=\frac{2\pi }{\sigma _k}+\gamma _k\left\| u_k\right\| _2^2 +o\left( \left\| u_k\right\| _2^{4}\right) , \end{aligned}$$

where ${\sigma _k}$ and $\gamma _k$ are given by (14).

Proof

For a detailed proof of this proposition, please refer to Section 3. $\square$

Applying Proposition 1, we have

$$\begin{aligned}&\int _{\Sigma }{\left| \nabla _gu_k \right| }^2{\mathrm {d}}v_g =\frac{1}{1+\alpha _k||u_k||_2^2}+\frac{\alpha _k||u_k||_2^2}{1+2\alpha _k||u_k||_2^2}+o\left( ||u_k||_2^{4} \right) \\&\quad =\left( 1-\alpha _k||u_k||_2^2+\alpha _k^2||u_k||_2^{4} \right) +\alpha _k||u_k||_2^2\left( 1-2\alpha _k||u_k||_2^2 \right) +o\left( ||u_k||_2^{4} \right) \\&\quad = 1-\alpha _k^2||u_k||_2^{4}+o\left( ||u_k||_2^{4} \right) , \end{aligned}$$

which together with (13) and (21) leads to $\alpha _k \rightarrow 0$ as $k \rightarrow +\infty$. This result contradicts with $\lim _{k \rightarrow +\infty }\alpha _k = \lambda _{1}(\Sigma )>0$ and ends the proof of Theorem 1.

3 Energy Estimates

In this section, we will prove Proposition 1 by analyzing the asymptotic behavior of $u_k$. We first choose a sequence of isothermal coordinate systems near blow-up point $x_0\in \partial \Sigma$. Define some notations for simplicity: $\mathbb { B }_{x}\left( r\right)$ denotes a ball centered at x with radius $r>0$, $\mathbb { B }_0^+\left( r\right) =\left\{ (x_1, x_2)\in \mathbb { B }_0\left( r\right) : x_2>0 \right\}$, ${\mathbb {R}}^2_+=\{(x_1,x_2)\in \mathbb { R } ^ 2:x_2>0\}$, $\nabla _{{\mathbb {R}}^2}$ and $\Delta _{{\mathbb {R}}^2}$, respectively, denote the standard gradient operator and the Laplacian operator on ${\mathbb {R}}^2$.

Lemma 1

For sufficiently large k and some fixed $\delta >0$, we can take an isothermal coordinate system $\left( U_k, \phi _k ;\left\{ x_{1}, x_2\right\} \right)$ around $x_k$, such that $\phi _k\left( x_k\right) =0$, $\phi _k (U_k)=\mathbb { B }_0^+\left( \delta \right)$, $\phi _k \left( U_k\cap \partial \Sigma \right) = \partial \mathbb { R } ^ { 2 }_+\cap \mathbb { B }_0\left( \delta \right)$ and

$$\begin{aligned} g_k= g\circ \phi _k^{-1}= e^{2f_k } \left( {\mathrm {d}}x_1^2 +{\mathrm {d}}x_2^2 \right) , \end{aligned}$$

(22)

where $f_k$ is a smooth function satisfying $f_k(0)=0$. Moreover, there holds

$$\begin{aligned} \Delta _{g_k}=- e^{ -2f_k } \Delta _{{\mathbb {R}}^2}. \end{aligned}$$

(23)

Proof

Following Lemma 4 in [37], we can take an isothermal coordinate system $\left( U,\phi ;\left\{ y_1, y_2\right\} \right)$ near the blow-up point $x_0\in \partial \Sigma$, such that $\phi (x_0) = 0$, $\phi (U)=\mathbb { B }_0^+\left( 2\delta \right)$ and $\phi (U\cap \partial \Sigma )= \partial \mathbb { R } ^ 2_+\cap \mathbb { B }_0\left( 2\delta \right)$ for some $\delta >0$. In such coordinates, the metric g has the representation

$$\begin{aligned} g= e^{2f} \left( {\mathrm {d}}y_1^2 +{\mathrm {d}}y_2^2 \right) , \end{aligned}$$

(24)

where f is a smooth function with $f(0)=0$. Moreover, there holds

$$\begin{aligned} \Delta _{g}=- e^{ -2f } \Delta _{{\mathbb {R}}^2}, \end{aligned}$$

(25)

We now take an isothermal coordinate system $\left( U_k, \phi _k ;\left\{ x_{1}, x_2\right\} \right)$ around $x_k$ for any fixed k, such that $\phi _k\left( x_k\right) =0$, $\phi _k (U_k)=\mathbb { B }_0^+\left( \delta \right)$, $\phi _k (U_k\cap \partial \Sigma )= \partial \mathbb { R } ^ { 2 }_+\cap \mathbb { B }_0\left( \delta \right)$ and $x=\phi _k \circ \phi ^{-1}(y)= e^{f\left( \phi (x_k)\right) } \left( y-\phi (x_k)\right)$. Let $g_k=g\circ \phi _k^{-1}$ and $f_k(x)=f(y)-f\left( \phi (x_k)\right)$. In view of $x_k \rightarrow x_0\in \partial \Sigma$ as $k \rightarrow +\infty$, there holds $U_k\subset U$ for sufficiently large k. Thus we have the coordinates $\left( U_k, \phi ;\left\{ y_1, y_2\right\} \right)$.

It is easy to know that $f_k$ is a smooth function and $f_k(0)=0$. According to ${\mathrm {d}}y= e^{f\left( \phi (x_k)\right) } {\mathrm {d}}x$ and (24), we obtain

$$\begin{aligned} g_k(x) =g(y)= e^{2f(y) } \left( e^{-2f\left( \phi (x_k)\right) } \left( {\mathrm {d}}x_1^2+{\mathrm {d}}x_2^2\right) \right) = e^{2f_k(x) } \left( {\mathrm {d}}x_1^2+{\mathrm {d}}x_2^2\right) , \end{aligned}$$

which gives (22). Combining this with (25), one has (23) immediately. $\square$

3.1 Gradient Estimates

In this section, we show the gradient estimate of $u_k$, which is similar to [10, 14, 23, 24, 35]. We first deal with the asymptotic behavior of the maximizers through blow-up analysis. Denote

$$\begin{aligned} r_k = \sqrt{\frac{ \lambda _ k }{ \beta _ k c _ k ^ { 2 } e^{ \sigma _ k c _ k ^2 } }}, \end{aligned}$$

(26)

where $\sigma _k$, $\lambda _k$ and $\beta _k$ are defined by (15). It follows from Lemma 3 in [39] that for any fixed positive integer $t_0$,

$$\begin{aligned} \lim _{k\rightarrow +\infty }r_k=0 \ \ {\mathrm {and}} \ \ \lim _{k\rightarrow +\infty }r_k^2c_k^{t_0}= 0. \end{aligned}$$

(27)

Moreover, there holds

$$\begin{aligned} \lim _{k\rightarrow +\infty }r_k^2 e^{tc_k^2 } = 0,\ \ \forall \, t\in (0,\, 2 \pi ). \end{aligned}$$

(28)

Hereafter the isothermal coordinates $\left( U_k, \phi _k ;\left\{ x_{1}, x_2\right\} \right)$ are constructed as in Lemma 1. On ${\mathbb {B}}_0\left( 2\delta \right)$, we define

$$\begin{aligned} \tilde{u}_k(x)=\left\{ \begin{aligned}&u_k \circ \phi _k^{-1}\left( x_{1}, x_{2}\right) ,&x_{2} \ge 0, \\&u_k \circ \phi _k^{-1}\left( x_{1},-x_{2}\right) ,&x_{2}<0, \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} \tilde{f}_k(x)=\left\{ \begin{aligned}&f_k \left( x_{1}, x_{2}\right) ,&x_{2} \ge 0, \\&f_k \left( x_{1},-x_{2}\right) ,&x_{2}<0. \end{aligned}\right. \end{aligned}$$

Denote $\mathcal {D}_k=\left\{ y\in {\mathbb {R}}^2: r_k y\in \mathbb { B }_0\left( 2\delta \right) \right\}$. Then one has $\mathcal {D}_k\rightarrow {\mathbb {R}}^2$ as $k \rightarrow +\infty$. On $\mathcal {D}_k$, we define two blowing up functions $\psi _k(y)= {\tilde{u} _k\left( r_k y\right) }/{c_k}$ and $\varphi _k(y)=c_k\left( \tilde{u}_k\left( r_k y\right) -c_k\right) .$ It follows from Lemma 4 in [39] that

$$\begin{aligned}&\lim _{ k\rightarrow +\infty }\psi _k(y) = 1 \ \ \mathrm {in} \ \ C_{loc}^{1}( {\mathbb {R}}^2 ),\ \ \ \ \end{aligned}$$

(29)

$$\begin{aligned}&\lim _{ k\rightarrow +\infty } \varphi _k(y)=\varphi (y) \ \ \mathrm {in} \ \ C_{loc}^{1}( {\mathbb {R}}^2 ), \end{aligned}$$

(30)

where

$$\begin{aligned} \varphi (y)=-\frac{1}{2 \pi } \log&\left( 1+\frac{\pi }{2}|y|^2\right) ,\end{aligned}$$

(31)

$$\begin{aligned}&\int _{{\mathbb {R}}^2_+} e^{4\pi \varphi (y) } {\mathrm {d}}y=1.\ \ \ \ \ \ \end{aligned}$$

(32)

We next prove a weak gradient estimate for $u_k$.

Lemma 2

For any $x \in \Sigma$ and fixed k, there holds

$$\begin{aligned} {\lambda _k}^{-1}{\beta _k} u_k^2(x) e^{ \sigma _k u_k^2(x) } \left( \mathrm {dist}_g\left( x, x_k\right) \right) ^2 \le C, \end{aligned}$$

where $\mathrm {dist}_g\left( \cdot ,\cdot \right)$ stands for the geodesic distance between two points and C is a constant depending only on $\Sigma$.

Proof

Suppose the lemma does not hold. Then we have

$$\begin{aligned} N_k=\max _{x \in \Sigma } {\lambda _k}^{-1}{\beta _k} u_k^2(x) e^{ \sigma _k u_k^2(x) } \left( \mathrm {dist}_g\left( x, x_k\right) \right) ^2\rightarrow +\infty \ \ \mathrm {as}\ \ k \rightarrow +\infty . \end{aligned}$$

(33)

Moreover, there exists a sequence of points $\left\{ x_{n_k}\right\} _{k=1}^{+\infty } \subset \Sigma$ such that

$$\begin{aligned} {\lambda _k}^{-1}{\beta _k} u_k^2\left( x_{n_k}\right) e^{\sigma _k u_k^2\left( x_{n_k}\right) } \left( \mathrm {dist}_g\left( x_{n_k}, x_k\right) \right) ^2=N_k. \end{aligned}$$

(34)

Denote

$$\begin{aligned} r_{n_k} = \sqrt{\frac{ \lambda _ k }{ \beta _ k u_k^2\left( x_{n_k}\right) e^{\sigma _k u_k^2\left( x_{n_k}\right) } }}. \end{aligned}$$

(35)

According to (17), (21), (33) and (34), one has

$$\begin{aligned} x_{n_k} \rightarrow x_0\in \partial \Sigma \ \ \mathrm { and }\ \ \left| u_k\left( x_{n_k}\right) \right| \rightarrow +\infty \ \mathrm { as } \ k \rightarrow +\infty . \end{aligned}$$

Using (15) and (35), we get for any $0<\xi <1$,

$$\begin{aligned} r_{n_k}^2 \le \frac{\int _{\Sigma } e^{(1-\xi )\sigma _ku_k^2 } {\mathrm {d}}v_g}{ \beta _ k e^{(1-\xi )\sigma _k u_k^2\left( x_{n_k}\right) } } \le \frac{C}{ \beta _ k e^{(1-\xi )\sigma _k u_k^2\left( x_{n_k}\right) } }, \end{aligned}$$

that is to say

$$\begin{aligned} \lim _{k\rightarrow +\infty }r_{n_k} = 0. \end{aligned}$$

(36)

In view of (16) and (27), we obtain $r_{n_k} \ge r_k$. It follows from (33), (34) and (35) that

$$\begin{aligned} \frac{\mathrm {dist}_g\left( x_{n_k}, x_k\right) }{r_{n_k}} \rightarrow +\infty \ \ \mathrm {as} \ \ k \rightarrow +\infty . \end{aligned}$$

(37)

Take an isothermal coordinate system $(U_{n_k}, \phi _{n_k} ;\{x_{1}, x_2\})$ near $x_{n_k},$ where $\phi _{n_k}(x_{n_k})=0$, $\phi _{n_k} (U_{n_k})=\mathbb { B }_0^+ (\delta )$ and $\phi _{n_k} \left( U_{n_k}\cap \partial \Sigma \right) = \partial \mathbb { R } ^ 2_+\cap \mathbb { B }_0\left( \delta \right)$. In such coordinate, it is clear that the metric $g_{n_k}= g\circ \phi _{n_k}^{-1} = e^{2f_{n_k} } \left( {\mathrm {d}}x_1^2 +{\mathrm {d}}x_2^2 \right)$, where $f_{n_k}$ is a smooth function with $f_{n_k} (0) = 0$. Moreover, we define on ${\mathbb {B}}_0\left( \delta \right)$,

$$\begin{aligned} \upsilon _{n_k}=\left\{ \begin{aligned}&u_k \circ \phi _{n_k}^{-1}\left( x_{1}, x_{2}\right) ,&x_{2} \ge 0, \\&u_k \circ \phi _{n_k}^{-1}\left( x_{1},-x_{2}\right) ,&x_{2}<0, \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} \tilde{f}_{n_k}(x)=\left\{ \begin{aligned}&f_{n_k} \left( x_{1}, x_{2}\right) ,&x_{2} \ge 0, \\&f_{n_k} \left( x_{1},-x_{2}\right) ,&x_{2}<0. \end{aligned}\right. \end{aligned}$$

Denote for $y \in \mathcal {D}_{n_k}=\left\{ y \in {\mathbb {R}}^2: r_{n_k} y \in {\mathbb {B}}_0\left( \delta \right) \right\} ,$

$$\begin{aligned} \psi _{n_k}(y)=\frac{\upsilon _{n_k}\left( r_{n_k} y\right) }{u_k\left( x_{n_k}\right) }. \end{aligned}$$

It follows from (14) and (23) that

$$\begin{aligned}&-\Delta _{{\mathbb {R}}^2} \psi _{n_k}(y)= e^{2\tilde{f}_{n_k}\left( r_{n_k} y\right) }\nonumber \\&\quad \left( \frac{\psi _{n_k}(y)}{u_k^2\left( x_{n_k}\right) } e^{\sigma _k\left( \upsilon _{n_k}^2\left( r_{n_k} y\right) -u_k^2\left( x_{n_k}\right) \right) } +r_{n_k}^2 \gamma _k \psi _{n_k}(y) -\frac{r_{n_k}^2\mu _k}{\lambda _ku_k\left( x_{n_k}\right) }\right) . \nonumber \\ \end{aligned}$$

(38)

From (37), there holds $r_{n_k}y\rightarrow 0$ as $k\rightarrow +\infty$ for any fixed $R>0$, which leads to for any $|y| \le R$, there hold

$$\begin{aligned} \left\{ \begin{aligned}&\left| \bar{x} _k\right| =\left( 1+o_k(1)\right) \left| r_{n_k} y-\phi _{n_k}(x_k)\right| ,\\&\mathrm {dist} _g\left( x_{n_k}, x_k\right) =\left( 1+o_k(1)\right) \left| \phi _{n_k}(x_k)\right| ,\\&\mathrm {dist}_g\left( \phi _k^{-1}\left( r_{n_k} y\right) ,x_k\right) =\left( 1+o_k(1)\right) \left| r_{n_k} y-\phi _{n_k}(x_k)\right| . \end{aligned} \right. \end{aligned}$$

According to this and (33), we get

$$\begin{aligned} {\lambda _k}^{-1}{\beta _k} u_k^2\left( \phi _{n_k}^{-1}\left( r_{n_k} y\right) \right) e^{\sigma _k u_k^2\left( \phi _{n_k}^{-1}\left( r_{n_k} y\right) \right) } \mathrm {dist}_g\left( \phi _k^{-1}\left( r_{n_k} y\right) , x_k\right) ^2 \le N_k,\nonumber \\ \end{aligned}$$

that is to say

$$\begin{aligned} \psi _{n_k}^2(y) e^{\sigma _ku_k^2\left( x_{n_k}\right) \left( \psi _{n_k}^2(y)-1\right) } \le 1+o_k(1). \end{aligned}$$

(39)

Then we have

$$\begin{aligned} \limsup _{k \rightarrow +\infty } \max _{|y| \le R}\left| \psi _{n_k}(y)\right| \le 1. \end{aligned}$$

Applying elliptic estimates to (38), we obtain

$$\begin{aligned} \lim _{ k\rightarrow +\infty }\psi _{n_k}(y)=1 \ \ \mathrm { in } \ \ C_{loc}^1\left( {\mathbb {R}}^2\right) . \end{aligned}$$

(40)

Denote

$$\begin{aligned} \varphi _{n_k}(y)=u_k\left( x_{n_k}\right) \left( \upsilon _{n_k}\left( r_{n_k} y\right) -u_k\left( x_{n_k}\right) \right) ,\ \ y \in \mathcal {D}_{n_k}. \end{aligned}$$

(41)

In view of (14) and (23), we have

$$\begin{aligned} -\Delta _{{\mathbb {R}}^2} \varphi _{n_k}(y)= & {} e^{2\tilde{f}_{n_k}\left( r_{n_k} y\right) } \left( \psi _{n_k}(y) e^{\sigma _k\left( \upsilon _{n_k}^2\left( r_{n_k} y\right) -u_k^2\left( x_{n_k}\right) \right) }\right. \nonumber \\&\left. +r_{n_k}^2 \gamma _k u_k^2\left( x_{n_k}\right) \psi _{n_k}(y)-r_{n_k}^2 {\lambda _k}^{-1}{\mu _k} u_k\left( x_{n_k}\right) \right) . \end{aligned}$$

(42)

It follows from (39) to (41) that for all $|y| \le R$,

$$\begin{aligned} \psi _{n_k}^2(y) e^{\sigma _k\left( \upsilon _{n_k}^{2}\left( r_{n_k}y \right) -u_{k}^{2}\left( x_{n_k} \right) \right) }= & {} \psi _{n_k}^2(y) e^{\sigma _k\varphi _{n_k}\left( y \right) \left( \psi _{n_k}\left( y \right) +1 \right) } \\= & {} (1+o_k(1)) e^{\varphi _{n_k}\left( y \right) \left( 4\pi +o_k\left( 1 \right) \right) } \\\le & {} 1+o_k\left( 1 \right) , \end{aligned}$$

which gives

$$\begin{aligned} \limsup _{k \rightarrow +\infty } \varphi _{n_k}(y) \le 0. \end{aligned}$$

(43)

From (40), there holds

$$\begin{aligned} r_{n_k}^2 u_k^2\left( x_{n_k}\right)= & {} \frac{2r_{n_k}^2}{\pi } \int _{{\mathbb {B}}^+_{0}(1)} u_k^2\left( x_{n_k}\right) {\mathrm {d}}y\\= & {} \frac{2+o_k(1)}{\pi } \int _{{\mathbb {B}}^+_0\left( {r_{n_k}}\right) } \upsilon _{n_k}^2(y) {\mathrm {d}}y\\\le & {} \frac{2+o_k(1)}{\pi } \int _{\Sigma } u_k^2 {\mathrm {d}}v_g. \end{aligned}$$

Combining this with (21), we have

$$\begin{aligned} \lim _{k \rightarrow +\infty }r_{n_k}^2 u_k^2\left( x_{n_k}\right) =0. \end{aligned}$$

(44)

Using (43), (44) and applying elliptic estimates to equation (42), we obtain

$$\begin{aligned}&\lim _{ k\rightarrow +\infty }\varphi _{n_k} \nonumber \\&\quad = \varphi \ \ \mathrm { in } \ \ C_{loc}^1\left( {\mathbb {R}}^2\right) , \end{aligned}$$

(45)

where $\varphi$ is defined by (31). In view of (31), (37) and (45), we have

$$\begin{aligned} 1+o_k(1)= & {} {\lambda _k}^{-1}{\beta _k} \int _{\Sigma } u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g \\\ge & {} {\lambda _k}^{-1}{\beta _k} \left( \int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( {R r_k}\right) \right) } u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g +\int _{\phi _{n_k}^{-1}\left( {\mathbb {B}}_0^+\left( R r_{n_k}\right) \right) } u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g\right) \\= & {} 2\left( 1+o_k(1)\right) \left( \int _{{\mathbb {B}}_0^+(R)} e^{4 \pi \varphi (y)} {\mathrm {d}}y+o_{R}(1)\right) . \end{aligned}$$

Letting $R\rightarrow +\infty$ and using (32), one has $1+o_k(1)\ge 4$, which is contradictory. Then the lemma follows. $\square$

We now prove a strong gradient estimate for $u_k$ as below.

Lemma 3

For any $x \in \Sigma$ and fixed k, there holds

$$\begin{aligned} \left| u_k(x) \Vert \nabla _g u_k(x)\right| \mathrm {dist}_g\left( x, x_k\right) \le C, \end{aligned}$$

where C is a constant depending only on $\Sigma$.

Proof

Suppose the lemma does not hold. Then we have

$$\begin{aligned} M_k=\max _{x \in \Sigma } \left| u_k(x) \Vert \nabla _g u_k(x)\right| \mathrm {dist}_g\left( x, x_k\right) \rightarrow +\infty \ \ \mathrm {as} \ \ k \rightarrow + \infty . \end{aligned}$$

(46)

Moreover, there exists a sequence of numbers $\left\{ x_{m_k}\right\} _{k=1}^{+\infty } \subset \Sigma$ such that

$$\begin{aligned} \left| u_k\left( x_{m_k}\right) \Vert \nabla _g u_k\left( x_{m_k}\right) \right| \mathrm {dist}_g\left( x_{m_k}, x_k\right) =M_k. \end{aligned}$$

(47)

In view of $u_k \rightarrow 0$ in $C_{loc}^1\left( \Sigma \backslash \left\{ x_{0}\right\} \right)$, we have $x_{m_k} \rightarrow x_{0}$ as $k \rightarrow + \infty$.

Take an isothermal coordinate system $\left( U_{m_k}, \phi _{m_k} ;\left\{ x_1, x_2\right\} \right)$ around $x_{m_k},$ where $\phi _{m_k}\left( x_{m_k}\right) =0$, $\phi _{m_k} (U_{m_k})=\mathbb { B }_0^+(\delta )$ and $\phi _{m_k} (U_{m_k}\cap \partial \Sigma )= \partial \mathbb { R } ^ { 2 }_+\cap \mathbb { B }_0(\delta )$. In such coordinate, the metric $g_{m_k}= g\circ \phi _{m_k}^{-1} = e^{2f_{m_k} } \left( {\mathrm {d}}x_1^2 +{\mathrm {d}}x_2^2 \right)$ and $f_{m_k}$ is a smooth function with $f_{m_k} (0) = 0$. Moreover, we define on ${\mathbb {B}}_0\left( \delta \right)$

$$\begin{aligned} \upsilon _{m_k}=\left\{ \begin{aligned}&u_k \circ \phi _{m_k}^{-1}\left( x_{1}, x_{2}\right) ,&x_{2} \ge 0, \\&u_k \circ \phi _{m_k}^{-1}\left( x_{1},-x_{2}\right) ,&x_{2}<0, \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} \tilde{f}_{m_k}(x)=\left\{ \begin{aligned}&f_{m_k} \left( x_{1}, x_{2}\right) ,&x_{2} \ge 0, \\&f_{m_k} \left( x_{1},-x_{2}\right) ,&x_{2}<0. \end{aligned}\right. \end{aligned}$$

In this coordinate system, we have

$$\begin{aligned} \left| y_{m_k}\right| \left| \upsilon _{m_k}(0) \Vert \nabla _{{\mathbb {R}}^2} \upsilon _{m_k}(0)\right| =\left( 1+o_k(1)\right) M_k, \end{aligned}$$

(48)

where $y_{m_k}=\phi _{m_k}(x_k)$. Denote $r_{m_k}=\left| y_{m_k}\right|$ and $y_k={y_{m_k}}/{r_{m_k}}$, then there hold

$$\begin{aligned} \lim _{ k \rightarrow +\infty }r_{m_k}=0 \ \ \ \mathrm { and }\ \, \lim _{ k \rightarrow +\infty }y_k=\bar{y} \end{aligned}$$

(49)

with $|\bar{y}|=1$. Define for $y \in \mathcal {D}_{m_k}=\left\{ y \in {\mathbb {R}}^2: r_{m_k} y \in {\mathbb {B}}_0\left( \delta \right) \right\}$,

$$\begin{aligned} \omega _{m_k}(y)=\upsilon _{m_k}\left( r_{m_k} y\right) . \end{aligned}$$

(50)

According to (14) and (23), we get

$$\begin{aligned} -\Delta _{{\mathbb {R}}^2} \omega _{m_k}(y) = e^{2\tilde{f}_{m_k}\left( r_{m_k} y\right) } r_{m_k}^2 \left( {\lambda _k}^{-1}{\beta _k} \omega _{m_k}(y) e^{\sigma _k \omega _{m_k}^2(y) } +\gamma _k \omega _{m_k}(y)-{\lambda _k}^{-1}{\mu _k}\right) . \end{aligned}$$

(51)

From Lemma 2, there exists a constant C depending only on $\Sigma$, such that for any $y \in \mathcal {D}_{m_k}$,

$$\begin{aligned} {\lambda _k}^{-1}{\beta _k} \upsilon _{m_k} ^2\left( r_{m_k} y\right) e^{\sigma _k \upsilon _{m_k}^2\left( r_{m_k} y\right) } r_{m_k}^2 \left| y-y_k\right| ^2\le C. \end{aligned}$$

(52)

In view of (49), for any fixed $R>0$, we have $\left| y-y_k\right| \ge {1}/{R}$ if $y \in \mathcal {D}_{m_k} \backslash {\mathbb {B}}_{\bar{y}}\left( {1}/{R}\right)$. Combining this, (50) with (52), we obtain for any $y \in \mathcal {D}_{m_k} \backslash {\mathbb {B}}_{\bar{y}}\left( {1}/{R}\right)$,

$$\begin{aligned} {\lambda _k}^{-1}{\beta _k} \omega _{m_k}^2(y) e^{\sigma _k \omega _{m_k}^2(y) } r_{m_k}^2 \frac{1}{R^2} \le C, \end{aligned}$$

which gives

$$\begin{aligned} {\lambda _k}^{-1}{\beta _k} \omega _{m_k}(y) e^{\sigma _k \omega _{m_k}^2(y) } r_{m_k}^2 \le C. \end{aligned}$$

(53)

Using a change of variables, we have for any $p>1$,

$$\begin{aligned} \int _{{\mathbb {B}}_0(R)}\left| r_{m_k}^2 \omega _{m_k}(y)\right| ^{p} {\mathrm {d}}y {=} \int _{{\mathbb {B}}_0\left( Rr_{m_k}\right) }\left| r_{m_k}^2 \upsilon _{m_k}\left( t\right) \right| ^{p}r_{m_k}^{-2} {\mathrm {d}}t {\le } \left( 1+o_k(1)\right) r_{m_k}^{2 p-2} \int _{\Sigma }\left| u_k\right| ^{p} {\mathrm {d}}v_g, \end{aligned}$$

which leads to

$$\begin{aligned} \lim _{ k \rightarrow +\infty }\int _{{\mathbb {B}}_0(R)}\left| r_{m_k}^2 \omega _{m_k}(y)\right| ^{p} {\mathrm {d}}y=0. \end{aligned}$$

(54)

In view of $M_k=\max _{x \in \Sigma } \mathrm {dist}_g\left( x, x_k\right) |u_k(x) | \left| \nabla _g u_k(x)\right|$, one has for any $y \in \mathcal {D}_{m_k} \backslash {\mathbb {B}}_{\bar{y}}\left( {1}/{R}\right)$,

$$\begin{aligned} \left| \nabla _{{\mathbb {R}}^2} \upsilon _{m_k}(r_{m_k} y)\right| \le (1+o_k(1))R M_k. \end{aligned}$$

According to this, (49) and (50), we obtain

$$\begin{aligned} \left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}^2(y)\right| \le C M_k, \end{aligned}$$

(55)

where C is a constant depending on R and k. It is easy to know that

$$\begin{aligned} \left| \omega _{m_k}(y)\right| \left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}(y)\right| = \left| y_{m_k}\right| \left| \upsilon _{m_k}(r_{m_k} y)\right| \left| \nabla _{{\mathbb {R}}^2} \upsilon _{m_k}(r_{m_k} y)\right| . \end{aligned}$$

Setting $y=0$ and using (48), we have

$$\begin{aligned} \left| \omega _{m_k}(0)\right| \left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}(0)\right| =\left( 1+o_k(1)\right) M_k. \end{aligned}$$

(56)

Let $S_k=\left| \omega _{m_k}(0)\right| +\left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}(0)\right|$. In view of (56), one has

$$\begin{aligned} M_k\le \left( 1+o_k(1)\right) S_k^2. \end{aligned}$$

(57)

From (46) and (47), there holds $S_k \rightarrow +\infty$ as $k \rightarrow +\infty .$ According to (55) and (57), we obtain for all $y \in {\mathbb {B}}_0(R) \backslash {\mathbb {B}}_{\bar{y}}\left( {1}/{R}\right)$,

$$\begin{aligned} \omega _{m_k}^2(y)\le \omega _{m_k}^2(0)+\frac{CM_k}{R} \le C S_k^2. \end{aligned}$$

(58)

Define $\psi _{m_k}(y)={\omega _{m_k}(y)}/{S_k}$. It follows from (58) that $\psi _{m_k}$ is bounded in $L^2\left( {\mathbb {R}}^2 \backslash \{\bar{y}\}\right)$. Combining (49)–(54) with $\left| {\lambda _k}^{-1}{\mu _k}\right| \le C$, we have $-\Delta _{{\mathbb {R}}^2} \omega _{m_k}(y)$ is bounded in $L^p\left( {\mathbb {R}}^2 \backslash \{\bar{y}\}\right)$. That is to say $-\Delta _{{\mathbb {R}}^2} \psi _{m_k}(y)$ is bounded in $L^p\left( {\mathbb {R}}^2 \backslash \{\bar{y}\}\right)$. Then by elliptic estimates, we obtain $\psi _{m_k} \rightarrow \psi$ as $k \rightarrow +\infty$ in $C_{loc}^{1}\left( {\mathbb {R}}^2 \backslash \{\bar{y}\}\right)$, where $\psi$ is a harmonic function in ${\mathbb {R}}^2 \backslash \{\bar{y}\} .$ For any fixed $R>0$, there holds

$$\begin{aligned} \int _{{\mathbb {B}}_0(R)}\left| \nabla _{{\mathbb {R}}^2} \psi _{m_k}(y)\right| ^2 {\mathrm {d}}y \le \frac{1}{S_k^2}\left( 1+o_k(1)\right) \int _{\Sigma }\left| \nabla _g u_k\right| ^2 {\mathrm {d}}v_g =o_k(1), \end{aligned}$$

(59)

which gives

$$\begin{aligned} \int _{{\mathbb {B}}_0(R) \backslash {\mathbb {B}}_{\bar{y}}\left( \frac{1}{R}\right) }\left| \nabla _{{\mathbb {R}}^2} \psi \right| ^2 {\mathrm {d}}y=\lim _{k \rightarrow +\infty } \int _{{\mathbb {B}}_0(R) \backslash {\mathbb {B}}_{\bar{y}}\left( \frac{1}{R}\right) }\left| \nabla _{{\mathbb {R}}^2} \psi _{m_k}\right| ^2 {\mathrm {d}}y=0. \end{aligned}$$

Then there holds $\lim _{k \rightarrow +\infty }\psi _{m_k}(y) = 1$ in $C_{loc}^{1}\left( {\mathbb {R}}^2 \backslash \{\bar{y}\}\right)$, that is to say $\omega _{m_k}(y) =\left( 1+o_k(1)\right) S_k.$ From (56), we have $\left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}(0)\right| >0$ for sufficiently large k. Then we can define for $y \in \mathcal {D}_{m_k}$,

$$\begin{aligned} \omega _{m_k}^{*}(y)=\frac{\omega _{m_k}(y)-\omega _{m_k}(0)}{\left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}(0)\right| }. \end{aligned}$$

(60)

It follows from DelaTorre–Mancini [7] that $\lim _{k \rightarrow +\infty }\omega _{m_k}^{*} = \omega ^{*}$ in $C_{loc}^{1}\left( {\mathbb {R}}^2 \backslash \{\bar{y}\}\right)$, where $\omega ^{*}$ is a harmonic function. Similar to (59), we obtain

$$\begin{aligned} \int _{{\mathbb {B}}_0(R) \backslash {\mathbb {B}}_{\bar{y}}\left( \frac{1}{R}\right) }\left| \nabla _{{\mathbb {R}}^2} \omega ^{*}(y)\right| ^2 {\mathrm {d}}y = \lim _{k \rightarrow + \infty } \int _{{\mathbb {B}}_0(R) \backslash {\mathbb {B}}_{\bar{y}}\left( \frac{1}{R}\right) }\left| \nabla _{{\mathbb {R}}^2} \omega _{m_k}^{*}(y)\right| ^2 {\mathrm {d}}y=0. \end{aligned}$$

Combining this with (60), we get $\omega ^{*} \equiv 0$ in ${\mathbb {R}}^2 \backslash \{\bar{y}\}$, which contradicts $\left| \nabla _{{\mathbb {R}}^2} \omega ^{*}(0)\right| =1$. Then we conclude the lemma. $\square$

3.2 Blow-up Analysis Near $x_0$

We now consider the local behavior of $u_k$ near $x_0$. In $(U_k, \phi _k ; \{x_1, x_2 \} )$, defined by Lemma 1, it follows from (30) and (31) that

$$\begin{aligned} \sigma _k\varphi _k\left( 2 \sigma _k^{-\frac{1}{2}} y\right) \rightarrow \varphi _{0}(y)\ \ \mathrm { in } \ \ C_{loc}^1\left( {\mathbb {R}}^2\right) , \end{aligned}$$

where

$$\begin{aligned} \varphi _{0}(y)=2\pi \varphi \left( \frac{2}{\sqrt{2\pi }}y\right) =-\log \left( 1+|y|^2\right) . \end{aligned}$$

(61)

In view of (14), (21)–(23) and (26)–(29), we obtain

$$\begin{aligned} -\Delta _{{\mathbb {R}}^2} \varphi _{0}=4 e^{2 \varphi _0 } \text{ in } {\mathbb {R}}^2. \end{aligned}$$

(62)

We define

$$\begin{aligned} \overline{\varphi }_k(y)=\varphi _{0}\left( \frac{\sqrt{\sigma _k}}{2 r_k} y\right) =-\log \left( 1+\frac{\sigma _k}{4 r_k^2}\left| y\right| ^2\right) , \ \ \forall \ y \in {\mathbb {R}}^2, \end{aligned}$$

(63)

which gives that

$$\begin{aligned} -\Delta _{{\mathbb {R}}^2}\varphi _k\left( y \right) =\frac{\sigma _k}{r_k^2} e^{2\varphi _k\left( y \right) } . \end{aligned}$$

(64)

For any fixed $\theta$ with $0<\theta <1$, we let

$$\begin{aligned} r_{\theta _k}= {2 r_k} \sqrt{ \frac{ e^{\theta \sigma _k c_k^2 } -1}{\sigma _k}}, \end{aligned}$$

(65)

which along with (16) and (28) leads to

$$\begin{aligned} r_{\theta _k} \rightarrow 0 \ \ \mathrm {as}\ \ k\rightarrow + \infty \end{aligned}$$

(66)

and

$$\begin{aligned} \frac{r_{\theta _k}}{r_k}\rightarrow +\infty \ \ \mathrm {as}\ \ k\rightarrow + \infty . \end{aligned}$$

(67)

It follows from (63) and (65) that

$$\begin{aligned} -\theta \sigma _k c_k^2=\overline{\varphi }_k\left( r_{\theta _k}\right) \le \overline{\varphi }_k(y) \le 0, \ \ \forall \ y \in {{\mathbb {B}}_0\left( r_{\theta _k}\right) }. \end{aligned}$$

(68)

Here and in the sequel, we slightly abuse a notation. If u is a function radially symmetric with respect to 0,  we write $u(|x|)=u(x)$. Let $\nu _k$ be the radially symmetric solution of

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta _{{\mathbb {R}}^2}\nu _k=\lambda _{k}^{-1}\beta _k\nu _k e^{\sigma _k\nu _k^2 } +\gamma _k\nu _k-\lambda _{k}^{-1}\mu _k, \\&\nu _k\left( 0 \right) =c_k, \end{aligned}\right. \end{aligned}$$

and $\eta _{0}$ be the radially symmetric solution of

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta _{{\mathbb {R}}^2} \eta _{0}=8 \eta _{0} e^{2 \varphi _0 } +4\left( \varphi _{0}^2+\varphi _{0}\right) e^{2 \varphi _0 } , \\&\eta _{0}(0)=0. \end{aligned}\right. \end{aligned}$$

It follows from Mancini–Martinazzi [21] that

$$\begin{aligned} \eta _{0}(y)=-\log |y|^2+O\left( \frac{\log |y|}{|y|^2}\right) \end{aligned}$$

(69)

and

$$\begin{aligned} -\int _{{\mathbb {R}}^2_+} \Delta _{{\mathbb {R}}^2} \eta _{0}\, {\mathrm {d}}y=2 \pi . \end{aligned}$$

(70)

Define

$$\begin{aligned} \eta _k(y)=\eta _{0}\left( \frac{\sqrt{\sigma _k}}{2 r_k} y\right) =-\log |y|^2-\log \frac{\sigma _k}{4 r_k^2} +O\left( \frac{\log |y|}{|y|^2}\right) , \ \ \forall \ y \in {\mathbb {R}}^2. \end{aligned}$$

(71)

According to (65), (69) and (71), we conclude $\eta _k(y)=O\left( c_k^2\right)$, for any $y \in {{\mathbb {B}}_0\left( r_{\theta _k}\right) }.$ Then using the same argument as the proof of Step 3.2 in [22], we can decompose $\nu _k$ as

$$\begin{aligned} \nu _k(|y|)=c_k+\frac{\overline{\varphi }_k(y)}{\sigma _kc_k}+\frac{\eta _k(y)}{\sigma _{k}^{2}c_{k}^{3}}+O\left( \frac{1-\overline{\varphi }_k(y)}{c_{k}^{5}} \right) , \ \ \forall \ y \in {{\mathbb {B}}_0\left( r_{\theta _k}\right) }, \end{aligned}$$

(72)

and

$$\begin{aligned} \nu _k^2(|y|)=c_k^2+\frac{2\overline{\varphi }_k(y)}{\sigma _k}+\frac{\overline{\varphi }_k^2(y)+2\eta _k(y)}{\sigma _k^2c_k^2}+O\left( \frac{1+\overline{\varphi }_k^2(y)}{c_k^4} \right) , \ \ \forall \ y \in {{\mathbb {B}}_0\left( r_{\theta _k}\right) }.\nonumber \\ \end{aligned}$$

(73)

Applying Proposition 3.1 in [11], we have $\left| \tilde{u}_k(y)-\nu _k(|y| )\right| \le C (c_k r_{\theta _k})^{-1}|y|$ for any $y \in {{\mathbb {B}}_0\left( r_{\theta _k}\right) }$. Then we decompose $\tilde{u}_k$ as the form

$$\begin{aligned} \tilde{u}_k\left( y \right) =\nu _k\left( |y| \right) +O\left( \frac{|y|}{c_kr_{\theta _k}} \right) . \end{aligned}$$

(74)

Applying (72) and (73), we conclude for any $y \in {{\mathbb {B}}_0\left( r_{\theta _k}\right) }$,

$$\begin{aligned} \lambda _k^{-1}{\beta _k}\nu _k e^{\sigma _k\nu _k^2 }=c_k\lambda _k^{-1}{\beta _k} e^{\sigma _kc_{k}^{2}+2\overline{\varphi }_k } \left( 1+\frac{\overline{\varphi }_{k}^{2}+\overline{\varphi }_k+2\eta _k}{\sigma _kc_{k}^{2}}+O\left( \frac{ e^{\overline{\varphi }_{k}^{2}/c_k^{2} } }{c_{k}^{4}} \right) \right) . \end{aligned}$$

(75)

In view of (68), we have on ${{\mathbb {B}}_0\left( r_{\theta _k}\right) }$,

$$\begin{aligned} \frac{\overline{\varphi }_k(y) }{ c_k^2} \le 0. \end{aligned}$$

(76)

From (26) and (74)–(76), there holds on ${{\mathbb {B}}_0\left( r_{\theta _k}\right) }$,

$$\begin{aligned} \lambda _k^{-1}{\beta _k} \tilde{u}_k e^{\sigma _k\tilde{u}_k^2 }= & {} \lambda _k^{-1}{\beta _k} \nu _k e^{\sigma _k\nu _k^2 } \left( 1+O\left( \frac{|y|}{r_{\theta _k}} \right) \right) \\= & {} c_k\lambda _k^{-1}{\beta _k} e^{\sigma _kc_k^2+2\overline{\varphi }_k } \left( 1+\frac{\overline{\varphi }_k^2+\overline{\varphi }_k+2\eta _k}{\sigma _kc_k^2}\right. \\&\left. +O\left( \frac{\left( 1+\overline{\varphi }_k^{4} \right) e^{{\overline{\varphi }_k^2}/{c_k^2} } }{c_k^{4}}+ \frac{\left( c_k^2+\overline{\varphi }_k^2\right) |y|}{c_k^2r_{\theta _k}} \right) \right) \\= & {} \frac{e^{2 \overline{\varphi }_k } }{r_k^2}\left( 1+\frac{\overline{\varphi }_k^2+2 \overline{\varphi }_k+2 \eta _k}{\sigma _k c_k^2}+O\left( \frac{\left( c_k^2+\overline{\varphi }_k^2\right) |y|}{c_k^2r_{\theta _k}} \right) \right) +O\left( \frac{e^{2 \overline{\varphi }_k } }{r_k^2 c_k^{4}}\right) , \end{aligned}$$

which leads to

$$\begin{aligned}&\int _{{\mathbb {B}}_0^+\left( r_{\theta _k} \right) }{\lambda _k^{-1}{\beta _k} }\tilde{u}_k^2 e^{\sigma _k\tilde{u}_k^2 } {\mathrm {d}}y =\int _{{\mathbb {B}}_0^+\left( r_{\theta _k} \right) }{\frac{ e^{2\overline{\varphi }_k} }{r_k^2}\left( 1+\frac{\overline{\varphi }_k^2+2\overline{\varphi }_k+2\eta _k}{\sigma _kc_k^2} \right) }{\mathrm {d}}y\nonumber \\&\quad +\int _{{\mathbb {B}}_0^+\left( r_{\theta _k} \right) }{\frac{e^{ 2\overline{\varphi }_k } }{r_k^2}}O\left( \frac{\left( c_k^2+\overline{\varphi }_k^2\right) |y|}{c_k^2r_{\theta _k}} \right) {\mathrm {d}}y +\int _{{\mathbb {B}}_0^+\left( r_{\theta _k} \right) }{O}\left( \frac{e^{2 \overline{\varphi }_k } }{r_k^2c_k^{4}} \right) {\mathrm {d}}y. \end{aligned}$$

(77)

To proceed, let us compute these three integrals separately. It first follows from (61) that $\int _{{\mathbb {R}}^2_+}e^{2\varphi _0 } \varphi _0 {\mathrm {d}}y =- {\pi }/{2},$ which along with (62)–(70) leads to

$$\begin{aligned}&\int _{{\mathbb {B}}_0^+\left( r_{\theta _k} \right) }{\frac{e^{2\overline{\varphi }_k } }{r_k^2}}\left( 1+\frac{\overline{\varphi }_k^2+2\overline{\varphi }_k+2\eta _k}{\sigma _kc_k^2} \right) {\mathrm {d}}y\nonumber \\&\quad =\frac{4}{\sigma _k}\int _{{\mathbb {B}}_0^+\left( \frac{\sqrt{\sigma _k}}{2r_k}r_{\theta _k} \right) }e^{2\varphi _0 } \left( 1+\frac{\varphi _{0}^2 +2\varphi _0 +2\eta _0}{\sigma _kc_k^2} \right) {\mathrm {d}}y \nonumber \\&\quad =\frac{4}{\sigma _k}\int _{{\mathbb {R}}^2_+}e^{2\varphi _0 } \left( 1+\frac{\varphi _{0}^2 +2\varphi _0 +2\eta _0}{\sigma _kc_k^2} \right) {\mathrm {d}}y+O\left( \frac{r^2_k}{r^2_{\theta _k}} \right) \nonumber \\&\quad =\frac{1}{\sigma _k^2c_k^2}\left( 4\int _{{\mathbb {R}}^2_+}e^{2\varphi _0 } \varphi _0 {\mathrm {d}}y- \int _{{\mathbb {R}}^2_+}\Delta _{{\mathbb {R}}^2}\eta _0{\mathrm {d}}y \right) +\frac{4}{\sigma _k}\int _{{\mathbb {R}}^2_+}e^{2\varphi _0 } {\mathrm {d}}y +O\left( \frac{r^2_k}{r^2_{\theta _k}} \right) \nonumber \\&\quad =\frac{2 \pi }{\sigma _k}+O\left( \frac{r^2_k}{r^2_{\theta _k}} \right) . \end{aligned}$$

(78)

Moreover, using a change of variables, we have

$$\begin{aligned}&\int _{{\mathbb {B}}_0^+\left( r_{\theta _k} \right) } {\frac{ e^{2\overline{\varphi }_k} }{r_k^2}}O\left( \frac{\left( c_k^2+\overline{\varphi }_k^2\right) |y|}{c_k^2r_{\theta _k}} \right) {\mathrm {d}}y \nonumber \\&\quad = \frac{4r_k}{\sigma _kr_{\theta _k}} \int _{{\mathbb {B}}_0^+\left( {\frac{\sqrt{\sigma _k}}{2r_k}r_{\theta _k}}\right) } e^{2\varphi _0 } O\left( \frac{\left( c_k^2+\varphi _{0}^2\right) |y|}{c_k^2} \right) {\mathrm {d}}y \nonumber \\&\quad =\frac{r_k}{r_{\theta _k}} \int _{{\mathbb {R}}^2_+}e^{2\varphi _0 } O\left( \frac{\left( c_k^2+\varphi _{0}^2\right) |y|}{c_k^2} \right) {\mathrm {d}}y+O\left( \frac{r_k^2}{r_{\theta _k}^2} \right) \nonumber \\&\quad =O\left( \frac{r_k}{r_{\theta _k}} \right) \end{aligned}$$

(79)

and

$$\begin{aligned} \int _{{\mathbb {B}}_0^+\left( r_{\theta _k}\right) } O\left( \frac{e^{2 \overline{\varphi }_k } }{r_k^2 c_k^{4}}\right) {\mathrm {d}}y=\frac{1}{c_k^{4}} \int _{{\mathbb {B}}_0^+\left( {\frac{\sqrt{\sigma _k}}{2r_k}r_{\theta _k}}\right) } O\left( e^{2 \varphi _{0} } \right) {\mathrm {d}}y= O\left( c_k^{-4} \right) . \end{aligned}$$

(80)

To sum up, it follows from (77)–(80) that

$$\begin{aligned} \int _{{\mathbb {B}}_0^+\left( r_{\theta _k}\right) }{\lambda _k^{-1}{\beta _k} }\tilde{u}_k^2e^{\sigma _k\tilde{u}_k^2 } {\mathrm {d}}y= & {} \frac{2\pi }{\sigma _k}+O\left( c_k^{-4} \right) . \end{aligned}$$

(81)

In view of (28) and (65), one has $r_{\theta _k}=O\left( c_k^{-4}\right)$. According to this and (81), we obtain

$$\begin{aligned} \int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) } \lambda _k^{-1}{\beta _k} u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g= & {} \int _{{\mathbb {B}}_0^+\left( r_{\theta _k}\right) } \lambda _k^{-1}{\beta _k} \tilde{u}_k^2 e^{\sigma _k \tilde{u}_k^2 } e^{2\tilde{f}_k } {\mathrm {d}}y \\= & {} \left( 1+O\left( r_{\theta _k}\right) \right) \left( \frac{2\pi }{\sigma _k}+O\left( c_k^{-4} \right) \right) \\= & {} \frac{2\pi }{\sigma _k}+O\left( c_k^{-4} \right) , \end{aligned}$$

which together with (19) leads to

$$\begin{aligned} \int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) } \lambda _k^{-1}{\beta _k} u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g=\frac{2\pi }{\sigma _k}+o\left( \left\| u_k\right\| _{2q}^{4}\right) . \end{aligned}$$

(82)

3.3 The Energy Estimate Away from ${x_0}$

To estimate the energy of $u_k$ away from $x_0$, we need the following lemma.

Lemma 4

Let $r_{s_k}$ be defined by

$$\begin{aligned} r_{s_k}=\sup \left\{ r: \tilde{u}_k(y) \ge \frac{1-\theta }{2} c_k, \ \ \forall \ r\ge |y|\right\} \end{aligned}$$

(83)

and $r_{\theta _k}$ be given as in (65). Then there holds $r_{s_k} \ge r_{\theta _k}.$

Proof

Suppose the lemma does not hold. Then we have for any k,

$$\begin{aligned} r_{s_k}<r_{\theta _k}. \end{aligned}$$

(84)

According to (26) and (31), we conclude

$$\begin{aligned} \frac{r_{s_k}}{r_k}\rightarrow +\infty \ \ \mathrm {as}\ \ k\rightarrow + \infty . \end{aligned}$$

(85)

In view of (14), we get

$$\begin{aligned} -\Delta _{{\mathbb {R}}^2} \tilde{u}_k(y)=e^{2\tilde{f}_k(y) } \left( \frac{\beta _k}{\lambda _k} \tilde{u}_k(y) e^{\sigma _k \tilde{u}_k^2(y) } +\gamma _k \tilde{u}_k(y)-\frac{\mu _k}{\lambda _k}\right) . \end{aligned}$$

(86)

Since (31), (83)-(86) and Lemma 3, we can use the argument of Proposition 3.1 in [11]. Then there holds

$$\begin{aligned} \left| \tilde{u}_k(y)-\nu _k(y)\right| \le C \frac{|y|}{c_k r_{s_k}}, \ \ \forall \ y \in {\mathbb {B}}_0\left( {r_{s_k}}\right) . \end{aligned}$$

(87)

It follows from (72) that $\nu _k (y)\ge \left( 1-\theta +o_k(1)\right) c_k$, for any $y \in {\mathbb {B}}_0\left( {r_{\theta _k}}\right) ,$ which together with (87) leads to

$$\begin{aligned} \tilde{u}_k(y) \ge \frac{2(1-\theta )}{3} c_k, \ \ \forall \ y \in {\mathbb {B}}_0\left( {r_{s_k}}\right) . \end{aligned}$$

This result contradicts with (83) and confirms the lemma. $\square$

Lemma 5

In the isothermal coordinate systems $\left( U_k, \phi _k ;\left\{ x_1, x_2\right\} \right)$ defined by Lemma 1, there exist some $q^{*}>1$ and a constant C such that

$$\begin{aligned} \int _{\Sigma \backslash \phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) } e^{q^{*} \sigma _k u_k^2 } {\mathrm {d}}v_g \le C, \end{aligned}$$

(88)

where $r_{\theta _k}$ is defined by (65).

Proof

For any fixed $0<\theta <1$, we denote

$$\begin{aligned} u_k^{*}(x)=\left\{ \begin{aligned}&u_k(x),&x \in \Sigma \backslash \phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) , \\&\min \left\{ u_k(x),\left( 1-\frac{\theta }{ 2}\right) c_k\right\} ,&x \in \phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) .\ \ \ \ \end{aligned}\right. \end{aligned}$$

(89)

It follows from Lemma 4 that $u_k^{*}(x)=\left( 1-{\theta }/{ 2}\right) c_k$, when $x \in \phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right)$. According to (21) and the fact of $0\le u_k^{*}\le u_k$, one has

$$\begin{aligned} \lim _{k\rightarrow +\infty }\bar{u}_k^{*}=0. \end{aligned}$$

(90)

We claim that

$$\begin{aligned} \lim _{k\rightarrow +\infty }\int _{\Sigma }\left| \nabla _g u_k^{*}\right| ^2 {\mathrm {d}}v_g=1-\frac{\theta }{2}. \end{aligned}$$

(91)

To confirm this claim, we set $\bar{u}_k^{*}=\int _{\Sigma } u_k^{*} {\mathrm {d}}v_g/|\Sigma |$. Then there holds $\int _{\Sigma }{\left( u_k^{*}-\bar{u}_k^{*} \right) }\,{\mathrm {d}}v_g=0$. Testing Eq. (14) by $u_k^{*}-\bar{u}_k^{*}$, we obtain

$$\begin{aligned} \int _{\Sigma }{\left| \nabla _gu_k^{*} \right| }^2{\mathrm {d}}v_g=\int _{\Sigma }{u_k^{*}\lambda _k^{-1}{\beta _k}u_k e^{\sigma _ku_k^2 } }{\mathrm {d}}v_g+o_k\left( 1 \right) . \end{aligned}$$

(92)

From (29), (30), (89), (90) and the fact of $r_{\theta _k}\ge r_k$, there holds

$$\begin{aligned} \int _{\Sigma }{u_k^{*}\lambda _k^{-1}{\beta _k} u_k e^{\sigma _ku_k^2 } }{\mathrm {d}}v_g= & {} \left( 1-\frac{\theta }{2} \right) \int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k} \right) \right) } \lambda _k^{-1}{\beta _k} c_ku_k e^{\sigma _ku_k^2 } {\mathrm {d}}v_g\!+\!\beta _k\!+\!o_k\left( 1 \right) \\\ge & {} \left( 1-\frac{\theta }{2}\right) \int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( R r_k\right) \right) }\lambda _k^{-1}{\beta _k} c_k u_k e^{\sigma _k u_k^2} {\mathrm {d}}v_g+o_k(1)\\= & {} \left( 1-\frac{\theta }{2}\right) \int _{{\mathbb {B}}_0^+(R)} e^{4 \pi \varphi (y)} {\mathrm {d}}y+o_k(1) \end{aligned}$$

for any fixed $R>0$. It follows from (32) and (92) that

$$\begin{aligned} \int _{\Sigma }{\left| \nabla _gu_k^{*} \right| }^2{\mathrm {d}}v_g \ge 1-\frac{\theta }{2}+o_k(1)+o_{R}(1), \end{aligned}$$

(93)

where $o_{R}(1) \rightarrow 0$ as $R \rightarrow +\infty$. We define $V_k=u_k-u_k^{*}$ and $\overline{V}_k=\int _{\Sigma }{V_k}{\mathrm {d}}v_g/|\Sigma |$. Then there hold $\int _ {\Sigma }{\left( V_k-\overline{V}_k \right) }\,{\mathrm {d}}v_g=0$ and

$$\begin{aligned} \int _{\Sigma }\left| \nabla _g V_k\right| ^2 {\mathrm {d}}v_g=1-\int _{\Sigma }\left| \nabla _g u_k^{*}\right| ^2 {\mathrm {d}}v_g+o_k(1). \end{aligned}$$

(94)

Testing equation (7) by $V_k-\overline{V}_k$, we obtain

$$\begin{aligned} \int _{\Sigma }\left| \nabla _g V_k\right| ^2 {\mathrm {d}}v_g=\int _{\Sigma }{V_k\lambda _k^{-1}{\beta _k} u_k {\sigma _ku_k^2 } }{\mathrm {d}}v_g+o_k\left( 1 \right) . \end{aligned}$$

(95)

From (29), (30), (32), (89) and the fact of $r_{\theta _k}\ge r_k$, there holds

$$\begin{aligned}&\int _{\Sigma }{V_k\lambda _k^{-1}{\beta _k} u_k e^{\sigma _ku_k^2 } }{\mathrm {d}}v_g\nonumber \\&\quad =\int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) }\lambda _k^{-1}{\beta _k} \left( u_k-\left( 1-\frac{\theta }{2} \right) c_k \right) u_k e^{\sigma _ku_k^2 } {\mathrm {d}}v_g+o_k\left( 1 \right) \nonumber \\&\quad \ge \frac{\theta }{2}\int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( R r_k\right) \right) }\lambda _k^{-1}{\beta _k} c_ku_k e^{\sigma _ku_k^2 } {\mathrm {d}}v_g+o_k\left( 1 \right) \nonumber \\&\quad = \frac{\theta }{2} \int _{{\mathbb {B}}_0^+(R)} e^{4 \pi \varphi (y)} {\mathrm {d}}y+o_k(1) \nonumber \\&\quad =\frac{\theta }{2}+o_k(1)+o_{R}(1), \end{aligned}$$

(96)

where $o_{R}(1) \rightarrow 0$ as $R \rightarrow +\infty .$ In view of (94)–(96), we have

$$\begin{aligned} \int _{\Sigma }{\left| \nabla _gu_k^{*} \right| }^2{\mathrm {d}}v_g \le 1-\frac{\theta }{2}+o_k(1)+o_{R}(1). \end{aligned}$$

(97)

To sum up, (91) is followed from (93) and (97).

For any fixed $0<\theta <1$, there holds $q_{1}$ satisfying $1<q_{1}<2 /(2-\theta )$. Combining (3) with (91), we have

$$\begin{aligned} \int _{\Sigma } e^{q_{1}\left( u_k^{*}-\bar{u}_k^{*}\right) ^2 } {\mathrm {d}}v_g \le C. \end{aligned}$$

It follows from (90) that

$$\begin{aligned} \int _{\Sigma } e^{q^{*} (u_k^*)^ 2 } {\mathrm {d}}v_g \le C, \end{aligned}$$

where $q^{*}$ satisfying $1<q^{*}<q_{1}$. According to the definition of $u_k^*$, we obtain (88) easily. $\square$

In view of (18), (19), Lemma 5 and Hölder’s inequality, we obtain that

$$\begin{aligned} \int _{\Sigma \backslash \phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) } \lambda _k^{-1}{\beta _k} u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g \le {C}\lambda _k^{-1}\left\| u_k\right\| _{2 q}^2 =C\lambda _k^{-1}{c_k^2} \frac{\left\| u_k\right\| _{2 q}^4}{c_k^2\left\| u_k\right\| _{2 q}^2} =o\left( \left\| u_k\right\| _{2 q}^{4}\right) \end{aligned}$$

(98)

for some $q>1$ with $1 / q+1 / q^{*}=1$.

3.4 Completion of the Proof of Proposition 1

We first give the following lemma.

Lemma 6

For any $q \ge 1$ and $1< p < 2$, we have $u_k /\left\| u_k\right\| _{2q}$ converges to $\omega _q$ weakly in $W^{1, p}(\Sigma )$ and strongly in $L^{r}(\Sigma )$, where $1<r<2 p /(2-p)$ and $\omega _q$ is a smooth solution of the equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta _g \omega =\lambda _{1}(\Sigma ) \omega , \\ \Vert \omega \Vert _{2q}=1. \end{array}\right. \end{aligned}$$

(99)

Moreover, there holds $\left\| u_k\right\| _{2q}^2=\left\| u_k\right\| _2^2 /\left( \left\| \omega _q\right\| _2^2+o_k(1)\right)$.

Proof

It follows from (14) that

$$\begin{aligned} \frac{\Delta _gu_k}{\left\| u_k\right\| _{2q}} =\frac{\Delta _g\left( c_k u_k\right) }{c_k\left\| u_k\right\| _{2q}} = \frac{{\lambda _k}^{-1}\beta _k c_k u_k e^{\sigma _k u_k^2} }{c_k\left\| u_k\right\| _{2q}} +\frac{\gamma _k c_ku_k}{c_k\left\| u_k\right\| _{2q}} -\frac{{\lambda _k}^{-1}{\mu _k c_k}}{c_k\left\| u_k\right\| _{2q}}. \end{aligned}$$

(100)

In view of (18) and (20), we conclude both ${\lambda _k}^{-1}{\mu _k c_k}$ and ${\lambda _k}^{-1}\beta _k c_k u_k e^{\sigma _k u_k^2}$ are bounded in $L^1(\Sigma )$. According to (19) and the Hölder inequality, we obtain $\int _{\Sigma } {|\Delta _gu_k|}/{\left\| u_k\right\| _{2q}}{\mathrm {d}}v_g\le C$ for any $q \ge 1$. Using Lemma 4.8 in [34], we have that $u_k /\left\| u_k\right\| _{2q}$ is bounded in $W^{1, p}(\Sigma )$ for any $1< p < 2$. Then there exists some function $\omega _q$ such that $u_k /\left\| u_k\right\| _{2q}$ converges to $\omega _q$ weakly in $W^{1, p}(\Sigma )$ and strongly in $L^{r}(\Sigma )$, where $1<r<2 p /(2-p)$. It follows from (100) that (99). Applying the regularity theory to (99), we have that $\omega _q$ is smooth.

In view of (99), one has $\left\| \omega _q\right\| _2>0$ for any $q\ge 1$. Then there holds

$$\begin{aligned} \int _{\Sigma } u_k^2 {\mathrm {d}}v_g=\left\| u_k\right\| _{2q}^2\int _{\Sigma } \frac{u_k^2}{\left\| u_k\right\| _{2q}^2} {\mathrm {d}}v_g=\left\| u_k\right\| _{2q}^2 \left( \left\| \omega _q\right\| _2^2+o_k(1)\right) , \end{aligned}$$

which leads to the lemma. $\square$

Testing Eq. (14) by $u_k\in \mathcal {S}$, we have

$$\begin{aligned} \int _{\Sigma }\left| \nabla _g u_k\right| ^2 {\mathrm {d}}v_g= & {} \int _{\phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) } \lambda _k^{-1}{\beta _k} u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g \\&+\int _{\Sigma \backslash \phi _k^{-1}\left( {\mathbb {B}}_0^+\left( r_{\theta _k}\right) \right) } \lambda _k^{-1}{\beta _k} u_k^2 e^{\sigma _k u_k^2} {\mathrm {d}}v_g +\int _{\Sigma } \gamma _k u_k^2 {\mathrm {d}}v_g. \end{aligned}$$

According to (82), (98) and (), we obtain

$$\begin{aligned} \int _{\Sigma }\left| \nabla _g u_k\right| ^2 {\mathrm {d}}v_g= & {} \frac{2\pi }{\sigma _k}+\gamma _k\left\| u_k\right\| _2^2+o\left( \left\| u_k\right\| _{2 q}^{4}\right) \end{aligned}$$

for some $q>1$. Applying Lemma 6, we have done the proof of Proposition 1.