Onofri-Type Inequalities for Singular Liouville Equations

Abstract

We study the blow-up behavior of minimizing sequences for the singular Moser–Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence of at most two singularities.

1 Introduction

Let $(\Sigma ,g)$ be a smooth, compact Riemannian surface; the standard Moser–Trudinger inequality (see [16, 22]) states that

$$\begin{aligned} \log \left( \frac{1}{|\Sigma |}\int _{\Sigma } e^{u-\overline{u}}dv_g\right) \le \frac{1}{16\pi }\int _{\Sigma } |\nabla _g u|^2 dv_g + C(\Sigma ,g)\qquad \forall \; u\in H^1(\Sigma ) \end{aligned}$$

(1)

where $C(\Sigma ,g)$ is a constant depending only on $\Sigma$ and $g$, and the coefficient $\frac{1}{16\pi }$ is optimal. A sharp version of (1) was proved by Onofri in [23] for the sphere endowed with the standard Euclidean metric $g_0$. He identified the sharp value of $C$ and the family of functions attaining equality, proving

$$\begin{aligned} \log \left( \frac{1}{4\pi }\int _{S^2} e^{u-\overline{u}} dv_{g_0}\right) \le \frac{1}{16\pi }\int _{S^2} |\nabla _{g_0} u|^2dv_{g_0} \end{aligned}$$

(2)

with equality holding if and only if the metric $e^{u}g$ has constant positive Gaussian curvature, or, equivalently, $u=\log |\det d \varphi |+c$ with $c\in {\mathbb {R}}$ and $\varphi$ a conformal diffeomorphism of $S^2$. Onofri’s inequality played an important role (see [12, 13]) in the variational approach to the equation

$$\begin{aligned} \Delta _{g_0} u +K \;e^{u}=1 \end{aligned}$$

which is connected to the classical problem of prescribing the Gaussian curvature of $S^2$. In this paper we will consider extensions of Onofri’s result in connection with the study of the more general equation

$$\begin{aligned} {-}\Delta _g v = \rho \left( \frac{K e^{v}}{\int _{\Sigma } K e^{v}dv_g}-\frac{1}{|\Sigma |} \right) -4\pi \sum _{i=1}^m \alpha _i\left( \delta _{p_i}-\frac{1}{|\Sigma |}\right) , \end{aligned}$$

(3)

where $K\in C^\infty (\Sigma )$ is a positive function, $\rho >0$, $p_1,\ldots , p_m\in \Sigma$ and $\alpha _1,\ldots ,\alpha _m \in (-1,+\infty )$. This is known as the singular Liouville equation and arises in several problems in Riemannian geometry and mathematical physics. When $(\Sigma ,g)=(S^2,g_0)$ and $\rho = 8\pi +4\pi \sum _{i=1}^m \alpha _i$, solutions of (3) provide metrics on $S^2$ with prescribed Gaussian curvature $K$ and conical singularities of angle $2\pi (1+\alpha _i)$ (or of order $\alpha _i$) in $p_i$, $i=1,\ldots ,m$ (see for example [3, 14, 27]). Equation (3) also appears in the description of Abelian Chern–Simons vortices in superconductivity and Electroweak theory [17, 25]. We refer to [4, 9–11, 21], for some recent existence results. Liouville equations also have applications in the description of holomorphic curves in ${\mathbb {C}}{\mathbb {P}}^n$ [6, 8] and in the nonabelian Chern–Simons theory which might have applications in high temperature superconductivity (see [26] and references therein). Denoting by $G_p$ the Green’s function at $p$, namely the solution of

$$\begin{aligned} {\left\{ \begin{array}{l} {-}\Delta _g G_{p} = \delta _p- \frac{1}{|\Sigma |}\\ \int _{\Sigma } G_p\; dv_g =0 \end{array} \right. }, \end{aligned}$$

the change of variables

$$\begin{aligned} u= v+ 4\pi \sum _{i=1}^m \alpha _i G_{p_i} \end{aligned}$$

transforms (3) into

$$\begin{aligned} {-}\Delta _g u = \rho \left( \frac{h e^{u}}{\int _{\Sigma } h e^{u}dv_g}-\frac{1}{|\Sigma |} \right) \end{aligned}$$

(4)

where

$$\begin{aligned} h= K\prod _{1\le i\le m} e^{-4\pi \alpha _i G_{p_i}} \end{aligned}$$

(5)

satisfies

$$\begin{aligned} h(p)\approx c_i\; d(p,p_i)^{2\alpha _i} \; \text{ for } p\approx p_i, \end{aligned}$$

(6)

with $c_i>0$.

In [27], studying curvature functions for surfaces with conical singularities, Troyanov proved that if $h\in C^\infty (\Sigma \backslash \{p_1,\ldots ,p_m\})$ is a positive function satisfying (6), then

$$\begin{aligned} \log \left( \frac{1}{|\Sigma |}\int _{\Sigma } h \; e^{u-\overline{u}}dv_g\right) \le \frac{1}{16\pi \displaystyle { \min \left\{ 1,1\!+\!\min _{1\le i\le m}\alpha _i\right\} } }\int _{\Sigma } |\nabla _g u|^2 dv_g+ C(\Sigma ,g,h). \end{aligned}$$

(7)

The optimal constant $C(\Sigma ,g,h)$ can be obtained by minimizing the functional

$$\begin{aligned} J_{\overline{\rho }}(u)=\frac{1}{2}\int _{\Sigma } |\nabla _g u|^2 dv_g+ \frac{\overline{\rho }}{|\Sigma |} \int _{\Sigma } u\; dv_g -\overline{\rho } \log \left( \frac{1}{|\Sigma |}\int _{\Sigma } he^{u}dv_g\right) , \end{aligned}$$

where $\overline{\rho } =\displaystyle {\min \left\{ 1,1+\min _{1\le i\le m}\alpha _i\right\} }$. In this paper we will assume non-existence of minimum points for $J_{\overline{\rho }}$ and exploit known blow-up results [1, 2, 5] to describe the behavior of a suitable minimizing sequence and compute $\displaystyle {\inf _{H^1(\Sigma )}J_{\overline{\rho }}}$. The same technique was used by Ding, Jost, Li and Wang [15] to give an existence result for (3) in the regular case. From their proof it follows that if $\alpha _i=0$ $\forall \; i$ and if there is no minimum for $J_{\overline{\rho }}$, then

$$\begin{aligned} \inf _{H^1(\Sigma )} J_{\overline{\rho }} = -8\pi \left( 1+\log \left( \frac{\pi }{|\Sigma |}\right) +\max _{p\in \Sigma } \left\{ 4\pi A(p)+ \log h(p)\right\} \right) \end{aligned}$$

where $A(p)$ is the value in $p$ of the regular part of $G_p$. Here we extend this result to the general case proving:

Theorem 1.1

Assume that $h$ satisfies (5) with $K\in C^\infty (\Sigma )$, $K>0$, $\alpha _i \in (-1,+\infty )\backslash \{0\}$, and that there is no minimum point of $J_{\overline{\rho }}$. If $\alpha :=\displaystyle {\min _{1\le i\le m} \alpha _i<0}$, then

$$\begin{aligned} \inf _{H^1(\Sigma )} J_{\overline{\rho }}&= -8\pi (1+\alpha ) \left( 1+\log \left( \frac{\pi }{|\Sigma |}\right) \right. \\&\left. +\max _{1\le i\le m,\alpha _i=\alpha }\left\{ 4\pi A(p_i)+ \log \left( \frac{K(p_i)}{1+\alpha }\prod _{j\ne i} e^{-4\pi \alpha _j G_{p_j}(p_i)}\right) \right\} \right) \end{aligned}$$

while if $\displaystyle {\alpha >0}$

$$\begin{aligned} \inf _{H^1(\Sigma )} J_{\overline{\rho }}= -8\pi \left( 1+\log \left( \frac{\pi }{|\Sigma |}\right) +\max _{p\in \Sigma \backslash \{p_1,\ldots ,p_m\}} \left\{ 4\pi A(p)+ \log h(p)\right\} \right) . \end{aligned}$$

In the last part of the paper we consider the case of the standard sphere with $K\equiv 1$ and at most two singularities. When $m=1$ a simple Kazdan–Warner type identity proves non-existence of solutions for (4). Thus, one can apply Theorem 1.1 to obtain the following sharp version of (7):

Theorem 1.2

If $h=e^{-4\pi \alpha _1 G_{p_1}}$ with $\alpha _1\ne 0$, then $\forall \; u\in H^1(S^2)$

$$\begin{aligned} \log \left( \frac{1}{4\pi }\int _{S^2} h e^{u-\overline{u}}dv_{g_0}\right)&< \frac{1}{16\pi \min \{1,1+\alpha _1\}}\int _{S^2} |\nabla u|^2dv_{g_0}\\&+\max \left\{ \alpha _1,-\log (1+\alpha _1)\right\} . \end{aligned}$$

The same non-existence argument works for $m=2$, $\min \{\alpha _1,\alpha _2\}<0$ and $\alpha _1 \ne \alpha _2$ if the singularities are located in two antipodal points.

Theorem 1.3

Assume $h=e^{-4\pi \alpha _1 G_{p_1}-4\pi \alpha _2 G_{p_2}}$ with $p_2= -p_1$, $\alpha _1= \min \{\alpha _1,\alpha _2\}<0$ and $\alpha _1\ne \alpha _2$; then $\forall \; u\in H^1(S^2)$

$$\begin{aligned} \log \left( \frac{1}{4\pi }\int _{S^2} h e^{u-\overline{u}}dv_{g_0}\right) < \frac{1}{16\pi (1+\alpha _1)}\int _{S^2} |\nabla u|^2dv_{g_0}+ \alpha _2-\log (1+\alpha _1). \end{aligned}$$

When $\alpha _1=\alpha _2<0$ Theorem 1.1 cannot be directly applied because (4) has solutions. However, it is possible to use a stereographic projection and a classification result in [24] to find an explicit expression for the solutions. In particular a direct computation allows to prove that all the solutions are minima of $J_{\overline{\rho }}$ and to find the value of $\displaystyle {\min _{H^1(S^2)}J_{\overline{\rho }}}$.

Theorem 1.4

Assume $h=e^{-4\pi \alpha \left( G_{p_1}+G_{p_2}\right) }$ with $\alpha <0$ and $p_1=-p_2$; then $\forall \; u\in H^1(S^2)$ we have

$$\begin{aligned} \log \left( \frac{1}{4\pi }\int _{S^2} h e^{u-\overline{u}}dv_{g_0}\right) \le \frac{1}{16\pi (1+\alpha )}\int _{S^2} |\nabla u|^2dv_{g_0}+ \alpha -\log (1+\alpha ). \end{aligned}$$

Moreover the following conditions are equivalent:

• $u$ realizes equality.

• If $\pi$ denotes the stereographic projection from $p_1$ then

  $$\begin{aligned} u\circ \pi ^{-1}(y)= 2\log \left( \frac{(1+|y|^2)^{1+\alpha }}{1+e^\lambda |y|^{2(1+\alpha )}} \right) +c \end{aligned}$$

  for some $\lambda ,c \in {\mathbb {R}}$.

• $h e^{u} g_0$ is a metric with constant positive Gaussian curvature and conical singularities of order $\alpha _i$ in $p_i$, $i=1,2$.

This is a generalization of Onofri’s inequality (2) for metrics with two conical singularities.

2 Preliminaries and Blow-Up Analysis

Let $(\Sigma ,g)$ be a smooth compact, connected, Riemannian surface and let $S:=\{p_1,\ldots ,p_m\}$ be a finite subset of $\Sigma$. Let us consider a function $h$ satisfying (5) with $K\in C^\infty (\Sigma )$, $K>0$ and $\alpha _i\in (-1,+\infty )\backslash \{0\}$. In order to distinguish the singular points of $h$ from the regular ones, we introduce a singularity index function

$$\begin{aligned} \beta (p):={\left\{ \begin{array}{cc} \alpha _i \quad \text{ if } \quad p=p_i\\ 0 \quad \text{ if } \quad p\notin S \end{array} \right. }. \end{aligned}$$

We will denote $\displaystyle {\alpha := \min _{p\in \Sigma } \beta (p)= \min \left\{ \min _{1\le i\le m}\alpha _i,0\right\} }$ the minimum singularity order. We shall consider the functional

$$\begin{aligned} J_\rho (u)=\frac{1}{2}\int _{\Sigma } |\nabla _g u|^2 dv_g + \frac{\rho }{|\Sigma |} \int _{\Sigma } u\; dv_g -\rho \log \left( \frac{1}{|\Sigma |}\int _{\Sigma } he^{u}dv_g\right) . \end{aligned}$$

(8)

Our goal is to give a sharp version of (7) finding the explicit value of

$$\begin{aligned} C(\Sigma ,g,h) = -\frac{1}{8\pi (1+\alpha )} \inf _{u\in H^1(\Sigma )} J_{8\pi (1+\alpha )} (u). \end{aligned}$$

(9)

To simplify the notation we will set $\overline{\rho }:=8\pi (1+\alpha )$, $\rho _\varepsilon = \overline{\rho }-\varepsilon$, $J_\varepsilon := J_{\rho _\varepsilon }$ and $J:= J_{\overline{\rho }}$. From (7) it follows that $\forall \; \varepsilon >0$ the functional $J_\varepsilon$ is coercive and, by direct methods, it is possible to find a function $u_\varepsilon \in H^1(\Sigma )$ satisfying

$$\begin{aligned} J_\varepsilon (u_\varepsilon )= \inf _{u\in H^1(\Sigma )}J_\varepsilon (u) \end{aligned}$$

(10)

and

$$\begin{aligned} {-}\Delta _g u_\varepsilon = \rho _\varepsilon \left( \frac{h e^{u_\varepsilon }}{\int _{\Sigma } h e^{u_\varepsilon }dv_g} -\frac{1}{|\Sigma |} \right) . \end{aligned}$$

(11)

Since $J_{\varepsilon }$ is invariant under addition of constants $\forall \; \varepsilon >0$, we may also assume

$$\begin{aligned} \int _{\Sigma } h\; e^{u_\varepsilon } dv_g = 1. \end{aligned}$$

(12)

Remark 2.1

$u_\varepsilon \in C^{0,\gamma }(\Sigma )\cap W^{1,s}(\Sigma )$ for some $\gamma \in (0,1)$ and $s>2$.

Proof

It is easy to see that $h\in L^q(\Sigma )$ for some $q >1$ ( $q=+\infty$ if $\alpha =0$ and $q< -\frac{1}{\alpha }$ for $\alpha <0)$. Applying locally Remarks 2 and 5 in [7] one can show that $u_\varepsilon \in L^{\infty }(\Sigma )$ so $-\Delta {u_\varepsilon }\in L^q(\Sigma )$ and by standard elliptic estimates $u_\varepsilon \in W^{2,q}(\Sigma )$. Since $q>1$ the conclusion follows by Sobolev’s embedding theorems.$\square$

The behavior of $u_\varepsilon$ is described by the following concentration-compactness result:

Proposition 2.1

Let $u_n$ be a sequence satisfying

$$\begin{aligned} {-}\Delta _g u_n = V_n e^{u_n} -\psi _n \end{aligned}$$

and

$$\begin{aligned} \int _{\Sigma } V_ne^{u_n} dv_g \le C_1, \end{aligned}$$

where $\Vert \psi _n\Vert _{L^s(\Sigma )}\le C_2$ for some $s>1$, and

$$\begin{aligned} V_n= K_n \prod _{1\le i\le m} e^{-4\pi \alpha _iG_{p_i}} \end{aligned}$$

with $K_n\in C^\infty (\Sigma )$, $0<a\le K_n\le b$ and $\alpha _i >-1$, $i=1,\ldots ,m$. Then there exists a subsequence $u_{n_k}$ of $u_n$ such that the following alternatives hold:

• 1. $u_{n_k}$ is uniformly bounded in $L^\infty (\Sigma )$;

• 2. $u_{n_k}\longrightarrow -\infty$ uniformly on $\Sigma$;

• 3. there exist a finite blow-up set $B =\{q_1,\ldots ,q_l\}\subseteq \Sigma$ and a corresponding family of sequences $\{q^j_k\}_{k\in {\mathbb {N}}}$, $j=1,\ldots , l$ such that $q_k^j\mathop {\longrightarrow }\limits ^{k\rightarrow \infty } q_j$ and $u_{n_k}(q_k^j)\mathop {\longrightarrow }\limits ^{k\rightarrow \infty } +\infty$ $j=1,\ldots ,l$. Moreover $u_{n_k}\mathop {\longrightarrow }\limits ^{k\rightarrow \infty } -\infty$ uniformly on compact subsets of $\Sigma \backslash B$ and $V_{n_k}e^{u_{n_k}} \rightharpoonup \sum _{j=1}^l \beta _j \delta _{q_j}$ weakly in the sense of measures where $\beta _j = 8\pi (1+\beta (q_j))$ for $j=1,\ldots ,l$.

A proof of Proposition 2.1 in the regular case can be found in [19] while the general case is a consequence of the results in [1, 5]. In our analysis we will also need the following local version of Proposition 2.1 proved by Li and Shafrir [20]:

Proposition 2.2

Let $\Omega$ be an open domain in ${\mathbb {R}}^2$ and $v_n$ be a sequence satisfying $\Vert e^{v_n}\Vert _{L^1(\Omega )}\le C$ and

$$\begin{aligned} {-}\Delta v_n = V_n e^{v_n} \end{aligned}$$

where $0\le V_n\in C_0(\overline{\Omega })$ and $V_n\longrightarrow V$ uniformly in $\overline{\Omega }$. If $v_n$ is not uniformly bounded from above on compact subset of $\Omega$, then $V_n e^{v_n} \rightharpoonup \displaystyle {8\pi \sum _{j=1}^lm_j \delta _{q_j}}$ as measures, with $q_j\in \Omega$ and $m_j\in \mathbb {N}^+$, $j=1,\ldots ,l$.

Applying Proposition 2.1 to $u_\varepsilon$ under the additional condition (12) we obtain that either $u_\varepsilon$ is uniformly bounded in $L^\infty (\Sigma )$ or its blow-up set contains a single point $p$ such that $\beta (p)= \alpha$. In the first case, one can use elliptic estimates to find uniform bounds on $u_\varepsilon$ in $W^{2,q}(\Sigma )$, for some $q>1$; consequently, a subsequence of $u_\varepsilon$ converges in $H^{1}(\Sigma )$ to a function $u\in H^1(\Sigma )$ that is a minimum point of $J$ and a solution of (4) for $\rho =\overline{\rho }$. We now focus on the second case, that is

$$\begin{aligned} \lambda _\varepsilon := \max _{\Sigma }u_\varepsilon = u_\varepsilon (p_\varepsilon ) \longrightarrow +\infty \quad \text{ and } \quad p_\varepsilon \longrightarrow p \quad \text{ with } \quad \beta (p)=\alpha . \end{aligned}$$

(13)

By Proposition 2.1 we also get:

Lemma 2.1

If $u_\varepsilon$ satisfies (11), (12) and (13), then, up to subsequences,

• 1. $\rho _\varepsilon h e^{u_\varepsilon } \rightharpoonup \overline{\rho }\; \delta _p$;

• 2. $u_\varepsilon \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}-\infty$ uniformly in $\Omega$, $\forall \;\Omega \subset \subset \Sigma \backslash \{p\}$;

• 3. $\overline{u}_\varepsilon \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}-\infty$;

• 4. There exist $\gamma \in (0,1)$, $s>2$ such that $u_\varepsilon -\overline{u_\varepsilon } \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0} \overline{\rho }\;G_p$ in $C^{0,\gamma }(\overline{\Omega })\cap W^{1,s}(\Omega )$ $\forall$ $\Omega \subset \subset \Sigma \backslash \{p\}$;

• 5. $\nabla u_\varepsilon$ is bounded in $L^q(\Sigma )$ $\forall \; q\in (1,2)$.

Proof

1., 2. and 3. are direct consequences of Proposition 2.1. To prove 4., we consider the Green’s representation formula

$$\begin{aligned} u_\varepsilon (x)-\overline{u}_\varepsilon =\rho _\varepsilon \int _{\Sigma } G_x(y) h(y)e^{u_\varepsilon (y)} dv_g(y). \end{aligned}$$

We stress that the Green’s function has the following properties:

• $\displaystyle {|G_x(y)|\le C_1 (1+ |\log d(x,y)|)}$ $\forall \; x,y \in \Sigma$, $x\ne y$.

• $\displaystyle {|\nabla ^x_g G_x(y)|}\le \frac{C_2}{d(x,y)}$ $\forall \; x,y\in \Sigma$, $x \ne y$.

• $G_x(y)= G_y(x)$ $\forall \; x,y\in \Sigma$, $x\ne y$.

Take $q>1$ such that $h\in L^q(\Sigma )$. The first property also yields

$$\begin{aligned} \sup _{x\in \Sigma } \Vert G_x\Vert _{L^{q'}(\Sigma )} \le C_3. \end{aligned}$$

(14)

Let us fix $\delta >0$ such that $B_{3\delta }(p)\subset \Sigma \backslash \Omega$ and take a cut-off function $\varphi$ such that $\varphi \equiv 1$ in $B_{\delta }(p)$ and $\varphi \equiv 0$ in $\Sigma \backslash B_{2\delta }(p)$.

$$\begin{aligned} u_\varepsilon (x)-\overline{u_\varepsilon }&= \rho _\varepsilon \int _{\Sigma } \varphi (y) G_x(y) h(y)e^{u_\varepsilon (y)} dv_g(y)\\&+\rho _\varepsilon \int _{\Sigma } (1-\varphi (y)) G_x(y) h(y)e^{u_\varepsilon (y)} dv_g(y). \end{aligned}$$

By (14) and 2. we have

$$\begin{aligned} \left| \int _{\Sigma } (1-\varphi (y)) G_x(y) {h}(y)e^{u_\varepsilon (y)} dv_g(y)\right|&\le \int _{\Sigma \backslash B_\delta (p)}\left| G_x(y)\right| {h}(y)e^{u_\varepsilon (y)} dv_g(y)\\&\le C_3 \Vert h\Vert _{L^q(\Sigma )} \Vert e^{u_\varepsilon }\Vert _{L^\infty (\Sigma \backslash B_\delta (p))} \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0} 0. \end{aligned}$$

By 1. and the smoothness of $\varphi G_x$ for $x\in \overline{ \Omega }$ and $y\in \Sigma$ we get

$$\begin{aligned} \int _{\Sigma } \varphi (y) G_x(y) {h}(y)e^{u_\varepsilon (y)} dv_g(y) \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0} \varphi (p)G_x(p)=G_p(x) \end{aligned}$$

uniformly for $x \in \Omega$. Similarly we have

$$\begin{aligned} \nabla _gu_\varepsilon (x)&= \rho _\varepsilon \int _{\Sigma } \varphi (y) \nabla _g^x G_x(y) {h}(y)e^{u_\varepsilon (y)} dv_g(y)\\&+\rho _\varepsilon \int _{\Sigma } (1-\varphi (y)) \nabla _g^x G_x(y) {h}(y)e^{u_\varepsilon (y)} dv_g(y) \end{aligned}$$

with

$$\begin{aligned} \int _{\Sigma } \varphi (y) \nabla ^x_g G_x(y) {h}(y)e^{u_\varepsilon (y)} dv_g(y) \mathop {\longrightarrow }\limits ^{k\rightarrow \infty } \nabla ^x_g G_p(x) \end{aligned}$$

uniformly in $\Omega$ and, assuming $q\in (1,2)$, by the Hardy–Littlewood–Sobolev inequality

$$\begin{aligned}&\int _{\Sigma } \left( \int _{\Sigma } (1-\varphi (y)) \nabla _g^x G_x(y) {h}(y)e^{u_\varepsilon (y)} dv_g(y)\right) ^s dv_g(x) \\&\quad \le C_2^s \int _{\Sigma } \left( \int _{\Sigma \backslash B_{\delta }(p)} \frac{{\;h}(y)e^{u_\varepsilon (y)\;}}{d(x,y)} dv_g(y)\right) ^s\; dv_g(x)\\&\quad \le C \Vert h\Vert _{L^q(\Sigma )}^s \Vert e^{u_n}\Vert _{L^\infty (\Sigma \backslash B_\delta (p))}^s \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0} 0 \end{aligned}$$

where

$$\begin{aligned} \frac{1}{s} = \frac{1}{q} - \frac{1}{2}. \end{aligned}$$

Note that $q>1$ implies $s>2$. Finally, to prove 5., we shall observe that for any $1<q<2$ there exists a positive constant $C_q$ such that

$$\begin{aligned} \int _{\Sigma }\varphi \; dv_g=0 \quad \text{ and } \quad \int _{\Sigma } |\nabla _g \varphi |^{q'}dv_g\le 1 \quad \Longrightarrow \quad \Vert \varphi \Vert _{\infty } \le C_q. \end{aligned}$$

Hence $\forall \; \varphi \in W^{1,q'}(\Sigma )$

$$\begin{aligned} \int _\Sigma \nabla _g u_\varepsilon \cdot \nabla _g \varphi \; dv_g = -\int _\Sigma \Delta u_\varepsilon \varphi \; dv_g \le C_q \Vert \Delta u_\varepsilon \Vert _{L^1(\Sigma ) }\le \tilde{C}_q \end{aligned}$$

so that

$$\begin{aligned} \Vert \nabla u_\varepsilon \Vert _{L^q}\le \sup \left\{ \int _{\Sigma }\nabla _g u_\varepsilon \cdot \nabla _g \varphi \; dv_g \;:\; \varphi \in W^{1,q'}(\Sigma ), \Vert \nabla \varphi \Vert _{L^{q'}}\le 1 \right\} \le \tilde{C}_q. \end{aligned}$$

$\square$

We now focus on the behavior of $u_\varepsilon$ near the blow-up point. First we consider the case $\alpha <0$. Let us fix a system of normal coordinates in a small ball $B_\delta (p)$, with $p$ corresponding to $0$ and $p_\varepsilon$ corresponding to $x_\varepsilon$. We define

$$\begin{aligned} \varphi _\varepsilon (x):= u_\varepsilon (t_\varepsilon x) -\lambda _\varepsilon ,\quad t_\varepsilon :=e^{-\frac{\lambda _\varepsilon }{2(1+\alpha )}}. \end{aligned}$$

(15)

Lemma 2.2

If $\alpha <0$, $\displaystyle { \frac{|x_\varepsilon |}{t_\varepsilon }}$ is bounded.

Proof

We define

$$\begin{aligned} \psi _\varepsilon (x) = u_\varepsilon (|x_\varepsilon | x)+2(1+\alpha )\log |x_\varepsilon | +s_\varepsilon (|x_\varepsilon |x) \end{aligned}$$

where $s_\varepsilon (x)$ is the solution of

$$\begin{aligned} {\left\{ \begin{array}{cl} {-}\Delta s_\varepsilon = \frac{\rho _\varepsilon }{|\Sigma |} &{} \quad \text{ in } \quad B_\delta (0) \\ s_\varepsilon =0 &{} \quad \text{ if } \quad |x|=\delta \end{array} \right. }. \end{aligned}$$

The function $\psi _\varepsilon$ satisfies

$$\begin{aligned} {-}\Delta \psi _\varepsilon = |x_\varepsilon |^{-2\alpha } \rho _\varepsilon h(|x_\varepsilon | x) e^{-s_\varepsilon (|x_\varepsilon |x)} e^{\psi _\varepsilon } = V_\varepsilon e^{\psi _\varepsilon } \end{aligned}$$

in $B_\frac{\delta }{|x_\varepsilon |}(0)$. We stress that, by standard elliptic estimates, $s_\varepsilon$ is uniformly bounded in $C^1(\overline{B_\delta })$ and that $G_p$ has the expansion

$$\begin{aligned} G_p(x) = -\frac{1}{2\pi } \log |x|+ A(p) + O(|x|) \end{aligned}$$

(16)

in $B_\delta (0)$. Thus

$$\begin{aligned}&|x_\varepsilon |^{-2\alpha } h(|x_\varepsilon | x)e^{-s_\varepsilon (|x_\varepsilon |x)} \\&\quad = |x_\varepsilon |^{-2\alpha } e^{2\alpha \log (|x_\varepsilon | |x|)-4\pi \alpha A(p)+O(|x_\varepsilon | |x|)} e^{-s_\varepsilon (|x_\varepsilon |x)} K(|x_\varepsilon | x)\\&\qquad \times \prod _{1\le i\le m,p_i\ne p} e^{-4\pi \alpha _i G_{p_i}(|x_\varepsilon | x)}\\&\quad = |x|^{2\alpha } e^{-4\pi \alpha A(p)} e^{O(|x_\varepsilon | |x|)}e^{-s_\varepsilon (|x_\varepsilon |x)} K(|x_\varepsilon | x)\\&\qquad \times \prod _{1\le i\le m,p_i\ne p} e^{-4\pi \alpha _i G_{p_i}(|x_\varepsilon | x)}= |x|^{2\alpha } \tilde{h}(|x_\varepsilon | x) \end{aligned}$$

where $\tilde{h}\in C^{1}(\overline{B_{\delta }})$. In particular $V_\varepsilon$ is uniformly bounded in $C^1_{loc}({\mathbb {R}}^2\backslash \{0\})$. If there existed a subsequence such that $\displaystyle { \frac{|x_\varepsilon |}{t_\varepsilon }}\longrightarrow +\infty$ then

$$\begin{aligned} \psi _\varepsilon \left( \frac{x_\varepsilon }{|x_\varepsilon |}\right) = 2(1+\alpha )\log \left( \frac{|x_\varepsilon |}{t_\varepsilon }\right) +s_\varepsilon (x_\varepsilon )\longrightarrow +\infty , \end{aligned}$$

so $\displaystyle {y_0:= \lim _{\varepsilon \rightarrow 0} \frac{x_\varepsilon }{|x_\varepsilon |}}$ would be a blow-up point for $\psi _\varepsilon$. Since $y_0\ne 0$, applying Proposition 2.2 to $\psi _\varepsilon$ in a small ball $B_r(y_0)$ we would get

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \int _{B_r(y_0)} V_\varepsilon e^{\psi _\varepsilon } dx \ge 8\pi . \end{aligned}$$

But this would be in contradiction to (12) since

$$\begin{aligned} \int _{B_r(y_0)} V_\varepsilon e^{\psi _\varepsilon } dx&= \int _{B_{r(y_0)}}\rho _\varepsilon \;|x_\varepsilon |^{-2\alpha } h(|x_\varepsilon | x) e^{-s_\varepsilon (|x_\varepsilon |x)} e^{\psi _\varepsilon }dx \\&\le \rho _\varepsilon \int _{B_\delta (p)} h e^{u_\varepsilon } dv_g \le 8\pi (1+\alpha )<8\pi . \end{aligned}$$

$\square$

Lemma 2.3

Assume $\alpha <0$. Then, possibly passing to a subsequence, $\varphi _\varepsilon$ converges uniformly on compact subsets of ${\mathbb {R}}^2$ and in $H^1_{loc}({\mathbb {R}}^2)$ to

$$\begin{aligned} \varphi _0(x):= -2\log \left( 1+\frac{\pi c(p) }{1+\alpha }|x|^{2(1+\alpha )}\right) \end{aligned}$$

where $\displaystyle {c(p)= K(p)e^{-4\pi \alpha A(p)} \prod _{1\le i\le m, p_i\ne p} e^{-4\pi \alpha _i G_{p_i}(p)}}$.

Proof

The function $\varphi _\varepsilon$ is defined in $B_\varepsilon =B_\frac{\delta }{t_\varepsilon }(0)$ and satisfies

$$\begin{aligned} {-}\Delta \varphi _\varepsilon = t_\varepsilon ^2 \rho _\varepsilon \left( h(t_\varepsilon x) e^{\varphi _\varepsilon } e^{\lambda _\varepsilon }- \frac{1}{|\Sigma |}\right) = t_\varepsilon ^{-2\alpha } \rho _\varepsilon h(t_\varepsilon x) e^{\varphi _\varepsilon }- \frac{t_\varepsilon ^2 \rho _\varepsilon }{|\Sigma |} \end{aligned}$$

and

$$\begin{aligned} t_\varepsilon ^{-2\alpha }\int _{B_\frac{\delta }{t_\varepsilon }} h(t_\varepsilon x) e^{\varphi _\varepsilon } \le 1. \end{aligned}$$

As in the previous proof we have

$$\begin{aligned} t_\varepsilon ^{-2\alpha } h(t_\varepsilon x)&= t_\varepsilon ^{-2\alpha } e^{2\alpha \log (t_\varepsilon |x|)-4\pi \alpha A(p)+O(t_\varepsilon |x|)} K(t_\varepsilon x) \prod _{1\le i\le m,p_i\ne p} e^{-4\pi \alpha _i G_{p_i}(t_\varepsilon x)} \\ \!&= \! |x|^{2\alpha } e^{-4\pi \alpha A(p)} e^{O(t_\varepsilon |x|)} K(t_\varepsilon x) \prod _{1\le i\le m,p_i\ne p} e^{-4\pi \alpha _i G_{p_i}(t_\varepsilon x)}\!\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}\! c(p)|x|^{2\alpha } \end{aligned}$$

in $L^q_{loc}({\mathbb {R}}^2)$ for some $q>1$. Fix $R>0$ and let $\psi _\varepsilon$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} {-}\Delta \psi _\varepsilon = t_\varepsilon ^{-2\alpha } \rho _\varepsilon h(t_\varepsilon x) e^{\varphi _\varepsilon }- \frac{t_\varepsilon ^2 \rho _\varepsilon }{|\Sigma |} &{} \text{ in } B_R(0) \\ \psi _\varepsilon =0 &{} \text{ su } \partial B_R(0) \end{array} \right. }. \end{aligned}$$

Since $\Delta \psi _\varepsilon$ is bounded in $L^q(B_R(0))$ with $q>1$, elliptic regularity shows that $\psi _\varepsilon$ is bounded in $W^{2,q}(B_R(0))$ and by Sobolev’s embeddings we may extract a subsequence such that $\psi _\varepsilon$ converges in $H^1(B_R(0))\cap C^{0,\lambda }(B_R(0))$. The function $\xi _\varepsilon = \varphi _\varepsilon -\psi _\varepsilon$ is harmonic in $B_R$ and bounded from above. Furthermore $\xi _\varepsilon \left( \frac{x_\varepsilon }{t_\varepsilon }\right) = -\psi _\varepsilon \left( \frac{x_\varepsilon }{t_\varepsilon }\right)$ is bounded from below, hence by Harnack inequality $\xi _\varepsilon$ is uniformly bounded in $C^{2}(\overline{B_{\frac{R}{2}}}(0))$. Thus $\varphi _\varepsilon$ is bounded in $W^{2,q}(B_{\frac{R}{2}})$ and we can extract a subsequence converging in $H^1(B_{\frac{R}{2}})\cap C^{0,\lambda }(B_\frac{R}{2})$. Using a diagonal argument we find a subsequence for which $\varphi _\varepsilon$ converges in $H^1_{loc}({\mathbb {R}}^2)\cap C^{0,\lambda }_{loc}({\mathbb {R}}^2)$ to a function $\varphi _0$ solving

$$\begin{aligned} {-}\Delta \varphi _0 = 8\pi (1+\alpha ) c(p)|x|^{2\alpha } e^{\varphi _0} \end{aligned}$$

on ${\mathbb {R}}^2$ with

$$\begin{aligned} \int _{{\mathbb {R}}^2} |x|^{2\alpha }e^{\varphi _0(x)} dx <\infty . \end{aligned}$$

The classification result in [24] yields

$$\begin{aligned} \varphi _0(x)=-2\log \left( 1+\frac{\pi e^{\lambda } c(p)}{1+\alpha } |x|^{2(1+\alpha )}\right) +\lambda \end{aligned}$$

for some $\lambda \in {\mathbb {R}}$. To conclude the proof it remains to note that, since $0$ is the unique maximum point of $\varphi _0$, the uniform convergence of $\varphi _\varepsilon$ implies $\frac{x_\varepsilon }{t_\varepsilon }\longrightarrow 0$ and $\lambda =0$.$\square$

As in [15], to give a lower bound on $J_\varepsilon (u_\varepsilon )$ we need the following estimate from below for $u_\varepsilon$:

Lemma 2.4

Fix $R>0$ and define $r_\varepsilon = t_\varepsilon R$. If $\alpha <0$ and $u_\varepsilon$ satisfies (11), (12), (13), then

$$\begin{aligned} u_\varepsilon \ge \overline{\rho }\; G_p-\lambda _\varepsilon -\overline{\rho }\; A(p) + 2 \log \left( \frac{ R^{2(1+\alpha )}}{1+\frac{\pi c(p)}{1+\alpha } R^{2(1+\alpha )}}\right) +o_\varepsilon (1) \end{aligned}$$

in $\Sigma \backslash B_{r_\varepsilon }(p)$, where $o_\varepsilon (1)$ is a function of $\varepsilon$ and $R$ such that $o_\varepsilon (1)\longrightarrow 0$ as $\varepsilon \rightarrow 0$.

Proof

$\forall \; C>0$ we have

$$\begin{aligned} {-}\Delta _g (u_\varepsilon -\overline{\rho }\; G_p-C) = \rho _\varepsilon \left( h e^{u_\varepsilon }-\frac{1}{|\Sigma |}\right) +\frac{\overline{\rho }}{|\Sigma |} = \rho _\varepsilon h e^{u_\varepsilon } +\frac{\varepsilon }{|\Sigma |}\ge 0. \end{aligned}$$

Let us consider normal coordinates near $p$. We know that

$$\begin{aligned} G_p(x)=-\frac{1}{2\pi } \log |x| + A(p)+O(|x|), \end{aligned}$$

so by Lemma 2.3 if $x=t_\varepsilon y$ with $|y| =R$ we have

$$\begin{aligned} u_\varepsilon (x)-\overline{\rho }\; G_p&= \varphi _\varepsilon (y)+\lambda _\varepsilon +4(1+\alpha )\log (t_\varepsilon R) -\overline{\rho }A(p)+O(t_\varepsilon R)\\&\ge -2\log \left( 1+\frac{\pi c(p)}{1+\alpha }R^{2(1+\alpha )}\right) \!-\!\lambda _\varepsilon \!+\!\log {R^{4(1+\alpha )}}\!-\!\overline{\rho }\; A(p)+o_\varepsilon (1). \end{aligned}$$

Thus, taking

$$\begin{aligned} C_{\varepsilon ,R}=- \lambda _\varepsilon -\overline{\rho }\; A(p) +2 \log \left( \frac{R^{2(1+\alpha )}}{1+\frac{\pi c(p)}{1+\alpha } R^{2(1+\alpha )}}\right) +o_\varepsilon (1) \end{aligned}$$

we have $u_\varepsilon -\overline{\rho } G_p -C_{\varepsilon ,R}\ge 0$ on $\partial B_{r_\varepsilon }(p)$ and the conclusion follows from the maximum principle.$\square$

As a consequence we also have

Lemma 2.5

If $u_\varepsilon$ and $t_\varepsilon$ are as above, then $t_\varepsilon ^2 \overline{u}_\varepsilon \longrightarrow 0$.

Proof

By Lemma 2.3

$$\begin{aligned} \int _{B_{t_\varepsilon }(p)} u_\varepsilon \; dv_g = t_\varepsilon ^2 \int _{B_1(0)} \varphi _\varepsilon (y)dy +\lambda _\varepsilon |B_{t_\varepsilon }|= o_\varepsilon (1). \end{aligned}$$

and by the previous lemma

$$\begin{aligned} \lambda _\varepsilon |\Sigma |\ge \int _{\Sigma \backslash B_{t_\varepsilon }(p)} u_\varepsilon \ge \overline{\rho } \int _{\Sigma \backslash B_{t_\varepsilon }(p)} G_p \;dv_g - \lambda _\varepsilon |\Sigma \backslash B_{t_\varepsilon }(p)| +O(1). \end{aligned}$$

Thus $\dfrac{|\overline{u}_\varepsilon |}{\lambda _\varepsilon }$ is bounded and, since $\lambda _\varepsilon t_\varepsilon ^2 =o_\varepsilon (1)$, we get the conclusion.$\square$

The case $\alpha =0$ can be studied in a similar way. The main difference is that, since we do not know whether $\frac{|x_\varepsilon |}{t_\varepsilon }$ is bounded, we have to center the scaling in $p_\varepsilon$ and not in $p$. Note that $\beta (p)=0$ means that $p\in \Sigma \backslash S$ is a regular point of $h$.

Lemma 2.6

Assume that $\alpha =0$ and that $u_\varepsilon$ satisfies (11), (12) and (13). In normal coordinates near $p$ define

$$\begin{aligned} \psi _\varepsilon (x)= u_\varepsilon (x_\varepsilon + t_\varepsilon x)-\lambda _\varepsilon \quad \text{ where } \quad t_\varepsilon = e^{-\frac{\lambda _\varepsilon }{2}}. \end{aligned}$$

Then

• 1. $\psi _\varepsilon$ converges in $C^1_{loc}({\mathbb {R}}^2)$ to

  $$\begin{aligned} \psi _0(x)= -2\log (1+\pi h(p) |x|^2) \end{aligned}$$

• 2. $\forall \;R>0$ one has

  $$\begin{aligned} u_\varepsilon \ge 8\pi G_{p_\varepsilon } - \lambda _\varepsilon -8\pi A(p) + 2\log \left( \frac{R^2}{1+\pi h(p) R^2}\right) +o_\varepsilon (1) \end{aligned}$$

  in $\Sigma \backslash B_{R t_\varepsilon }(p_\varepsilon );$

• 3. $t_\varepsilon ^2 \overline{u}_\varepsilon \rightarrow 0$.

3 A Lower Bound

In this section and in the next one we present the proof of Theorem 1.1. We begin by giving an estimate from below of $\displaystyle {\inf _{H^1(\Sigma )}J}$. As before we consider $u_\varepsilon$ satisfying (10), (11), (12), and (13). Again we will focus on the case $\alpha <0$ since the computation for $\alpha =0$ is equivalent to the one in [15]. We consider normal coordinates in a small ball $B_\delta (p)$ and assume that $G_p$ has the expansion (16) in $B_\delta (p)$. Let $t_\varepsilon$ be defined as in (15), then $\forall \;R>0$ we shall consider the decomposition

$$\begin{aligned} \displaystyle {\int _{\Sigma }|\nabla _g u_\varepsilon |^2dv_g\!=\! \int _{\Sigma \backslash B_\delta (p)} |\nabla _g u_\varepsilon |^2dv_g \!+\! \int _{B_\delta \backslash B_{r_\varepsilon }(p)} |\nabla _g u_\varepsilon |^2dv_g \!+\! \int _{B_{r_\varepsilon }(p)}|\nabla _g u_\varepsilon |^2dv_g}. \end{aligned}$$

Throughout this section, $o_\delta (1)$ (and $o_R(1)$) will denote a function depending only on $\delta$ (resp. $R$) which converges to 0 as $\delta \rightarrow 0$ (resp. $R\rightarrow \infty$), while the notation $o_\varepsilon (1)$ will be used for functions of $\varepsilon , \delta$ and $R$ such that, for fixed $\delta$ and $R$, $o_\varepsilon (1) \longrightarrow 0$ as $\varepsilon \rightarrow 0$.

On $\Sigma \backslash B_\delta (p)$ we can use Lemma 2.1 and an integration by parts to obtain:

$$\begin{aligned} \int _{\Sigma \backslash B_\delta } |\nabla _g u_\varepsilon |^2 dv_g&= \overline{\rho }^2\int _{\Sigma \backslash B_\delta }|\nabla _g G_p|^2 dv_g +o_\varepsilon (1) \nonumber \\&= -\frac{\overline{\rho }^2}{|\Sigma |} \int _{\Sigma \backslash B_\delta } G_p\; dv_g - \overline{\rho }^2\int _{\partial B_\delta } G_p \frac{\partial G_p}{ \partial n} \; d\sigma _g+o_\varepsilon (1) \nonumber \\&= - \overline{\rho }^2\int _{\partial B_\delta } G_p \frac{\partial G_p}{ \partial n} d\sigma _g +o_\varepsilon (1)+o_\delta (1). \end{aligned}$$

(17)

On $B_{r_\varepsilon }(p)$ the convergence result for the scaling (15) stated in Lemma 2.3 yields

$$\begin{aligned} \int _{B_{r_\varepsilon }} |\nabla _g u_\varepsilon |^2 dv_g \!&= \! \int _{B_R(0)} |\nabla \varphi _0|^2 dx +o_\varepsilon (1)\!=\! 2\overline{\rho }\left( \log \left( 1+\frac{\pi \;c(p)}{1+\alpha } R^{2(1+\alpha )}\right) \!-\!1\right) \nonumber \\&+\,o_\varepsilon (1)+o_R(1). \end{aligned}$$

(18)

For the remaining term we can use (11) and Lemma 2.1 to obtain

$$\begin{aligned} \int _{B_\delta \backslash B_{r_\varepsilon }} |\nabla _g u_\varepsilon |^2 dv_g&= \rho _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } u_\varepsilon dv_g - \frac{\rho _\varepsilon }{|\Sigma |}\int _{{B_\delta \backslash B_{r_\varepsilon }}} u_\varepsilon dv_g\nonumber \\&+ \int _{\partial B_\delta } u_\varepsilon \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g- \int _{\partial { B_{r_\varepsilon }}} u_\varepsilon \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g \nonumber \\&= \rho _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } u_\varepsilon dv_g - \frac{\rho _\varepsilon }{|\Sigma |}\int _{{B_\delta \backslash B_{r_\varepsilon }}} u_\varepsilon dv_g\nonumber \\&+\, \overline{u}_\varepsilon \int _{\partial B_\delta } \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g - \int _{\partial { B_{r_\varepsilon }}} u_\varepsilon \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g \nonumber \\&+\, \overline{\rho }^2\int _{\partial B_\delta } G_p \frac{\partial G_p}{ \partial n}d\sigma _g +o_\varepsilon (1). \end{aligned}$$

(19)

By Lemma 2.4 and (12) we get

$$\begin{aligned} \rho _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } u_\varepsilon dv_g&\ge \rho _\varepsilon \overline{\rho } \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } G_p dv_g - \rho _\varepsilon \lambda _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } dv_g\nonumber \\&+\,O(1)\int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } dv_g \nonumber \\&= \rho _\varepsilon \overline{\rho } \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } G_p dv_g - \rho _\varepsilon \lambda _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } dv_g\nonumber \\&+\,o_\varepsilon (1)+o_R(1). \end{aligned}$$

(20)

Again by (11) and Lemma 2.1

$$\begin{aligned} \rho _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } G_p dv_g&= \int _{B_\delta \backslash B_{r_\varepsilon }} G_p \left( -\Delta u_\varepsilon +\frac{\rho _\varepsilon }{|\Sigma |}\right) dv_g \nonumber \\&= - \frac{1}{|\Sigma |}\int _{B_\delta \backslash B_{r_\varepsilon }} u_\varepsilon dv_g +\int _{\partial ( B_\delta \backslash B_{r_\varepsilon })} u_\varepsilon \frac{\partial G_p}{ \partial n} -\, G_p \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g\nonumber \\&+o_\varepsilon (1)+ o_\delta (1) \nonumber \\&= - \frac{1}{|\Sigma |}\int _{B_\delta \backslash B_{r_\varepsilon }} u_\varepsilon dv_g +\overline{u}_\varepsilon \int _{\partial B_\delta } \frac{\partial G_p}{ \partial n} d\sigma _g\nonumber \\&+\int _{\partial B_{r_\varepsilon }} G_p \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g - \int _{\partial B_{r_\varepsilon }} u_\varepsilon \frac{\partial G_p}{ \partial n} d\sigma _g \nonumber \\&+ \;o_\varepsilon (1) +o_\delta (1), \end{aligned}$$

(21)

and

$$\begin{aligned} \rho _\varepsilon \lambda _\varepsilon \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{u_\varepsilon } dv_g&= - \lambda _\varepsilon \int _{\partial B_\delta \backslash B_{r_\varepsilon }} \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g +\frac{\rho _\varepsilon \lambda _\varepsilon }{|\Sigma |} \left( Vol(B_\delta )-Vol(B_{r_\varepsilon })\right) \nonumber \\ \!&= \! -\lambda _\varepsilon \!\int _{\partial B_\delta }\! \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g \!+\! \lambda _\varepsilon \!\int _{\partial B_{r_\varepsilon }} \frac{\partial u_\varepsilon }{ \partial n}\! d\sigma _g\!+\! \frac{\rho _\varepsilon \lambda _\varepsilon }{|\Sigma |} Vol(B_\delta ) \! +\!o_\varepsilon (1).\qquad \end{aligned}$$

(22)

Using (19), (20), (21) and (22) we get

$$\begin{aligned} \int _{B_\delta \backslash B_{r_\varepsilon }}|\nabla _g u_\varepsilon |^2dv_g&\ge - (16\pi (1+\alpha )-\varepsilon ) \frac{1}{|\Sigma |}\int _{B_\delta \backslash B_{r_\varepsilon }} u_\varepsilon \; dv_g - \frac{\rho _\varepsilon \lambda _\varepsilon }{|\Sigma |}Vol(B_\delta ) \nonumber \\&+\, \overline{\rho }\; \overline{u}_\varepsilon \int _{\partial B_\delta } \frac{\partial G_p}{ \partial n} d\sigma _g+ \lambda _\varepsilon \int _{\partial B_\delta } \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g + \overline{u}_\varepsilon \int _{\partial B_{\delta }} \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g \nonumber \\&+\, \overline{\rho }^2 \int _{\partial B_\delta } G_p \frac{\partial G_p}{ \partial n} d\sigma _g -\overline{\rho }\int _{\partial B_{r_\varepsilon }} u_\varepsilon \frac{\partial G_p}{ \partial n} d\sigma _g\nonumber \\&\quad -\int _{\partial B_{r_\varepsilon }} \Big ( u_\varepsilon - \overline{\rho }\;G_p +\lambda _\varepsilon \Big ) \frac{\partial u_\varepsilon }{ \partial n}d\sigma _g \nonumber \\&+\, o_\varepsilon (1)+o_\delta (1) +o_R(1). \end{aligned}$$

(23)

By Lemmas 2.1 and 2.5 we can say that

$$\begin{aligned} \int _{B_\delta \backslash B_{r_\varepsilon }}u_\varepsilon dv_g&= \int _{B_\delta \backslash B_{r_\varepsilon }} (u_\varepsilon -\overline{u}_\varepsilon ) dv_g + \overline{u}_\varepsilon (Vol (B_\delta )-Vol(B_{r_\varepsilon }))\\&= \overline{u}_\varepsilon Vol(B_\delta ) + o_\delta (1)+o_\varepsilon (1). \end{aligned}$$

Using Green’s formula we find

$$\begin{aligned} \overline{u}_\varepsilon \int _{\partial B_\delta } \frac{\partial G_p}{ \partial n} d\sigma _g = - \overline{u}_\varepsilon \int _{\Sigma \backslash B_\delta } \Delta _g G_p \; dv_g =- \overline{ u}_\varepsilon \left( 1-\frac{Vol(B_\delta )}{|\Sigma |}\right) . \end{aligned}$$

Similarly

$$\begin{aligned} \int _{\partial B_\delta } \frac{\partial u_\varepsilon }{ \partial n}d\sigma _g&= -\int _{\Sigma \backslash B_\delta } \Delta u_\varepsilon \; dv_g = \int _{\Sigma \backslash B_\delta } \rho _\varepsilon \left( h e^{u_\varepsilon }-\frac{1}{|\Sigma |}\right) dv_g\\&\ge -\rho _\varepsilon \left( 1-\frac{Vol(B_\delta )}{|\Sigma |}\right) \end{aligned}$$

and

$$\begin{aligned} \overline{u}_\varepsilon \int _{\partial B_\delta } \frac{\partial u_\varepsilon }{ \partial n}d\sigma _g&= \overline{u}_\varepsilon \rho _\varepsilon e^{\overline{u}_\varepsilon } \int _{\Sigma \backslash B_\delta (p)} h \;e^{u_\varepsilon -\overline{u}_\varepsilon }dv_g-\overline{u}_\varepsilon \rho _\varepsilon \left( 1-\frac{Vol(B_\delta )}{|\Sigma |}\right) \\&= - \overline{u}_\varepsilon \rho _\varepsilon \left( 1-\frac{Vol(B_\delta )}{|\Sigma |}\right) +o_\varepsilon (1). \end{aligned}$$

Lemma 2.3 yields

$$\begin{aligned} \int _{\partial B_{r_\varepsilon }} u_\varepsilon \frac{\partial G_p}{ \partial n} d\sigma _g&= \lambda _\varepsilon \int _{\partial B_{r_\varepsilon }} \frac{\partial G_p}{ \partial n} d\sigma _g + t_\varepsilon \int _{\partial B_R(0)}\varphi _\varepsilon \frac{\partial G_p}{ \partial n}(t_\varepsilon x)(1+o_\varepsilon (1))d\sigma \\&= - \lambda _\varepsilon \left( 1-\frac{Vol(B_{r_\varepsilon })}{|\Sigma |}\right) +t_\varepsilon \int _{\partial B_R(0)} \varphi _0 \left( -\frac{1}{2\pi t_\varepsilon R} +O(1)\right) d\sigma \\&= -\lambda _\varepsilon +2 \log \left( 1+ \frac{\pi \; c(p)}{1+\alpha } R^{2(1+\alpha )}\right) + o_\varepsilon (1) \end{aligned}$$

and the estimate in Lemma 2.4 gives

$$\begin{aligned}&-\int _{\partial B_{r_\varepsilon }} \Big ( u_\varepsilon - \overline{\rho }\;G_p +\lambda _\varepsilon \Big ) \frac{\partial u_\varepsilon }{ \partial n} d\sigma _g \\&\quad \ge \left( 2 \log \left( \frac{ R^{2(1+\alpha )}}{1+ \frac{ \pi c(p)}{(1+\alpha )} R^{2(1+\alpha )}}\right) - \overline{\rho } A(p)\right) \frac{8 \pi ^2 c(p) R^{2 (1+\alpha )} }{ \left( 1+\frac{\pi c(p) R^{2 (1+\alpha )}}{1+\alpha }\right) } +o_\varepsilon (1) \\&\quad = - \overline{\rho }^2A(p) - 2 \; \overline{\rho } \; \log \left( \frac{\pi c(p)}{1+\alpha }\right) +o_\varepsilon (1) +o_R(1). \end{aligned}$$

Hence

$$\begin{aligned} \int _{B_\delta \backslash B_{r_\varepsilon }}|\nabla _g u_\varepsilon |^2dv_g&\ge -(16\pi (1+\alpha )-\varepsilon )\overline{u}_\varepsilon +\varepsilon \lambda _\varepsilon + \overline{\rho }^2 \int _{\partial B_\delta } G_p \frac{\partial G_p}{ \partial n} d\sigma _g \nonumber \\&-\, 2 \overline{\rho } \log \left( 1+\frac{\pi c(p)}{1+\alpha } R^{2(1+\alpha )}\right) \!-\! \overline{\rho }^2A(p) - 2 \overline{\rho }\log \left( \frac{\pi c(p)}{1+\alpha }\right) \nonumber \\&+\, o_\varepsilon (1)+o_\delta (1)+o_R(1). \end{aligned}$$

(24)

By (17), (18) and (24) we can therefore conclude

$$\begin{aligned} \int _{\Sigma }|\nabla _g u_\varepsilon |^2 dv_g&\ge -(16\pi (1+\alpha )-\varepsilon )\overline{u}_\varepsilon +\varepsilon \lambda _\varepsilon -\overline{\rho }^2A(p)-2\overline{\rho }\log \left( \frac{\pi c(p)}{1+\alpha }\right) -2\overline{\rho } \\&+ \;o_\varepsilon (1)+\;o_\delta (1) +\;o_R(1), \end{aligned}$$

so that

$$\begin{aligned} J_\varepsilon (u_\varepsilon )&\ge \frac{\varepsilon }{2}(\lambda _\varepsilon -\overline{u}_\varepsilon ) -\frac{\overline{\rho }^2}{2}A(p)-\overline{\rho }\log \left( \frac{\pi c(p)}{1+\alpha }\right) -\overline{\rho }+\rho _\varepsilon \log |\Sigma |\\&+\;o_\varepsilon (1)+ o_\delta (1) + o_R(1) \\&\ge -\,\overline{\rho } \left( 4\pi (1+\alpha ) A(p)+1+\log \left( \frac{\pi c(p)}{1+\alpha }\right) -\log |\Sigma |\right) \\&+\;o_\varepsilon (1)+o_\delta (1) + o_R(1). \end{aligned}$$

As $\varepsilon ,\delta \rightarrow 0$ and $R\rightarrow \infty$ we obtain

$$\begin{aligned} \inf _{H^1(\Sigma )} J&\ge -\overline{\rho } \left( 4\pi (1+\alpha ) A(p)+1+\log \left( \frac{\pi c(p)}{1+\alpha }\right) -\log {|\Sigma |}\right) \nonumber \\ \!&= \! -\overline{\rho }\left( \! 1\!+\!\log \frac{\pi }{|\Sigma |} \!+\!4\pi A(p)\!+\!\log \left( \frac{K(p)}{1\!+\!\alpha }\prod _{q\in S, q\ne p} e^{-4\pi \beta (q) G_q(p)}\right) \!\right) . \quad \qquad \end{aligned}$$

(25)

Using Lemma 2.6 it is possible to prove that (25) holds even for $\alpha =0$. About the blow-up point $p$ we only know that $\beta (p)=\alpha$, so we have proved

Proposition 3.1

If $J$ has no minimum point, then

$$\begin{aligned} \inf _{H^1(\Sigma )} J&\ge -\overline{\rho } \left( 1+\log \frac{\pi }{|\Sigma |}+\max _{p\in \Sigma ,\beta (p)=\alpha } \left\{ 4\pi A(p)\right. \right. \\&\left. \left. + \log \left( \frac{K(p)}{1+\alpha }\prod _{q\in S, q\ne p} e^{-4\pi \beta (q) G_q(p)}\right) \right\} \right) . \end{aligned}$$

Notice that, if $\alpha <0$, the set

$$\begin{aligned} \left\{ p \in \Sigma \;:\; \beta (p)=\alpha \right\} =\left\{ p_i\;:\; i\in \{1,\ldots ,m\},\;\alpha _i=\alpha \right\} \end{aligned}$$

is finite, while if $\alpha =0$

$$\begin{aligned} \left\{ p \in \Sigma \;:\; \beta (p)=\alpha \right\} =\Sigma \backslash S. \end{aligned}$$

Although this set is not finite, the maximum in the above expression is still well defined since the function

$$\begin{aligned} p\longmapsto 4\pi A(p)+ \log \left( K(p)\prod _{q\in S} e^{-4\pi \beta (q) G_q(p)}\right) = 4\pi A(p)+ \log h(p) \end{aligned}$$

is continuous on $\Sigma \backslash S$ and approaches $-\infty$ near $S$.

4 An Estimate from Above

In order to complete the proof of Theorem 1.1 we need to exhibit a sequence $\varphi _\varepsilon \in H^1(\Sigma )$ such that

$$\begin{aligned}&J(\varphi _\varepsilon )\longrightarrow -\overline{\rho } \left( 1+\log \frac{\pi }{|\Sigma |}+\max _{p\in \Sigma ,\beta (p)=\alpha } \left\{ 4\pi A(p)\right. \right. \\&\quad \left. \left. + \log \left( \frac{K(p)}{1+\alpha }\prod _{q\in S, q\ne p} e^{-4\pi \beta (q) G_q(p)}\right) \right\} \right) . \end{aligned}$$

Let us define $r_\varepsilon :=\gamma _\varepsilon \varepsilon ^\frac{1}{2(1+\alpha )}$ where $\gamma _\varepsilon$ is chosen so that

$$\begin{aligned} \gamma _\varepsilon \rightarrow +\infty ,\quad r_\varepsilon ^2 \log \varepsilon \longrightarrow 0, \quad r_\varepsilon ^2 \log \big (1+\gamma _\varepsilon ^{2(1+\alpha )}\big )\longrightarrow 0. \end{aligned}$$

(26)

Let $p\in \Sigma$ be such that $\beta (p)=\alpha$ and

$$\begin{aligned}&4\pi A(p)+ \log \left( \frac{K(p)}{1+\alpha }\prod _{q\in S, q\ne p} e^{-4\pi \beta (q) G_q(p)}\right) \\&\quad =\max _{\xi \in \Sigma ,\beta (\xi )=\alpha } \left\{ 4\pi A(\xi )+ \log \left( \frac{K(\xi )}{1+\alpha }\prod _{q\in S, q\ne \xi } e^{-4\pi \beta (q) G_q(\xi )}\right) \right\} \end{aligned}$$

and consider a cut-off function $\eta _\varepsilon$ such that $\eta _\varepsilon \equiv 1$ in $B_{r_\varepsilon }(p)$, $\eta _\varepsilon \equiv 0$ in $\Sigma \backslash B_{2 r_\varepsilon }(p)$ and $|\nabla _g \eta _\varepsilon |=O(r_\varepsilon ^{-1})$. Define

$$\begin{aligned} \varphi _\varepsilon (x)= {\left\{ \begin{array}{cc} -2\log \big (\varepsilon +r^{2(1+\alpha )}\big )+\log \varepsilon &{}\quad r\le r_\varepsilon \\ \overline{\rho } \left( G_p-\eta _\varepsilon \sigma \right) +C_\varepsilon +\log \varepsilon &{}\quad r\ge r_\varepsilon \end{array} \right. } \end{aligned}$$

where $r=d(x,p)$, $\sigma (x)=O(r)$ is defined by

$$\begin{aligned} G_p(x) = -\frac{1}{2\pi } \log r +A(p) + \sigma (x), \end{aligned}$$

(27)

and

$$\begin{aligned} C_\varepsilon = -2\log \left( \frac{1 +\gamma _\varepsilon ^{2(1+\alpha )}}{\gamma _\varepsilon ^{2(1+\alpha )}}\right) -\overline{\rho } \; A(p). \end{aligned}$$

In the case $\alpha _i=0$ $\forall \;i$, a similar family of functions was used in [15] to give an existence result for (4) by proving, under some strict assumptions on $h$, that

$$\begin{aligned} \inf _{H^1(\Sigma )} J_{\overline{\rho }}< -8\pi \left( 1+\log \left( \frac{\pi }{|\Sigma |}\right) +\max _{p\in \Sigma } \left\{ 4\pi A(p)+ \log h(p)\right\} \right) . \end{aligned}$$

Here we only prove a weak inequality but we have no extra assumptions on $h$. Taking normal coordinates in a neighborhood of $p$ it is simple to verify that

$$\begin{aligned} \int _{B_{r_\varepsilon }}|\nabla _g \varphi _\varepsilon |^2dv_g&= 16\pi (1+\alpha )\left( \log \left( 1+\gamma _\varepsilon ^{2(1+\alpha )}\right) + \frac{1}{1+\gamma _\varepsilon ^{2(1+\alpha )}}-1\right) +o_\varepsilon (1)\\&= 16\pi (1+\alpha )\left( \log \left( 1+\gamma _\varepsilon ^{2(1+\alpha )}\right) -1\right) +o_\varepsilon (1). \end{aligned}$$

By our definition of $\varphi _\varepsilon$

$$\begin{aligned} \int _{\Sigma \backslash B_{r_\varepsilon }}|\nabla _g \varphi _\varepsilon |^2dv_g&= \overline{\rho }^2\left( \int _{\Sigma \backslash B_{r_\varepsilon }} |\nabla _g G_p|^2dv_g + \int _{\Sigma \backslash B_{r_\varepsilon }} |\nabla _g (\eta _\varepsilon \sigma )|^2dv_g \right. \\&\left. -\, 2\int _{\Sigma \backslash B_{r_\varepsilon }} \nabla _g G_p \cdot \nabla _g (\eta _\varepsilon \sigma )\; dv_g \right) \end{aligned}$$

and by the properties of $\eta _\varepsilon$

$$\begin{aligned} \int _{\Sigma \backslash B_{r_\varepsilon }} |\nabla _g (\eta _\varepsilon \sigma )|^2 dv_g&= \int _{B_{2r_\varepsilon }\backslash B_{r_\varepsilon }} |\nabla _g \eta _\varepsilon |^2 \sigma ^2 + 2 \eta _\varepsilon \sigma \; \nabla _g \eta _\varepsilon \cdot \nabla _g \sigma + \eta _\varepsilon ^2|\nabla _g \sigma |^2 \; dv_g\\&= O(r_\varepsilon ^2). \end{aligned}$$

Hence, integrating by parts and using (27), one has

$$\begin{aligned} \int _{\Sigma \backslash B_{r_\varepsilon }}|\nabla _g \varphi _\varepsilon |^2dv_g&= \overline{\rho } ^2 \left( \int _{\Sigma \backslash B_{r_\varepsilon }} |\nabla G_p|^2 dv_g \right. \\&\left. -\, 2\int _{\Sigma \backslash B_{r_\varepsilon }} \nabla _g G_p \cdot \nabla _g ( \eta _\varepsilon \sigma ) \;dv_g \right) +o_\varepsilon (1)\\&= - \overline{\rho } ^2 \left( \frac{1}{|\Sigma |} \int _{\Sigma \backslash B_{r_\varepsilon }} (G_p-2\eta _\varepsilon \sigma ) \; dv_g\right. \\&\left. +\int _{\partial B_{r_\varepsilon }} (G_p-2\eta _\varepsilon \sigma ) \frac{\partial G_p}{ \partial n} d\sigma _g \right) +o_\varepsilon (1)\\&= -\overline{\rho } ^2 \int _{\partial B_{r_\varepsilon }} (G_p-2\sigma ) \frac{\partial G_p}{ \partial n} d\sigma _g +o_\varepsilon (1)\\&= -\overline{\rho }^2 \int _{\partial B_{r_\varepsilon }} \quad \left( -\frac{1}{2\pi }\log (r_\varepsilon ) +A(p)-\sigma \right) \\&\times \left( -\frac{1}{2\pi r_\varepsilon }+ \nabla \sigma \right) \left( 1+O(r_\varepsilon ^2)\right) d\sigma +o_\varepsilon (1)\\&= -\overline{\rho } ^2 \int _{\partial B_{r_\varepsilon }} \left( \frac{\log r_\varepsilon }{4\pi ^2 r_\varepsilon }\!-\!\frac{1}{2\pi r_\varepsilon } A(p)\!+\!O(\log r_\varepsilon ) \!+\! O(1) \right) d\sigma \!+\!o_\varepsilon (1)\\&= -\frac{\overline{\rho } ^2 }{2\pi } \log \left( \gamma _\varepsilon \varepsilon ^\frac{1}{2(1+\alpha )}\right) +\overline{\rho }^2 A(p) +o_\varepsilon (1)\\&= -2\overline{\rho } \left( \log \gamma _\varepsilon ^{2(1+\alpha )} + \log \varepsilon - 4 \pi (1+\alpha )A(p)\right) +o_\varepsilon (1). \end{aligned}$$

Thus

$$\begin{aligned} \int _{\Sigma }|\nabla _g \varphi _\varepsilon |^2 dv_g&= 2\overline{\rho }\left( \log \left( \frac{1+\gamma _\varepsilon ^{2(1+\alpha )}}{\gamma _\varepsilon ^{2(1+\alpha )}}\right) -1+ 4\pi (1+\alpha )A(p)- \log \varepsilon \right) +o_\varepsilon (1)\nonumber \\&= -2\overline{\rho } \left( 1- 4\pi (1+\alpha )A(p)+ \log \varepsilon \right) +o_\varepsilon (1). \end{aligned}$$

(28)

Similarly one has

$$\begin{aligned} \int _{B_{r_\varepsilon }} \varphi _\varepsilon \; dv_g&= |B_{r_\varepsilon }| \log \varepsilon -4\pi \int _{0}^{r_\varepsilon } r \log \left( \varepsilon + r^{2(1+\alpha )}\right) (1+o_\varepsilon (1))dr \\ \!&= \!|B_{r_\varepsilon }| \log \varepsilon -2\pi r_\varepsilon ^2 \log \varepsilon -4\pi \int _0^{r_\varepsilon } r \log \left( 1+\frac{r^{2(1+\alpha )}}{\varepsilon }\right) (1+o_\varepsilon (1))dr \\&= O \big (r_\varepsilon ^2 \log \varepsilon \big ) -4\pi \int _0^{1} r_\varepsilon ^2 s \log \left( 1+\gamma _\varepsilon ^{2(1+\alpha )} s^{2(1+\alpha )}\right) (1+o_\varepsilon (1))dr\\&= O \big (r_\varepsilon ^2 \log \varepsilon \big ) + O \left( r_\varepsilon ^2 \log \left( 1+\gamma _\varepsilon ^{2(1+\alpha )}\right) \right) =o_\varepsilon (1) \end{aligned}$$

and

$$\begin{aligned} \int _{\Sigma \backslash B_{r_\varepsilon }} \varphi _\varepsilon \; dv_g&= \overline{\rho } \int _{\Sigma \backslash B_{r_\varepsilon }} (G_p-\eta _\varepsilon \sigma ) dv_g+( C_\varepsilon +\log \varepsilon ) |\Sigma \backslash B_{r_\varepsilon }(p)|\\&= |\Sigma | \log \varepsilon -\overline{\rho } |\Sigma |A(p) +o_\varepsilon (1) \end{aligned}$$

so that

$$\begin{aligned} \frac{1}{|\Sigma |}\int _{\Sigma }\varphi _\varepsilon dv_g = \log \varepsilon -\overline{\rho }\; A(p)+o_\varepsilon (1). \end{aligned}$$

(29)

To compute the integral of the exponential term we fix a small $\delta >0$ and observe that

$$\begin{aligned} \int _\Sigma h e^{\varphi _\varepsilon } dv_g&= \tilde{h}(p) \int _{B_{r_\varepsilon }} e^{-4\pi \alpha G_p} e^{\varphi _\varepsilon } dv_g+ \int _{B_{r_\varepsilon }}\left( \tilde{h}-\tilde{h}(p)\right) e^{-4\pi \alpha G_p} e^{\varphi _\varepsilon } dv_g \\&+ \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{\varphi _\varepsilon }dv_g+ \int _{\Sigma \backslash B_\delta } h e^{\varphi _\varepsilon } dv_g \end{aligned}$$

where $\displaystyle {\tilde{h} = h\; e^{4\pi \alpha G_p}= K \prod _{q\in S,q\ne p}}e^{-4\pi \beta (q) G_q}$. For the first term we have

$$\begin{aligned} \int _{B_{r_\varepsilon }} e^{-4\pi \alpha G_p} e^{\varphi _\varepsilon } dv_g&= \varepsilon \int _{B_{r_\varepsilon }} e^{2\alpha \log r -4\pi \alpha A(p) -4\pi \alpha \sigma } e^{-2\log \big (\varepsilon +r^{2(1+\alpha )}\big )} dv_g \nonumber \\&= \varepsilon e^{-4\pi \alpha A(p)} \int _{B_{r_\varepsilon }} \frac{r^{2\alpha }}{\big (\varepsilon + r^{2(1+\alpha )}\big )^2}(1+o_\varepsilon (1)) dv_g\nonumber \\&= \frac{\pi e^{-4\pi \alpha A(p)}}{ 1+\alpha } \frac{\gamma _\varepsilon ^{2(1+\alpha )}}{1+\gamma _\varepsilon ^{2(1+\alpha )}}(1 +o_\varepsilon (1)) \nonumber \\&= \frac{\pi e^{-4\pi \alpha A(p)}}{ 1+\alpha } +o_\varepsilon (1). \end{aligned}$$

(30)

Since $\tilde{h}$ is smooth in a neighborhood of $p$ we obtain

$$\begin{aligned} \int _{B_{r_\varepsilon }}\left( \tilde{h}-\tilde{h}(p)\right) e^{-4\pi \alpha G_p} e^{\varphi _\varepsilon } dv_g = o_\varepsilon (1) \int _{B_{r_\varepsilon }} e^{-4\pi \alpha G_p} e^{\varphi _\varepsilon } dv_g = o_\varepsilon (1) \end{aligned}$$

(31)

and

$$\begin{aligned} \left| \int _{B_\delta \backslash B_{r_\varepsilon }} h e^{\varphi _\varepsilon } dv_g \right|&= \left| \int _{B_\delta \backslash B_{r_\varepsilon }}\tilde{h} e^{-4\pi \alpha G_p }e^{\varphi _\varepsilon } dv_g \right| \le \sup _{B_\delta } |\tilde{h}|\int _{B_\delta \backslash B_{r_\varepsilon }} e^{-4\pi \alpha G_p }e^{\varphi _\varepsilon }dv_g \nonumber \\&= \varepsilon e^{C_\varepsilon } \sup _{B_\delta }|\tilde{h}|\int _{B_\delta \backslash B_{r_\varepsilon }} e^{4\pi (2+\alpha ) G_p } e^{-\overline{\rho } \eta _\varepsilon \sigma } dv_g \nonumber \\&= O(\varepsilon )\int _{B_\delta \backslash B_{r_\varepsilon }} e^{4\pi (2+\alpha ) G_p } dx = O(\varepsilon ) \int _{B_\delta \backslash B_{r_\varepsilon }} \frac{1}{|x|^{2(2+\alpha )}} dx\nonumber \\&= O(\varepsilon )\left( \frac{1}{r_\varepsilon ^{2(1+\alpha )}}-\frac{1}{\delta ^{2(1+\alpha )}}\right) = O\left( \frac{1}{\gamma _\varepsilon ^{2(1+\alpha )}}\right) +O(\varepsilon )\nonumber \\&= o_\varepsilon (1). \end{aligned}$$

(32)

Finally

$$\begin{aligned} \int _{\Sigma \backslash B_\delta } h e^{\varphi _\varepsilon } dv_g = \varepsilon e^{C_\varepsilon } \int _{\Sigma \backslash B_\delta } h e^{\overline{\rho } G_p} dv_g = O(\varepsilon ) \end{aligned}$$

(33)

so by (30), (31), (32) and (33) we have

$$\begin{aligned} \int _{\Sigma } h e^{\varphi _\varepsilon } dv_g = \frac{\pi \tilde{h}(p) e^{-4\pi \alpha A(p)}}{ 1+\alpha } +o_\varepsilon (1). \end{aligned}$$

(34)

Using (28), (29) and (34) we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}J(\varphi _\varepsilon )&= -\overline{\rho }\left( 1+4\pi A(p)+ \log \left( \frac{1}{|\Sigma |} \frac{\pi \tilde{h}(p) }{ 1+\alpha } \right) \right) \\&= -\overline{\rho }\left( 1+\log \frac{\pi }{|\Sigma |}+\max _{\xi \in \Sigma ,\beta (\xi )=\alpha } \left\{ 4\pi A(\xi )\right. \right. \\&\left. \left. + \log \left( \frac{K(\xi )}{1+\alpha }\prod _{q\in S, q\ne \xi } e^{-4\pi \beta (q) G_q(\xi )}\right) \right\} \right) . \end{aligned}$$

This, together with Proposition 3.1, completes the proof of Theorem 1.1.

5 Onofri’s Inequalities on $S^2$

In this section we will consider the special case of the standard sphere $(S^2,g_0)$ with $m\le 2$ and $K\equiv 1$. We fix $\alpha _1,\alpha _2\in {\mathbb {R}}$ with $-1<\alpha _1\le \alpha _2$ and as before we consider the singular weight

$$\begin{aligned} h= e^{-4\pi \alpha _1 G_{p_1}-4\pi \alpha _2 G_{p_2}}. \end{aligned}$$

In order to apply Theorem 1.1 and obtain sharp versions of (7), we need to study the existence of minimum points for the functional $J$. Let us fix a system of coordinates $(x_1,x_2,x_3)$ on ${\mathbb {R}}^3$ such that $p_1=(0,0,1)$. When $h\in C^1(S^2)$, the Kazdan–Warner identity (see [18]) states that any solution of (4) has to satisfy

$$\begin{aligned} \int _{S^2} \nabla h \cdot \nabla x_i \; e^u\; dv_{g_0}= \left( 2-\frac{\rho }{4\pi }\right) \int _{S^2} h e^{u} x_i \;dv_{g_0}\quad i=1,2,3. \end{aligned}$$

We claim that if $p_2=-p_1$ the same identity holds, at least in the $x_3$-direction, even when $h$ is singular.

Lemma 5.1

Let $u$ be a solution of (4) on $S^2$, then there exist $C,\delta _0>0$ such that

$$\begin{aligned} \begin{array}{ll} \bullet \quad |\nabla u(x)|\le C d(x,p_i)^{2\alpha _i +1} \quad if\quad \alpha _i<- \frac{1}{2};\\ \bullet \quad |\nabla u(x)|\le C \left( -\log d(x,p_i)\right) \quad if \quad \alpha _i=- \frac{1}{2};\\ \bullet \quad |\nabla u(x)|\le C \quad if \quad \alpha _i>- \frac{1}{2}; \end{array} \end{aligned}$$

for $0<d(x,p_i)<\delta _0$, $\;i=1,2$.

Proof

Let us fix $0<r_0<\frac{1}{2}\min \{\frac{\pi }{2},d(p_1,p_2)\}$ and $i\in \{1,2\}$. If $\alpha _i>- \frac{1}{2}$ then, by standard elliptic regularity, $u \in C^1(\overline{B_{r_0}(p_i)})$ and the conclusion holds for $\delta _0=r_0$ and $C=\Vert \nabla u\Vert _{L^\infty (B_{r_0}(p_i))}$. Let us now assume $\alpha _i \le -\frac{1}{2}$. We know that $h(y)\le C_1 d(y,p_i)^{2\alpha _i}$ for $y\in B_{2r_0}(p_i)$ so, if $\delta _0 < r_0$, by Green’s representation formula we have

$$\begin{aligned} |\nabla u|(x)&\le \rho e^{\Vert u \Vert _\infty } \int _{S^2} \frac{h(y)}{d(x,y)} dv_{g_0}(y)\le \frac{\rho e^{\Vert u\Vert _\infty } \Vert h\Vert _{L^1(S^2)}}{r_0} \\&+ \rho e^{\Vert u \Vert _\infty } C_1 \int _{B_{r_0}(x)} \frac{d(y,p_i)^{2\alpha _i}}{d(x,y)}dv_{g_0}(y). \end{aligned}$$

Let $\pi$ be the stereographic projection from the point $-p_i$. It is easy to check that there exist $C_2,C_3>0$ such that

$$\begin{aligned} C_2 \; d(q,q') \le |\pi (q)-\pi (q')|\le C_3\; d(q,q') \end{aligned}$$

$\forall \; q,q'\in B_{\frac{\pi }{2}}(p_i)$. Thus we have

$$\begin{aligned} \int _{B_{r_0}(x)} \frac{d(y,p_i)^{2\alpha _i}}{d(x,y)}dv_{g_0}(y) \!&\le \! \int _{B_{\frac{\pi }{2}}(p_i)} \frac{d(y,p_i)^{2\alpha _i}}{d(x,y)}dv_{g_0}(y) \!\le \! C_4 \int _{\{|z|\le 1\}} \frac{|z|^{2\alpha _i}}{|\pi (x)-z|} dz \\&= C_4 |\pi (x)|^{2\alpha _i+1}\int _{\left\{ |z|\le \frac{1}{|\pi (x)|}\right\} } \frac{|z|^{2\alpha _i}}{\left| \frac{\pi (x)}{|\pi (x)|}-z\right| }dz\\&\le C_5 d(x,p_i)^{2\alpha _i+1} \int _{\left\{ |z|\le \frac{1}{|\pi (x)|}\right\} } \frac{|z|^{2\alpha _i}}{\left| \frac{\pi (x)}{|\pi (x)|}-z\right| } dz. \end{aligned}$$

Notice that

$$\begin{aligned} \int _{\left\{ |z|\le \frac{1}{|\pi (x)|}\right\} } \frac{|z|^{2\alpha _i}}{\left| \frac{\pi (x)}{|\pi (x)|}-z\right| } dz&\le \frac{1}{2^{2\alpha _i}} \int _{\left\{ \left| \frac{\pi (x)}{|\pi (x)|}-z\right| \le \frac{1}{2}\right\} }\frac{1}{\left| \frac{\pi (x)}{|\pi (x)|}-z\right| } dz \\&+\,2 \int _{\{|z|\le 2\}} |z|^{2\alpha _i}dz + 2\int _{\left\{ 2\le |z|\le \frac{1}{|\pi (x)|}\right\} } |z|^{2\alpha _i-1} dz \\&\le C_6+ 2 \int _{\left\{ 2\le |z|\le \frac{1}{|\pi (x)|}\right\} } |z|^{2\alpha _i-1} dz. \end{aligned}$$

If $\alpha _i<-\frac{1}{2}$

$$\begin{aligned} \int _{\left\{ 2\le |z|\le \frac{1}{|\pi (x)|}\right\} } |z|^{2\alpha _i-1} dz \le C_7, \end{aligned}$$

while if $\alpha _i=-\frac{1}{2}$

$$\begin{aligned} \int _{\left\{ 2\le |z|\le \frac{1}{|\pi (x)|}\right\} } |z|^{2\alpha _i-1} dz = 2\pi \log \left( \frac{1}{2|\pi (x)|}\right) \le C_8 \left( - \log d(x,p_i) \right) . \end{aligned}$$

Thus we get the conclusion for $\delta _0$ sufficiently small.$\square$

In any case there exists $s\in [0,1)$ such that

$$\begin{aligned} |\nabla u(x)|\le C d(x,p_i)^{-s} \left( -\log d(x,p_i)\right) \end{aligned}$$

(35)

for $0<d(x,p_i)<\delta _0$, $\;i=1,2$.

Proposition 5.1

If $p_2=-p_1$ then any solution of (4) satisfies

$$\begin{aligned} \int _{S^2} \nabla h \cdot \nabla x_3 \; e^u \; dv_{g_0} = \left( 2-\frac{\rho }{4\pi }\right) \int _{S^2} h e^{u} x_3 \; dv_{g_0}. \end{aligned}$$

Proof

Without loss of generality we may assume

$$\begin{aligned} \int _{S^2} h e^{u} dv_{g_0}=1. \end{aligned}$$

(36)

Let us denote $S_\delta = S^2\backslash B_{\delta }(p_1)\cup B_\delta (p_2)$. Since $u$ is smooth in $S_\delta$, multiplying (4) by $\nabla u\cdot \nabla x_3$ and integrating on $S_\delta$ we have

$$\begin{aligned} -\int _{S_\delta } \Delta u \; \nabla u\cdot \nabla x_3 \, dv_{g_0}=\rho \int _{S_\delta } \left( h\; e^{u} -\frac{1}{4\pi }\right) \nabla u \cdot \nabla x_3 \; dv_{g_0} \end{aligned}$$

(37)

Integrating by parts we obtain

$$\begin{aligned} -\int _{S_\delta } \Delta u \; \nabla u\cdot \nabla x_3\; dv_{g_0}&= \int _{S_\delta } \nabla u \cdot \nabla (\nabla u \cdot \nabla x_3)dv_{g_0}\\&+\sum _{i=1}^2 \int _{\partial B_\delta (p_i)}\nabla u \cdot \nabla x_3 \frac{\partial u}{ \partial n} d\sigma _{g_0} \end{aligned}$$

and by (35)

$$\begin{aligned} \left| \int _{\partial B_\delta (p_i)}\nabla u \cdot \nabla x_3 \; \frac{\partial u}{ \partial n}\; d\sigma _{g_0}\right| \!\le \! \int _{\partial B_{\delta }(p_i)} |\nabla u|^2 |\nabla x_3| d\sigma _{g_0} \!=\! O(\delta ^{2(1-s)} \log ^2 \delta )=o_\delta (1). \end{aligned}$$

Using the identities

$$\begin{aligned} \nabla u \cdot \nabla (\nabla u \cdot \nabla x_3)= \frac{1}{2}\nabla \left( |\nabla u|^2 \cdot \nabla x_3\right) -x_3|\nabla u|^2 \end{aligned}$$

and

$$\begin{aligned} {-}\Delta x_3 = 2x_3, \end{aligned}$$

and applying again (35) to estimate the boundary term, we get

$$\begin{aligned} -\int _{S_\delta } \Delta u \; \nabla u\cdot \nabla x_3 \; dv_{g_0}&= \int _{S_\delta } \frac{1}{2}\nabla |\nabla u|^2 \cdot \nabla x_3 \; dv_{g_0} -\int _{S_\delta } x_3 |\nabla u|^2 dv_{g_0}+o_\delta (1)\\&= -\frac{1}{2}\int _{S_\delta } \Delta x_3 \; |\nabla u|^2 dv_{g_0} -\sum _{i=1}^2\int _{\partial B_\delta (p_i)} |\nabla u|^2 \frac{\partial x_3}{ \partial n} d\sigma _{g_0} \\&-\int _{S_\delta } x_3 |\nabla u|^2 dv_{g_0} =o_\delta (1). \end{aligned}$$

Thus (37) becomes

$$\begin{aligned} \int _{S_\delta } h e^{u}\nabla u \cdot \nabla x_3 \; dv_{g_0} - \frac{1}{4\pi } \int _{S_\delta } \nabla u \cdot \nabla x_3\; dv_{g_0}=o_\delta (1). \end{aligned}$$

(38)

Moreover

$$\begin{aligned} \int _{S_\delta } \nabla u \cdot \nabla x_3\; dv_{g_0}&= -\int _{S_\delta } \Delta u \; x_3\; dv_{g_0} -\sum _{i=1}^2 \int _{\partial B_\delta (p_i) } x_3 \frac{\partial u}{ \partial n} \; d\sigma _{g_0}\\&= \rho \int _{S_\delta } \left( h e^u-\frac{1}{4\pi }\right) x_3 \;dv_{g_0}+ O(\delta ^{1-s}(-\log \delta ))\\&= \rho \int _{S_\delta } h e^u x_3 \;dv_{g_0}+o_\delta (1) \end{aligned}$$

and

$$\begin{aligned} \int _{S_\delta } h e^{u} \; \nabla u \cdot \nabla x_3\; dv_{g_0}&= \int _{S_\delta } \nabla e^{u} \cdot h \nabla x_3 \; dv_{g_0} = - \int _{S_\delta } e^u {{\mathrm{div}}}(\;h \nabla x_3) dv_{g_0}\\&-\sum _{i=1}^2\int _{\partial B_\delta (p_i)} h e^{u} \frac{\partial x_3}{ \partial n}\; d\sigma _{g_0}\\&= -\int _{S_\delta } \nabla h \cdot \nabla x_3 \; e^{u} \; dv_{g_0} + 2 \int _{S_\delta } h e^u x_3 dv_{g_0} +O \big (\delta ^{2(1+\alpha )}\big ). \end{aligned}$$

Thus by (38) we have

$$\begin{aligned} \int _{S_\delta } \nabla h \cdot \nabla x_3 \; e^{u}\; dv_{g_0} = \left( 2-\frac{\rho }{4\pi }\right) \int _{S_\delta } h e^u x_3\; dv_{g_0}+o_\delta (1). \end{aligned}$$

Since $u$ is continuous on $S^2$ and $h, \nabla h \cdot \nabla x_3\in L^1(S^2)$ as $\delta \rightarrow 0$ we get the conclusion.$\square$

Remark 5.1

In this proof there is no need to assume $K\equiv 1$.

Assuming $p_1=(0,0,1)$ and $p_2=(0,0,-1)$, one may easily verify that

$$\begin{aligned} G_{p_1}(x)=-\frac{1}{4\pi } \log (1-x_3) -\frac{1}{4\pi } \log \left( \frac{e}{2}\right) \end{aligned}$$

and

$$\begin{aligned} G_{p_2}(x)=-\frac{1}{4\pi } \log (1+x_3) -\frac{1}{4\pi } \log \left( \frac{e}{2}\right) , \end{aligned}$$

so that

$$\begin{aligned} \nabla h \cdot \nabla x_3 = -4\pi h (\alpha _1 \nabla G_1+ \alpha _2 \nabla G_2)\cdot \nabla x_3= (\alpha _2 -\alpha _1)h - (\alpha _1+\alpha _2)h x_3. \end{aligned}$$

Thus we can rewrite the identity in Proposition 5.1 as

$$\begin{aligned} \alpha _2-\alpha _1 = \left( 2-\frac{\rho }{4\pi } +\alpha _1+\alpha _2\right) \int _{S^2} h e^u x_3 \;dv_{g_0}. \end{aligned}$$

(39)

Proof of Theorem 1.2

Assume $m=1$ (i.e., $\alpha _2=0$). We claim that equation (4) has no solutions for $\rho = \overline{\rho } = 8\pi (1+\min \{0,\alpha _1\})$, unless $\alpha _1 =0$. Indeed if $u$ were a solution of (4) satisfying (36), then applying (39) with $\rho = \overline{\rho }$ we would get

$$\begin{aligned} -\alpha _1 = \left( \alpha _1-2\min \{0,\alpha _1\}\right) \int _{S^2} h e^u x_3\;dv_{g_0} \end{aligned}$$

so that, if $\alpha _1\ne 0$,

$$\begin{aligned} \left| \int _{S^2} h e^{u} x_3\; dv_{g_0} \right| =1. \end{aligned}$$

This contradicts (4). In particular we proved non-existence of minimum points for $J_{\overline{\rho }}$ so we can exploit Theorem 1.1 and (9) to prove that (7) holds with

$$\begin{aligned} C= \max _{p\in S^2 ,\beta (p)=\alpha } \left\{ \log \left( \frac{1}{1+\alpha }\prod _{q\in S, q\ne p} e^{-4\pi \beta (q) G_q(p)}\right) \right\} . \end{aligned}$$

If $\alpha _1<0$ one has

$$\begin{aligned} C=-\log (1+\alpha _1). \end{aligned}$$

If $\alpha _1>0$,

$$\begin{aligned} C= \max _{p\in S^2\backslash \{p_1\} } \left\{ -4\pi \alpha _1 G_{p_1}(p)\right\} =-4\pi \alpha _1 G_{p_1}(p_2) = \alpha _1. \end{aligned}$$

$\square$

Proof of Theorem 1.3

As in the previous proof, applying (39) with $\rho = \overline{\rho } =8\pi (1+\alpha _1)$, we obtain that any critical point of (4) for which (36) holds has to satisfy

$$\begin{aligned} \alpha _2-\alpha _1= (\alpha _2 -\alpha _1)\int _{S^2} h e^u x_3 dv_{g_0}. \end{aligned}$$

Since $\alpha _1 \ne \alpha _2$ one has

$$\begin{aligned} \int _{S^2} h e^u x_3 dv_{g_0}=1 \end{aligned}$$

which is impossible. Thus $J_{\overline{\rho }}$ has no critical points and by Theorem 1.1 one has

$$\begin{aligned} C=\log \left( \frac{1}{1+\alpha _1}e^{-4\pi \alpha _2 G_{p_2}(p_1)}\right) = \alpha _2-\log (1+\alpha _1). \end{aligned}$$

$\square$

Now we assume $\alpha _1=\alpha _2<0$. In this case identity (39) gives no useful condition. Let us denote by $\pi$ the stereographic projection from the point $p_1$. It is easy to verify that $u$ satisfies (4) and (36) if and only if

$$\begin{aligned} v:= u\circ \pi ^{-1} +(1+\alpha )\log \left( \frac{4}{(1+|y|^2)^2}\right) +2\alpha \log \left( \frac{e}{2}\right) \end{aligned}$$

solves

$$\begin{aligned} {-}\Delta _{{\mathbb {R}}^2} v = 8\pi (1+\alpha )|y|^{2\alpha } e^{v} \end{aligned}$$

(40)

in ${\mathbb {R}}^2$ and

$$\begin{aligned} \int _{{\mathbb {R}}^2} |y|^{2\alpha } e^{v} dy =1. \end{aligned}$$

As we pointed out in the proof of Lemma 2.3 and Eq. (40) has a one-parameter family of solutions:

$$\begin{aligned} v_\lambda (y)= -2 \log \left( 1+\frac{\pi }{1+\alpha } e^l |y|^{2(1+\alpha )}\right) \end{aligned}$$

$l\in {\mathbb {R}}$. Thus we have a corresponding family $\{u_{\lambda ,c}\}$ of critical points of $J_{\overline{\rho }}$ given by the expression

$$\begin{aligned} u_{\lambda ,c}\circ \pi ^{-1}(y) = 2\log \left( \frac{\big (1+|y|^2 \big )^{1+\alpha }}{1+ \lambda |y|^{2(1+\alpha )}}\right) +c, \end{aligned}$$

(41)

$c\in {\mathbb {R}},\lambda >0$. A priori we do not know whether these critical points are minima for $J_{\overline{\rho }}$ (as it happens for $\alpha =0$), so a direct application of 1.1 is not possible. However, we can still get the conclusion by comparing $J_{\overline{\rho }}(u_{\lambda ,c})$ with the blow-up value provided by Theorem 1.1.

Proof of Theorem 1.4

Let us first compute $J(u_{\lambda ,c})$. Let $\varphi _t:S^2\longrightarrow S^2$ be the conformal transformation defined by $\pi (\varphi _t (\pi ^{-1}(y)))= t y$. It is not difficult to prove that $\forall \; t>0$

$$\begin{aligned} J_{\overline{\rho }}(u)=J_{\overline{\rho }}(u \circ \varphi _t + (1+\alpha )\log |\det d\varphi _t|); \end{aligned}$$

in particular, since

$$\begin{aligned} u_{\lambda ,c}= u_{1,0}\circ \varphi _{\lambda ^\frac{1}{2(1+\alpha )}} +(1+\alpha ) \log |\det \varphi _{\lambda ^\frac{1}{2(1+\alpha )}}| +c-\log \lambda , \end{aligned}$$

we have that $J(u_{\lambda ,c})$ does not depend on $\lambda$ and $c$. Thus we may assume $\lambda =1$ and $c=0$. A simple computation shows that

$$\begin{aligned} \int _{S^2} h \; e^{u_{1,0}} dv_{g_0}= 4 e^{2\alpha } \int _{{\mathbb {R}}^2}\frac{|y|^{2\alpha }}{\left( 1+|y|^{2(1+\alpha )}\right) ^2} dy = \frac{4 e^{2\alpha } \pi }{1+\alpha }. \end{aligned}$$

(42)

Since $u_{1,0}(p_1)=0$ and $u_{1,0}$ solves

$$\begin{aligned} {-}\Delta u_{1,0} = \omega \; h \; e^{u_{1,0}} -2(1+\alpha ) \quad \text{ with } \quad \omega :=2(1+\alpha )^2 e^{-2\alpha } \end{aligned}$$

one has

$$\begin{aligned} \int _{S^2} u_{1,0} \; dv_{g_0} = 4\pi \int _{S^2} \Delta u_{1,0}\; G_{p_1} dv_{g_0}= -4\pi \omega \int _{S^2} h e^{u_{1,0}} G_{p_1} dv_{g_0} \end{aligned}$$

and

$$\begin{aligned}&\frac{1}{2}\int _{S^2}|\nabla u_{1,0}|^2dv_{g_0} +2(1+\alpha ) \int _{S^2} u_{1,0}\; dv_{g_0}\nonumber \\&\quad = \frac{1}{2} \omega \int _{S^2} h e^{u_{1,0}} u_{1,0} \; dv_{g_0} + (1+\alpha )\int _{S^2} u_{1,0}\; dv_{g_0}\nonumber \\&\quad = \frac{\omega }{2}\int _{S^2} h e^{u_{1,0}} ( u_{1,0}-\overline{\rho } G_{p_1}) dv_{g_0}. \end{aligned}$$

(43)

Since

$$\begin{aligned} G_{p_1}(\pi ^{-1}(y)):= \frac{1}{4\pi } \log (1+|y|^2) -\frac{1}{4\pi } \end{aligned}$$

we get

$$\begin{aligned} \int _{S^2} h e^{u_{1,0}} ( u_{1,0}-\overline{\rho } G_{p_1})&= 2(1+\alpha )\int _{S^2} h e^{u_{1,0}} dv_{g_0}\nonumber \\&-\,8e^{2\alpha }\int _{{\mathbb {R}}^2} \frac{|y|^{2\alpha } \log \left( {1+|y|^{2(1+\alpha )}}\right) }{\left( 1+|y|^{2(1+\alpha )}\right) ^2} dy\nonumber \\&= 8\pi e^{2\alpha }-\frac{8 \pi e^{2\alpha }}{1+\alpha } \int _{0}^{+\infty } \frac{\log (1+s)}{(1+s)^2}ds = \frac{8\pi \alpha e^{2\alpha }}{1+\alpha }.\qquad \end{aligned}$$

(44)

Using (42), (43) and (44) we obtain

$$\begin{aligned} J(u_{\lambda ,c})=J(u_{1,0})= 8\pi (1+\alpha )\left( \log (1+\alpha )-\alpha \right) \qquad \forall \; \lambda >0, c \in {\mathbb {R}}. \end{aligned}$$

To conclude the proof it is sufficient to observe that $u_{\lambda ,c}$ have to be minimum points for $J_{\overline{\rho }}$ that is

$$\begin{aligned} \inf _{H^1(S^2)} J_{\overline{\rho }} = 8\pi (1+\alpha )\left( \log (1+\alpha )-\alpha \right) . \end{aligned}$$

Indeed if this were false then $J_{\overline{\rho }}$ would have no minimum points but, by Theorem 1.1, we would get

$$\begin{aligned} \inf _{H^1(S^2)} J_{\overline{\rho }} = 8\pi (1+\alpha )\left( \log (1+\alpha )-\alpha \right) = J(u_{\lambda ,c}). \end{aligned}$$

This is clearly a contradiction.$\square$

Remark 5.2

There is no need to assume $p_1=-p_2$.

Indeed given two arbitrary points $p_1,p_2\in S^2$ with $p_1\ne p_2$ it is always possible to find a conformal diffeomorphism $\varphi :S^2 \longrightarrow S^2$ such that $\varphi ^{-1}(p_1) = -\varphi ^{-1}(p_2)$. Moreover one has

$$\begin{aligned} J_{\overline{\rho }}(u)= \widetilde{J_{\overline{\rho }}}(u\circ \varphi + (1+\alpha )\log |\det d \varphi |)+c_{\alpha ,p_1,p_2} \end{aligned}$$

$\forall \; u \in H^1(S^2)$, where $\widetilde{J}$ is the Moser–Trudinger functional associated to

$$\begin{aligned} \widetilde{h}= e^{-4\pi \alpha G_{\varphi ^{-1}(p_1)}-4\pi \alpha G_{\varphi ^{-1}(p_2)}}. \end{aligned}$$

and $c_{\alpha ,p_1,p_2}$ is an explicitly known constant depending only on $\alpha$, $p_1$ and $p_2$. In particular one can still compute $\min _{H^1(S^2)} J_{\overline{\rho }}$ and describe the minimum points of $J_{\overline{\rho }}$ in terms of $\varphi$ and the family (41).