On Gaussian curvature equation in $\mathbb{R}^{2}$ with

prescribed nonpositive curvature

Huyuan<EMAIL_ADDRESS>
Department of Mathematics, Jiangxi Normal University,
Nanchang, Jiangxi 330022, PR China
Dong<EMAIL_ADDRESS><EMAIL_ADDRESS>
Center for PDEs, School of Mathematical Sciences, East China Normal
University,
Shanghai Key Laboratory of PMMP, Shanghai 200062, PR China

IECL, UMR 7502, University of Lorraine, 57050 Metz, France
Feng<EMAIL_ADDRESS>
Center for PDEs, School of Mathematical Sciences, East China Normal
University,
Shanghai Key Laboratory of PMMP, Shanghai 200062, PR China

Dedicated to Professor Weiming Ni’s seventieth birthday

###### Abstract

The purpose of this paper is to study the solutions of

$\Delta u+K(x)e^{2u}=0\quad{\rm in}\;\;\mathbb{R}^{2}$

with $K\leq 0$. We introduce the following quantity:

$\alpha_{p}(K)=\sup\left\\{\alpha\in\mathbb{R}:\,\int_{\mathbb{R}^{2}}|K(x)|^{p}(1+|x|)^{2\alpha
p+2(p-1)}dx<+\infty\right\\},\quad\forall\;p\geq 1.$

Under the assumption $({\mathbb{H}}_{1})$: $\alpha_{p}(K)>-\infty$ for some
$p>1$ and $\alpha_{1}(K)>0$, we show that for any $0<\alpha<\alpha_{1}(K)$,
there is a unique solution $u_{\alpha}$ with
$u_{\alpha}(x)=\alpha\ln|x|+c_{\alpha}+o\big{(}|x|^{-\frac{2\beta}{1+2\beta}}\big{)}$
at infinity and $\beta\in(0,\,\alpha_{1}(K)-\alpha)$. Furthermore, we show an
example $K_{0}\leq 0$ such that $\alpha_{p}(K_{0})=-\infty$ for any $p>1$ and
$\alpha_{1}(K_{0})>0$, for which we study the asymptotic behavior of
solutions. In particular, we prove the existence of a solution $u_{*}$ such
that $u_{*}-\alpha_{*}\ln|x|=O(1)$ at infinity for some $\alpha_{*}>0$, but
who does not converge to a constant at infinity. This example exhibits a new
phenomenon of solutions with logarithmic growth and non-uniform behavior at
infinity.

Key words: Gaussian curvature; Conformal metrics; Asymptotic behavior.

MSC 2010: 35R06, 35A01, 35J66.

##  1 Introduction

In this paper, we study the solutions of

$\Delta u+K(x)e^{2u}=0\quad{\rm in}\;\;\mathbb{R}^{2},$ (1.1)

where $K$ is a nonpositive, non trivial and locally bounded function in
$\mathbb{R}^{2}$.

Equation (1.1) finds its origin from conformal geometry: Given a Riemannian
surface $(\mathcal{M},g)$ and a function $K$ on $\mathcal{M}$, we may ask
whether there is a new Riemannian metric $g_{1}$, conformal to $g$, such that
$K$ is the Gaussian curvature of $g_{1}$. In other words, we need to find a
function $u$ on $\mathcal{M}$ such that $g_{1}=e^{2u}g$ and $u$ is a solution
of the elliptic equation

$\Delta_{g}u-k_{g}+Ke^{2u}=0\quad{\rm on}\;\;\mathcal{M}.$ (1.2)

Here $\Delta_{g}$ is the Laplace-Beltrami operator, $k_{g}$ is the Gaussian
curvature on $({\mathcal{M}},g)$. In the Euclidean case
$\mathcal{M}=\mathbb{R}^{2}$ and $g=|dx|^{2}$, (1.2) reduces to equation
(1.1).

The equation (1.1) has been studied extensively, especially for the
nonpositive, nonconstant curvature case, see [5, 9, 8, 6, 3] and the
references therein. The nonexistence of entire solution in $\mathbb{R}^{2}$
with $K\equiv-1$ was given by Ahlfors [1] in 1938, and then improved by
Sattinger [9], Ni [8] for $K\leq 0$ in $\mathbb{R}^{2}$ and
$K(x)\leq-C|x|^{-2}$ at infinity with $C>0$. The first general existence
result for (1.1) is due to Ni [8], he proved that if $K\leq 0$, $K\not\equiv
0$ in $\mathbb{R}^{2}$ with $|K(x)|\leq C|x|^{-\ell}$ at infinity for some
$\ell>2$, then for any $\alpha\in(0,\beta)$ with
$\beta=\min\\{8,\frac{\ell-2}{3}\\}$, there exists a solution $u$ satisfying
$u(x)=\alpha\ln|x|+O(1)$ at infinity. Later on, it was proved by Cheng-Ni [3]
that if $K\leq 0$ in $\mathbb{R}^{2}$, $K(x)\sim-|x|^{-\ell}$ at infinity with
$\ell>2$, then for any $\alpha\in(0,\,\frac{\ell-2}{2})$, (1.1) has a unique
solution satisfying

$\displaystyle u_{\alpha}(x)=\alpha\ln|x|+c_{\alpha}+o(1)\;\;\mbox{as
}\;|x|\to\infty,\quad\mbox{with }c_{\alpha}\in\mathbb{R}.$ (1.3)

The solutions set has layer structure, that is $u_{\beta}<u_{\alpha}$ in
$\mathbb{R}^{2}$, for any $0<\beta<\alpha<\alpha_{1}(K)$. Moreover, a unique
maximal solution $U$ exists such that
$U(x)=\frac{\ell-2}{2}\ln|x|-\ln\ln|x|+O(1)$ at infinity.

For general curvature functions $K$, the following quantity was introduced by
Cheng-Ni in [4],

$\alpha_{1}(K)=\sup\left\\{\alpha\in\mathbb{R}:\,\int_{\mathbb{R}^{2}}|K(x)|(1+|x|)^{2\alpha}dx<+\infty\right\\}$
(1.4)

with the convention $\sup\emptyset=-\infty$. When $K$ is controlled
polynomially at infinity, using $\alpha_{1}(K)$, Cheng-Ni characterized the
existence to problem (1.1): Assume that $K\leq 0$ satisfies

$\alpha_{1}(K)>0$ and $|K(x)|<C(|x|^{m}+1)$ in $\mathbb{R}^{2}$ for some
$m>0$. $None$

Then for any $\alpha\in(0,\alpha_{1}(K))$, (1.1) has a unique solution such
that

$u_{\alpha}=\alpha\ln|x|+O(1)\quad{\rm as}\ \ |x|\to\infty.$ (1.5)

Later on, by employing a variational method with weighted Sobolev spaces
(firstly applied by McOwen in [7]), Cheng-Lin [2] succeeded to handle the
existence of problem (1.1) without any growth control for $K$: Assume that
$\alpha_{1}(K)>0$ for a nonpositive, locally bounded function $K$.

  * •

For any $\alpha\in(0,\alpha_{1}(K))$, equation (1.1) possesses a unique
solution $u_{\alpha}$ satisfying

$\displaystyle\int_{\mathbb{R}^{2}}Ke^{2u}dx=-2\alpha\pi\quad\mbox{and}\quad\int_{\mathbb{R}^{2}\backslash
B_{1}}\Big{|}\nabla(u-\alpha\ln|x|)\Big{|}^{2}dx<+\infty.$ (1.6)

  * •

Moreover, for any $0<\beta<\alpha<\alpha_{1}(K)$, there holds
$u_{\beta}<u_{\alpha}$ in $\mathbb{R}^{2}$.

  * •

If $\alpha_{1}(K)<\infty$, then
$\lim_{\alpha\to\alpha_{1}(K)}u_{\alpha}=u_{\alpha_{1}}$ is a solution to
(1.1) satisfying (1.6) with $\alpha=\alpha_{1}(K)$.

The quantity

$\mathcal{C}(u):=\int_{\mathbb{R}^{2}}Ke^{2u}dx$

is called the total curvature of the conformal metric $g_{u}=e^{2u}|dx|^{2}$.
It plays an important role in the analysis of (1.1) and has a notable meaning
in geometry. There are many interesting discussions in [2]. For example, by
Theorem 1.4 in [2], when $K\leq 0$, a solution $u$ to (1.1) is logarithmically
up bounded at infinity if and only if ${\mathcal{C}}(u)$ is finite and the
metric $e^{2u}|dx|^{2}$ is complete.

However, to claim the asymptotic behavior like (1.5), Cheng-Lin need
additional assumptions like $({\mathbb{H}}_{0})$. A naturel question is to
understand the asymptotic behavior of solutions provided by variational method
in [2]. We will point out that any solution satisfying (1.5) is just a Cheng-
Lin type solution, see Proposition 2.1 below. Moreover, we want to prove the
existence of solutions satisfying (1.5) without any growth assumption on the
curvature function $K$.

Another naturel question is to ask if there exist solutions with logarithmic
growth but anisotropic behavior at infinity, more precisely, solutions of
(1.1) satisfying (1.5) but not (1.3). Our purpose here is to answer these
questions.

In fact, we propose to replace the pointwise growth assumption
$(\mathbb{H}_{0})$ by an integral estimate. To this end, consider the
following quantity related to $K$, which generalizes $\alpha_{1}(K)$:

$\alpha_{p}(K)=\sup\left\\{\alpha\in\mathbb{R}:\,\int_{\mathbb{R}^{2}}|K(x)|^{p}(1+|x|)^{2\alpha
p+2(p-1)}dx<+\infty\right\\},\quad\forall\;p\geq 1.$ (1.7)

We use always the convention $\sup\emptyset=-\infty$. Our first main result
states as follows.

######  Theorem 1.1.

Let $K\leq 0$ be locally bounded in $\mathbb{R}^{2}$. Assume that

$\alpha_{1}(K)>0$ and there exists $p>1$ such that $\alpha_{p}(K)>-\infty$.
$None$

Then for any $\alpha\in(0,\,\alpha_{1}(K))$, problem (1.1) possesses a unique
solution $u_{\alpha}$ satisfying (1.5). Moreover, the total curvature of
$u_{\alpha}$ is $-2\alpha\pi$; and there exists $c_{\alpha}\in\mathbb{R}$ such
that for any $\beta\in(0,\alpha_{1}(K)-\alpha)$,

$u_{\alpha}(x)=\alpha\ln|x|+c_{\alpha}+o\left(|x|^{-\frac{2\beta}{1+2\beta}}\right)\quad{\rm
as}\;|x|\to\infty.$ (1.8)

######  Remark 1.1.

Combining with Proposition 2.1 below, Theorem 1.1 means that under
$(\mathbb{H}_{1})$, for any $\alpha\in(0,\,\alpha_{1}(K))$, the unique
solution satisfying (1.6) given by [2] has the asymptotic behavior (1.5).

######  Remark 1.2.

Assumption $(\mathbb{H}_{1})$ is weaker than $(\mathbb{H}_{0})$, see Lemma 2.2
below.

Our construction is based on the understanding of $\alpha_{p}(K)$. Indeed,
under $({\mathbb{H}}_{1})$, we can prove that $\alpha_{p}(K)$ are positive for
$p>1$ but nearby. This observation allows us to construct suitable super and
sub solutions of (1.1). Let $w_{0}$ be a positive smooth radial function such
that

$w_{0}(x)=\ln|x|\quad\ {\rm in}\ \,\mathbb{R}^{2}\setminus B_{1},$ (1.9)

where $B_{1}=\\{x\in\mathbb{R}^{2},|x|<1\\}.$ In the following, $B_{r}(x)$
stands for the standard open ball with center $x\in\mathbb{R}^{2}$ and the
radius $r$; $B_{r}$ denotes always $B_{r}(0)$. We will look for a solution
$u_{\alpha}$ to (1.1) in the form

$u_{\alpha}=\alpha w_{0}+v_{\alpha},$

where $v_{\alpha}$ resolves

$-\Delta v=Ke^{2\alpha w_{0}}e^{2v}+\alpha\Delta w_{0}\quad{\rm
in}\;\;\mathbb{R}^{2}.$

The crucial point is to know whether the above equation has a bounded
solution, or equivalently, whether $Ke^{2\alpha w_{0}}e^{2v}+\alpha\Delta
w_{0}$ belongs to the Kato’s class. If the answer is affirmative, the Perron’s
method can be employed to get a solution of (1.1).

Our second concern is to understand solutions to (1.1) with an exotic
$K_{0}\leq 0$ such that

$\alpha_{p}(K_{0})=-\infty\;\;\mbox{for any
}\;p>1\quad\mbox{and}\quad\alpha_{1}(K_{0})>0.$ (1.10)

We construct here an explicit model $K_{0}$ as follows. Consider the sequence
$a_{n}:=(n,0)_{n\geq 2}\in\mathbb{R}^{2}$. Let

$K_{0}(x)=-\sum^{\infty}_{n=2}r_{n}^{-2}n^{-\ell}\eta_{0}\left(\frac{x-a_{n}}{r_{n}}\right)\quad\mbox{in
}\mathbb{R}^{2},$ (1.11)

where

$\displaystyle\ell>q>1,\quad r_{n}=e^{-n^{q}}\;\;\mbox{for all }\;n\geq 2$

and $\eta_{0}$ is a smooth radial cut-off function satisfying

$\chi_{B_{\frac{1}{2}}}\leq\eta_{0}\leq\chi_{B_{1}}.$ (1.12)

Here $\chi_{\Omega}$ means the usual characteristic function of $\Omega$.
Observe that $r_{n}\leq e^{-2}<\frac{1}{4}$ for any $n\geq 2$, so

$\displaystyle\int_{\mathbb{R}^{2}}|K_{0}(x)|^{p}(1+|x|)^{2\alpha
p+2(p-1)}dx=\sum^{\infty}_{n=2}\int_{B_{r_{n}}(a_{n})}|K_{0}(x)|^{p}(1+|x|)^{2\alpha
p+2(p-1)}dx=:\sum^{\infty}_{n=2}I_{n}.$

Direct computation implies then: for $p\geq 1$, $\alpha\in\mathbb{R}$, (as $n$
goes to $\infty$)

$\displaystyle I_{n}$ $\displaystyle:=r_{n}^{-2p}n^{\ell
p}\int_{B_{r_{n}}(a_{n})}(1+|x|)^{2\alpha
p+2(p-1)}\eta_{0}\left(\frac{x-a_{n}}{r_{n}}\right)^{2\alpha p+2(p-1)}dx$
$\displaystyle\sim r_{n}^{-2p}n^{2\alpha p+2(p-1)+\ell
p}\int_{B_{r_{n}}(a_{n})}\eta_{0}\left(\frac{x-a_{n}}{r_{n}}\right)^{2\alpha
p+2(p-1)}dx$ $\displaystyle=M_{p}r_{n}^{2-2p}n^{2\alpha p+2(p-1)+\ell p}$
$\displaystyle=M_{p}e^{2(p-1)n^{q}}n^{2(\alpha+1-\ell)p-2},$

with $M_{p}=\displaystyle\int_{\mathbb{R}^{2}}\eta_{0}^{p}(z)dz>0$. We can
verify easily (1.10) and $\alpha_{1}(K_{0})=\frac{\ell-1}{2}$.

The following result exhibits clearly the asymptotic behavior of Cheng-Lin’s
solutions to (1.1) with $K=K_{0}$.

######  Theorem 1.2.

Assume that $\ell>q>1$ and $K=K_{0}$ given by (1.11). By [2], for any
$0<\alpha<\frac{\ell-1}{2}$, there is a unique solution of (1.1) $u_{\alpha}$
satisfying (1.6). Denote $\alpha_{*}=\frac{\ell-q}{2}$, we have:

  1. (i)

For any $0<\alpha<\alpha_{*}$, $u_{\alpha}$ satisfying (1.3).

  2. (ii)

For $\alpha=\alpha_{*}$, the solution denoted by $u_{*}$ satisfies (1.5), but
the remainder term $u_{\alpha_{*}}-\alpha_{*}\ln|x|$ does not converge to any
constant as $|x|\to\infty$.

  3. (iii)

For any $\alpha_{*}<\alpha<\frac{\ell-1}{2}$, the remainder term
$u_{\alpha}-\alpha\ln|x|$ is unbounded at infinity.

To our best knowledge, this is a first example of $K_{0}$ without any symmetry
or growth assumption, where we show a complete picture of asymptotic behavior
for solutions provided in [2]. This is also a first example of solution to
(1.1) with nonpositive Gaussian curvature, logarithmic growth but non uniform
behavior at infinity.

The rest of our paper is organized as follows. In Section 2, we show the link
between solutions satisfying (1.5) and that provided by Cheng-Lin, we show
also some properties of $\alpha_{p}(K)$ and the relationship between
$({\mathbb{H}}_{0})$ and $({\mathbb{H}}_{1})$. In Section 3, we prove Theorem
1.1. Section 4 is addressed to the study of solutions with $K_{0}$. In the
following, $C$ or $C_{i}$ denote always generic positive constants, their
values could be changed from one line to another.

##  2 Preliminaries

First, we indicate the relationship between solutions satisfying (1.5) and
solutions given by [2].

######  Proposition 2.1.

Let $u$ be a continuous solution to (1.1) satisfying (1.5) with
$\alpha<\alpha_{1}(K)$. Then $u$ is the Cheng-Lin’s solution, i.e. $u$
satisfies (1.6).

Proof. We involve function $w_{0}$ given by (1.9). So $\Delta w_{0}(x)=0$ for
$|x|>1$ and

$\displaystyle\int_{\mathbb{R}^{2}}\Delta w_{0}dx=2\pi.$ (2.1)

We claim that

$\int_{\mathbb{R}^{2}}|\nabla(u-\alpha
w_{0})|^{2}dx<\infty\quad\mbox{and}\quad\int_{\mathbb{R}^{2}}Ke^{2u}dx=-2\alpha\pi.$
(2.2)

Denote $v=u-\alpha w_{0}$, we prove first $v\in L^{2}(\mathbb{R}^{2})$. To
this end, set $\eta_{\rho}(t)=\eta_{0}(t/\rho)$ for $\rho>0$, where $\eta_{0}$
is a smooth function satisfying (1.12). Observe that

$-\Delta v=Ke^{2\alpha w_{0}}e^{2v}+\alpha\Delta w_{0}\quad{\rm
in}\;\;\mathbb{R}^{2}.$ (2.3)

As $v$ is bounded in $\mathbb{R}^{2}$ and $\alpha<\alpha_{1}(K)$, there holds

$\displaystyle 2\int_{\mathbb{R}^{2}}\eta_{\rho}^{2}|\nabla v|^{2}dx$
$\displaystyle=\int_{\mathbb{R}^{2}}\eta_{\rho}^{2}\left(\Delta v^{2}-2v\Delta
v\right)dx$
$\displaystyle=\int_{\mathbb{R}^{2}}v^{2}\Delta\eta_{\rho}^{2}dx-2\int_{\mathbb{R}^{2}}\eta_{\rho}^{2}v\Delta
vdx$
$\displaystyle\leq\|v\|_{\infty}^{2}\int_{\mathbb{R}^{2}}|\Delta\eta_{\rho}^{2}|dx+2\|v\|_{\infty}\int_{\mathbb{R}^{2}}|\Delta
v|dx$ $\displaystyle\leq
C_{1}+2\|v\|_{\infty}\int_{\mathbb{R}^{2}}\left(|K|e^{2\alpha
w_{0}+2\|v\|_{\infty}}+|\alpha\Delta w_{0}|\right)dx\leq C_{2}.$

Here $C_{i}$ are positive constants independent on $\rho$. Passing
$\rho\to\infty$, we see that $\nabla v\in L^{2}(\mathbb{R}^{2})$.

Furthermore,

$\displaystyle\left|\int_{\mathbb{R}^{2}}\eta_{\rho}\Delta vdx\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{2}}\nabla\eta_{\rho}\cdot\nabla
vdx\right|$ $\displaystyle\leq\left(\int_{B_{2\rho}\setminus B_{\rho}}|\nabla
v|^{2}dx\right)^{\frac{1}{2}}\left(\int_{B_{2\rho}\setminus
B_{\rho}}|\nabla\eta_{\rho}|^{2}dx\right)^{\frac{1}{2}}$
$\displaystyle=\|\nabla\eta_{0}\|_{L^{2}(\mathbb{R}^{2})}\left(\int_{B_{2\rho}\setminus
B_{\rho}}|\nabla v|^{2}dx\right)^{\frac{1}{2}}.$

Remark that $\Delta v\in L^{1}(\mathbb{R}^{2})$ and $\nabla v\in
L^{2}(\mathbb{R}^{2})$, we obtain, taking $\rho\to\infty$,

$\int_{\mathbb{R}^{2}}\Delta vdx=0.$

Combining (2.3) and (2.1), we obtain the second part of (2.2). $\Box$

We give some elementary consequences of $(\mathbb{H}_{1})$ and
$(\mathbb{H}_{0})$.

######  Lemma 2.1.

Assume that $(\mathbb{H}_{1})$ holds true with $p_{0}>1$. Then for any
$p\in[1,p_{0})$, $\alpha_{p}(K)\geq\alpha_{p_{0}}(K)$ and
$\alpha_{p}(K)\to\alpha_{1}(K)$ as $p\to 1^{+}$.

Proof. For simplicity, we denote $\alpha_{p}(K)$ by $\alpha_{p}$. Take any
$\beta<\alpha<\alpha_{p_{0}}$ and $p\in[1,\,p_{0})$. Denote
$\tau_{\alpha}=-2+\frac{2p_{0}p}{p_{0}-p}(\beta-\alpha)<-2$, then

$\tau_{\alpha}\left(1-\frac{p}{p_{0}}\right)+\Big{[}2\alpha
p_{0}+2(p_{0}-1)\Big{]}\frac{p}{p_{0}}=2\beta p+2(p-1).$

By Hölder’s inequality, as $\alpha<\alpha_{p_{0}}$,

$\displaystyle\int_{\mathbb{R}^{2}}|K(x)|^{p}(1+|x|)^{2\beta p+2(p-1)}dx$
$\displaystyle\leq$
$\displaystyle\;\left(\int_{\mathbb{R}^{2}}|K(x)|^{p_{0}}(1+|x|)^{2\alpha
p_{0}+2(p_{0}-1)}dx\right)^{\frac{p}{p_{0}}}\left(\int_{\mathbb{R}^{2}}(1+|x|)^{\tau_{\alpha}}dx\right)^{1-\frac{p}{p_{0}}}<\infty.$

It means that $\alpha_{p}\geq\alpha_{p_{0}}$ for any $p\in[1,\,p_{0})$. We see
also that the function $p\mapsto\alpha_{p}$ is non-increasing in $[1,p_{0}]$.

If $\alpha_{p_{0}}=\alpha_{1}$, then $\alpha_{p}\equiv\alpha_{p_{0}}$ for
$p\in[1,p_{0}]$ and we are done. Now assume that $\alpha_{p_{0}}<\alpha_{1}$.
Let $\varepsilon\in(0,\alpha_{1}-\alpha_{p_{0}})$, take

$\lambda=\alpha_{1}-\varepsilon,\quad\beta_{2}<\alpha_{p_{0}}<\lambda<\beta_{1}<\alpha_{1},$

and consider

$h(p):=\beta_{1}(p_{0}-p)+\beta_{2}p_{0}(p-1)-\lambda p(p_{0}-1)\quad\mbox{in
}\;[1,p_{0}].$

Readily, $h(1)=(\beta_{1}-\lambda)(p_{0}-1)>0$ and
$h(p_{0})=(\beta_{2}-\lambda)(p_{0}-1)p_{0}<0.$ Hence, there exists
$p_{*}\in(1,p_{0})$ such that $h(p_{*})=0$, or equivalently

$\displaystyle
2\beta_{1}\theta+\Big{[}2\beta_{2}p_{0}+2(p_{0}-1)\Big{]}(1-\theta)=2\lambda
p_{*}+2(p_{*}-1)\quad\mbox{where
}\;\theta:=\frac{p_{0}-p_{*}}{p_{0}-1}\in(0,1).$

As $\theta+p_{0}(1-\theta)=p_{*}$, there holds

$\displaystyle\int_{\mathbb{R}^{2}}|K(x)|^{p_{*}}(1+|x|)^{2\lambda
p_{*}+2(p_{*}-1)}dx$ $\displaystyle\leq$
$\displaystyle\;\left(\int_{\mathbb{R}^{2}}|K(x)|(1+|x|)^{2\beta_{1}}dx\right)^{\theta}\left(\int_{\mathbb{R}^{2}}|K(x)|^{p_{0}}(1+|x|)^{2\beta_{2}p_{0}+2(p_{0}-1)}dx\right)^{1-\theta}<\infty.$

Thus we get $\alpha_{p_{*}}\geq\lambda$, so
$\alpha_{1}-\varepsilon\leq\alpha_{p}\leq\alpha_{1}$ for $p\in[1,p_{*}]$,
which means $\lim_{p\to 1^{+}}\alpha_{p}=\alpha_{1}$. $\Box$

######  Remark 2.1.

By Lemmas 2.1, we see that $({\mathbb{H}}_{1})$ is equivalent to
$({\mathbb{H}}_{1}^{\prime})$: There exists $p>1$ such that $\alpha_{p}(K)>0$.

######  Lemma 2.2.

If $K$ satisfies $({\mathbb{H}}_{0})$, then there exists $p>1$ such that
$\alpha_{p}(K)>0$.

Proof. Fix some $\alpha\in(0,\alpha_{1})$ and choose $p>1$ but nearby such
that

$\beta:=\alpha+\left(\frac{m}{2}+\alpha+1\right)(p-1)<\alpha_{1}.$

Then

$\displaystyle|K(x)|^{p-1}(1+|x|)^{2(\alpha+1)(p-1)}\leq
C(1+|x|)^{m(p-1)+2(\alpha+1)(p-1)}=C(1+|x|)^{2\beta-2\alpha},$

so that

$\displaystyle\int_{\mathbb{R}^{2}}|K(x)|^{p}(1+|x|)^{2\alpha p+2(p-1)}dx$
$\displaystyle=\int_{\mathbb{R}^{2}}|K(x)|(1+|x|)^{2\alpha}\Big{[}|K(x)|^{p-1}(1+|x|)^{2(\alpha+1)(p-1)}\Big{]}dx$
$\displaystyle\leq C\int_{\mathbb{R}^{2}}|K(x)|(1+|x|)^{2\beta}dx<\infty.$

The proof is completed. $\Box$

We end this section by a simple fact of uniqueness, which can be derived from
Theorem 4.1 in [3]. For the convenience of readers, we give a short proof
here.

######  Lemma 2.3.

Let $K\leq 0$, then for any $\alpha\in\mathbb{R}$, there exists at most one
continuous solution to (1.1) satisfying (1.5).

Let $\widetilde{u}_{\alpha}$ and $u_{\alpha}$ be two solutions of (1.1)
satisfying (1.5) with $K\leq 0$. Denote $v=\widetilde{u}_{\alpha}-u_{\alpha}$,
then

$\displaystyle-\Delta v^{2}=-2v\Delta v-2|\nabla v|^{2}\leq-2v\Delta
v=2Ke^{2u_{\alpha}}v\left(e^{2v}-1\right)\leq 0,$

that is, $v^{2}$ is bounded and subharmonic in $\mathbb{R}^{2}$, hence $v^{2}$
is a constant. So $0=-\Delta v=Ke^{2u_{\alpha}}(e^{2v}-1)$ in
$\mathbb{R}^{2}$, which implies that $v\equiv 0$ and the uniqueness follows.

##  3 Proof of Theorem 1.1

We begin with a result which plays an important role to obtaining the
asymptotic behavior (1.8).

######  Proposition 3.1.

Let

$\displaystyle f\in\dot{L}^{1}(\mathbb{R}^{2}):=\left\\{f\in
L^{1}(\mathbb{R}^{2}),\;\int_{\mathbb{R}^{2}}f(x)dx=0\right\\}.$

Assume that $\alpha_{p_{0}}(f)>0$ with $p_{0}>1$, then there exists a unique
ground state solution to $-\Delta w=f$ in $\mathbb{R}^{2}$. Moreover, $w\in
C(\mathbb{R}^{2})$ and

$\lim_{|x|\to\infty}w(x)|x|^{\frac{2\beta}{1+2\beta}}=0\quad{\rm for\
any}\;\beta\in(0,\alpha_{1}(f)).$ (3.1)

Proof. The uniqueness of the ground state solution is a trivial consequence of
Liouville’s Theorem. So we prove only the existence of a solution satisfying
(3.1). By $\alpha_{p_{0}}(f)>0$ and Lemma 2.1, we get $\alpha_{1}(f)>0$ and
$f(z)\ln(2+|z|)\in L^{1}(\mathbb{R}^{2})$. As $f\in\dot{L}^{1}\cap
L^{p_{0}}_{loc}$, consider

$\displaystyle
w(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\ln|x-y|f(y)dy\quad\mbox{in
}\mathbb{R}^{2}.$

Obviously, $-\Delta w=f$ in $\mathbb{R}^{2}$, $w\in C(\mathbb{R}^{2})$ and for
any $x\neq 0$,

$\displaystyle 2\pi
w(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\ln\frac{|x|}{|x-y|}f(y)dy=\sum_{i=1}^{3}\int_{\Omega_{i}}\ln\frac{|x|}{|x-y|}f(y)dy=:\sum_{i=1}^{3}J_{i}(x),$

where $\Omega_{1}=B_{R}$, $R\geq 2$, $\Omega_{2}=B_{\frac{|x|}{2}}(x)$ and
$\Omega_{3}=\mathbb{R}^{2}\backslash(\Omega_{1}\cup\Omega_{2})$. For $|x|\geq
4R>0,$

$\displaystyle|J_{1}(x)|$ $\displaystyle\leq
C\int_{B_{R}}\frac{|y|}{|x|}|f(y)|dy\leq\frac{CR}{|x|}\|f\|_{L^{1}(\mathbb{R}^{2})}.$

Let $e_{x}=\frac{x}{|x|}$, $p_{0}^{\prime}=\frac{p_{0}}{p_{0}-1}$ and fix
$0<\beta<\alpha_{p_{0}}(f)$, we get

$\displaystyle|J_{2}(x)|$
$\displaystyle=\left|\int_{\Omega_{2}}\ln\frac{|x|}{|x-y|}f(y)dy\right|$
$\displaystyle\leq\left(\int_{\Omega_{2}}|f(y)|^{p_{0}}(1+|y|)^{2\beta
p_{0}+2(p_{0}-1)}dy\right)^{\frac{1}{p_{0}}}\left(\int_{\Omega_{2}}\left|\ln\frac{|x|}{|x-y|}\right|^{p_{0}^{\prime}}(1+|y|)^{-2-2\beta
p_{0}^{\prime}}dy\right)^{\frac{1}{p_{0}^{\prime}}}$ $\displaystyle\leq
C\left(\int_{B_{\frac{1}{2}}(e_{x})}\big{|}\ln|e_{x}-z|\big{|}^{p_{0}^{\prime}}|z|^{-2-2\beta
p_{0}^{\prime}}|x|^{-2\beta
p_{0}^{\prime}}dz\right)^{\frac{1}{p_{0}^{\prime}}}$
$\displaystyle=C|x|^{-2\beta}.$

Let $|x|\geq 4R\geq 8$. In $\Omega_{3}$, as $\min(|x-y|,|x|,|y|)\geq R\geq 2$,
we can check that

$\forall\;y\in\Omega_{3},\qquad\left|\ln\frac{|x|}{|x-y|}\right|\leq\ln|y|,$

since $\frac{1}{|y|}\leq\frac{|x|}{|x-y|}\leq|y|$. Then there holds

$\displaystyle|J_{3}(x)|$ $\displaystyle\leq\int_{\Omega_{3}}|f(y)|\ln|y|dy$
$\displaystyle\leq\left(\int_{\Omega_{3}}|f(y)|^{p_{0}}(1+|y|)^{2\beta
p_{0}+2(p_{0}-1)}dy\right)^{\frac{1}{p_{0}}}\left(\int_{\Omega_{3}}(\ln|y|)^{p_{0}^{\prime}}|y|^{-2-2\beta
p_{0}^{\prime}}dy\right)^{\frac{1}{p_{0}^{\prime}}}$ $\displaystyle\leq
C\left(\int_{\mathbb{R}^{2}\backslash
B_{R}}(\ln|y|)^{p_{0}^{\prime}}|y|^{-2-2\beta
p_{0}^{\prime}}dy\right)^{\frac{1}{p_{0}^{\prime}}}$ $\displaystyle\leq
CR^{-2\beta}\ln R.$

Notice that all the constants $C$ involved are independent of $|x|\geq 4R$ and
$R\geq 2$. Taking $R=|x|^{\frac{1}{1+2\beta}}$, there holds

$|w(x)|\leq C|x|^{-\frac{2\beta}{1+2\beta}}\ln|x|\quad{\rm for}\ |x|\;{\rm
large}.$ (3.2)

This ends the proof of (3.1), because we can get (3.2) with any
$p\in(1,p_{0})$ and any $\beta<\alpha_{p}(f)$. $\Box$

We are now in position to prove Theorem 1.1.

Proof of Theorem 1.1. Let $\alpha\in(0,\alpha_{1}(K))$ and $w_{0}$ be given by
(1.9), there exists then a unique $t_{\alpha}\in\mathbb{R}$ such that

$g_{\alpha}:=Ke^{2\alpha w_{0}}e^{2t_{\alpha}}+\alpha\Delta
w_{0}\in\dot{L}^{1}(\mathbb{R}^{2}).$

Indeed, as $K\leq 0$ in $\mathbb{R}^{2}$ and $\alpha>0$, the unique choice is
given by

$t_{\alpha}=\frac{1}{2}\ln\left(-\frac{2\alpha\pi}{\displaystyle\int_{\mathbb{R}^{2}}Ke^{2\alpha
w_{0}}dx}\right).$

On the other hand, using $(\mathbb{H}_{1})$ and Lemma 2.1, there exists $p>1$
such that $\alpha<\alpha_{p}(K)$. As $\alpha\Delta w_{0}$ is compactly
supported, it’s not difficult to see that

$\alpha_{p}(g_{\alpha})=\alpha_{p}(K)-\alpha>0.$

Using Proposition 3.1, we get a continuous ground state solution to
$-\Delta\widetilde{w}=g_{\alpha}$ in $\mathbb{R}^{2}$.

We conclude by super-sub solution method. Let $\overline{v}=\alpha
w_{0}+t_{\alpha}+\widetilde{w}+\|\widetilde{w}\|_{\infty}$. As $K\leq 0$, it’s
easy to see that

$-\Delta\overline{v}=-\alpha\Delta w_{0}-\Delta\widetilde{w}=Ke^{2(\alpha
w_{0}+t_{\alpha})}\geq Ke^{2\overline{v}}.$

Hence $\overline{v}$ is a supersolution to equation (1.1). Similarly,
$\underline{v}=\alpha
w_{0}+t_{\alpha}+\widetilde{w}-\|\widetilde{w}\|_{\infty}$ is a subsolution to
(1.1). Obviously $\overline{v}\geq\underline{v}$ in $\mathbb{R}^{2}$. Using
the standard Perron’s method, there exists a solution $u$ to (1.1) such that
$\underline{v}\leq u\leq\overline{v}$ in $\mathbb{R}^{2}$. Clearly, $u$
satisfies (1.5).

By Proposition 2.1, there holds $Ke^{2u}+\alpha\Delta
w_{0}\in\dot{L}^{1}(\mathbb{R}^{2})$. Applying Lemma 2.1, there exists $p>1$
such that

$\alpha_{p}\left(Ke^{2u}+\alpha\Delta w_{0}\right)=\alpha_{p}(K)-\alpha>0.$

By Proposition 3.1, we get a continuous solution of $-\Delta
w=Ke^{2u}+\alpha\Delta w_{0}$ in $\mathbb{R}^{2}$, which satisfies (3.1) for
any $0<\beta<\alpha_{1}(K)-\alpha$. Therefore, $v=u-\alpha w_{0}-w$ is a
bounded harmonic function over $\mathbb{R}^{2}$, hence $v$ is a constant
$c_{\alpha}$. Finally $u=\alpha w_{0}+c_{\alpha}+w$, so (1.8) holds trues for
any $0<\beta<\alpha_{1}(K)-\alpha$. $\Box$

Similarly, using the super-sub solution method and Proposition 2.1, we can
claim

######  Proposition 3.2.

Assume that for $K\leq 0$, $\alpha>0$ and $t\in\mathbb{R}$, there exists a
bounded solution to $-\Delta v=Ke^{2\alpha w_{0}+2t}+\alpha\Delta w_{0}$ in
$\mathbb{R}^{2}$, then the equation (1.1) admits a unique solution
$u_{\alpha}$ satisfying (1.5). Moreover, the total curvature of $u_{\alpha}$
is $-2\alpha\pi$.

##  4 Exotic curvature $K_{0}$ and anisotropic behavior solutions

Here we prove Theorem 1.2. We divide our study by several steps.

###  4.1 Solution satisfying (1.5) for $\alpha\leq\alpha_{*}$.

Remark that we cannot apply Theorem 1.1 in [2] or its proof, since they don’t
provide the asymptotic behavior. Indeed, we will construct directly solutions
satisfying (1.5), and Proposition 2.1 ensures then they are solutions given by
[2]. Remark also that we cannot apply the results of Section 3, since
$\alpha_{p}(K_{0})=-\infty$ for any $p>1$.

Recall that $\alpha_{1}(K_{0})=\frac{\ell-1}{2}>0$ and $w_{0}$ is given by
(1.9). Let $0<\alpha\leq\frac{\ell-q}{2}$ and

$\displaystyle
v_{\alpha}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}K_{0}(y)e^{2\alpha
w_{0}(y)}\ln|x-y|dy.$

It is easy to see that $v_{\alpha}$ is well defined, continuous and locally
bounded in $\mathbb{R}^{2}$. Let

$\displaystyle n\geq 2\quad\mbox{and}\quad x\in
B_{n+\frac{1}{2}}\setminus\overline{B}_{n-\frac{1}{2}}.$ (4.1)

We decompose $v_{\alpha}(x)$ as follows,

$\displaystyle 2\pi v_{\alpha}(x)$
$\displaystyle=\sum^{\infty}_{k=2,\,k\not=n}r_{k}^{-2}k^{-\ell}\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|x-y|dy$
$\displaystyle\hskip
14.22636pt+r_{n}^{-2}n^{-\ell}\int_{B_{r_{n}}(a_{n})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|x-y|\,dy$
$\displaystyle=:E_{1}(x)+E_{2}(x).$

Denote

$\displaystyle
s_{k}:=\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)dy\leq
C(k+1)^{2\alpha}r_{k}^{2},\quad\forall\;k\geq 2.$

Consider first $k\leq\frac{n}{2}$ and $k\geq 2$. Notice that there exists
$C>0$ such that

$\Big{|}\ln|x-y|-\ln|x|\Big{|}\leq C\frac{|y|}{|x|},\qquad\forall\;|x|\geq
2|y|>0.$

For $x$ satisfying (4.1), there holds then

$\displaystyle\left|\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|x-y|\,dy-
s_{k}\ln|x|\right|$
$\displaystyle=\left|\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln\frac{|x-y|}{|x|}dy\right|$
$\displaystyle\leq\frac{C}{n}(k+1)^{2\alpha+1}\int_{B_{r_{k}}(a_{k})}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)dy$
$\displaystyle=\frac{C}{n}k^{2\alpha+1}r_{k}^{2}.$

Let now $k\neq n$, $k>\frac{n}{2}$. As $|y|\geq\frac{7}{4}$ and
$|x-y|\geq\frac{1}{4}$ in $B_{r_{k}}(a_{k})$, there exists $C>1$ (independent
of $k,n\geq 2$, $k\neq n$) such that

$\forall\;y\in
B_{r_{k}}(a_{k}),\quad|y|^{-C}\leq\frac{\frac{1}{4}}{|y|+\frac{1}{4}}\leq\frac{|x-y|}{|y|+|x-y|}\leq\frac{|x-y|}{|x|}\leq
1+|y|\leq|y|^{C}.$

We obtain

$\displaystyle\left|\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|x-y|\,dy-
s_{k}\ln|x|\right|$
$\displaystyle=\left|\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln\frac{|x-y|}{|x|}dy\right|$
$\displaystyle\leq
C\left|\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|y|dy\right|$
$\displaystyle\leq C(k+1)^{2\alpha}r_{k}^{2}\ln(k+1).$

Therefore, for $x$ in (4.1),

$\displaystyle\left|E_{1}(x)-\left(\sum_{k=2}^{\infty}r_{k}^{-2}k^{-\ell}s_{k}\right)\ln|x|\right|$
$\displaystyle\leq$
$\displaystyle\;\frac{C}{n}\sum^{[\frac{n}{2}]}_{k=2}k^{2\alpha-\ell+1}+C\sum_{k>[\frac{n}{2}],k\neq
n}k^{2\alpha-\ell}\ln k+r_{n}^{-2}n^{-\ell}s_{n}\ln|x|$ $\displaystyle\leq$
$\displaystyle\;\frac{C}{n}\sum^{[\frac{n}{2}]}_{k=2}k^{2\alpha-\ell+1}+C\sum_{k=[\frac{n}{2}]+1}^{\infty}k^{2\alpha-\ell}\ln
k\quad\rightarrow 0,\;\;\mbox{as }n\to\infty.$

Here $[\frac{n}{2}]$ is the integer part of $\frac{n}{2}$. Finally, as
$2\alpha-\ell+1\leq 1-q<0$, there holds

$\displaystyle\lim_{|x|\to\infty}\Big{|}E_{1}(x)-A_{\alpha}\ln|x|\Big{|}=0,\quad\mbox{where
}\;A_{\alpha}=\sum_{k=2}^{\infty}r_{k}^{-2}k^{-\ell}s_{k}\in(0,\infty).$ (4.2)

On the other hand, for $x$ satisfying (4.1),

$\displaystyle\begin{split}|E_{2}(x)|&\leq
Cr_{n}^{-2}n^{2\alpha-\ell}\int_{B_{r_{n}}(a_{n})}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\times(-\ln|x-y|)dy\\\
&=-Cn^{2\alpha-\ell}\int_{B_{1}}\eta_{0}(z)\ln|x-a_{n}-r_{n}z|dz\\\ &\leq-
Cn^{2\alpha-\ell}\int_{B_{1}}\eta_{0}(z)\ln|r_{n}z|dz\\\
&=Cn^{2\alpha-\ell}(|\ln r_{n}|+1)\\\ &\leq Cn^{2\alpha-\ell+q}.\end{split}$
(4.3)

For the third line, we used the symmetric decreasing rearrangement argument,
as $\eta_{0}$ and $-\ln|r_{n}z|$ are radial non-increasing positive functions
in $B_{1}$.

Combining (4.2)–(4.3), $v_{\alpha}$ is a solution to $-\Delta
v_{\alpha}=K_{0}e^{2\alpha w_{0}}$ in $\mathbb{R}^{2}$ such that

$\displaystyle v_{\alpha}(x)=\frac{A_{\alpha}}{2\pi}\ln|x|+O(1),\;\;\mbox{as
}|x|\to\infty.$

Taking

$t_{\alpha}=\frac{1}{2}\ln\frac{2\alpha\pi}{A_{\alpha}},$

$h_{\alpha}=e^{2t_{\alpha}}v_{\alpha}-\alpha w_{0}$ is clearly a bounded
solution to

$\displaystyle-\Delta h=K_{0}e^{2\alpha w_{0}+2t_{\alpha}}+\alpha\Delta
w_{0}\quad\mbox{in }\;\mathbb{R}^{2}.$

By Propositions 3.2, for any $\alpha\in(0,\,\frac{\ell-q}{2}]$, equation (1.1)
with $K=K_{0}$ admits a unique solution satisfying (1.5), and whose total
curvature is equal to $-2\alpha\pi$.

###  4.2 Uniform behavior for $0<\alpha<\alpha_{*}$.

For $\alpha\in(0,\frac{\ell-q}{2})$, we have a continuous solution
$u_{\alpha}$ to (1.1) satisfying (1.5). Rewrite $u_{\alpha}=\alpha
w_{0}+\xi_{\alpha}$ with $\xi_{\alpha}\in L^{\infty}(\mathbb{R}^{2})$. Let

$\displaystyle\widetilde{v}_{\alpha}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}K_{0}(y)e^{2\alpha
w_{0}(y)+2\xi_{\alpha}(y)}\ln|x-y|dy.$

By decomposing $\widetilde{v}_{\alpha}$ as above and using

$\displaystyle\widetilde{s}_{k}:=\int_{B_{r_{k}}(a_{k})}e^{2\xi_{\alpha}(y)}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)dy\quad\mbox{for
}\;k\geq 2,$

the similar estimates to (4.2)–(4.3) hold true for $\widetilde{v}_{\alpha}$,
which yield that (as now $2\alpha-\ell+q<0$)

$\displaystyle\lim_{|x|\to\infty}\left[\widetilde{v}_{\alpha}(x)-\frac{\widetilde{A}_{\alpha}}{2\pi}\ln|x|\right]=0,\quad\mbox{where
}\;\widetilde{A}_{\alpha}=\sum_{k=2}^{\infty}r_{k}^{-2}k^{-\ell}\widetilde{s}_{k}\in(0,+\infty).$
(4.4)

So $u_{\alpha}-\widetilde{v}_{\alpha}$ is a harmonic function in
$\mathbb{R}^{2}$ and $(u_{\alpha}-\widetilde{v}_{\alpha})(x)=O(\ln|x|)$ at
infinity, hence $u_{\alpha}-\widetilde{v}_{\alpha}$ is a constant in
$\mathbb{R}^{2}$. The behavior (4.4) means that $u_{\alpha}$ satisfies (1.3).

###  4.3 Anisotropic behavior for $\alpha=\alpha_{*}$.

Let $\alpha=\alpha_{*}=\frac{\ell-q}{2}$, and $u_{*}$ be the unique solution
to (1.1) satisfying (1.5), hence $\xi_{*}:=u_{*}-\alpha_{*}w_{0}$ is uniformly
bounded in $\mathbb{R}^{2}$. However, we will show that $\xi_{*}$ does not
converge to a constant at infinity. Define

$\displaystyle 2\pi v_{*}(x)$
$\displaystyle=-\int_{\mathbb{R}^{2}}K_{0}(y)e^{2\alpha_{*}w_{0}(y)+2\xi_{*}(y)}\ln|x-y|dy$
$\displaystyle=\sum^{\infty}_{k=2,\,k\not=n}r_{k}^{-2}k^{-\ell}\int_{B_{r_{k}}(a_{k})}e^{2\xi_{*}(y)}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|x-y|\,dy$
$\displaystyle\hskip
14.22636pt+r_{n}^{-2}n^{-\ell}\int_{B_{r_{n}}(a_{n})}e^{2\xi_{*}(y)}|y|^{2\alpha}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|x-y|dy$
$\displaystyle=:F_{1}(x)+F_{2}(x).$

We see that

$\displaystyle-\Delta
v_{*}=K_{0}e^{2\alpha_{*}w_{0}+2\xi_{*}}=K_{0}e^{2u_{*}}\;\;\mbox{in
}\;\mathbb{R}^{2}\quad\mbox{and}\quad v_{*}(x)=O(\ln|x|)\;\;\mbox{as
}\;|x|\to\infty.$

The logarithmic control of $v_{*}$ at infinity can be obtained as above, using
$\xi_{*}\in L^{\infty}(\mathbb{R}^{2})$.

We shall estimate the difference between $v_{*}(a_{n})$ and $v_{*}(-a_{n})$.
The estimation of $F_{1}$ is very similar to $E_{1}$ in the above. Let $n\geq
2$. If $k\leq\frac{n}{2}$ and $k\geq 2$, there holds

$\displaystyle\left|\int_{B_{r_{k}}(a_{k})}e^{2\xi_{*}(y)}|y|^{2\alpha_{*}}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\Big{[}\ln|a_{n}-y|-\ln|a_{n}+y|\Big{]}dy\right|$
$\displaystyle\leq$
$\displaystyle\;\frac{C}{n}e^{2\|\xi_{*}\|_{\infty}}(k+1)^{2\alpha_{*}}\int_{B_{r_{k}}(a_{k})}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)|y|\,dy$
$\displaystyle\leq$ $\displaystyle\;\frac{C}{n}k^{2\alpha_{*}+1}r_{k}^{2}.$

Let $k>\frac{n}{2}$ and $k\not=n$, we have

$\displaystyle\left|\int_{B_{r_{k}}(a_{k})}e^{2\xi_{*}(y)}|y|^{2\alpha_{*}}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\Big{[}\ln|a_{n}-y|-\ln|a_{n}+y|\Big{]}dy\right|$
$\displaystyle\leq$ $\displaystyle\;C(k+1)^{2\alpha_{*}}(\ln n+\ln
k)\int_{B_{r_{k}}(a_{k})}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\,dy$
$\displaystyle\leq$ $\displaystyle\;Ck^{2\alpha_{*}}r_{k}^{2}\ln k.$

As $2\alpha_{*}-\ell=-q<-1$, we get then

$\displaystyle|F_{1}(a_{n})-F_{1}(-a_{n})|$
$\displaystyle\leq\frac{C}{n}\sum^{[\frac{n}{2}]}_{k=2}k^{2\alpha_{*}-\ell+1}+C\sum^{\infty}_{k=[\frac{n}{2}]+1}k^{2\alpha_{*}-\ell}\ln
k\;\;\rightarrow 0,\quad{\rm as}\;\;n\to\infty.$ (4.5)

Now we estimate $F_{2}(\pm a_{n})$:

$\displaystyle 0\leq F_{2}(-a_{n})$ $\displaystyle\leq
Cr_{n}^{-2}n^{-\ell+2\alpha_{*}}\int_{B_{r_{n}}(a_{n})}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|a_{n}+y|dy\leq
Cn^{-\ell+2\alpha}\ln(2n+1)$

and

$\displaystyle-F_{2}(a_{n})$
$\displaystyle=-r_{n}^{-2}n^{-\ell}\int_{B_{r_{n}}(a_{n})}e^{2\xi_{*}(y)}|y|^{2\alpha_{*}}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|a_{n}-y|dy$
$\displaystyle\geq-e^{-2\|\xi_{*}\|_{\infty}}r_{n}^{-2}n^{-\ell}(n-1)^{2\alpha_{*}}\int_{B_{r_{n}}(a_{n})}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|a_{n}-y|dy$
$\displaystyle\geq Cn^{-\ell+2\alpha_{*}}\int_{B_{1}}\eta_{0}(z)\Big{[}-\ln
r_{n}-\ln|z|\Big{]}dz$ $\displaystyle\geq-Cn^{-\ell+2\alpha_{*}}\ln r_{n}$
$\displaystyle=Cn^{-\ell+2\alpha_{*}+q}=C>0.$

Therefore $\liminf_{n\to\infty}|F_{2}(a_{n})-F_{2}(-a_{n})|\geq C>0$.
Combining with (4.5), we get

$\displaystyle\liminf_{n\to\infty}|v_{*}(a_{n})-v_{*}(-a_{n})|\geq\frac{C}{2\pi}>0.$

Furthermore, as $-\Delta(v_{*}-u_{*})=0$ in $\mathbb{R}^{2}$ and $v_{*}-u_{*}$
is $O(\ln|x|)$ at infinity, the standard Liouville’s theorem implies that
$v_{*}-u_{*}$ is constant in $\mathbb{R}^{2}$. Consequently,

$\displaystyle\liminf_{n\to\infty}|\xi_{*}(a_{n})-\xi_{*}(-a_{n})|=\liminf_{n\to\infty}|v_{*}(a_{n})-v_{*}(-a_{n})|\geq\frac{C}{2\pi}>0.$

The above estimate means that $\xi_{*}$ does not converge to a constant at
infinity.

###  4.4 Unbounded remainder term for $\alpha_{*}<\alpha<\alpha_{1}(K_{0})$.

Recall that by Theorem 1.1 in [2], for any
$\alpha\in\left(\frac{\ell-q}{2},\,\frac{\ell-1}{2}\right)$, problem (1.1)
possesses a unique solution $u_{\alpha}$ satisfying (1.6). We shall prove that
$\xi_{\alpha}:=u_{\alpha}-\alpha w_{0}$ is unbounded.

Suppose the contrary, assume that $\xi_{\alpha}$ is bounded in
$\mathbb{R}^{2}$. Let

$\eta_{\alpha}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}K_{0}(y)e^{2u_{\alpha}(y)}\ln|x-y|dy\quad\mbox{in
}\;\mathbb{R}^{2},$

then we have

$\displaystyle 2\pi\eta_{\alpha}(x)$
$\displaystyle=\sum^{\infty}_{k=2,\,k\not=n}r_{k}^{-2}k^{-\ell}\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}e^{2\xi_{\alpha}}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|x-y|dy$
$\displaystyle\hskip
14.22636pt+r_{n}^{2}n^{-\ell}\int_{B_{r_{n}}(a_{n})}|y|^{2\alpha}e^{2\xi_{\alpha}}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|x-y|dy$
$\displaystyle=:G_{1}(x)+G_{2}(x).$

We will estimate $\eta_{\alpha}(a_{n})$. For $k\neq n$, $k\geq 2$,

$\displaystyle\left|\int_{B_{r_{k}}(a_{k})}|y|^{2\alpha}e^{2\xi_{\alpha}}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\ln|a_{n}-y|\,dy\right|$
$\displaystyle\leq$
$\displaystyle\;Ce^{2\|\xi_{\alpha}\|_{\infty}}(k+1)^{2\alpha}\int_{B_{r_{k}}(a_{k})}\eta_{0}\left(\frac{y-a_{k}}{r_{k}}\right)\big{(}\ln
n+\ln|y|\big{)}dy$ $\displaystyle\leq$
$\displaystyle\;Ck^{2\alpha}r_{k}^{2}\big{(}\ln n+\ln k\big{)}.$

Since $\alpha<\frac{\ell-1}{2}$, there exists $C>0$ such that
$|G_{1}(a_{n})|\leq C\ln n$ for $n$ large. On the other hand,

$\displaystyle-G_{2}(a_{n})$
$\displaystyle\geq-e^{-2\|\xi_{\alpha}\|_{\infty}}r_{n}^{-2}n^{-\ell}(n-1)^{2\alpha}\int_{B_{r_{n}}(a_{n})}\eta_{0}\left(\frac{y-a_{n}}{r_{n}}\right)\ln|a_{n}-y|dy$
$\displaystyle\geq-Cn^{-\ell+2\alpha}\ln r_{n}$
$\displaystyle=C_{3}n^{-\ell+2\alpha+q}.$

Combining the estimations of $G_{1}(a_{n})$ and $G_{2}(a_{n})$, we get, as
$\alpha>\frac{\ell-q}{2}$,

$\displaystyle\lim_{n\to\infty}\frac{\eta_{\alpha}(a_{n})}{\ln n}=-\infty.$
(4.6)

Moreover, remark that for $x$ satisfying (4.1) and $y\in B_{r_{n}}(a_{n})$,
there holds $|x-y|\leq 3|x|$. Similar estimations as above yield that there
exists $C>0$ such that

$\eta_{\alpha}(x)\leq C\ln(2+|x|)\;\;\mbox{in }\;\mathbb{R}^{2}.$

As $\widetilde{v}:=\eta_{\alpha}-u_{\alpha}=\eta_{\alpha}-\alpha
w_{0}-\xi_{\alpha}$ is harmonic over $\mathbb{R}^{2}$, and
$\widetilde{v}(x)\leq C\ln(2+|x|)$ in $\mathbb{R}^{2}$, $\widetilde{v}$ is
constant by Liouville’s theorem. However, $\eta_{\alpha}=\alpha
w_{0}+\xi_{\alpha}+C$ with $\xi_{\alpha}\in L^{\infty}$ contradicts the
estimate (4.6). This means that $\xi_{\alpha}$ cannot be bounded in
$\mathbb{R}^{2}$. $\Box$

Acknowledgements. H.C. is supported by NSFC (No. 11726614 and 11661045), and
Jiangxi Provincial Natural Science Foundation (No. 20161ACB20007). F.Z. and
D.Y. are supported by Science and Technology Commission of Shanghai
Municipality (STCSM), grant No. 18dz2271000. F.Z. is also supported by NSFC
(No. 11726613 and 11431005).

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