Nonexistence of extremals for an inequality of Adimurthi-Druet on a closed Riemann surface

TL;DR

This paper proves that for certain parameters, extremals do not exist for a specific inequality of Adimurthi-Druet on closed Riemann surfaces, extending previous results and clarifying the inequality's attainability conditions.

Contribution

The paper establishes the nonexistence of extremals for the Adimurthi-Druet inequality on closed Riemann surfaces for a range of parameters, complementing earlier findings.

Findings

Extremals do not exist for the inequality when \\alpha \\in (\\\alpha^*, \\\lambda_1(\\\Sigma))

Identifies a threshold \\alpha^*<\\lambda_1(\\\Sigma) for nonattainability

Extends previous results by Mancini-Thizy and the authors' earlier work.

Abstract

Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(\Sigma,g)$. Precisely, if $\lambda_1(\Sigma)$ is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number $\alpha^\ast<\lambda_1(\Sigma)$ such that for all $\alpha\in (\alpha^\ast,\lambda_1(\Sigma))$, the supremum $$\sup_{u\in W^{1,2}(\Sigma,g),\,\int_\Sigma udv_g=0,\,\|\nabla_gu\|_2\leq 1}\int_\Sigma \exp(4\pi u^2(1+\alpha\|u\|_2^2))dv_g$$ can not be attained by any $u\in W^{1,2}(\Sigma,g)$ with $\int_\Sigma udv_g=0$ and $\|\nabla_gu\|_2\leq 1$, where $W^{1,2}(\Sigma,g)$ denotes the usual Sobolev space and $\|\cdot\|_2=(\int_\Sigma|\cdot|^2dv_g)^{1/2}$ denotes the $L^2(\Sigma,g)$-norm. This complements our earlier result in [39].

Equations

$$\underset{u\in W^{1,2}(\Sigma ,g),{\int }_{\Sigma }ud{v}_g=0,\Vert {\nabla }_{g}u{\Vert }_2\le 1}{\sup }{\int }_{\Sigma }\exp (4\pi u^2(1+\alpha \Vert u{\Vert }_2^2))d{v}_g$$

$$\Vert u{\Vert }_{W_0^{1,p}(\Omega )}={\left({\int }_{\Omega }|\nabla u{|}^{p}dx\right)}^{1/p}$$

$$\underset{u\in W_0^{1,n}(\Omega ),\Vert u{\Vert }_{W_0^{1,n}(\Omega )}\le 1}{\sup }{\int }_{\Omega }\exp ({\alpha }_n|u{|}^{\frac{n}{n-1}})dx<\infty ,$$

$$\underset{u\in W_0^{1,2}(\Omega ),\Vert u{\Vert }_{W_0^{1,2}(\Omega )}\le 1}{\sup }{\int }_{\Omega }\exp (4\pi u^2(1+\alpha \Vert u{\Vert }_2^2))dx<\infty ;$$

$$\underset{u\in W_0^{1,n}(\Omega ),\Vert u{\Vert }_{W_0^{1,n}(\Omega )}\le 1}{\sup }{\int }_{\Omega }\exp ({\alpha }_n|u{|}^{\frac{n}{n-1}}(1+\alpha \Vert u{\Vert }_n^n{)}^{\frac{1}{n-1}})dx<\infty ,$$

$$\underset{u\in W_0^{1,2}(\Omega ),\Vert \nabla u{\Vert }_2^2-\alpha \Vert u{\Vert }_2^2\le 1}{\sup }{\int }_{\Omega }\exp (4\pi u^2)dx<\infty .$$

$$\underset{u\in W_0^{1,n}(\Omega ),\Vert \nabla u{\Vert }_n^n-\alpha \Vert u{\Vert }_n^n\le 1}{\sup }{\int }_{\Omega }\exp ({\alpha }_n|u{|}^{\frac{n}{n-1}})dx<\infty ;$$

$$\underset{u\in W^{1,n}(M),{\int }_{M}ud{v}_g=0,{\int }_M|{\nabla }_{g}u{|}^{n}d{v}_g\le 1}{\sup }{\int }_M\exp ({\alpha }_n|u{|}^{\frac{n}{n-1}})d{v}_g<\infty ,$$

$$\underset{u\in W^{1,2}(\Sigma ),{\int }_{\Sigma }ud{v}_g=0,{\int }_{\Sigma }|{\nabla }_{g}u{|}^{2}d{v}_g\le 1}{\sup }{\int }_{\Sigma }\exp (4\pi u^2(1+\alpha \Vert u{\Vert }_2^2))d{v}_g<\infty$$

$$\underset{u\in W^{1,2}(\Sigma ),{\int }_{\Sigma }ud{v}_g=0,{\int }_{\Sigma }|{\nabla }_{g}u{|}^{2}d{v}_g-\alpha {\int }_{\Sigma }{u}^{2}d{v}_g\le 1}{\sup }{\int }_{\Sigma }\exp (4\pi u^2)d{v}_g<\infty ;$$

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

$${\lambda }_1(\Sigma )=\underset{u\in \mathcal{H},u\not\equiv 0}{\inf }\Vert {\nabla }_{g}u{\Vert }_2^2/\Vert u{\Vert }_2^2,$$

$${\Lambda }_{\alpha }(\Sigma )=\underset{u\in \mathcal{H},\Vert {\nabla }_{g}u{\Vert }_2\le 1}{\sup }{\int }_{\Sigma }{e}^{4\pi u^2(1+\alpha \Vert u{\Vert }_2^2)}d{v}_g.$$

$${\int }_{\Sigma }|{\nabla }_g{u}_k{|}^{2}d{v}_g=1-\frac{3}{2}{\alpha }_k^2\Vert u_k{\Vert }_2^4+o(\Vert u_k{\Vert }_2^4),$$

$${\Lambda }_{{\alpha }_k}(\Sigma )={\int }_{\Sigma }{e}^{4\pi u_k^2(1+{\alpha }_k\Vert u_k{\Vert }_2^2)}d{v}_g.$$

$$\underset{k\to \infty }{\lim }{\Lambda }_{{\alpha }_k}(\Sigma )={\Lambda }_{{\lambda }_1(\Sigma )}(\Sigma )=+\infty .$$

$$\left\{\begin{array}[]{lll}\Delta_{g}u_{k}=\frac{\beta_{k}}{\lambda_{k}}u_{k}e^{\sigma_{k}u_{k}^{2}}+\gamma_{k}u_{k}-\mu_{k}\,\,\,{\rm in}\,\,\,\Sigma\\[5.16663pt] u_{k}\in\mathcal{H},\,\,\,\|\nabla_{g}u_{k}\|_{2}=1\\[5.16663pt] \beta_{k}=\frac{1+\alpha_{k}\|u_{k}\|_{2}^{2}}{1+2\alpha_{k}\|u_{k}\|_{2}^{2}},\,\,\,\lambda_{k}=\int_{\Sigma}u_{k}^{2}e^{\sigma_{k}u_{k}^{2}}dv_{g}\\[5.16663pt] \sigma_{k}=4\pi(1+\alpha_{k}\|u_{k}\|_{2}^{2}),\,\,\,\gamma_{k}=\frac{\alpha_{k}}{1+2\alpha_{k}\|u_{k}\|_{2}^{2}}\\[5.16663pt] \mu_{k}=\frac{\beta_{k}}{{\rm Vol}_{g}(\Sigma)}\frac{1}{\lambda_{k}}\int_{\Sigma}u_{k}e^{\sigma_{k}u_{k}^{2}}dv_{g},\end{array}\right.$$

$$c_k=u_k(x_k)=\underset{\Sigma }{\max }|u_k|\to +\infty ,x_k\to x_0\in \Sigma$$

$${\int }_{\Sigma }{u}_k^2{e}^{{\sigma }_k{u}_k^2}d{v}_g\ge {\int }_{\Sigma }(e^{{\sigma }_k{u}_k^2}-1)d{v}_g,$$

$${\lambda }_k\to +\infty \mathrm{as}k\to \infty .$$

$$u_k\to 0\mathrm{in}{C}_{\mathrm{loc}}^1(\Sigma \setminus \{x_0\}).$$

$${\int }_{\Sigma }|{\nabla }_g{u}_k{|}^{2}d{v}_g=\frac{1}{1+{\alpha }_k\Vert u_k{\Vert }_2^2}+\frac{{\alpha }_k\Vert u_k{\Vert }_2^4}{1+2{\alpha }_k\Vert u_k{\Vert }_2^2}+o(\Vert u_k{\Vert }_2^4),$$

$$1$$

$$(B_{x_0}(\delta ),\varphi ;\{x^1,x^2\}),$$

$$\tilde{g}(x)=({\varphi }^{-1}{)}^{\ast }g(x)=\exp (f(x))(d{x^1}^2+d{x^2}^2),$$

$$\exp (f(x))=1+O(|x|)=1+O\left({\mathrm{dist}}_g({\varphi }^{-1}(x),x_0)\right).$$

$${\nabla }_{\tilde{g}}=\exp (-f){\nabla }_{{\mathbb{R}}^2},{\Delta }_{\tilde{g}}=-\exp (-f){\Delta }_{{\mathbb{R}}^2},$$

$$g_k(y)=({\varphi }_k^{-1}{)}^{\ast }g(y)=\exp (f_k(y))(d{y^1}^2+d{y^2}^2),$$

$$\exp (f_k(y))=1+O(|y|)=1+O\left({\mathrm{dist}}_g({\varphi }_k^{-1}(y),x_k)\right),$$

$${\nabla }_{g_k}=e^{-f_k}{\nabla }_{{\mathbb{R}}^2},{\Delta }_{g_k}=-e^{-f_k}{\Delta }_{{\mathbb{R}}^2}.$$

Full text

[TABLE]

can not be attained by any $u\in W^{1,2}(\Sigma,g)$ with $\int_{\Sigma}udv_{g}=0$ and $\|\nabla_{g}u\|_{2}\leq 1$, where $W^{1,2}(\Sigma,g)$ denotes the usual Sobolev space and $\|\cdot\|_{2}=(\int_{\Sigma}|\cdot|^{2}dv_{g})^{1/2}$ denotes the $L^{2}(\Sigma,g)$-norm. This complements our earlier result in [39].

††journal: ***

1 Introduction

Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^{n}$, $n\geq 2$, and $W_{0}^{1,p}(\Omega)$ be the usual Sobolev space, namely, the completion of $C_{0}^{\infty}(\Omega)$ under the norm

[TABLE]

for some $p\geq 1$. By the Sobolev embedding theorem, when $p<n$, $W_{0}^{1,p}(\Omega)$ is continuously embedded in $L^{\frac{np}{n-p}}(\Omega)$; when $p>n$, $W_{0}^{1,p}(\Omega)$ is continuously embedded in $C^{1-\frac{n}{p}}(\Omega)$. However $W_{0}^{1,n}(\Omega)$ can not be embedded in $L^{\infty}(\Omega)$. This limit case was solved respectively by Yudovich [44], Pohozaev [34], Peetre [33], Trudinger [37] and Moser [31]. Precisely there holds

[TABLE]

where $\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$ and $\omega_{n-1}$ denotes the area of the unit sphere in $\mathbb{R}^{n}$. Moreover, $\alpha_{n}$ is the best constant in the sense that if $\alpha_{n}$ is replaced by any larger number $\alpha$, the integrals in (1) are still finite, but the supremum is infinite. The existence of extremals for the supremum in (1) was first obtained by Carleson-Chang [6] in the case that $\Omega$ is a unit ball in $\mathbb{R}^{n}$, then extended by Struwe [35] to a domain close to a unit ball in the sense of measure, by Flucher [18] to a general domain in $\mathbb{R}^{2}$, and by Lin [24] to an arbitrary domain in $\mathbb{R}^{n}$.

In literature, (1) is known as the Trudinger-Moser inequality. It was improved by Adimurthi-Druet [1] as follows: Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^{2}$, and $\lambda_{1}(\Omega)$ be the first eigenvalue of the Laplace-Beltrami operator with respect to the Dirichlet boundary condition. Then for any $\alpha<\lambda_{1}(\Omega)$, there holds

[TABLE]

moreover, the above supremum is infinite if $\alpha\geq\lambda_{1}(\Omega)$, where $\|\cdot\|_{p}=(\int_{\Omega}|\cdot|^{p}dx)^{1/p}$. This result was generalized by the author [38] to the higher dimensional case. In particular, $\Omega$ is assumed to be a smooth bounded domain in $\mathbb{R}^{n}$, $n\geq 3$. Define $\lambda_{n}(\Omega)=\inf_{u\in W_{0}^{1,n}(\Omega),\,u\not\equiv 0}\|\nabla u\|_{n}^{n}/\|u\|_{n}^{n}$. Then for any $\alpha<\lambda_{n}(\Omega)$, there holds

[TABLE]

where $\alpha_{n}$ is defined as in (1). Moreover, the supremum in (3) can be attained by some $u\in W_{0}^{1,n}(\Omega)$ with $\|u\|_{W_{0}^{1,n}(\Omega)}=1$ for all $\alpha<\lambda_{n}(\Omega)$. But in the case $n=2$, it was shown by Lu-Yang [25] that extremals of the supremum in (2) exist only for sufficiently small $\alpha>0$. However, among other results, an inequality stronger than (2) was obtained by Tintarev [36]. Namely, if $\alpha<\lambda_{1}(\mathbb{B}_{R})$ with $|\mathbb{B}_{R}|=|\Omega|$, where $|\cdot|$ denotes the Lebesgue measure of a set, then

[TABLE]

Later, it was shown by the author [40] that (4) holds for all $\alpha<\lambda_{1}(\Omega)$, that extremals of the supremum in (4) exist for all $\alpha<\lambda_{1}(\Omega)$, and that similar results are still valid when higher order eigenvalues of the Laplace-Beltrami operator are taken into account. Recently a higher dimensional version of (4) was established by Nguyen [32], say for any $\alpha<\lambda_{n}(\Omega)$,

[TABLE]

also the corresponding extremals exist. One can check that (5) is stronger than (3). Nevertheless one may ask whether or not extremals of (2) exist for $\alpha$ sufficiently close to $\lambda_{1}(\Omega)$. Based on works of Malchiodi-Martinazzi [26], Mancini-Martinazzi [27] and Druet-Thizy [16], it was Mancini-Thizy [28] who gave a negative answer. Namely, when $\alpha$ is sufficiently close to $\lambda_{1}(\Omega)$, the supremum in (2) has no extremal.

Trudinger-Moser inequalities were introduced on Riemannian manifolds by Aubin [4], Cherrier [8] and Fontana [19]. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold, $\alpha_{n}$ be as in (1) and $W^{1,n}(M,g)$ be the usual Sobolev space. Then there holds

[TABLE]

where $\nabla_{g}$ denotes the gradient operator and $dv_{g}$ stands for the volume element. Based on works of Ding-Jost-Li-Wang [13] and Adimurthi-Struwe [2], Li [22, 23] was able to prove the existence of extremals for the supremum in (6). In [39], the author improved (6) in the case $n=2$ as follows: Assuming that $(\Sigma,g)$ is a closed Riemann surface, we get an analog of (2), namely

[TABLE]

for any $\alpha<\lambda_{1}(\Sigma)=\inf_{u\in W^{1,2}(\Sigma,g),\int_{\Sigma}udv_{g}=0,\|u\|_{2}=1}\|\nabla_{g}u\|_{2}^{2}$; moreover the above supremum is infinite when $\alpha\geq\lambda_{1}(\Sigma)$. Furthermore there exists some $\alpha_{\ast}>0$ such that extremals exist when $\alpha<\alpha_{\ast}$. Concerning (4), we also have its analog [40], say for any $\alpha<\lambda_{1}(\Sigma)$,

[TABLE]

in addition, extremals exist for any $\alpha<\lambda_{1}(\Sigma)$. Also (8) is stronger than (7). In view of [28], we suspect that extremals of the supremum in (7) do not exist for $\alpha$ sufficiently close to $\lambda_{1}(\Sigma)$. Our aim is to confirm this suspicion. Define a function space

[TABLE]

Then $\lambda_{1}(\Sigma)$ can be equivalently written as

[TABLE]

where $\|\cdot\|_{p}=(\int_{\Sigma}|\cdot|^{p}dv_{g})^{1/p}$. We denote for simplicity

[TABLE]

Our main result reads

Theorem 1**.**

Let $(\Sigma,g)$ be a closed Riemann surface, $\lambda_{1}(\Sigma)$ and $\Lambda_{\alpha}(\Sigma)$ be defined as in (10) and (11) respectively. There exists some $\alpha^{\ast}<\lambda_{1}(\Sigma)$, such that if $\alpha^{\ast}<\alpha<\lambda_{1}(\Sigma)$, then the supremum $\Lambda_{\alpha}$ has no extremal.

The proof of Theorem 1 is essentially based on the method of energy estimate, which was used by Mancini-Martinazzi [27] and Mancini-Thizy [28]. Let us describe its outline. Suppose Theorem 1 dose not hold. There would exist a sequence of numbers $\alpha_{k}$ increasingly tending to $\lambda_{1}(\Sigma)$ such that $\Lambda_{\alpha_{k}}=\int_{\Sigma}e^{4\pi u_{k}^{2}(1+\alpha_{k}\|u_{k}\|_{2}^{2})}dv_{g}$ for some function $u_{k}\in\mathcal{H}$ with $\|\nabla_{g}u_{k}\|_{2}=1$. Clearly $u_{k}$ satisfies the corresponding Euler-Lagrange equation (see (14) below). Moreover $u_{k}\rightarrow 0$ strongly in $L^{p}(\Sigma,g)$ for any $p>0$. By performing local blow-up analysis and global analysis on $u_{k}$, we obtain

[TABLE]

which contradicts $\|\nabla_{g}u_{k}\|_{2}=1$ for sufficiently large $k$, since $\alpha_{k}\rightarrow\lambda_{1}(\Sigma)$. It should be remarked that since $u_{k}$ changes sign in our case, we must re-establish a gradient estimate on $u_{k}$ instead of Druet’s original one [15]; moreover, when performing local blow-up analysis, we must take into account the fact that $u_{k}$ may change sign near blow-up points. Here we choose a sequence of isothermal coordinates instead of a fixed isothermal coordinate. Such coordinates greatly simplify the energy calculation. When estimating the energy on domains away from blow-up points, we use the Hölder inequality and the global convergence of $u_{k}/\|u_{k}\|_{p}$ for any $p\geq 2$. This is different from ([28], Step 3.4).

Before ending this introduction, we mention related works such as de Souza-do Ó [9, 14], Ishiwata [20], Martinazzi [29], Martinazzi-Struwe [30], Lamm-Robert-Struwe [21], Adimurthi-Yang [3], del Pino-Musso-Ruf [10, 11], Yang [42] and Figueroa-Musso [17]. The remaining part of this paper will be organized as follows: In Section 2, we prove Theorem 1 by using an energy estimate (Proposition 3); In Section 3, we prove Proposition 3 by using blow-up analysis. Hereafter we do not distinguish sequence and subsequence; moreover, we often denote various constants by the same $C$.

2 Proof of Theorem 1

Let $\mathcal{H}$, $\lambda_{1}(\Sigma)$ and $\Lambda_{\alpha}(\Sigma)$ be defined as in (9), (10) and (11) respectively. In this section, we shall prove Theorem 1 by contradiction. Suppose the contrary. There would be a sequence of numbers $(\alpha_{k})$ increasingly converging to $\lambda_{1}(\Sigma)$, such that $\Lambda_{\alpha_{k}}(\Sigma)$ can be attained by some function $u_{k}\in\mathcal{H}$ with $\|\nabla_{g}u_{k}\|_{2}=1$, namely

[TABLE]

Obviously $(\Lambda_{\alpha_{k}}(\Sigma))$ is an increasing sequence with respect to $k$. Since the supremum in (7) is infinite when $\alpha\geq\lambda_{1}(\Sigma)$, there holds

[TABLE]

By a straightforward calculation, $u_{k}$ satisfies the following Euler-Lagrange equation:

[TABLE]

where $\nabla_{g}$ and $\Delta_{g}$ represent the gradient and the Laplace-Beltrami operator respectively. Denote $c_{k}=\max_{\Sigma}|u_{k}|$. Since $-u_{k}$ is also a maximizer of $\Lambda_{\alpha_{k}}(\Sigma)$, in view of (13), we can assume up to a subsequence,

[TABLE]

as $k\rightarrow\infty$. Note that

[TABLE]

which together with (12) and (13) leads to

[TABLE]

To proceed, we observe an energy concentration phenomenon of $u_{k}$, namely

Lemma 2**.**

$u_{k}$* converges to [math] weakly in $W^{1,2}(\Sigma,g)$, strongly in $L^{p}(\Sigma)$ for any $p\geq 1$, and almost everywhere in $\Sigma$. Moreover, $|\nabla_{g}u_{k}|^{2}dv_{g}\rightharpoonup\delta_{x_{0}}$ in the sense of measure. As a consequence, there holds $\sigma_{k}\rightarrow 4\pi$, $\beta_{k}\rightarrow 1$ and $\gamma_{k}\rightarrow\lambda_{1}(\Sigma)$ as $k\rightarrow\infty$.*

Since the proof of Lemma 2 is an obvious analog of that of ([39], Lemma 4.3), we omit it, but refer the reader to [39] for details. In view of Lemma 2, we have by applying elliptic estimates to (14) that

[TABLE]

We now state the following energy estimate:

Proposition 3**.**

Let $(u_{k})$ satisfy (12) and particularly satisfy (14). Then we have up to a subsequence,

[TABLE]

where $o(\|u_{k}\|_{2}^{4})/\|u_{k}\|_{2}^{4}\rightarrow 0$ as $k\rightarrow\infty$.

The proof of Proposition 3 will be postponed to the subsequent section. Assuming this, we conclude Theorem 1 as follows:

Completion of the proof of Theorem 1. It follows from Lemma 2 that $\|u_{k}\|_{2}\rightarrow 0$ as $k\rightarrow\infty$. Keeping in mind $\|\nabla_{g}u_{k}\|_{2}=1$, we have by (18) that

[TABLE]

which is a contradiction since $\alpha_{k}\rightarrow\lambda_{1}(\Sigma)>0$ as $k\rightarrow\infty$. $\hfill\Box$

3 Energy estimate

In this section, we prove Proposition 3 by analyzing the global and local asymptotic behavior of $u_{k}$.

3.1 Isothermal coordinates

We begin with the choice of a sequence of isothermal coordinate systems near blow-up points. It is well known (see for example Bers [5], Lecture 3) that there exists an isothermal coordinate system near any point of a Riemann surface. Let $(\Sigma,g)$ be the Riemann surface given as in Theorem 1. In particular, for $x_{0}$ given by (15), there exists a real number $\delta>0$ and an isothermal coordinate system

[TABLE]

where $B_{x_{0}}(\delta)\subset\Sigma$ denotes a geodesic ball centered at $x_{0}$ with radius $\delta$, $\phi(x_{0})=(0,0)$. In this coordinate system, the metric $g$ can be represented by

[TABLE]

where $x=(x^{1},x^{2})$, $f$ is smooth on $\phi(B_{x_{0}}(\delta))\subset\mathbb{R}^{2}$, $f(0)=0$ and

[TABLE]

Here and in the sequel, ${\rm dist}_{g}(\cdot,\cdot)$ stands for the geodesic distance between two points of $\Sigma$. Moreover, in the above coordinate system, the gradient operator and the Laplace-Beltrami operator read as

[TABLE]

where $\nabla_{\mathbb{R}^{2}}$ and $\Delta_{\mathbb{R}^{2}}$ denote the standard gradient operator and Laplacian operator on $\mathbb{R}^{2}$ respectively. Recall $c_{k}=u_{k}(x_{k})=\max_{\Sigma}|u_{k}|\rightarrow+\infty$ and $x_{k}\rightarrow x_{0}$ as $k\rightarrow\infty$ (see (15) above). Though in many cases of the current topic (see for examples [22, 39, 41, 17]) the isothermal coordinates (19), especially its properties (21) and (22), had been well used, sometimes (see Section 3.4 below) a sequence of isothermal coordinates will be essentially needed, namely

Lemma 4**.**

There exists some integer $k_{0}$ such that for any $k\geq k_{0}$, one can find an isothermal coordinate system $(B_{x_{k}}(\delta/2),\phi_{k};\{y^{1},y^{2}\})$ satisfying $\phi_{k}(x_{k})=0$ and

[TABLE]

where $f_{k}:\phi_{k}(B_{x_{k}}(\delta/2))\rightarrow\mathbb{R}$ is a smooth function with $f_{k}(0)=0$, $|\nabla^{2}_{\mathbb{R}^{2}}f_{k}|\leq C$,

[TABLE]

$C^{-1}|y|\leq{\rm dist}_{g}(\phi_{k}^{-1}(y),x_{k}\leq C|y|$* and $|O(|y|)|\leq C|y|$ for some constant $C$ independent of $k$. Moreover, there holds*

[TABLE]

Proof. Near the point $x_{0}$, we first take an isothermal coordinate system $(B_{x_{0}}(\delta),\phi;\{x^{1},x^{2}\})$ given as in (19). Since $x_{k}\rightarrow x_{0}$ as $k\rightarrow\infty$, there must be an integer $k_{0}$ such that if $k\geq k_{0}$, then $B_{x_{k}}(\delta/2)\subset B_{x_{0}}(\delta)$. Thus we have the coordinates $(B_{x_{k}}(\delta/2),\phi;\{x^{1},x^{2}\})$, in which $\widetilde{x}_{k}=\phi(x_{k})$ and the metric $g$ can be represented by (20). Now we take another coordinate system around $x_{k}$, namely $(B_{x_{k}}(\delta/2),\phi_{k};\{y^{1},y^{2}\})$ satisfying

[TABLE]

Set $f_{k}(y)=f(x)-f(\widetilde{x}_{k})$ and $g_{k}=(\phi_{k}^{-1})^{\ast}g$. Clearly $\phi_{k}(x_{k})=0$; $f_{k}:\phi_{k}(B_{x_{k}}(\delta/2))\rightarrow\mathbb{R}$ is smooth, $f_{k}(0)=0$, and $|\nabla^{2}_{\mathbb{R}^{2}}f_{k}|$ is uniformly bounded on $\phi_{k}(B_{x_{k}}(\delta/2))$, in particular, (24) holds. Moreover, since $dy=\exp(f(\widetilde{x}_{k})/2)dx$, we have

[TABLE]

This gives (23). Finally (25) follows from (23) immediately. $\hfill\Box$

3.2 Global analysis

From now on, in the above isothermal coordinates $(B_{x_{k}}(\delta/2),\phi_{k};\{y^{1},y^{2}\})$, we write $\widetilde{x}=\phi_{k}(x)$ for all $x\in B_{x_{k}}(\delta/2)$ and $\widetilde{u}=u\circ\phi_{k}^{-1}$ for all functions $u:B_{x_{k}}(\delta/2)\rightarrow\mathbb{R}$. Let

[TABLE]

where $\lambda_{k}$, $\beta_{k}$ and $\sigma_{k}$ are defined as in (14). Using the same argument as in the proof of ([39], Lemma 4.4), we have that for any fixed real number $a<4\pi$,

[TABLE]

It follows from (14), elliptic estimates and a result of Chen-Li [7] that

[TABLE]

For any $\theta\in(0,1)$, we recall the truncation $u_{k,\theta}=\min\{u_{k},\theta c_{k}\}$ defined by [22, 39]. An obvious analog of ([39], Lemma 4.5) gives that

[TABLE]

To understand the asymptotic behavior of $u_{k}$ away from $x_{0}$, we need the following:

Lemma 5**.**

$(i)$* $c_{k}^{2}/\lambda_{k}=o_{k}(1)$; $(ii)$ $\int_{\Sigma}\frac{\beta_{k}}{\lambda_{k}}c_{k}|u_{k}|\exp{(\sigma_{k}u_{k}^{2})}dv_{g}=1+o_{k}(1)$, $\int_{\Sigma}\frac{\beta_{k}}{\lambda_{k}}c_{k}u_{k}\exp{(\sigma_{k}u_{k}^{2})}dv_{g}=1+o_{k}(1)$.*

Proof. Let $\theta\in(0,1)$ be fixed. In view of (29) and the fact that $\int_{\Sigma}u_{k,\theta}dv_{g}=o_{k}(1)$, Fontana’s inequality (6) implies that $\exp{(\sigma_{k}u_{k,\theta}^{2})}$ is bounded in $L^{p}(\Sigma,g)$ for some $p>1$. This together with Lemma 2 leads to

[TABLE]

It then follows that

[TABLE]

Since $\exp{(\sigma_{k}u_{k,\theta}^{2})}$ is bounded in $L^{p}(\Sigma,g)$ and $u_{k}$ converges to [math] in $L^{q}(\Sigma,g)$ for any fixed $q>1$, we conclude by using the Hölder inequality that

[TABLE]

Combining (30), (31) and the definition of $\lambda_{k}$ (see (14)), we have

[TABLE]

This together with (13) concludes $(i)$.

We now prove $(ii)$. Given any $\theta\in(0,1)$. It follows from $(i)$ and (30) that

[TABLE]

As a consequence,

[TABLE]

which leads to

[TABLE]

Nevertheless, by the definition of $\lambda_{k}$, it is obvious to see

[TABLE]

Since $\theta\in(0,1)$ is arbitrary, we have by combining (32) and (33) that

[TABLE]

The other equality in $(ii)$ can be derived in the same way. $\hfill\Box$

Lemma 6**.**

There holds ${(\log\lambda_{k})}/{c_{k}^{2}}=o_{k}(1)$.

Proof. Given any $\nu$, $0<\nu<4\pi$. Since $\|\nabla_{g}u_{k}\|_{2}=1$, we have by Fontana’s inequality (6) that

[TABLE]

for some constant $C>0$ depending on $\nu$. Hence

[TABLE]

which leads to

[TABLE]

This together with (15) and (16) implies that

[TABLE]

Since $0<\nu<4\pi$ is arbitrary, we get the desired result. $\hfill\Box$

Moreover, we have the following:

Lemma 7**.**

For any $p\geq 2$, there holds $c_{k}\|u_{k}\|_{p}\rightarrow\infty$ as $k\rightarrow\infty$.

Proof. By the Hölder inequality, it suffices to prove $c_{k}\|u_{k}\|_{2}\rightarrow\infty$. Suppose not. There would exist some constant $C$ such that $c_{k}\|u_{k}\|_{2}\leq C$. Hence $4\pi\alpha_{k}u_{k}^{2}\|u_{k}\|_{2}^{2}\leq C$ on $\Sigma$. This together with Fontana’s inequality (6) leads to

[TABLE]

which contradicts (13). $\hfill\Box$

Furthermore, we describe the global convergence of $u_{k}$ as follows:

Lemma 8**.**

For any $p\geq 2$ and any $r>1$, there holds up to a subsequence ${u_{k}}/{\|u_{k}\|_{p}}\rightarrow\psi_{p}$ in $L^{r}(\Sigma,g)$, where $\psi_{p}$ is a smooth solution of the equation

[TABLE]

Proof. By (14), we have

[TABLE]

In view of Lemma 5, both $\mu_{k}c_{k}$ and $\|\frac{\beta_{k}}{\lambda_{k}}c_{k}u_{k}\exp{(\sigma_{k}u_{k}^{2})}\|_{1}$ are bounded. Since $\gamma_{k}$ is also bounded, we conclude from Lemma 7 that $h_{k}/(c_{k}\|u_{k}\|_{p})$ is bounded in $L^{1}(\Sigma,g)$ for any $p\geq 2$. In view of (35), one has

[TABLE]

For any fixed $r>1$, we take $q<2$ such that $r<2q/(2-q)$. In view of (36), using the Green representation formula as in the proof of ([43], Lemma 2.10), we have that $u_{k}/\|u_{k}\|_{p}$ is bounded in $W^{1,q}(\Sigma,g)$. Hence there exists some function $\psi_{p}$ such that $u_{k}/\|u_{k}\|_{p}$ converges to $\psi_{p}$ weakly in $W^{1,q}(\Sigma,g)$, and strongly in $L^{r}(\Sigma,g)$. Clearly $\psi_{p}$ is a distributional solution of (34). Applying the regularity theory to (34), we have that $\psi_{p}$ is smooth. This ends the proof of the lemma. $\hfill\Box$

A comparison between $\|u_{k}\|_{p}$ and $\|u_{k}\|_{2}$ reads

Lemma 9**.**

For any $p\geq 2$, we have $\|u_{k}\|_{p}^{2}=\|u_{k}\|_{2}^{2}/(\|\psi_{p}\|_{2}^{2}+o_{k}(1))$.

Proof.

By Lemma 8,

[TABLE]

which leads to

[TABLE]

Note that $\|\psi_{p}\|_{2}>0$ since $\psi_{p}\not\equiv 0$. We get the desired result. $\hfill\Box$

To proceed, we need gradient estimates for $u_{k}$, which are analogs of [15, 29, 30, 21, 41]. The difference is that $u_{k}$ changes sign in our case.

3.3 Gradient estimates

Recalling $x_{k}\rightarrow x_{0}$ as $k\rightarrow\infty$, we first prove a weak gradient estimate for $u_{k}$ as below.

Lemma 10**.**

There exists a constant $C$ depending only on $(\Sigma,g)$ such that for all $x\in\Sigma$ and all $k$,

[TABLE]

Proof. Suppose not. There would exist $(x_{1,k})\subset\Sigma$ such that

[TABLE]

Let $r_{1,k}>0$ satisfy

[TABLE]

It follows easily from (16), (17), (38) and (39) that $|u_{k}(x_{1,k})|\rightarrow+\infty$, $r_{1,k}\rightarrow 0$ and $x_{1,k}\rightarrow x_{0}$. By (26) and (39), we have $r_{1,k}\geq r_{k}$. Also (38) leads to

[TABLE]

as $k\rightarrow\infty$. Take an isothermal coordinate system $(B_{x_{1,k}}(\delta/2),\phi_{k};\{y^{1},y^{2}\})$ around $x_{1,k}$, which is constructed as in Lemma 4, where $x_{k}$ is replaced by $x_{1,k}$. In particular, $\widetilde{x}_{1,k}=\phi_{k}(x_{1,k})=0$ and $\widetilde{x}_{k}=\phi_{k}(x_{k})\in\phi_{k}(B_{x_{1,k}}(\delta/2)$. Define for $y\in\mathcal{D}_{k}=\{y\in\mathbb{R}^{2}:r_{1,k}y\in\phi_{k}(B_{x_{1,k}}(\delta/2)\}$,

[TABLE]

where $\widetilde{u}_{k}=u_{k}\circ\phi_{k}^{-1}$. It follows from (14) and (25) that

[TABLE]

Let $R$ be any fixed positive number. Since ${\rm dist}_{g}(x_{1,k},x_{k})=(1+o_{k}(1))|\widetilde{x}_{k}|$ and $|r_{1,k}y-\widetilde{x}_{k}|=(1+o_{k}(1)){\rm dist}_{g}(\phi_{k}^{-1}(r_{1,k}y),x_{k})$ for all $|y|\leq R$, we have by (38) and (40) that

[TABLE]

and that for all $|y|\leq R$,

[TABLE]

This leads to

[TABLE]

Since $R>0$ is arbitrary, having in mind (43), we obtain by applying elliptic estimates to (41),

[TABLE]

Define for $y\in\mathcal{D}_{k}$,

[TABLE]

Then by (14) and (25),

[TABLE]

In view of (42) and (44), one has $\exp{(\sigma_{k}(2+o_{k}(1))\varphi_{k}(y))}\leq 1+o_{k}(1)$ for all $|y|\leq R$. This together with the fact $\sigma_{k}\rightarrow 4\pi$ implies

[TABLE]

It follows from (44) that

[TABLE]

which together with Lemma 2 gives

[TABLE]

Combining (46) and (47), we obtain by applying elliptic estimates to the equation (45),

[TABLE]

where $\varphi$ is defined as in (28). In view of (28) and (48), we have by noticing that $\beta_{k}=1+o_{k}(1)$,

[TABLE]

which is impossible if $R$ is chosen sufficiently large since $\int_{\mathbb{R}^{2}}\exp{(8\pi\varphi(y))}dy=1$. This confirms (37) and completes the proof of the lemma. $\hfill\Box$

We next prove a strong gradient estimate similar to that of DelaTorre-Mancini [12], namely.

Lemma 11**.**

There exists some constant $C$ such that for all $x\in\Sigma$ and all $k$,

[TABLE]

Proof. Suppose (49) does not hold. Then $L_{k}=\sup_{x\in\Sigma}{\rm dist}_{g}(x,x_{k})|u_{k}(x)||\nabla_{g}u_{k}(x)|\rightarrow+\infty$ as $k\rightarrow\infty$. Take $x_{2,k}$ such that

[TABLE]

Noting that $u_{k}\rightarrow 0$ in $C^{2}_{\rm loc}(\Sigma\setminus\{x_{0}\})$, we have $x_{2,k}\rightarrow x_{0}$ as $k\rightarrow\infty$, where $x_{0}$ is given as in (15). Take an isothermal coordinate system $(B_{x_{2,k}}(\delta/2),\phi_{k};\{y^{1},y^{2}\})$ around $x_{2,k}$, constructed as in Lemma 4. In particular, $\phi_{k}(x_{2,k})=0$, $\widetilde{x}_{k}=\phi_{k}(x_{k})\in\phi_{k}(B_{x_{2,k}}(\delta/2))\subset\mathbb{R}^{2}$. In this coordinate system, we have

[TABLE]

where $\widetilde{u}_{k}=u_{k}\circ\phi_{k}^{-1}$. Denote $s_{k}=|\widetilde{x}_{k}|$ and $y_{k}=\widetilde{x}_{k}/s_{k}$. Assume $y_{k}\rightarrow\overline{y}$ as $k\rightarrow\infty$, where $|\overline{y}|=1$. Define $v_{k}(y)=\widetilde{u}_{k}(s_{k}y)$ for $y\in\mathcal{U}_{k}=\{y\in\mathbb{R}^{2}:s_{k}y\in\phi_{k}(B_{x_{2,k}}(\delta/2))\}$. By (14) and (25),

[TABLE]

By Lemma 10, there exists a constant $C$ such that for all $y\in\mathcal{U}_{k}$,

[TABLE]

This implies that for any fixed $R>0$, there exists some constant $C$ depending on $R$ satisfying

[TABLE]

Moreover, by a change of variables

[TABLE]

Also, we have $|\nabla_{\mathbb{R}^{2}}v_{k}^{2}(y)|\leq CL_{k}$ in $\mathcal{U}_{k}\setminus\mathbb{B}_{\overline{y}}(\frac{1}{R})$. Hence for all $y\in\mathbb{B}_{0}(R)\setminus\mathbb{B}_{\overline{y}}(\frac{1}{R})$

[TABLE]

where $M_{k}=|v_{k}(0)|+|\nabla v_{k}(0)|$, since by (50), $L_{k}=(1+o_{k}(1))|v_{k}(0)||\nabla_{\mathbb{R}^{2}}v_{k}(0)|\leq(1+o_{k}(1))M_{k}^{2}$. Obviously $M_{k}\rightarrow\infty$ as $k\rightarrow\infty$. Define $\overline{v}_{k}(y)=v_{k}(y)/M_{k}$. In view of (51)-(54) and the fact that $\mu_{k}$ is bounded, we conclude by using elliptic estimates that $\overline{v}_{k}\rightarrow v$ in $C^{1}_{\rm loc}(\mathbb{R}^{2}\setminus\{\overline{y}\})$, where $v$ is a harmonic function in $\mathbb{R}^{2}\setminus\{\overline{y}\}$. Since for any $R>0$,

[TABLE]

we obtain

[TABLE]

Hence $v\equiv C$ for some constant $C$ and $|\nabla_{\mathbb{R}^{2}}\overline{v}_{k}(0)|\rightarrow|\nabla_{\mathbb{R}^{2}}v(0)|=0$. Thus $|\nabla_{\mathbb{R}^{2}}{v}_{k}(0)|=o(M_{k})$ and

[TABLE]

Since $L_{k}=(1+o_{k}(1))|v_{k}(0)||\nabla_{\mathbb{R}^{2}}v_{k}(0)|\rightarrow+\infty$ as $k\rightarrow\infty$, we have $|\nabla_{\mathbb{R}^{2}}v_{k}(0)|>0$ for sufficiently large $k$. Define for $y\in\mathcal{U}_{k}$,

[TABLE]

By (51), we have

[TABLE]

On one hand, by definition of $L_{k}$ and (56), we have $v^{\ast}_{k}\in L^{\infty}_{\rm loc}(\mathbb{R}^{2}\setminus\{\overline{y}\})$. On the other hand, by $L_{k}\rightarrow+\infty$, analogs of (52) and (53), we conclude that $\Delta_{\mathbb{R}^{2}}v^{\ast}_{k}(y)\rightarrow 0$ in $L^{p}_{\rm loc}(\mathbb{R}^{2}\setminus\{\overline{y}\})$ for any $p>1$. Then elliptic estimates leads to $v_{k}^{\ast}\rightarrow v^{\ast}$ in $C^{1}_{\rm loc}(\mathbb{R}^{2}\setminus\{\overline{y}\})$, where $v^{\ast}$ is a harmonic function. Similarly as (55), we have

[TABLE]

Noting that $v_{k}^{\ast}(0)=0$, we conclude $v^{\ast}\equiv 0$ in $\mathbb{R}^{2}\setminus\{\overline{y}\}$. This contradicts $|\nabla_{\mathbb{R}^{2}}v^{\ast}(0)|=1$ and completes the proof of the lemma. $\hfill\Box$

3.4 Local blow-up analysis

We now consider the local behavior of $u_{k}$ near $x_{0}$. In the isothermal coordinate system $(B_{x_{k}}(\delta/2),\phi_{k};\{y^{1},y^{2}\})$ defined as in Lemma 4, for convenience, we rewrite (28) by

[TABLE]

where $\varphi_{0}(y)=-\log(1+|y|^{2})$ satisfies

[TABLE]

We define

[TABLE]

For any fixed $\tau$ with $0<\tau<1$, we let $r_{k,\tau}>0$ be such that $\varphi_{k}(r_{k,\tau})=-\tau\sigma_{k}c_{k}^{2}$, which leads to

[TABLE]

In view of (27), one has $r_{k,\tau}\rightarrow 0$ as $k\rightarrow\infty$. Here and in the sequel, we slightly abuse a notation. If $u$ is a function radially symmetric with respect to [math], we write $u(r)=u(x)$ with $|x|=r$.

Let $S_{k}$ be the radially symmetric solution of

[TABLE]

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

[TABLE]

According to [15], $\eta_{0}$ satisfies

[TABLE]

and

[TABLE]

Set

[TABLE]

For the decomposition of $S_{k}$, we have the following:

Lemma 12**.**

For any $z_{k}\in\mathbb{B}_{0}(r_{k,\tau})$, there holds

[TABLE]

Proof. Let $w_{1,k}$ be given by

[TABLE]

and define $\rho_{1,k}$ by

[TABLE]

By (57), we have

[TABLE]

Also, by (58), (59) and (61), we get

[TABLE]

which together with (64) and (65) leads to

[TABLE]

By (63), (65) and (66), we have on $\mathbb{B}_{0}(\rho_{1,k})$ that

[TABLE]

that

[TABLE]

and that

[TABLE]

Moreover, we have

[TABLE]

Then using the same argument as the proof of ([28], Step 3.2), and noting that Lemma 6 is an obvious substitution for ([28], (3.16)), we obtain

[TABLE]

This together with (64) implies $\rho_{1,k}=r_{k,\tau}$ and yields the desired result. $\hfill\Box$

Let

[TABLE]

We claim that up to a subsequence

[TABLE]

For otherwise, we suppose for all $k$,

[TABLE]

By (26) and (28), there holds

[TABLE]

Clearly we have for all $y\in\mathbb{B}_{0}(\rho_{k})$

[TABLE]

In view of (28), (69), (71), (72), Lemma 11 and the fact that $f_{k}(0)=0$, using an argument of ([16], Proposition 3.1), we obtain

[TABLE]

By Lemma 12, $S_{k}(y)\geq(1-\tau+o_{k}(1))\sqrt{\sigma_{k}}\,c_{k}$ for all $y\in\mathbb{B}_{0}(r_{k,\tau})$. This together with (73) leads to

[TABLE]

which contradicts (69), the definition of $\rho_{k}$. Hence our claim (70) holds. Using again the argument of ([16], Proposition 3.1), we conclude

[TABLE]

In view of (57), (59) and (61), we have that

[TABLE]

We have by applying the inequality $|\exp(t)-1-t|\leq t^{2}\exp{(|t|)}$ for all $t\in\mathbb{R}$ that

[TABLE]

by employing (67) and (68) that

[TABLE]

and by using (74) that

[TABLE]

Since $\varphi_{k}/c_{k}^{2}\geq-\tau$ on $\mathbb{B}_{0}(r_{k,\tau})$, if $\vartheta$ is a real number satisfying $1<\vartheta<2-\tau$, then

[TABLE]

Hence we have by (26) that

[TABLE]

In the same way, we calculate on $\mathbb{B}_{0}(r_{k,\tau})$,

[TABLE]

Note that

[TABLE]

which together with (57) and (60) leads to

[TABLE]

Moreover we have that

[TABLE]

and that

[TABLE]

Combining (75)-(78), we obtain

[TABLE]

since $r_{k}/r_{k,\tau}=O(1/c_{k}^{4})$. It then follows that

[TABLE]

Here we used the fact $r_{k,\tau}=O(1/c_{k}^{4})$, which is due to (27) and (58).

3.5 Energy estimate away from $x_{0}$

To estimate the energy of $u_{k}$ away from $x_{0}$, we compute the integral

[TABLE]

where $\phi_{k}$ is given as in the isothermal coordinate system constructed in Lemma 4. The following observation is very important for this purpose.

Lemma 13**.**

There exist some $p^{\ast}>1$ and a constant $C>0$ such that

[TABLE]

Proof.

Define

[TABLE]

In view of (62) and (74), there exists a constant $\delta_{k}>0$ such that if $x\in\phi_{k}^{-1}(\mathbb{B}_{0}{({r_{k,\tau}})})$ satisfies ${\rm dist}_{g}(x,\partial\phi_{k}^{-1}(\mathbb{B}_{0}{({r_{k,\tau}})}))\leq\delta_{k}$, then there holds $u_{k}^{\ast}(x)=u_{k}(x)$. In particular, $0\leq\nu_{k}=u_{k}-u_{k}^{\ast}\in W_{0}^{1,2}(\phi_{k}^{-1}(\mathbb{B}_{0}(r_{k,\tau})))$ and $u_{k}^{\ast}\in W^{1,2}(\Sigma,g)$. We claim the following:

[TABLE]

By Lemma 2, we have for any $q\geq 1$,

[TABLE]

As a consequence,

[TABLE]

Testing (14) by $u_{k}^{\ast}-\overline{u_{k}^{\ast}}$, we have by (28) and (82),

[TABLE]

where $o_{R}(1)\rightarrow 0$ as $R\rightarrow+\infty$. Testing (14) by $\nu_{k}-\overline{\nu_{k}}$, we have in the same way

[TABLE]

Moreover,

[TABLE]

Combining (84), (85) and (86), we conclude our claim (81).

Let $p_{1}$ be any fixed number satisfying $1<p_{1}<1/(1-\tau/2)$. Then it follows from (81) and Fontana’s inequality (6) that

[TABLE]

In view of (83), one can choose a number $p^{\ast}$ such that $1<p^{\ast}<p_{1}$ and

[TABLE]

This particularly implies (80) and completes the proof of the lemma. $\hfill\Box$

As a consequence of Lemma 13, we obtain by using the Hölder inequality, $(i)$ of Lemma 5, and Lemma 7 that

[TABLE]

where $1/q+1/p^{\ast}=1$.

3.6 Completion of the proof of Proposition 3

Testing the equation (14) by $u_{k}$, in view of (79), (87) and Lemma 7, we have

[TABLE]

This together with Lemma 9 gives (18), as desired. $\hfill\Box$

Acknowledgements. This work is partly supported by the National Science Foundation of China (Grant Nos. 11471014 and 11761131002).