Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

TL;DR

This paper investigates the existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surfaces, using blow-up analysis to determine conditions for the functional's infimum and solution existence.

Contribution

It provides new criteria for the existence of solutions to Kazdan-Warner equations by analyzing the functional's behavior and employing blow-up analysis techniques.

Findings

If <(), the infimum of the functional is calculated using blow-up analysis.

A sufficient condition for the existence of solutions to the Kazdan-Warner equation is established.

The functional is unbounded below when  or >8, indicating no minimizers in these cases.

Abstract

Let $(\Sigma,g)$ be a compact Riemannian surface without boundary and $\lambda_1(\Sigma)$ be the first eigenvalue of the Laplace-Beltrami operator $\Delta_g$. Let $h$ be a positive smooth function on $\Sigma$. Define a functional $$J_{\alpha,\beta}(u)=\frac{1}{2}\int_\Sigma(|\nabla_gu|^2-\alpha u^2)dv_g-\beta\log\int_\Sigma he^udv_g$$ on a function space $\mathcal{H}=\left\{u\in W^{1,2}(\Sigma): \int_\Sigma udv_g=0\right\}$. If $\alpha<\lambda_1(\Sigma)$ and $J_{\alpha,8\pi}$ has no minimizer on $\mathcal{H}$, then we calculate the infimum of $J_{\alpha,8\pi}$ on $\mathcal{H}$ by using the method of blow-up analysis. As a consequence, we give a sufficient condition under which a Kazdan-Warner equation has a solution. If $\alpha\geq \lambda_1(\Sigma)$, then $\inf_{u\in\mathcal{H}}J_{\alpha,8\pi}(u)=-\infty$. If $\beta>8\pi$, then for any $\alpha\in\mathbb{R}$, there holds $\inf_{u\in\mathcal{H}}J_{\alpha,\beta}(u)=-\infty$. Moreover, we consider the same problem in the case that $\alpha$ is large, where higher order eigenvalues are involved.

Equations

$$J_{\alpha ,\beta }(u)=\frac{1}{2}{\int }_{\Sigma }(|{\nabla }_{g}u{|}^2-\alpha u^2)d{v}_g-\beta \log {\int }_{\Sigma }h{e}^{u}d{v}_g$$

$$\mathcal{H}=\left\{u\in W^{1,2}(\Sigma ):{\int }_{\Sigma }ud{v}_g=0\right\}.$$

$$J_{\beta }(u)=\frac{1}{2}{\int }_{\Sigma }|{\nabla }_{g}u{|}^{2}d{v}_g-\beta \log {\int }_{\Sigma }h{e}^{u}d{v}_g,$$

$$\underset{u\in \mathcal{H}}{\inf }{J}_{8\pi }(u)=-8\pi -8\pi \log \pi -4\pi \underset{p\in \Sigma }{\max }(A_p+2\log h(p)),$$

$$\{\begin{array}{l}{\Delta }_g{G}_p=8\pi {\delta }_p-\frac{8\pi }{{\mathrm{Vol}}_g(\Sigma )}\\ {\int }_{\Sigma }{G}_{p}d{v}_g=0,\end{array}$$

$${\Delta }_{g}u=\frac{8\pi h{e}^u}{{\int }_{\Sigma }h{e}^{u}d{v}_g}-\frac{8\pi }{{\mathrm{Vol}}_g(\Sigma )}.$$

$${\lambda }_1(\Sigma )=\underset{u\in \mathcal{H},{\int }_{\Sigma }{u}^{2}d{v}_g=1}{\inf }{\int }_{\Sigma }|{\nabla }_{g}u{|}^{2}d{v}_g.$$

$$\Vert u{\Vert }_{1,\alpha }={\left({\int }_{\Sigma }(|{\nabla }_{g}u{|}^2-\alpha u^2)d{v}_g\right)}^{1/2}$$

$$\underset{u\in \mathcal{H},\Vert u{\Vert }_{1,\alpha }\le 1}{\sup }{\int }_{\Sigma }{e}^{4\pi u^2}d{v}_g<\infty$$

$$\frac{1}{2}\Vert u{\Vert }_{1,\alpha }^2-8\pi \log {\int }_{\Sigma }h{e}^{u}d{v}_g\ge -C.$$

$$J_{\alpha ,\beta }(u)=\frac{1}{2}{\int }_{\Sigma }(|{\nabla }_{g}u{|}^2-\alpha u^2)d{v}_g-\beta \log {\int }_{\Sigma }h{e}^{u}d{v}_g.$$

$$\underset{u\in \mathcal{H}}{\inf }{J}_{\alpha ,8\pi }(u)=-8\pi -8\pi \log \pi -4\pi \underset{p\in \Sigma }{\max }(A_p+2\log h(p)),$$

$$\{\begin{array}{l}{\Delta }_g{G}_p-\alpha G_p=8\pi {\delta }_p-\frac{8\pi }{{\mathrm{Vol}}_g(\Sigma )}\\ {\int }_{\Sigma }{G}_{p}d{v}_g=0;\end{array}$$

$${\Delta }_{g}u-\alpha u=\frac{8\pi h{e}^u}{{\int }_{\Sigma }h{e}^{u}d{v}_g}-\frac{8\pi }{{\mathrm{Vol}}_g(\Sigma )},$$

$$\underset{u\in \mathcal{H}}{\inf }{J}_{\alpha ,8\pi }(u)\ne -8\pi -8\pi \log \pi -4\pi \underset{p\in \Sigma }{\max }(A_p+2\log h(p)),$$

$$E_{{\lambda }_{\ell }(\Sigma )}=\left\{u\in \mathcal{H}:{\Delta }_{g}u={\lambda }_{\ell }(\Sigma )u\right\},$$

$$E_{\ell }^{\perp }=\left\{u\in \mathcal{H}:{\int }_{\Sigma }uvd{v}_g=0,\forall v\in E_{\ell }\right\}.$$

$$\underset{u\in E_{\ell }^{\perp }}{\inf }{J}_{\alpha ,8\pi }(u)=-8\pi -8\pi \log \pi -4\pi \underset{p\in \Sigma }{\max }(A_{\alpha ,p}+2\log h(p)),$$

$$\{\begin{array}{l}{\Delta }_g{G}_{\alpha ,p}-\alpha G_{\alpha ,p}=8\pi {\delta }_p-\frac{8\pi }{{\mathrm{Vol}}_g(\Sigma )}\\ {\int }_{\Sigma }{G}_{\alpha ,p}vd{v}_g=0,\forall v\in E_{\ell };\end{array}$$

$$\underset{u\in E_{\ell }^{\perp }}{\inf }{J}_{\alpha ,8\pi }(u)\ne -8\pi -8\pi \log \pi -4\pi \underset{p\in \Sigma }{\max }(A_{\alpha ,p}+2\log h(p)),$$

$${\int }_{\Sigma }|{\nabla }_g{u}_0{|}^{2}d{v}_g={\lambda }_1(\Sigma ){\int }_{\Sigma }{u}_0^{2}d{v}_g$$

$${\int }_{\Sigma }(|{\nabla }_g{u}_0{|}^2-\alpha u_0^2)d{v}_g\le 0.$$

$$u_0(x)\ge u_0(x_0)/2>0,\forall x\in B_{\delta }(x_0).$$

$$J_{\alpha ,8\pi }(t{u}_0)$$

$$M_{k}=M_{k}(x,r)=\left\{\begin{array}[]{lll}\log k&{\rm when}&\rho\leq rk^{-\frac{1}{4}}\\[6.45831pt] 4\log\frac{r}{\rho}&{\rm when}&rk^{-\frac{1}{4}}<\rho\leq r\\[6.45831pt] 0&{\rm when}&\rho>r,\end{array}\right.$$

$${\int }_{\Sigma }|{\nabla }_g{M}_k{|}^{2}d{v}_g=(1+O(r))8\pi \log k,$$

$${\int }_{\Sigma }{M}_k^{j}d{v}_g=O(1),j=1,2.$$

$$\widetilde{M}_{k}=\widetilde{M}_{k}(x,r)=\left\{\begin{array}[]{lll}M_{k}(x,r)&{\rm for}&x\in B_{2i_{g}(\Sigma)}(p)\\[6.45831pt] t_{k}\zeta(x)&{\rm for}&x\in\Sigma\setminus B_{2i_{g}(\Sigma)}(p),\end{array}\right.$$

$$t_k=-\frac{1}{{\int }_{\Sigma }\zeta d{v}_g}{\int }_{\Sigma }{M}_{k}d{v}_g=O(1).$$

$${\int }_{\Sigma }|{\nabla }_g{\tilde{M}}_k{|}^{2}d{v}_g-\alpha {\int }_{\Sigma }{\tilde{M}}_k^{2}d{v}_g=(1+O(r))8\pi \log k+O(1)$$

$${\int }_{\Sigma }h{e}^{{\tilde{M}}_k}d{v}_g\ge {\int }_{B_{r{k}^{-1/4}}(p)}h{e}^{M_k}d{v}_g\ge (\underset{\Sigma }{\min }h)\pi r^2\sqrt{k}(1+o_k(1)).$$

Full text

[TABLE]

on a function space $\mathcal{H}=\left\{u\in W^{1,2}(\Sigma):\int_{\Sigma}udv_{g}=0\right\}$. If $\alpha<\lambda_{1}(\Sigma)$ and $J_{\alpha,8\pi}$ has no minimizer on $\mathcal{H}$, then we calculate the infimum of $J_{\alpha,8\pi}$ on $\mathcal{H}$ by using the method of blow-up analysis. As a consequence, we give a sufficient condition under which a Kazdan-Warner equation has a solution. If $\alpha\geq\lambda_{1}(\Sigma)$, then $\inf_{u\in\mathcal{H}}J_{\alpha,8\pi}(u)=-\infty$. If $\beta>8\pi$, then for any $\alpha\in\mathbb{R}$, there holds $\inf_{u\in\mathcal{H}}J_{\alpha,\beta}(u)=-\infty$. Moreover, we consider the same problem in the case that $\alpha$ is large, where higher order eigenvalues are involved.

††journal: ***

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1 Introduction and main results

Let $(\Sigma,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(\Sigma)$ be the usual Sobolev space. Define a function space

[TABLE]

Let $h$ be a positive smooth function on $\Sigma$ and $J_{\beta}:W^{1,2}(\Sigma)\rightarrow\mathbb{R}$ be a functional defined by

[TABLE]

where $\nabla_{g}u$ denotes the gradient of $u$ and $dv_{g}$ denotes the volume element of $(\Sigma,g)$. In view of the Trudinger-Moser inequality due to Fontana [7], $J_{\beta}$ has a minimizer on $\mathcal{H}$ for any $\beta<8\pi$; while in the case $\beta=8\pi$, the situation becomes subtle. Using a method of blow-up analysis, Ding-Jost-Li-Wang [4] proved that if $J_{8\pi}$ has no minimizer on $\mathcal{H}$, then

[TABLE]

where $A_{p}=\lim_{x\rightarrow p}(G_{p}(x)+4\log r)$ is a constant, $r$ denotes the geodesic distance between $x$ and $p$, $G_{p}$ is a Green function satisfying

[TABLE]

and $\Delta_{g}$ is the Laplace-Beltrami operator. Moreover, they give a geometric condition under which $J_{8\pi}$ has a minimizer on $\mathcal{H}$. Clearly the minimizer is a solution of a Kazdan-Warner equation [8], namely

[TABLE]

We let $\lambda_{1}(\Sigma)$ be the first eigenvalue of $\Delta_{g}$, say

[TABLE]

It follows from the Poincaré inequality that if $\alpha<\lambda_{1}(\Sigma)$, then

[TABLE]

defines a Sobolev norm on $\mathcal{H}$. In a previous work [15], using the method of blow-up analysis, we proved the following: for any $\alpha<\lambda_{1}(\Sigma)$, there holds

[TABLE]

and the supremum is attained. As a consequence of (6), there exists some constant $C$ depending only on $(\Sigma,g)$ and $\alpha<\lambda_{1}(\Sigma)$ such that for all $u\in\mathcal{H}$,

[TABLE]

This improves the Trudinger-Moser inequality of the weak form, namely (7) in the case $\alpha=0$. We refer the reader to [1, 14, 10, 5, 6, 13, 16, 17, 18] for related works involving the norms $\|u\|_{1,\alpha}$.

Our aim in this paper is to achieve an analog of (3). More precisely, we consider functionals

[TABLE]

Obviously, when $\alpha=0$, $J_{\alpha,\beta}$ reduces to $J_{\beta}$ defined as in (2). Our first result reads

Theorem 1**.**

*Let $(\Sigma,g)$ be a compact Riemannian surface without boundary, $h$ be a positive smooth function on $\Sigma$, and $\mathcal{H}$, $\lambda_{1}(\Sigma)$ and $J_{\alpha,\beta}$ be defined as in (1), (5) and (8) respectively. Then we have the following three assertions:

$(i)$ If $\alpha<\lambda_{1}(\Sigma)$ and $J_{\alpha,8\pi}$ has no minimizer in $\mathcal{H}$, then there holds*

[TABLE]

where $A_{p}=\lim_{x\rightarrow p}(G_{p}(x)+4\log r)$ is a constant, $r$ denotes the geodesic distance between $x$ and $p$, $G_{p}$ is a Green function satisfying

[TABLE]

$(ii)$* If $\alpha\geq\lambda_{1}(\Sigma)$, then $\inf_{u\in\mathcal{H}}J_{\alpha,8\pi}(u)=-\infty;$

$(iii)$ If $\beta>8\pi$, then for any $\alpha\in\mathbb{R}$, we have $\inf_{u\in\mathcal{H}}J_{\alpha,\beta}(u)=-\infty$.*

Since the Euler-Lagrange equation of a minimum point of $J_{\alpha,8\pi}$ on $\mathcal{H}$ is

[TABLE]

an application of $(i)$ of Theorem 1 is the following:

Corollary 2**.**

For any $\alpha<\lambda_{1}(\Sigma)$, if

[TABLE]

then the Kazdan-Warner equation (10) has a solution $u\in\mathcal{H}$.

As in [15], we consider the case that $\alpha$ is allowed to be larger than $\lambda_{1}(\Sigma)$. Precisely, we let $\lambda_{1}(\Sigma)<\lambda_{2}(\Sigma)<\cdots$ be all distinct eigenvalues of $\Delta_{g}$, $E_{\lambda_{\ell}(\Sigma)}$ be the eigenfunction space with respect to $\lambda_{\ell}(\Sigma)$, namely

[TABLE]

and $E_{\ell}=E_{\lambda_{1}(\Sigma)}\oplus E_{\lambda_{2}(\Sigma)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Sigma)}$, $\ell=1,2,\cdots$. Define

[TABLE]

Now we state an analog of Theorem 1 as follows:

Theorem 3**.**

*Let $(\Sigma,g)$, $h$, $\mathcal{H}$ and $J_{\alpha,\beta}$ be as in Theorem 1, $\lambda_{\ell}(\Sigma)$ be the $\ell$-th eigenvalue of the Laplace-Beltrami operator, $E_{\ell}$ and $E_{\ell}^{\perp}$ be defined as in (13). Then we have the following three assertions:

$(i)$ If $\alpha<\lambda_{\ell+1}(\Sigma)$ and $J_{\alpha,8\pi}$ has no minimizer in $E_{\ell}^{\perp}$, then there holds*

[TABLE]

where $A_{\alpha,p}=\lim_{x\rightarrow p}(G_{\alpha,p}(x)+4\log r)$ is a constant, $r$ denotes the geodesic distance between $x$ and $p$, $G_{\alpha,p}$ is a Green function satisfying

[TABLE]

$(ii)$* If $\alpha\geq\lambda_{\ell+1}(\Sigma)$, then $\inf_{u\in E_{\ell}^{\perp}}J_{\alpha,8\pi}(u)=-\infty;$

$(iii)$ If $\beta>8\pi$, then for any $\alpha\in\mathbb{R}$, we have $\inf_{u\in E_{\ell}^{\perp}}J_{\alpha,\beta}(u)=-\infty$.*

Similar to Corollary 2, we have the following:

Corollary 4**.**

For any $\alpha<\lambda_{\ell+1}(\Sigma)$, if

[TABLE]

then the Kazdan-Warner equation (10) has a solution $u\in E_{\ell}^{\perp}$.

We remark that any geometric hypothesis under which (11) or (15) holds would be extremely interesting. When $\alpha=0$, a geometric condition was given by Ding-Jost-Li-Wang [4]. Generally it is difficult to be obtained possibly because the Green function has at most $C^{1,\gamma}$-regularity for some $0<\gamma<1$ in presence of $\alpha$.

Now we describe our method. For the proof of $(ii)$ and $(iii)$ of Theorems 1 and 3, we shall construct suitable function sequences. To prove $(i)$ of Theorems 1 and 3, we use the blow-up scheme proposed by Ding-Jost-Li-Wang [4]. Our analysis is different from that of Ding-Jost-Li-Wang [4] at least in three points: Let $u_{\epsilon}$ be a minimizer of $J_{\alpha,8\pi(1-\epsilon)}$ and $u_{\epsilon}(x_{\epsilon})=\max_{\Sigma}u_{\epsilon}\rightarrow+\infty$. One is that before understanding the exact asymptotic behavior of $u_{\epsilon}$ near the blow-up point, we must prove $u_{\epsilon}(x)/u_{\epsilon}(x_{\epsilon})=1+o_{\epsilon}(1)$ in $B_{Rr_{\epsilon}}(x_{\epsilon})$, where $r_{\epsilon}\rightarrow 0$ is an appropriate sequence of positive numbers, $R>0$ is fixed, and $o_{\epsilon}(1)\rightarrow 0$ uniformly in $B_{Rr_{\epsilon}}(x_{\epsilon})$; The other is that in the process of deriving lower bound of $J_{\alpha,8\pi}$, we estimate the energy $\int_{\Sigma}|\nabla_{g}u_{\epsilon}|^{2}dv_{g}$ on two regions $B_{Rr_{\epsilon}}$ and $\Sigma\setminus B_{Rr_{\epsilon}}$ instead of three regions $B_{Rr_{\epsilon}}$, $B_{\delta}\setminus B_{Rr_{\epsilon}}$ and $\Sigma\setminus B_{\delta}$, which simplifies the calculation in [4]; The third is in the final step (test function computation), we construct a sequence of test functions different from that of [4].

Before ending this introduction, we mention several related works also based on the blow-up scheme in [4]. Ni [12] considered the mean field equation with critical parameter in a planar domain. Zhou [19] obtained existence of solution to the mean field equation for the equilibrium turbulence. Liu-Wang [9] studied the equation (4) with an extra drifting term $\nabla\phi\cdot\nabla u$. Mancini [11] proved an Onofri inequality.

Throughout this paper, $o_{\epsilon}(1)\rightarrow 0$ as $\epsilon\rightarrow 0$, $o_{R}(1)\rightarrow 0$ as $R\rightarrow\infty$, and so on. We do not distinguish sequence and subsequence and often denote various constants by the same $C$. The remaining part of this paper is organized as follows: In Section 2, we prove Theorem 1; In Section 3, we give the proof of Theorem 3.

2 Proof of Theorem 1

The proof of $(ii)$ and $(iii)$ is easy and will be shown first.

2.1 Proof of $(ii)$ of Theorem 1

Let $\alpha\geq\lambda_{1}(\Sigma)$ be fixed and $E_{\lambda_{1}(\Sigma)}$ be the eigenfunction space defined as in (12). Take $0\not\equiv u_{0}\in E_{\lambda_{1}(\Sigma)}$. Obviously we have

[TABLE]

and thus

[TABLE]

Since $\int_{\Sigma}u_{0}dv_{g}=0$, there exists $x_{0}\in\Sigma$ and $\delta>0$ such that

[TABLE]

It follows from (16) and (17) that for any $t>0$,

[TABLE]

Hence $J_{\alpha,8\pi}(tu_{0})\rightarrow-\infty$ as $t\rightarrow+\infty$ and the desired result follows immediately. $\hfill\Box$

2.2 Proof of $(iii)$ of Theorem 1

Let $\beta>8\pi$ be fixed and $i_{g}(\Sigma)$ be the injectivity radius of $(\Sigma,g)$. Fix some point $p\in\Sigma$. Let $r$, $0<r<i_{g}(\Sigma)/2$, be a real number to be determined later. Take a sequence of functions

[TABLE]

where $x\in\Sigma$, $\rho$ denotes the geodesic distance between $x$ and $p$, and $k=2,3\cdots$. One calculates

[TABLE]

Choose $\zeta\in C_{0}^{1}(\Sigma\setminus B_{2i_{g}(\Sigma)}(p))$ satisfying $\zeta\geq 0$ and $\zeta\not\equiv 0$ on $\Sigma\setminus B_{2i_{g}(\Sigma)}(p)$. We set

[TABLE]

where $t_{k}\in\mathbb{R}$ is chosen so that $\widetilde{M}_{k}\in\mathcal{H}$. As a consequence,

[TABLE]

It follows from (18)-(20) that

[TABLE]

and

[TABLE]

Combining (21) and (22), we have

[TABLE]

Note that $\beta>8\pi$. If $r$ is chosen sufficiently small, then we conclude

[TABLE]

This completes the proof of $(iii)$ of Theorem 1. $\hfill\Box$

In the remaining part of this section, we always assume $\alpha<\lambda_{1}(\Sigma)$. Since the proof of $(i)$ of Theorem 1 is very long, we sketch its outline as follows: Step 1. For any $\epsilon$, $0<\epsilon<1$, there exists a minimizer $u_{\epsilon}\in\mathcal{H}\cap C^{1}(\Sigma)$ for the subcritical functional $J_{\alpha,8\pi(1-\epsilon)}$. By assumption that $J_{\alpha,8\pi}$ has no minimizer on $\mathcal{H}$, we have $u_{\epsilon}(x_{\epsilon})=\max_{\Sigma}u_{\epsilon}\rightarrow+\infty$ as $\epsilon\rightarrow 0$. Step 2. We prove that $u_{\epsilon}(\exp_{x_{\epsilon}}(r_{\epsilon}x))-\max_{\Sigma}u_{\epsilon}\rightarrow\varphi(x)$ in $C^{1}_{\rm loc}(\mathbb{R}^{2})$, where $\exp_{x_{\epsilon}}$ denotes the exponential map on $(\Sigma,g)$, $r_{\epsilon}$ is an appropriate scale, and $\varphi$ can be explicitly written out via a classification result of Chen-Li [3]. Moreover, assuming $x_{\epsilon}\rightarrow p$, we show that $u_{\epsilon}\rightarrow G_{p}$ weakly in $W^{1,q}(\Sigma)$ for any $1<q<2$ and in $C^{1}_{\rm{loc}}(\Sigma\setminus\{p\})$, where $G_{p}$ is a Green function on $(\Sigma,g)$. Step 3. Applying the maximum principle to $u_{\epsilon}-G_{x_{\epsilon}}$ and using the asymptotic behavior of $u_{\epsilon}$ derived in Step 2, we obtain a lower bound of $J_{\alpha,8\pi}$ on $\mathcal{H}$. Step 4. We construct a sequence of functions $\phi_{\epsilon}\in\mathcal{H}$ such that $J_{\alpha,8\pi}(\phi_{\epsilon})$ converges to the lower bound obtained in Step 3.

2.3 Minimizers for subcritical functionals

We first prove that $\inf_{u\in\mathcal{H}}J_{\alpha,\beta}(u)$ is attained for any $\beta<8\pi$. Precisely we have

Proposition 5**.**

For any $0<\epsilon<1$, there exists some function $u_{\epsilon}\in\mathcal{H}\cap C^{1}(\Sigma)$ such that

[TABLE]

Moreover, $u_{\epsilon}$ satisfies the Euler-Lagrange equation

[TABLE]

Proof. Let $0<\epsilon<1$ be fixed. Take $u_{j}\in\mathcal{H}$ such that

[TABLE]

as $j\rightarrow\infty$. Noting that

[TABLE]

we obtain by (25) and (6),

[TABLE]

Hence $u_{j}$ is bounded in $\mathcal{H}$. We can assume without loss of generality that $u_{j}$ converges to $u_{\epsilon}$ weakly in $\mathcal{H}$, strongly in $L^{q}(\Sigma)$ for any $q>0$ and almost everywhere in $\Sigma$. Clearly

[TABLE]

Moreover an analog of (26) implies that $e^{|u_{j}|}$ is bounded in $L^{p}(\Sigma)$ for any $p>0$. This together with the mean value theorem and the Hölder inequality,

[TABLE]

Combining (27) and (28), we conclude (23).

Using a method of Lagrange multiplier, one easily gets (24), the Euler-Lagrange equation of the minimizer $u_{\epsilon}$. Applying elliptic estimates to (24), we have $u_{\epsilon}\in C^{1}(\Sigma)$. $\hfill\Box$

Lemma 6**.**

$\liminf_{\epsilon\rightarrow 0}\lambda_{\epsilon}>0$.

Proof. One may conclude the lemma by using the Jensen inequality. But we prefer a contradiction argument as below. Clearly

[TABLE]

If $\liminf_{\epsilon\rightarrow 0}\lambda_{\epsilon}=0$, then up to a subsequence

[TABLE]

which contradicts (29). Thus we get the desired result. $\hfill\Box$

Lemma 7**.**

There holds

[TABLE]

Proof. Though the proof may be obvious for experts, we give the details here for reader’s convenience. On one hand, for any $\eta>0$, there exists some $u_{\eta}\in\mathcal{H}$ such that

[TABLE]

Obviously we have

[TABLE]

Hence

[TABLE]

Since $\eta>0$ is arbitrary, we have

[TABLE]

On the other hand,

[TABLE]

Extracting diagonal sequence, we obtain

[TABLE]

Combining (30) and (31), we get the desired result. $\hfill\Box$

Lemma 8**.**

If $\lambda_{\epsilon}$ is bounded, then $u_{\epsilon}$ is bounded in $\mathcal{H}$ and a minimizer for the functional $J_{\alpha,8\pi}$ exists on the function space $\mathcal{H}$.

Proof. Since $\lambda_{\epsilon}$ is a bounded sequence, it follows from (29) that

[TABLE]

Hence $u_{\epsilon}$ is bounded in $\mathcal{H}$ and thus $e^{|u_{\epsilon}|}$ is bounded in $L^{p}(\Sigma)$ for any $p>0$. Applying elliptic estimates to the equation (24), in view of Lemma 6, we have that $u_{\epsilon}$ converges to some $u_{0}\in\mathcal{H}$ in $C^{1}(\Sigma)$. By Lemma 7, we have

[TABLE]

Therefore $u_{0}$ is a minimizer of $J_{\alpha,8\pi}$. $\hfill\Box$

Denote

[TABLE]

Proposition 9**.**

If $c_{\epsilon}$ is bounded from above, then $J_{\alpha,8\pi}$ has a minimizer in $\mathcal{H}$.

Proof. Multiplying both sides of the equation (24) by $u_{\epsilon}$, we have by using Lemma 6, the assumption that $c_{\epsilon}$ is bounded from above and the Sobolev embedding theorem,

[TABLE]

This implies that $u_{\epsilon}$ is bounded in $\mathcal{H}$. Applying elliptic estimates to (24), we conclude that $u_{\epsilon}$ converges to a minimizer of $J_{\alpha,8\pi}$ in $C^{1}(\Sigma)$. $\hfill\Box$

2.4 Blow-up analysis

We now analyze the asymptotic behavior of $u_{\epsilon}$. By our assumption that $J_{\alpha,8\pi}$ has no minimizer on $\mathcal{H}$, in view of Lemma 8 and Proposition 9, we have

[TABLE]

The convergence of $u_{\epsilon}$ will be described in the following proposition.

Proposition 10**.**

Assume $\alpha<\lambda_{1}(\Sigma)$ and $J_{\alpha,8\pi}$ has no minimizer in $\mathcal{H}$. Let $u_{\epsilon}$ be a sequence of solutions to the equation (24). Let $c_{\epsilon}=u_{\epsilon}(x_{\epsilon})$ be defined as in (32) and assume that $x_{\epsilon}\rightarrow p\in\Sigma$. If we define

[TABLE]

and

[TABLE]

then

[TABLE]

Moreover, $u_{\epsilon}$ converges to a Green function $G_{p}$ weakly in $W^{1,q}(\Sigma)$ for any $1<q<2$, strongly in $L^{r}(\Sigma)$ for all $0<r<2q/(2-q)$, and in $C^{1}_{\rm loc}(\Sigma\setminus\{p\})$, where $G_{p}$ satisfies

[TABLE]

The proof of Proposition 10 will be divided into several lemmas.

Lemma 11**.**

Let $r_{\epsilon}$ be defined as in (35). For any $\gamma<1/2$, there holds $r_{\epsilon}^{2}e^{\gamma c_{\epsilon}}\rightarrow 0$. In particular, $r_{\epsilon}c_{\epsilon}^{q}\rightarrow 0$ for any $q>0$.

Proof. Multiplying both sides of the equation (24) by $u_{\epsilon}$, we have

[TABLE]

In view of the Trudinger-Moser inequality (6), we estimate

[TABLE]

It follows that

[TABLE]

This together with (33) gives the desired result. $\hfill\Box$

Let $0<\delta<i_{g}(\Sigma)$ be fixed and $i_{g}(\Sigma)$ be the injectivity radius of $(\Sigma,g)$. For $y\in\mathbb{B}_{\delta r_{\epsilon}^{-1}}(0)$, the Euclidean ball of center [math] and radius $\delta r_{\epsilon}^{-1}$, we set

[TABLE]

Clearly $g_{\epsilon}\rightarrow g_{0}$, the standard Euclidean metric, in $C^{2}_{\rm loc}(\mathbb{R}^{2})$ as $\epsilon\rightarrow 0$. Note that $\psi_{\epsilon}\leq\psi_{\epsilon}(0)=1$. Concerning the asymptotic behavior of $\psi_{\epsilon}$, we have the following:

Lemma 12**.**

$\psi_{\epsilon}\rightarrow 1$* in $C^{1}_{\rm loc}(\mathbb{R}^{2})$.*

Proof. In view of (24), (39) and (40), we have

[TABLE]

Let $q>1$ be any fixed number. By (38) and the Sobolev embedding theorem, we have

[TABLE]

Let $B_{r}(x)$ be a geodesic ball centered at $x\in\Sigma$ with radius $r$. It follows from a change of variables, (42) and Lemma 11 that

[TABLE]

as $\epsilon\rightarrow 0$. Therefore we conclude that $\Delta_{g_{\epsilon}}\psi_{\epsilon}(y)$ converges to [math] in $L^{q}_{\rm loc}(\mathbb{R}^{2})$ for any $q>1$. Noting that $\psi_{\epsilon}(y)\leq 1$ for all $y\in\mathbb{B}_{\delta r_{\epsilon}^{-1}}(0)$ and applying elliptic estimates to (41), we obtain $\psi_{\epsilon}\rightarrow\psi$ in $C^{1}_{\rm loc}(\mathbb{R}^{2})$ for some $\psi$ satisfying

[TABLE]

where $\Delta_{\mathbb{R}^{2}}$ denotes the usual Laplacian operator on $\mathbb{R}^{2}$. Then the Liouville theorem leads to $\psi(y)\equiv 1$ for $y\in\mathbb{R}^{2}$. This completes the proof of the lemma. $\hfill\Box$

Let $\varphi_{\epsilon}(y)$ be defined as in (34) for $y\in\mathbb{B}_{\delta r_{\epsilon}^{-1}}(0)$. To prove (36), we calculate on $\mathbb{B}_{\delta r_{\epsilon}^{-1}}(0)$,

[TABLE]

An obvious analog of (43) implies that $r_{\epsilon}^{2}{u}_{\epsilon}(\exp_{x_{\epsilon}}(r_{\epsilon}y))\rightarrow 0$ in $L^{q}_{\rm loc}(\mathbb{R}^{2})$ as $\epsilon\rightarrow 0$ for any $q>1$. Note that $\varphi_{\epsilon}(y)\leq\varphi_{\epsilon}(0)=0$. In view of Lemma 12, we have by applying elliptic estimates to (44), $\varphi_{\epsilon}\rightarrow\varphi$ in $C^{1}_{\rm loc}(\mathbb{R}^{2})$ as $\epsilon\rightarrow 0$, where $\varphi$ satisfies

[TABLE]

A result of Chen-Li [3] implies that $\varphi$ can be written as in (36) and thus

[TABLE]

Lemma 13**.**

There holds $\lambda_{\epsilon}^{-1}he^{u_{\epsilon}}\rightharpoonup\delta_{p}$ in sense of measure, where $\delta_{p}$ denotes the Dirac measure centered at $p$.

Proof. By a change of variables, we have

[TABLE]

This together with (45) leads to

[TABLE]

Hence

[TABLE]

Combining (46) and (47), we have for any $\eta\in C^{0}(\Sigma)$,

[TABLE]

This gives the desired result. $\hfill\Box$

Lemma 14**.**

If $u\in C^{2}(\Sigma)$ is a solution of $\Delta_{g}u=f$, then for any $1<q<2$, there exists some constant $C$ depending only on $(\Sigma,g)$ and $q$ such that

[TABLE]

Proof. Without loss of generality we assume $\int_{\Sigma}udv_{g}=0$. Let $G(x,y)$ be the standard Green function satisfying $\Delta_{g,y}G(x,y)=\delta_{x}(y)-1/V$ and $\int_{\Sigma}G(x,y)dv_{g,y}=0$, where $\delta_{x}(y)$ denotes the Dirac measure centered at $x$, and $V$ is the area of $\Sigma$. Clearly

[TABLE]

It follows from [7] that $|\nabla_{g,x}G(x,y)|\leq Cr(x,y)^{-1}$ for some constant $C$ depending only on $(\Sigma,g)$, where $r(x,y)$ stands for the geodesic distance between $x$ and $y$. Let $1<q<2$. One calculates by using the Hölder inequality

[TABLE]

Hence $\|\nabla_{g}u\|_{L^{q}(\Sigma)}\leq C\|f\|_{L^{1}(\Sigma)}$ for some constant $C$ depending only on $(\Sigma,g)$ and $q$. $\hfill\Box$

Lemma 15**.**

For any $1<q<2$, there exists some constant $C$ such that $\|\nabla_{g}u_{\epsilon}\|_{L^{q}(\Sigma)}\leq C$.

Proof. Clearly (24) gives

[TABLE]

In view of Lemma 14, it suffices to prove

[TABLE]

Noting that $\lambda_{\epsilon}^{-1}\int_{\Sigma}he^{u_{\epsilon}}dv_{g}=1$, we only need to prove that $u_{\epsilon}$ is bounded in $L^{1}(\Sigma)$. For otherwise we can assume $\|u_{\epsilon}\|_{L^{1}(\Sigma)}\rightarrow+\infty$ as $\epsilon\rightarrow 0$. Set $v_{\epsilon}=u_{\epsilon}/\|u_{\epsilon}\|_{L^{1}(\Sigma)}$. Then $\|v_{\epsilon}\|_{L^{1}(\Sigma)}=1$ and $\Delta_{g}v_{\epsilon}=f_{\epsilon}/\|u_{\epsilon}\|_{L^{1}(\Sigma)}$. Obviously $f_{\epsilon}/\|u_{\epsilon}\|_{L^{1}(\Sigma)}$ is bounded in $L^{1}(\Sigma)$. Given any $1<q<2$. It follows from Lemma 14 that $v_{\epsilon}$ is bounded in $W^{1,q}(\Sigma)$. One can assume up to a subsequence, $v_{\epsilon}$ converges to $v$ weakly in $W^{1,q}(\Sigma)$, strongly in $L^{r}(\Sigma)$ for any $0<r<2q/(2-q)$, and almost everywhere in $\Sigma$. Moreover, $v$ is a distributional solution of

[TABLE]

This leads to $v\equiv 0$ contradicting the fact that $\|v\|_{L^{1}(\Sigma)}=\lim_{\epsilon\rightarrow 0}\|v_{\epsilon}\|_{L^{1}(\Sigma)}=1$. Therefore $\|u_{\epsilon}\|_{L^{1}(\Sigma)}$ must be bounded and thus (49) holds. $\hfill\Box$

Combining Lemmas 13 and 15, we obtain for all $1<q<2$ and $0<r<2q/(2-q)$,

[TABLE]

where $G_{p}$ is a distributional solution of (37). Applying elliptic estimates to (37), we have that $G_{p}$ takes the form

[TABLE]

where $r$ denotes the geodesic distance between $x$ and $p$, $\psi\in C^{1}(\Sigma)$ and $\psi(p)=0$. To complete the proof of Proposition 10, we also need the following:

Lemma 16**.**

$u_{\epsilon}\rightarrow G_{p}$* in $C^{1}_{\rm loc}(\Sigma\setminus\{p\})$ as $\epsilon\rightarrow 0$.*

Proof. For any domain $\Omega\subset\subset\Sigma\setminus\{p\}$, let $u_{\epsilon}^{(1)}$ be a solution of

[TABLE]

By Lemma 13, $\lambda_{\epsilon}^{-1}he^{u_{\epsilon}}$ converges to [math] in $L^{1}(\Omega)$ as $\epsilon\rightarrow 0$. A result of Brezis-Merle [2] implies that for any $r>0$, there exists some constant $C$ such that

[TABLE]

Setting $u_{\epsilon}^{(2)}=u_{\epsilon}-u_{\epsilon}^{(1)}$, we have on $\Omega$,

[TABLE]

In view of Lemma 15, $u_{\epsilon}$ is bounded in $L^{r}(\Sigma)$ for any $r>0$. For any $\Omega^{\prime}\subset\subset\Omega$, applying elliptic estimates to (52), we have that $u_{\epsilon}^{(2)}$ is uniformly bounded in $\Omega^{\prime}$. This together with (51) leads to $\|f_{\epsilon}\|_{L^{r}(\Omega^{\prime})}\leq C$ for some $r>2$, where $f_{\epsilon}$ is defined as in (48). Then we get the desired result by applying elliptic estimates to (48). $\hfill\Box$

2.5 Lower bound estimate

In this subsection, we shall derive a lower bound of $J_{\alpha,8\pi}$ on $\mathcal{H}$. Let $G_{x_{\epsilon}}$ be the Green function satisfying $\int_{\Sigma}G_{x_{\epsilon}}dv_{g}=0$ and

[TABLE]

Clearly $G_{x_{\epsilon}}$ can be represented by

[TABLE]

where $r$ denotes the geodesic distance between $x_{\epsilon}$ and $x$. Moreover, there holds $A_{x_{\epsilon}}\rightarrow A_{p}$ as $\epsilon\rightarrow 0$. Similar to [4], we have the following maximum principle.

Lemma 17**.**

For any $\epsilon>0$ and $R>0$, there exists a constant $L_{\epsilon}$ such that for all $y\in\Sigma\setminus B_{Rr_{\epsilon}}(x_{\epsilon})$, there holds

[TABLE]

Proof. Note that

[TABLE]

In view of Proposition 10 and the formula (54), we have on $\partial B_{Rr_{\epsilon}}(x_{\epsilon})$ that

[TABLE]

The desired result follows from the maximum principle immediately. $\hfill\Box$

For any fixed $R>0$, we have

[TABLE]

By Proposition 10 we have

[TABLE]

It follows from (24) that

[TABLE]

Lemma 17 leads to

[TABLE]

We estimate three terms on the right hand side of (2.5) respectively. Using (24) and (53), we obtain

[TABLE]

It also follows from (24) that

[TABLE]

In view of (47), one has

[TABLE]

Here, in (60), we use the fact that $A_{x_{\epsilon}}$ is bounded. Inserting (2.5)-(60) into (2.5), we get the lower bound estimate of ${\lambda_{\epsilon}^{-1}}\int_{\Sigma\setminus{B_{Rr_{\epsilon}}(x_{\epsilon})}}{u_{\epsilon}he^{u_{\epsilon}}}dv_{g}$. Then inserting this lower bound to (2.5), we obtain the estimate of $\int_{\Sigma\setminus{B_{Rr_{\epsilon}}(x_{\epsilon})}}|\nabla_{g}u_{\epsilon}|^{2}dv_{g}$ as below.

[TABLE]

Using Proposition 10 and Lemma 17, one has

[TABLE]

In view of (50) and Proposition 10, we have

[TABLE]

Then it follows by Lemma 11 that

[TABLE]

Also Proposition 10 and Lemma 11 lead to

[TABLE]

In view of (50) and Lemma 11, we have

[TABLE]

It follows from Lemma 6, Proposition 10 and Lemma 11 that

[TABLE]

and

[TABLE]

Hence we have by inserting (2.5)-(67) into (2.5),

[TABLE]

Combining (2.5) and (2.5), we have

[TABLE]

Letting $\epsilon\rightarrow 0$ first and then $R\rightarrow+\infty$, one has

[TABLE]

2.6 Test function computation

In this subsection, we construct a sequence of functions $(\phi_{\epsilon})_{\epsilon>0}$ satisfying

[TABLE]

where

[TABLE]

Suppose that $2\log h(p)+A_{p}=\max_{x\in\Sigma}\left(2\log h(x)+A_{x}\right)$. Let $r=r(x)$ be the geodesic distance between $x$ and $p$. We set

[TABLE]

where $\eta\in C_{0}^{\infty}(B_{2R\epsilon}(p))$ is a cut-off function, $\eta\equiv 1$ in $B_{R{\epsilon}}(p)$, $|\nabla_{g}\eta(x)|\leq\frac{4}{R{\epsilon}}$ for all $x\in B_{2R\epsilon}(p)$, $\psi$ is defined as in (50),

[TABLE]

and $R=R(\epsilon)$ satisfying $R\rightarrow+\infty$ and $(R{\epsilon})^{2}\log R\rightarrow 0$ as $\epsilon\rightarrow 0$.

A straightforward calculation shows

[TABLE]

Moreover we have

[TABLE]

Using (50) one has

[TABLE]

By (53) and (50) we have

[TABLE]

where we have used the facts $\int_{B_{R{\epsilon}}(p)}G^{2}_{p}dv_{g}=o_{\epsilon}(1)$ and $\int_{B_{R{\epsilon}}(p)}G_{p}dv_{g}=o_{\epsilon}(1)$.

Moreover one can easily get that

[TABLE]

that

[TABLE]

and that

[TABLE]

where we have used (53) and (50). Inserting (74)-(78) into (2.6) we obtain

[TABLE]

Combining (72) and (79), we have

[TABLE]

Clearly

[TABLE]

and

[TABLE]

We now estimate $\int_{\Sigma}he^{\phi_{\epsilon}}dv_{g}$. Choosing $\delta>0$ sufficiently small and noting that $G_{p}$ has the expression (50) in $B_{\delta}(p)$, we have

[TABLE]

A straightforward calculation gives

[TABLE]

and

[TABLE]

Also one has

[TABLE]

and

[TABLE]

Inserting (84)-(87) into (2.6), we have

[TABLE]

where $C$ is a constant depending only on $(\Sigma,g)$, $\max_{\Sigma}h$ and $\delta$. Hence

[TABLE]

Combining (80), (81), (82) and (88), we have

[TABLE]

This implies (70), which together with (2.5) completes the proof of $(i)$ of Theorem 1. $\hfill\Box$

3 Proof of Theorem 3

In this section we prove Theorem 3 by using similar method of the proof of Theorem 1. Let $\{e_{i}\}_{i=1}^{m_{\ell}}\subset\mathcal{H}\cap C^{1}(\Sigma)$ be an orthonormal basis of $E_{\ell}=E_{\lambda_{1}(\Sigma)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Sigma)}$, namely $E_{\ell}={\rm span}\{e_{1},\cdots,e_{m_{\ell}}\}$ and

[TABLE]

for all $i,j=1,\cdots,m_{\ell}$. Note that $\mathcal{H}=E_{\ell}\oplus E_{\ell}^{\perp}$. $v\in E_{\ell}^{\perp}$ if and only if $v\in\mathcal{H}$ and $\langle v,e_{i}\rangle=0$ for all $i=1,\cdots,m_{\ell}$.

Proof of $(i)$ of Theorem 3. Let $\alpha<\lambda_{\ell+1}$ be fixed. Using the argument of Subsections 2.3-2.5 with minor modifications, we obtain an analog of (2.5) as the following: If $J_{\alpha,8\pi}$ has no minimizer in $E_{\ell}^{\perp}$, then

[TABLE]

where $A_{\alpha,x}=\lim_{y\rightarrow x}(G_{\alpha,x}(y)+4\log r)$ is a constant, $r$ denotes the geodesic distance between $y$ and $x$, $G_{\alpha,x}$ is a Green function satisfying (14). Assume

[TABLE]

It follows from elliptic estimates that $G_{\alpha,p}$ can be written as

[TABLE]

where $\psi_{\alpha}\in C^{1}(\Sigma)$ and $r$ denotes the geodesic distance between $x$ and $p$. We now prove that if $J_{\alpha,8\pi}$ has no minimizer in $E_{\ell}^{\perp}$, then

[TABLE]

Similar to (71), we set

[TABLE]

where $\eta\in C_{0}^{\infty}(B_{2R\epsilon}(p))$ is a cut-off function, $\eta\equiv 1$ in $B_{R{\epsilon}}(p)$, $|\nabla_{g}\eta(x)|\leq\frac{4}{R{\epsilon}}$ for all $x\in B_{2R\epsilon}(p)$,

[TABLE]

and $R=R(\epsilon)$ satisfying $R\rightarrow+\infty$ and $(R{\epsilon})^{2}\log R\rightarrow 0$ as $\epsilon\rightarrow 0$. Similar to (2.6), we derive

[TABLE]

where

[TABLE]

Define a new sequence of functions

[TABLE]

where $\{e_{i}\}_{i=1}^{m_{\ell}}$ is an orthonormal basis on $E_{\ell}\subset\mathcal{H}$. Clearly $\psi_{\alpha,\epsilon}\in E_{\ell}^{\perp}$.

Noting that

[TABLE]

we have

[TABLE]

for any $k=1,\cdots,m_{\ell}$. This together with (93) leads to

[TABLE]

Also we calculate

[TABLE]

and

[TABLE]

for some constant $C$ depending only on $(\Sigma,g)$ and $G_{\alpha,p}$. Hence we have

[TABLE]

Denote $\phi_{\alpha,\epsilon}^{(\ell)}=\sum_{k=1}^{m_{\ell}}\langle\phi_{\alpha,\epsilon}-\bar{\phi}_{\alpha,\epsilon},e_{k}\rangle e_{k}$. Noting that $e_{k}\in C^{1}(\Sigma)$ for $k=1,\cdots,m_{\ell}$, we have by combining (94) and (95) that

[TABLE]

Obviously (94) leads to

[TABLE]

In view of (96) and (97), we have

[TABLE]

Noting that

[TABLE]

and recalling (94), we get

[TABLE]

Similarly we have

[TABLE]

It follows from (98)-(100) that

[TABLE]

This together with (92) leads to

[TABLE]

Comparing (101) with (90), we conclude (91) under the assumption that $J_{\alpha,8\pi}$ has no minimizer on $E_{\ell}^{\perp}$. This completes the proof of $(i)$ of Theorem 3. $\hfill\Box$

Proof of $(ii)$ of Theorem 3. Since the proof is completely analogous to that of $(ii)$ of Theorem 1, we omit the details but leave it to the interested reader. $\hfill\Box$

Proof of $(iii)$ of Theorem 3. Let $\widetilde{M}_{k}\in\mathcal{H}$ be defined as in (20). We set

[TABLE]

Obviously $M_{\ell,k}\in E_{\ell}^{\perp}$. One can check that

[TABLE]

and

[TABLE]

for some positive constants $C_{1}$ and $C_{2}$ depending only on $(\Sigma,g)$, $\alpha$ and $r$. For any $\beta>8\pi$, we take $r>0$ such that $8\pi(1+C_{1}r)<\beta$. Then we have $J_{\alpha,\beta}(M_{\ell,k})\rightarrow-\infty$ as $k\rightarrow+\infty$. This gives the desired result. $\hfill\Box$

Remark 18**.**

When we were students, we learned from Professor Jiayu Li that they mistyped $x_{\epsilon}$ as $p$ at some places in [4]. The first place is in Lemma 2.5, one needs choose a local normal coordinate system around $x_{\epsilon}$. The second place is Lemma 2.9, it should be written as: In $\Sigma\setminus B_{r_{\epsilon}}(x_{\epsilon})$, we have $u_{\epsilon}\geq G_{x_{\epsilon}}-c_{\epsilon}-2\log(\frac{1+\pi h(p)R^{2}}{R^{2}})-A_{x_{\epsilon}}+o_{\epsilon}(1)$, where $o_{\epsilon}(1)\rightarrow 0$ as $\epsilon\rightarrow 0$. And the third place is in the proof of Lemma 2.2, the integral $\int_{\Sigma}|\nabla u_{\epsilon}|^{2}dv_{g}$ should be divided into $\int_{\Sigma\setminus B_{\delta}(x_{\epsilon})}|\nabla u_{\epsilon}|^{2}dv_{g}+\int_{B_{\delta}(x_{\epsilon})\setminus B_{r_{\epsilon}}(x_{\epsilon})}|\nabla u_{\epsilon}|^{2}dv_{g}+\int_{B_{r_{\epsilon}}(x_{\epsilon})}|\nabla u_{\epsilon}|^{2}dv_{g}$ and then be estimated respectively.

Acknowledgements. Y. Yang is supported by the National Science Foundation of China (Grant Nos.11171347 and 11471014). X. Zhu is supported by the National Science Foundation of China (Grant Nos. 41275063 and 11401575).