# Classification of solutions to the isotropic
horospherical $p$-Minkowski problem in hyperbolic plane

Haizhong Li  Department of Mathematical Sciences, Tsinghua University, Beijing
100084, P.R. China<EMAIL_ADDRESS>and  Yao Wan  Department of
Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In [LX], the first author and Xu introduced and studied the horospherical
$p$-Minkowski problem in hyperbolic space $\mathbb{H}^{n+1}$. In particular,
they established the uniqueness result for solutions to this problem when the
prescribed function is constant and $p\geq-n$. This paper focuses on the
isotropic horospherical $p$-Minkowski problem in hyperbolic plane
$\mathbb{H}^{2}$, which corresponds to the equation

$\varphi^{-p}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma\quad\text{on}\
\mathbb{S}^{1},$ (0.1)

where $\gamma$ is a positive constant. We provide a classification of
solutions to (0.1) for $p\geq-7$, as well as a nonuniqueness result of
solutions to (0.1) for $p<-7$. Furthermore, we extend this problem to the
isotropic horospherical $q$-weighted $p$-Minkowski problem in hyperbolic plane
and derive some uniqueness and nonuniqueness results.

###### Key words and phrases:

Horospherical $p$-Minkowski problem; Hyperbolic plane; Fully nonlinear ODE;
Uniqueness; Nonuniqueness

###### 2020 Mathematics Subject Classification:

52A55; 53A04; 34C25

##  1\. Introduction

The classical Brunn-Minkowski theory focuses on the convex geometry and
geometric analysis of convex bodies by use of the Minkowski sum, including
geometric measures and geometric inequalities of convex bodies, see, for
example Schneider’s book [Sch14]. In recent decades, this theory has been
extended to the $L_{p}$ Brunn-Minkowski theory and the dual Brunn-Minkowski
theory, see e.g. [CW06, HLY16, HZ18, HLY05, Lut75, Lut93, Lut96, LYZ04, LYZ18]
and references therein. Furthermore, the $L_{p}$ dual Minkowski problem is a
common generalization of the $L_{p}$ Minkowski problem and the dual Minkowski
problem. Building upon the work of Ben Andrews [Ben03], who classified the
solutions to the planar isotropic $L_{p}$ Minkowski problem, we provided a
classification of solutions to the planar isotropic $L_{p}$ dual Minkowski
problem in [LW22].

For the theory of convex geometry in hyperbolic space, the first author and Xu
[LX] introduced the hyperbolic $p$-sum, a sum of two sets in hyperbolic space
for any $p>0$. This leads to the development of the horospherical $p$-Brunn-
Minkowski theory in hyperbolic space. The authors defined the horospherical
$k$-th $p$-surface area measure associated with a smooth horospherically
convex bounded domain in hyperbolic space, and proposed the following
horospherical $p$-Minkowski problem for $k=0$:

######  Problem 1.1 (The horospherical $p$-Minkowski problem).

Let $p$ be a real number. Given a smooth positive function $f(z)$ defined on
$\mathbb{S}^{n}$, what are necessary and sufficient conditions for $f$, such
that there exists a smooth uniformly h-convex bounded domain
$K\subset\mathbb{H}^{n+1}$ for which its horospherical $p$-surface area
measure has density $f$, i.e.

$\displaystyle dS_{p}(K,z)=f(z)d\sigma.$

That is, finding a smooth positive solution to

$\displaystyle\varphi^{-p}(z)\det(A[\varphi(z)])=f(z),$

such that the matrix

$\displaystyle
A_{ij}[\varphi(z)]=D_{j}D_{i}\varphi-\frac{1}{2}\frac{|D\varphi|^{2}}{\varphi}\sigma_{ij}+\frac{1}{2}\left(\varphi-\frac{1}{\varphi}\right)\sigma_{ij}>0$

for all $z\in\mathbb{S}^{n}$, where $\sigma_{ij}$ denotes the standard metric
on $\mathbb{S}^{n}$ and $D$ denotes the Levi-Civita connection with respect to
$\sigma_{ij}$.

By using a new volume preserving flow, the first author and Xu [LX] solved the
existence of solutions to the horospherical $p$-Minkowski problem for all
$p\in(-\infty,+\infty)$ when the given measure is even. Afterwards, the
horospherical $p$-Brunn-Minkowski theory was further investigated by [LW23,
LW23-10, HWZ], etc.

In particular, when $f(z)$ is constant and $n=1$, this problem reduces to the
isotropic horospherical $p$-Minkowski problem in hyperbolic plane, which
corresponds to the following second-order ordinary differential equation

$\varphi^{-p}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma\quad\text{on}\
\mathbb{S}^{1},$ (1.1)

where $\gamma$ is a positive constant. In [LX], the first author and Xu
obtained the uniqueness of solutions to (1.1) for $p\geq-1$.

######  Theorem 1.1 ([LX]).

  1. (1)

If $p>1$, then the solution to (1.1) must be constant, i.e.
$\varphi(\theta)\equiv c$, where $c$ solves the equation

$c^{1-p}-c^{-1-p}=2\gamma.$ (1.2)

Denote $\gamma_{0}=\frac{(p-1)^{\frac{p-1}{2}}}{(p+1)^{\frac{p+1}{2}}}$. When
$0<\gamma<\gamma_{0}$, (1.2) has exactly two solutions. When
$\gamma=\gamma_{0}$, (1.2) has a unique solution. When $\gamma>\gamma_{0}$,
(1.2) has no solution.

  2. (2)

If $p=1$, then the solution to (1.1) must be constant that satisfies (1.2).
When $0<\gamma<\frac{1}{2}$, (1.2) has a unique solution. When
$\gamma\geq\frac{1}{2}$, (1.2) has no solution.

  3. (3)

If $-1<p<1$, then the solution to (1.1) must be constant that satisfies (1.2).
For each $\gamma>0$, (1.2) has a unique solution.

  4. (4)

If $p=-1$, then the solutions to (1.1) are given by

$\varphi(\theta)=\alpha\cos\left(\theta-\theta_{0}\right)+\beta,$ (1.3)

where $\theta_{0}\in\mathbb{R},\
\alpha\in(-\sqrt{1+2\gamma},\sqrt{1+2\gamma})$ and
$\beta=\sqrt{1+2\gamma-\alpha^{2}}$.

In this paper, by applying the method employed in [Ben03, LW23], we establish
the following uniqueness of solutions to (1.1) for $-7\leq p<-1$, and the
nonuniqueness of solutions to (1.1) for $p<-7$ and $\gamma>\gamma_{p}$.

######  Theorem 1.2.

  1. (i)

If $-7\leq p<-1$, then the unique solution to (1.1) is $\varphi(\theta)\equiv
c$, where $c$ solves the equation (1.2).

  2. (ii)

If $-17\leq p<-7$ and
$\gamma>\gamma_{p}=\frac{-3}{1+p}\left(\frac{7+p}{1+p}\right)^{\frac{p-1}{2}}$,
then there exist at least two solutions to (1.1).

  3. (iii)

Generally, let $k$ and $l$ be two integers such that $1\leq l<k$. If
$1-2(k+1)^{2}\leq p<1-2k^{2}$ and $\gamma>\gamma_{p,l}$, where $\gamma_{p,l}$
is given by

$\displaystyle\gamma_{p,l}=\frac{1-(l+1)^{2}}{1+p}\left(\frac{2(l+1)^{2}-1+p}{1+p}\right)^{\frac{p-1}{2}},$

then there exist at least $(l+1)$ solutions to (1.1). Furthermore, for any
integer $2\leq j\leq l+1$, there exists a non-constant solution that
corresponds to a h-convex curve $\Gamma_{j,p,\gamma}$ with $j$-fold symmetry
in $\mathbb{H}^{2}$.

$p$-axis$\gamma$-axis$-17$$-16$$-15$$-14$$-13$$-12$$-11$$-10$$-9$$-8$$-7$$-6$$-5$$-4$$-3$$-2$$-1$$1$$2$$-1$$1$$10$$20$$30$$40$$0$$p=-7$$\gamma_{p}=\frac{-3}{1+p}\left(\frac{7+p}{1+p}\right)^{\frac{p-1}{2}}$
Figure 1. Number $k$ of solutions to (1.1) Grey domain: $k=0$. Blue domain:
$k=1$. Red domain: $k=2$. Blue domain with shadow: $k\geq 1$. Red domain with
shadow: $k\geq 2$. Yellow line: $k=\infty$.

In a similar manner as in the case of Euclidean spaces, by modifying the
powers of weight $\cosh
r=\frac{|D\varphi|^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}$, we
naturally extend Problem 1.1 to the horospherical $q$-weighted $p$-Minkowski
problem:

######  Problem 1.2 (The horospherical $q$-weighted $p$-Minkowski problem).

Let $p$ and $q$ be two real numbers. Given a positive function $f(z)$ defined
on $\mathbb{S}^{n}$, what are necessary and sufficient conditions for $f$,
such that there exists a smooth positive solution $\varphi(z)$ satisfying

$\displaystyle\varphi^{-p}(z)\left(\frac{|D\varphi|^{2}}{2\varphi}(z)+\frac{\varphi(z)+\varphi^{-1}(z)}{2}\right)^{q-n}\det(A[\varphi(z)])=f(z),$

such that $A[\varphi(z)]>0$ for all $z\in\mathbb{S}^{n}$.

Note that when $q=n$, Problem 1.2 reduces to the horospherical $p$-Minkowski
problem; when $q=n+1$, Problem 1.2 reduces to Problem 11.1 in [LX].

Now we consider the isotropic horospherical $q$-weighted $p$-Minkowski problem
in hyperbolic plane, which corresponds to the equation

$\varphi^{-p}\left(\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}\right)^{q-1}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma\quad\text{on}\
\mathbb{S}^{1},$ (1.4)

where $\gamma$ is a positive constant. In this paper, we establish the
following uniqueness result.

######  Theorem 1.3.

Suppose either

  1. (i)

$p\geq-1,\ q\leq 1$ and at least one of these inequalities is strict, or

  2. (ii)

$p<-1,\ q\geq 1$ and $q-p\leq 8$.

Then the solution to (1.4) must be constant, i.e. $\varphi(\theta)\equiv c$,
where $c$ solves the equation

$c^{-p}\left(\frac{c+c^{-1}}{2}\right)^{q-1}\left(\frac{c-c^{-1}}{2}\right)=\gamma.$

On the other hand, we also prove the nonuniqueness of solutions to (1.4).

######  Theorem 1.4.

Let $l\geq 1$ be an integer. If $p<-1,\ q\geq 1,\ q-p>2(l+1)^{2}$ and
$\gamma>\gamma_{p,q,l}$, where $\gamma_{p,q,l}$ is given by

$\displaystyle\gamma_{p,q,l}=2^{-q}(u_{\gamma}^{2}+u_{\gamma}^{-2})^{q-1}(u_{\gamma}^{2-2p}-u_{\gamma}^{-2-2p}),$

where $u_{\gamma}\in(1,\infty)$ solves the equation

$\displaystyle(1-p)+(1+p)u_{\gamma}^{-4}+(q-1)\frac{(1-u_{\gamma}^{-4})^{2}}{1+u_{\gamma}^{-4}}=2(l+1)^{2},$
(1.5)

then there exist at least $(l+1)$ solutions to (1.4). Furthermore, for any
integer $2\leq j\leq l+1$, there exists a non-constant solution that
corresponds to a h-convex curve $\Gamma_{j,p,q,\gamma}$ with $j$-fold symmetry
in $\mathbb{H}^{2}$.

$p$-axis $q$-axis $-8$ $-7$ $-6$ $-5$ $-4$ $-3$ $-2$ $-1$ $1$ $2$ $3$ $4$ $5$
$6$ $7$ $8$ $-2$ $-1$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $0$ $q=1$$p=-1$$q-p=8$
Figure 2. Whether or not there exist non-constant solutions to (1.4) Grey
domain: Non-existence for $\gamma>0$. Black domain: Existence for
$\gamma>\gamma_{p,q,1}$.

######  Remark 1.5.

Let $\psi(u_{\gamma})$ denote the left hand side of (1.5). Since $p<-1$ and
$q\geq 1$, then $\psi$ is strictly increasing with respect to $u_{\gamma}$.
Note that $\psi$ tends to $2<2(l+1)^{2}$ as $u_{\gamma}$ tends to 1, and also
tends to $q-p>2(l+1)^{2}$ as $u_{\gamma}$ tends to infinity. Therefore, there
exists a unique solution $u_{\gamma}$ to (1.5), and $\gamma_{p,q,l}$ is well-
defined.

The paper is organized as follows. In Section 2, we construct a period
function $\Theta_{p}(\gamma,E)$ which is closely related to the solution to
(1.1), and give two alternative expressions $\Theta_{p}[u_{-},u_{+}]$ and
$\Theta_{p}\\{\alpha,r\\}$. In Section 3, we study special values and
asymptotic behaviours of $\Theta_{p}$. In Section 4, we prove that when
$p\leq-1$, then $\Theta_{p}\\{\alpha,r\\}$ is monotone increasing in $p$ and
in $\alpha$, respectively. In Section 5, we prove Theorem 1.2. In Section 6,
by using the same method to the equation (1.4), we give the proofs of Theorem
1.3 and Theorem 1.4.

######  Acknowledgments.

This work was supported by NSFC Grant No.11831005.

##  2\. The period function

Let $\varphi(\theta)$ be a smooth positive solution to the isotropic
horospherical $p$-Minkowski problem in hyperbolic plane, that is,
$\varphi(\theta)$ satisfies

$\varphi^{-p}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma,$
(2.1)

where $\gamma$ is a positive constant. Due to Theorem 1.1, we only consider
the case where $p\leq-1$ and $u(\tau)$ is not constant from now on.

Define $u(\tau)=\sqrt{\varphi(2\tau)}$, then (2.1) is equivalent to

$u^{1-2p}(u_{\tau\tau}+u-u^{-3})=2\gamma.$ (2.2)

Since $p<0$, then (2.2) can be rewritten as

$\displaystyle(u_{\tau}^{2}+u^{2}+u^{-2})_{\tau}=\frac{2\gamma}{p}(u^{2p})_{\tau}.$

Thus a first integral of (2.2) is given by

$\displaystyle u_{\tau}^{2}+u^{2}+u^{-2}-\frac{2\gamma}{p}u^{2p}=E,$ (2.3)

for some constant $E$.

Since the solution $\varphi$ to (2.1) is a $2\pi$-periodic positive function
of $\theta$, then the corresponding solution $u$ to (2.2) is a $\pi$-periodic
positive function of $\tau$, and its maximum $u_{+}$ and minimum values
$u_{-}$ are determined by $p,\gamma$ and $E$. Define a function

$E_{\gamma}(u)=u^{2}+u^{-2}-\frac{2\gamma}{p}u^{2p},$

then it follows from (2.3) that $E=E_{\gamma}(u_{\pm})$. Note that

$E_{\gamma}^{\prime}(u)=2u-2u^{-3}-4\gamma u^{2p-1},$

and

$E_{\gamma}^{\prime\prime}(u)=2+6u^{-4}-4\gamma(2p-1)u^{2p-2}>0,$

thus $E_{\gamma}(u)$ is convex, and decreasing for $u\in(0,u_{\gamma})$ and
increasing for $u\in(u_{\gamma},+\infty)$, where $u_{\gamma}$ satisfies
$E_{\gamma}^{\prime}(u_{\gamma})=0$. Moreover, $E_{\gamma}(u)$ has the minimum
value $E_{\gamma}^{*}=E_{\gamma}(u_{\gamma})$, and $E_{\gamma}(u)$ tends to
infinity as $u$ tends to zero or infinity. Consequently, we obtain

######  Lemma 2.1.

Let $u_{+}>u_{-}$ be the two positive roots of the equation $E_{\gamma}(u)=E$,
where $E>E_{\gamma}^{*}$. Then $u_{+}$ increases to infinity and $u_{-}$
decreases to zero as $E$ increases to infinity.

Furthermore, if $\gamma_{1}>\gamma_{2}>0$ and
$E>E_{\gamma_{1}}^{*},E>E_{\gamma_{2}}^{*}$, denote $u_{+}^{(i)},u_{-}^{(i)}$
by the two roots of the equation $E_{\gamma_{i}}(u)=E$ for $i=1,2$, then
$u_{+}^{(2)}>u_{+}^{(1)}>u_{-}^{(1)}>u_{-}^{(2)}>0$.

Let $\tau_{-},\tau_{+}\in[0,\pi)$ be two consecutive critical points such that
$u(\tau_{\pm})=u_{\pm}$. Without loss of generality, we assume
$\tau_{+}>\tau_{-}$. Using (2.3), we have

$u_{\tau}=\sqrt{E+\frac{2\gamma}{p}u^{2p}-u^{2}-u^{-2}}>0$

for $\tau\in(\tau_{-},\tau_{+})$. Then

$\tau_{+}-\tau_{-}=\int_{\tau_{-}}^{\tau_{+}}d\tau=\int_{u_{-}}^{u_{+}}\frac{du}{u_{\tau}}.$
(2.4)

Thus we consider the period function

$\Theta_{p}(\gamma,E):=\int_{u_{-}}^{u_{+}}\frac{du}{\sqrt{E+\frac{2\gamma}{p}u^{2p}-u^{2}-u^{-2}}},$
(2.5)

where $u_{-}$ and $u_{+}$ satisfy $E_{\gamma}(u_{-})=E_{\gamma}(u_{+})=E$.

If $u(\tau)$ is a $\pi$-periodic non-constant solution to (2.2), then it must
have an even number of critical points in $[0,\pi)$. It follows from (2.4) and
(2.5) that there exists a positive integer $m$ such that
$\Theta_{p}(\gamma,E)=\frac{\pi}{2m}$. We obtain

######  Lemma 2.2.

  1. (i)

If $\frac{\pi}{4}<\Theta_{p}(\gamma,E)<\frac{\pi}{2}$ for any
$E>E_{\gamma}^{*}$, then the solution to (2.1) must be constant. Moreover, the
solution to (2.1) is unique.

  2. (ii)

If $E>E_{\gamma}^{*}$ and $\Theta_{p}(\gamma,E)=\frac{\pi}{2m}$ for some
integer $m$, then there exists a $\frac{2\pi}{m}$-periodic non-constant
solution to (2.1).

For convenience, we provide two new expressions of $\Theta_{p}$. First, using
$E_{\gamma}(u_{-})=E_{\gamma}(u_{+})=E$, we get

$-\frac{2\gamma}{p}=\frac{(u_{+}^{2}-u_{-}^{2})(u_{+}^{-2}u_{-}^{-2}-1)}{u_{+}^{2p}-u_{-}^{2p}},$
(2.6)

and

$E=\frac{u_{+}^{2p}(u_{-}^{2}+u_{-}^{-2})-u_{-}^{2p}(u_{+}^{2}+u_{+}^{-2})}{u_{+}^{2p}-u_{-}^{2p}}.$
(2.7)

It follows from $p\leq-1$ and (2.6) that $\gamma>0$ if and only if
$u_{+}u_{-}>1$. We can then consider $\Theta_{p}$ as a function of $u_{-}$ and
$u_{+}$

$\Theta_{p}[u_{-},u_{+}]:=\Theta_{p}(\gamma(u_{-},u_{+}),E(u_{-},u_{+}))=\int_{u_{-}}^{u_{+}}\frac{du}{\sqrt{G_{p}(u,u_{-},u_{+})}},$
(2.8)

where

$\begin{split}G_{p}(u,u_{-},u_{+})=\frac{u_{-}^{2p}-u^{2p}}{u_{-}^{2p}-u_{+}^{2p}}(u_{+}^{2}+u_{+}^{-2})+\frac{u^{2p}-u_{+}^{2p}}{u_{-}^{2p}-u_{+}^{2p}}(u_{-}^{2}+u_{-}^{-2})-u^{2}-u^{-2}\end{split}$
(2.9)

for $(u_{-},u_{+})$ satisfying $u_{+}>u_{-}>0$ and $u_{+}u_{-}>1$.

Next, let $r=\frac{u_{+}}{u_{-}}\in(1,\infty)$ and
$\alpha=\frac{1}{u_{+}u_{-}}\in(0,1)$, and make the change of variable
$s=\frac{u}{u_{-}}$. We define

$\Theta_{p}\\{\alpha,r\\}:=\Theta_{p}[\alpha^{-\frac{1}{2}}r^{-\frac{1}{2}},\alpha^{-\frac{1}{2}}r^{\frac{1}{2}}]=\int_{1}^{r}\frac{ds}{\sqrt{F_{p}(s,\alpha,r)}},$
(2.10)

where

$\begin{split}F_{p}(s,\alpha,r)&=\frac{1-s^{2p}}{1-r^{2p}}(r^{2}+\alpha^{2})+\frac{s^{2p}-r^{2p}}{1-r^{2p}}(1+\alpha^{2}r^{2})-s^{2}-\alpha^{2}r^{2}s^{-2}\\\
&=\frac{1-s^{2p}}{1-r^{2p}}(r^{2}-1)-(s^{2}-1)-\alpha^{2}\left[\frac{1-s^{2p}}{1-r^{2p}}(r^{2}-1)-r^{2}(1-s^{-2})\right]\end{split}$
(2.11)

for $(\alpha,r)$ satisfying $0<\alpha<1<r$.

In the following sections, we will study the asymptotic behavior and
monotonicity of the period function $\Theta_{p}$.

##  3\. Special values and asymptotic behaviors

Let $p\leq-1$. Denote $D_{1}=\\{(\gamma,E):\ \gamma>0,\ E>E_{\gamma}^{*}\\}$
and $D_{2}=\\{(\alpha,r):\ 0<\alpha<1<r\\}$.

######  Theorem 3.1.

$\Theta_{p}(\cdot,\cdot)$ and $\Theta_{p}\\{\cdot,\cdot\\}$ are continuous on
$D_{1}$ and $D_{2}$, respectively. $\Theta_{p}$ is continuous with respect to
$p$, and has following limiting values:

$\displaystyle\lim\limits_{p\to-\infty}\Theta_{p}\\{\alpha,r\\}=\arccos\sqrt{\frac{1-\alpha^{2}}{r^{2}-\alpha^{2}}},$
(3.1) $\displaystyle\lim\limits_{r\to
1}\Theta_{p}\\{\alpha,r\\}=\frac{\pi}{\sqrt{(2-2p)+(2+2p)\alpha^{2}}},$ (3.2)
$\displaystyle\lim\limits_{r\to+\infty}\Theta_{p}\\{\alpha,r\\}=\frac{\pi}{2},$
(3.3) $\displaystyle\lim\limits_{\gamma\to
0}\Theta_{p}(\gamma,E)=\frac{\pi}{2},$ (3.4)
$\displaystyle\lim\limits_{E\to+\infty}\Theta_{p}(\gamma,E)=\frac{\pi}{2},$
(3.5)
$\displaystyle\lim\limits_{E\to(E_{\gamma}^{*})^{+}}\Theta_{p}(\gamma,E)=\frac{\pi}{\sqrt{(2-2p)+(2+2p)u_{\gamma}^{-4}}},$
(3.6)

where $u_{\gamma}\in(1,+\infty)$ is given by

$u_{\gamma}^{2-2p}-u_{\gamma}^{-2-2p}=2\gamma.$ (3.7)

Moreover, we also have

$\Theta_{-1}(\gamma,E)=\frac{\pi}{2},\quad\quad(\gamma,E)\in D_{1},$ (3.8)
$\Theta_{p}\\{1,r\\}=\frac{\pi}{2},\quad\quad\quad\quad r>1.$ (3.9)

The following lemma is needed in the proof of (3.6), see [Per10, Corollary
4.2].

######  Lemma 3.2 ([Per10]).

Let $\delta$ be a positive real number and let
$f:(u_{0}-\delta,u_{0}+\delta)\to\mathbb{R}$ be a smooth function such that
$f(u_{0})=f^{\prime}(u_{0})=0$ and $f^{\prime\prime}(u_{0})=-2a<0$. If for any
small $c>0$, $u_{1}(c)<u_{0}<u_{2}(c)$ are such that
$f(u_{1}(c))+c=0=f(u_{2}(c))+c$, then

$\lim\limits_{c\to
0^{+}}\int_{u_{1}(c)}^{u_{2}(c)}\frac{du}{\sqrt{f(u)+c}}=\frac{\pi}{\sqrt{a}}.$

For the sake of completeness, let us give a proof of Lemma 3.2.

###### Proof of Lemma 3.2.

For any $b>a$, we define the function $h(u)=f^{\prime}(u)+2b(u-u_{0})$. Since
$h^{\prime}(u_{0})=2(b-a)>0$, there exists a $\epsilon(b)\in(0,\delta)$ such
that $h^{\prime}(u)>0$ for any $u\in[u_{0},u_{0}+\epsilon]$. Then it follows
from $h(u_{0})=0$ that $h(u)>0$ for $u\in(u_{0},u_{0}+\epsilon]$.

For any small $c>0$ such that $u_{2}(c)<u_{0}+\epsilon$, the function

$g(u)=f(u)+c+b(u-u_{2}(c))(u+u_{2}(c)-2u_{0})$

satisfies $g(u_{2}(c))=0$ and $g^{\prime}(u)=h(u)>0$. Thus $g(u)<0$ for
$u\in[u_{0},u_{2}(c))$. We get

$0<f(u)+c<b(u_{2}(c)-u)(u+u_{2}(c)-2u_{0}),\quad\text{for any}\
u\in(u_{0},u_{2}(c))$

and then

$\frac{\pi}{2\sqrt{b}}=\int_{u_{0}}^{u_{2}(c)}\frac{du}{\sqrt{b(u_{2}(c)-u)(u+u_{2}(c)-2u_{0})}}<\int_{u_{0}}^{u_{2}(c)}\frac{du}{\sqrt{f(u)+c}}.$

On the other hand, for any $0<b<a$ and small $c>0$, the same argument shows

$\int_{u_{0}}^{u_{2}(c)}\frac{du}{\sqrt{f(u)+c}}<\int_{u_{0}}^{u_{2}(c)}\frac{du}{\sqrt{b(u_{2}(c)-u)(u+u_{2}(c)-2u_{0})}}=\frac{\pi}{2\sqrt{b}}.$

Therefore, we obtain $\lim\limits_{c\to
0^{+}}\int_{u_{0}}^{u_{2}(c)}\frac{du}{\sqrt{f(u)+c}}=\frac{\pi}{2\sqrt{a}}.$
The similar calculation holds for
$\int_{u_{1}(c)}^{u_{0}}\frac{du}{\sqrt{f(u)+c}}$, and then we complete the
proof of this lemma. ∎

###### Proof of Theorem 3.1.

At first, it follows from (2.10) and (2.11) that

$\displaystyle\lim\limits_{p\to-\infty}\Theta_{p}\\{\alpha,r\\}=\int_{1}^{r}\frac{ds}{\sqrt{(r^{2}-s^{2})(1-\alpha^{2}s^{-2})}}=\arccos\sqrt{\frac{1-\alpha^{2}}{r^{2}-\alpha^{2}}}.$

Now let $\beta>0$, and set $s^{\beta}=x$, where

$\displaystyle v=\frac{r^{\beta}-1}{2}z+\frac{r^{\beta}+1}{2},\quad\quad
z\in(-1,1).$

Then the change of variables formula implies that

$\displaystyle\Theta_{p}\\{\alpha,r\\}=\int_{-1}^{1}\frac{dz}{\sqrt{J_{p}(\alpha,r,\beta,z)}},$

where

$J_{p}(\alpha,r,\beta,z)=\frac{4\beta^{2}}{(r^{\beta}-1)^{2}}x^{\frac{2(\beta-1)}{\beta}}\left[\frac{1-x^{\frac{2p}{\beta}}}{1-r^{2p}}(r^{2}+\alpha^{2})+\frac{x^{\frac{2p}{\beta}}-r^{2p}}{1-r^{2p}}(1+\alpha^{2}r^{2})-x^{\frac{2}{\beta}}-\alpha^{2}r^{2}x^{-\frac{2}{\beta}}\right].$

When $r$ is close to $1$, a direct computation using Taylor expansions yields

$\displaystyle\begin{split}J_{p}(\alpha,r,\beta,z)&=\left((2-2p)+(2+2p)\alpha^{2}\right)(1-z^{2})\\\
&\quad\cdot\left[1+\frac{z}{2}\left(\beta-\frac{4-2p}{3}-\frac{4(1+p)\alpha^{2}}{(3-3p)+(3+3p)\alpha^{2}}\right)(r-1)+o(r-1)\right],\end{split}$

from which it follows that

$\displaystyle\lim\limits_{r\to
1}\Theta_{p}\\{\alpha,r\\}=\int_{-1}^{1}\frac{dz}{\sqrt{\left((2-2p)+(2+2p)\alpha^{2}\right)(1-z^{2})}}=\frac{\pi}{\sqrt{(2-2p)+(2+2p)\alpha^{2}}}.$

When $r$ tends to infinity, let $y=\frac{z+1}{2}\in(0,1)$ and choose
$\beta=1$, we have

$J_{p}(\alpha,r,1,z)=4\left[1-y^{2}+(2-2y)\frac{1}{r}+o\left(\frac{1}{r}\right)\right],\quad
p<-1,$

from which it follows that

$\lim\limits_{r\to+\infty}\Theta_{p}\\{\alpha,r\\}=\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}=\frac{\pi}{2}.$

Next, let $w=u_{+}-u_{-}$ and $u=wt+u_{-}$ for $t\in(0,1)$. By
$E_{\gamma}(u_{\pm})=E$, we have

$\frac{u_{+}}{\sqrt{E}}\to
1,\quad\frac{u_{-}}{\left(\frac{-p}{2\gamma}E\right)^{\frac{1}{2p}}}\to 1\
\text{and}\ \frac{w}{\sqrt{E}}\to 1,\quad\text{as}\ E\to+\infty,$

from which it follows that

$\displaystyle\lim\limits_{E\to+\infty}\Theta_{p}(\gamma,E)$
$\displaystyle=\lim\limits_{E\to+\infty}\int_{0}^{1}\frac{dt}{\sqrt{\frac{E}{w^{2}}-\left(t+\frac{u_{-}}{w}\right)^{2}-w^{-4}\left(t+\frac{u_{-}}{w}\right)^{-2}+\frac{2\gamma}{p}w^{2p-2}\left(t+\frac{u_{-}}{w}\right)^{2p}}}$
$\displaystyle=\int_{0}^{1}\frac{dt}{\sqrt{1-t^{2}}}=\frac{\pi}{2}.$

On the other hand, we choose
$f(u)=E_{\gamma}^{*}+\frac{2\gamma}{p}u^{2p}-u^{2}-u^{-2}$ and
$c=E-E_{\gamma}^{*}$ in Lemma 3.2, then $u_{0}=u_{\gamma},\ u_{1}(c)=u_{-}$
and $u_{2}(c)=u_{+}$. It follows from $f^{\prime}(u_{\gamma})=0$ that
$u_{\gamma}$ satisfies (3.7). Thus we have

$\displaystyle a$
$\displaystyle=-\frac{1}{2}f^{\prime\prime}(u_{\gamma})=1+3u_{\gamma}^{-4}+2\gamma(2p-1)u_{\gamma}^{2p-2}$
$\displaystyle=(2-2p)+(2+2p)u_{\gamma}^{-4}.$

Using Lemma 3.2, we obtain

$\displaystyle\lim\limits_{E\to(E_{\gamma}^{*})^{+}}\Theta_{p}(\gamma,E)=\frac{\pi}{\sqrt{a}}=\frac{\pi}{\sqrt{(2-2p)+(2+2p)u_{\gamma}^{-4}}}.$

Finally, (3.4), (3.8) and (3.9) can be deduced by direct calculation. ∎

##  4\. Monotonicity of $\Theta_{p}\\{\alpha,r\\}$

Let $p\leq-1$ and $0<\alpha<1<r$. By (2.10) and (2.11), we have

$\Theta_{p}\\{\alpha,r\\}=\int_{1}^{r}\frac{ds}{\sqrt{F_{p}(s,\alpha,r)}},$
(4.1)

where

$\displaystyle F_{p}(s,\alpha,r)$
$\displaystyle=\frac{s^{2p}-1}{r^{2p}-1}(r^{2}+\alpha^{2})+\frac{r^{2p}-s^{2p}}{r^{2p}-1}(1+\alpha^{2}r^{2})-s^{2}-\alpha^{2}r^{2}s^{-2}$
(4.2)
$\displaystyle=\frac{s^{2p}-1}{r^{2p}-1}(r^{2}-1)-(s^{2}-1)-\alpha^{2}\left[\frac{s^{2p}-1}{r^{2p}-1}(r^{2}-1)-r^{2}(1-s^{-2})\right].$
(4.3)

###  4.1. Monotonicity in $p$

Given $1<s<r$, we denote

$H(p)=\frac{s^{2p}-1}{r^{2p}-1}\in(0,1).$

######  Lemma 4.1.

$H(p)$ is monotone decreasing in $p<0$.

###### Proof.

Taking the derivative of $H(p)$ yields

$\displaystyle
H^{\prime}(p)=\frac{1}{p(r^{2p}-1)^{3}(s^{2p}-1)}\left(\frac{s^{2p}\log
s^{2p}}{s^{2p}-1}-\frac{r^{2p}\log r^{2p}}{r^{2p}-1}\right).$

Let $h(t)=\frac{t\log t}{t-1}$ for $0<t<1$. Taking the derivative of $h(t)$
yields

$\displaystyle h^{\prime}(t)=\frac{t-1-\log t}{(t-1)^{2}}>0.$

Since $0<r^{2p}<s^{2p}<1$, we have $h(r^{2p})<h(s^{2p})$ and therefore
$H^{\prime}(p)<0$. We complete the proof of Lemma 4.1. ∎

Since $(r^{2}-1)(1-\alpha^{2})>0$ for $0<\alpha<1<r$, by (4.1) and (4.3),
Lemma 4.1 immediately deduces the following theorem.

######  Theorem 4.2.

Let $p\leq-1$. $\Theta_{p}\\{\alpha,r\\}$ is monotone increasing in $p$.

This together with (3.8) yields an upper bound for $\Theta_{p}\\{\alpha,r\\}$.

######  Corollary 4.3.

Let $p<-1$. $\Theta_{p}\\{\alpha,r\\}<\frac{\pi}{2}$ for any $0<\alpha<1<r$.

###  4.2. Monotonicity in $\alpha$

Using Lemma 4.1, we have

$\frac{s^{2p}-1}{r^{2p}-1}(r^{2}-1)-(s^{2}-1)>0,$ (4.4)

and

$\frac{s^{2p}-1}{r^{2p}-1}(r^{2}-1)-r^{2}(1-s^{-2})>0,$ (4.5)

for $1<s<r$ and $p<-1$. It follows from (4.3), (4.4) and (4.5) that
$F_{p}(s,\alpha,r)$ is monotone decreasing in $\alpha\in(0,1)$ for any $1<s<r$
and $p<-1$. Therefore, we obtain

######  Theorem 4.4.

Let $0<\alpha<1$. $\Theta_{p}\\{\alpha,r\\}$ is monotone increasing in
$\alpha$.

Combining this with (3.8), we can conclude Corollary 4.3 again. Moreover,
denote

$\hat{\Theta}_{p}(r):=\Theta_{p}\\{0,r\\}=\int_{1}^{r}\frac{ds}{\sqrt{\frac{s^{2p}-1}{r^{2p}-1}(r^{2}-1)-(s^{2}-1)}}.$

Making the change of variable
$s^{\beta}=x=\frac{r^{\beta}-1}{2}z+\frac{r^{\beta}+1}{2}$ with
$\beta=\frac{4-2p}{3}$ yields

$\hat{\Theta}_{p}(r)=\int_{-1}^{1}\frac{dz}{\sqrt{\hat{J}_{p}(r,z)}},$

where

$\hat{J}_{p}(r,z)=\frac{4\beta^{2}}{(r^{\beta}-1)^{2}}\left(\frac{r^{2}-r^{2p}}{1-r^{2p}}x^{\frac{2(\beta-1)}{\beta}}-x^{2}-\frac{r^{2}-1}{1-r^{2p}}x^{\frac{2(\beta-1+p)}{\beta}}\right).$

Note that $\hat{\Theta}_{p}(r)$ is the period function
$\Theta\left(\frac{1}{1-2p},r\right)$ given by Ben Andrews [Ben03]. We recall
the properties of $\hat{\Theta}_{p}(r)$ as follows:

######  Proposition 4.5 ([Ben03]).

Let $p\leq-1$. Then $\hat{\Theta}_{p}(\cdot)$ is continuous on $(1,\infty)$,
and has the following limiting values:

$\displaystyle\lim\limits_{r\to\infty}\hat{\Theta}_{p}(r)=\frac{\pi}{2},$
(4.6) $\displaystyle\lim\limits_{r\to
1}\hat{\Theta}_{p}(r)=\frac{\pi}{\sqrt{2-2p}}.$ (4.7)

######  Proposition 4.6 ([Ben03]).

Let $p<-1$ and $r>1$. Then $\frac{\partial}{\partial r}\hat{J}_{p}(r,z)<0$ for
any $z\in(-1,1)$. Furthermore, $\hat{\Theta}_{p}(r)$ is monotone increasing in
$r$.

Consequently, we deduce the following corollary.

######  Corollary 4.7.

Let $p<-1$.
$\Theta_{p}\\{\alpha,r\\}>\hat{\Theta}_{p}(r)>\frac{\pi}{\sqrt{2-2p}}$ for any
$0<\alpha<1<r$.

##  5\. Proof of Theorem 1.2

At first, if $-7\leq p<-1$, using Corollary 4.3 and Corollary 4.7, we have

$\displaystyle\frac{\pi}{2}>\Theta_{p}\\{\alpha,r\\}>\frac{\pi}{\sqrt{2-2p}}>\frac{\pi}{4},$

for any $0<\alpha<1<r$. Therefore, if $\gamma>0$ and $E>E_{\gamma}^{*}$, then

$\displaystyle\Theta_{p}(\gamma,E)=\Theta_{p}\\{\alpha(\gamma,E),r(\gamma,E)\\}\in\left(\frac{\pi}{4},\frac{\pi}{2}\right).$

It follows from Lemma 2.2 (i) that the solution to (1.1) must be constant.
Moreover, the solution to (1.1) is unique. We prove Theorem 1.2 (i).

Next, if $1-2(k+1)^{2}\leq p<1-2k^{2}$ and $\gamma>\gamma_{p,l}$ for $1\leq
l<k$, then it follows from (3.5), (3.6) and (3.7) that

$\displaystyle\lim\limits_{E\to+\infty}\Theta_{p}(\gamma,E)=\frac{\pi}{2}$

and

$\displaystyle\lim\limits_{E\to(E_{\gamma}^{*})^{+}}\Theta_{p}(\gamma,E)<\frac{\pi}{2(l+1)}.$

Therefore, there exist at least $l$ values $\\{E_{j}\\}_{j=1}^{l}$ such that
$E_{j}>E_{\gamma}^{*}$ and $\Theta_{p}(\gamma,E_{j})=\frac{\pi}{2(j+1)}$. It
follows from Lemma 2.2 (ii) that there exist at least $(l+1)$ solutions to
(1.1). We complete the proof of Theorem 1.2.

##  6\. Proofs of Theorem 1.3 and Theorem 1.4

###  6.1. Proof of Theorem 1.3 (i)

Let $\varphi(\theta)$ be the solution to (1.4), i.e.

$\varphi^{-p}\left(\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}\right)^{q-1}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma,\quad\text{on}\
\mathbb{S}^{1},$ (6.1)

where $\gamma$ is a positive constant. The case $q=1$ of Theorem 1.3 (i) was
shown in Theorem 1.1. Recall that the following Heintze-Karcher type
inequality in $\mathbb{H}^{2}$ is crucial in the proof of Theorem 1.1 in [LX].

######  Lemma 6.1 ([LX]).

Let $\Omega$ be a smooth uniformly h-convex bounded domain in
$\mathbb{H}^{2}$, and $\varphi(\theta)$ be its exponential horospherical
support function. Then we have

$\int_{\partial\Omega}\left(\frac{\cosh
r-\tilde{u}}{\tilde{\kappa}}-\tilde{u}\right)d\mu\geq 0,$ (6.2)

which is equivalent to

$\int_{\mathbb{S}^{1}}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{\varphi}\right)\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)d\theta\geq
0.$ (6.3)

Equality holds if and only if $\Omega$ is a geodesic ball, i.e.
$\varphi(\theta)$ is given by (1.3).

Now we can give the proof. If $p\geq-1$ and $q<1$, by (6.1) and (6.3), we have

$\displaystyle 0$
$\displaystyle\leq\int_{\mathbb{S}^{1}}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{\varphi}\right)\varphi^{p}\left(\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}\right)^{1-q}d\theta$
$\displaystyle=\int_{\mathbb{S}^{1}}\left(\frac{\varphi_{\theta}}{\varphi}\right)_{\theta}\varphi^{1+p}\left(\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}\right)^{1-q}d\theta$
$\displaystyle=-(1+p)\int_{\mathbb{S}^{1}}\varphi_{\theta}^{2}\varphi^{p-1}\left(\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}\right)^{1-q}d\theta$
$\displaystyle\quad-(1-q)\int_{\mathbb{S}^{1}}\varphi_{\theta}^{2}\varphi^{1+p}\left(\varphi,\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi+\varphi^{-1}}{2}\right)^{-q}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^{2}}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)d\theta$
$\displaystyle\leq 0.$

Hence the above inequalities are both equalities, and then
$\varphi_{\theta}\equiv 0$ on $\mathbb{S}^{1}$. This completes the proof of
Theorem 1.3 (i).

###  6.2. The period function $\Theta_{p,q}$

By Theorem 1.3 (i), we only consider the case where $p\leq-1,\ q\geq 1$ and
$\varphi(\theta)$ is not constant from now on. Define
$u(\tau)=\sqrt{\varphi(2\tau)}$, then (6.1) is equivalent to

$u^{1-2p}(u_{\tau}^{2}+u^{2}+u^{-2})^{q-1}(u_{\tau\tau}+u-u^{-3})=2^{q}\gamma.$
(6.4)

Since $p<0<q$, (6.4) can be rewritten as

$\displaystyle\left(\frac{1}{q}(u_{\tau}^{2}+u^{2}+u^{-2})^{q}\right)_{\tau}=\frac{2^{q}\gamma}{p}(u^{2p})_{\tau}.$

Thus a first integral of (6.4) is given by

$\displaystyle\frac{1}{q}(u_{\tau}^{2}+u^{2}+u^{-2})^{q}-\frac{2^{q}\gamma}{p}u^{2p}=E.$

for some constant $E$.

For the case $p\leq-1$ and $q\geq 1$, the argument is roughly the same as the
case $q=1$. Let
$\bar{E}_{\gamma}(u)=\frac{1}{q}(u^{2}+u^{-2})^{q}-\frac{2^{q}\gamma}{p}u^{2p}$,
it follows that $\bar{E}_{\gamma}(u)$ is convex, and decreasing for
$u\in(0,u_{\gamma})$ and increasing for $u\in(u_{\gamma},+\infty)$, where
$u_{\gamma}$ satisfies $\bar{E}_{\gamma}^{\prime}(u_{\gamma})=0$. Moreover,
$\bar{E}_{\gamma}(u)$ has the minimum value
$\bar{E}_{\gamma}^{*}=\bar{E}_{\gamma}(u_{\gamma})$, and $\bar{E}_{\gamma}(u)$
tends to infinity as $u$ tends to zero or infinity.

For any $E>\bar{E}_{\gamma}^{*}$, we define the period function

$\Theta_{p,q}(\gamma,E):=\int_{u_{-}}^{u_{+}}\frac{du}{u_{\tau}}=\int_{u_{-}}^{u_{+}}\frac{du}{\sqrt{\left(qE+\frac{2^{q}q\gamma}{p}u^{2p}\right)^{\frac{1}{q}}-u^{2}-u^{-2}}},$
(6.5)

where $u_{-}$ and $u_{+}$ satisfy
$\bar{E}_{\gamma}(u_{-})=\bar{E}_{\gamma}(u_{+})=E$. Since $u(\tau)$ is
$\pi$-periodic, we obtain

######  Lemma 6.2.

  1. (i)

If $\frac{\pi}{4}<\Theta_{p,q}(\gamma,E)<\frac{\pi}{2}$ for any
$E>\bar{E}_{\gamma}^{*}$, then the solution to (6.1) must be constant.
Moreover, the solution to (6.1) is unique.

  2. (ii)

If $E>\bar{E}_{\gamma}^{*}$ and $\Theta_{p,q}(\gamma,E)=\frac{\pi}{2m}$ for
some integer $m$, then there exists a $\frac{2\pi}{m}$-periodic non-constant
solution to (6.1).

We reparameterize $\Theta_{p,q}$. Let $r=\frac{u_{+}}{u_{-}}>1$ and
$\alpha=\frac{1}{u_{+}u_{-}}$. It follows from $\bar{E}_{\gamma}(u_{\pm})=E$
that

$-\frac{2^{q}q\gamma}{p}=\frac{(u_{+}^{2}+u_{+}^{-2})^{q}-(u_{-}^{2}+u_{-}^{-2})^{q}}{u_{-}^{2p}-u_{+}^{2p}}=\frac{(r^{2}+\alpha^{2})^{q}-(1+\alpha^{2}r^{2})^{q}}{(\alpha
r)^{q-p}(1-r^{2p})},$ (6.6)

and

$qE=\frac{u_{-}^{2p}(u_{+}^{2}+u_{+}^{-2})^{q}-u_{+}^{2p}(u_{-}^{2}+u_{-}^{-2})^{q}}{u_{-}^{2p}-u_{+}^{2p}}=\frac{(r^{2}+\alpha^{2})^{q}-r^{2p}(1+\alpha^{2}r^{2})^{q}}{(\alpha
r)^{q}(1-r^{2p})}.$ (6.7)

Since $p\leq-1,\ q\geq 1$ and (6.6), we can conclude that $\gamma>0$ if and
only if $0<\alpha<1$. Making the change of variable $s=\frac{u}{u_{-}}$, we
define

$\Theta_{p,q}\\{\alpha,r\\}:=\Theta_{p,q}(\gamma(\alpha,r),E(\alpha,r))=\int_{1}^{r}\frac{ds}{\sqrt{F_{p,q}(s,\alpha,r)}},$
(6.8)

where

$\begin{split}F_{p,q}(s,\alpha,r)&=\left[\frac{1-s^{2p}}{1-r^{2p}}(r^{2}+\alpha^{2})^{q}+\frac{s^{2p}-r^{2p}}{1-r^{2p}}(1+\alpha^{2}r^{2})^{q}\right]^{\frac{1}{q}}-s^{2}-\alpha^{2}r^{2}s^{-2}\end{split}$
(6.9)

for $(\alpha,r)$ satisfying $0<\alpha<1<r$.

###  6.3. Properties of $\Theta_{p,q}$

In this subsection, we assume $p\leq-1$ and $q\geq 1$. Denote
$\bar{D}_{1}=\\{(\gamma,E):\ \gamma>0,\ E>\bar{E}_{\gamma}^{*}\\}$ and
$\bar{D}_{2}=\\{(\alpha,r):\ 0<\alpha<1,\ r>1\\}$. Since the proof of the
following properties of $\Theta_{p,q}$ follows similarly, we briefly present
it.

######  Theorem 6.3.

$\Theta_{p,q}(\cdot,\cdot)$ and $\Theta_{p,q}\\{\cdot,\cdot\\}$ are continuous
on $\bar{D}_{1}$ and $\bar{D}_{2}$, respectively. $\Theta_{p,q}$ is continuous
with respect to $p$ and $q$, and has following limiting values:

$\displaystyle\lim\limits_{p\to-\infty}\Theta_{p,q}\\{\alpha,r\\}=\arccos\sqrt{\frac{1-\alpha^{2}}{r^{2}-\alpha^{2}}},$
(6.10)
$\displaystyle\lim\limits_{q\to+\infty}\Theta_{p,q}\\{\alpha,r\\}=\arccos\sqrt{\frac{1-\alpha^{2}}{r^{2}-\alpha^{2}}},$
(6.11) $\displaystyle\lim\limits_{r\to
1}\Theta_{p,q}\\{\alpha,r\\}=\frac{\pi}{\sqrt{(2-2p)+(2+2p)\alpha^{2}+(2q-2)\frac{(1-\alpha^{2})^{2}}{1+\alpha^{2}}}},$
(6.12)
$\displaystyle\lim\limits_{r\to+\infty}\Theta_{p,q}\\{\alpha,r\\}=\frac{\pi}{2},$
(6.13) $\displaystyle\lim\limits_{\gamma\to
0}\Theta_{p,q}(\gamma,E)=\frac{\pi}{2},$ (6.14)
$\displaystyle\lim\limits_{E\to+\infty}\Theta_{p,q}(\gamma,E)=\frac{\pi}{2},$
(6.15)
$\displaystyle\lim\limits_{E\to(\bar{E}_{\gamma}^{*})^{+}}\Theta_{p,q}(\gamma,E)=\frac{\pi}{\sqrt{(2-2p)+(2+2p)u_{\gamma}^{-4}+(2q-2)\frac{(1-u_{\gamma}^{-4})^{2}}{1+u_{\gamma}^{-4}}}},$
(6.16)

where $u_{\gamma}\in(1,\infty)$ is given by

$u_{\gamma}^{2-2p}-u_{\gamma}^{-2-2p}=2^{q}\gamma(u_{\gamma}^{2}+u_{\gamma}^{-2})^{1-q}.$
(6.17)

We will use the following lemma from [PP15, Per10] in the proof of (6.16)

######  Lemma 6.4 ([PP15, Per10]).

Let $f(u,E)$ and $g(u,E)$ be smooth functions such that

$f(u_{0},E_{0})=\frac{\partial f}{\partial
u}(u_{0},E_{0})=0\quad\text{and}\quad\frac{\partial^{2}f}{\partial
u^{2}}(u_{0},E_{0})=-2A<0.$

Suppose that $u_{-}=u_{-}(E)$ and $u_{+}=u_{+}(E)$ satisfy

$f(u_{-},E)=f(u_{+},E)=0,\quad u_{-}<u_{0}<u_{+},$

with

$f(u,E)>0\ \text{for all}\ u\in(u_{-},u_{+})\quad\text{and}\quad
u_{-},u_{+}\to u_{0}\ \text{as}\ E\to(E_{0})^{+}.$

Then we have

$\displaystyle\lim\limits_{E\to(E_{0})^{+}}\int_{u_{-}}^{u_{+}}\frac{g(u,E)du}{\sqrt{f(u,E)}}=g(u_{0},E_{0})\frac{\pi}{\sqrt{A}}.$

###### Proof of Theorem 6.3.

We will only provide the proofs of (6.12) and (6.16), as the others can be
derived similarly to the proof of Theorem 3.1.

Let $\beta>0$, and set $s^{\beta}=x$, where

$\displaystyle x=\frac{r^{\beta}-1}{2}z+\frac{r^{\beta}+1}{2},\quad\quad
z\in(-1,1).$

Then the change of variables formula implies that

$\displaystyle\Theta_{p,q}\\{\alpha,r\\}=\int_{-1}^{1}\frac{dz}{\sqrt{J_{p,q}(\alpha,r,\beta,z)}},$

where

$\begin{split}J_{p,q}(\alpha,r,\beta,z)&=\frac{4\beta^{2}}{(r^{\beta}-1)^{2}}x^{\frac{2(\beta-1)}{\beta}}\left[\left(\frac{1-x^{\frac{2p}{\beta}}}{1-r^{2p}}(r^{2}+\alpha^{2})^{q}+\frac{x^{\frac{2p}{\beta}}-r^{2p}}{1-r^{2p}}(1+\alpha^{2}r^{2})^{q}\right)^{\frac{1}{q}}\right.\\\
&\quad\quad\left.-x^{\frac{2}{\beta}}-\alpha^{2}r^{2}x^{-\frac{2}{\beta}}\right].\end{split}$

When $r$ is close to $1$, a direct computation using Taylor expansions yields

$\begin{split}J_{p,q}(\alpha,r,\beta,z)&=\left((2-2p)+(2+2p)\alpha^{2}+(2q-2)\frac{(1-\alpha^{2})^{2}}{1+\alpha^{2}}\right)(1-z^{2})\\\
&\quad\cdot\left[1+\frac{z}{2}\left(\beta-\frac{2(2q-p)}{3}-\frac{4\alpha^{2}\mu_{p,q}(\alpha,z)}{3}\right)(r-1)+o(r-1)\right],\end{split}$

where

$\displaystyle\mu_{p,q}(\alpha,z)=\frac{(p+1)-(q-1)\left(\frac{4(2+\alpha^{2})}{(1+\alpha^{2})^{2}}-(1+p)\frac{3-\alpha^{2}}{1+\alpha^{2}}+(q-1)\frac{2(1-\alpha^{2})^{2}}{1+\alpha^{2}}\right)}{(1-p)+(1+p)\alpha^{2}+(q-1)\frac{(1-\alpha^{2})^{2}}{1+\alpha^{2}}}.$

Thus it follows that

$\displaystyle\lim\limits_{r\to 1}\Theta_{p,q}\\{\alpha,r\\}$
$\displaystyle=\int_{-1}^{1}\frac{dz}{\sqrt{\left((2-2p)+(2+2p)\alpha^{2}+(2q-2)\frac{(1-\alpha^{2})^{2}}{1+\alpha^{2}}\right)(1-z^{2})}}$
$\displaystyle=\frac{\pi}{\sqrt{(2-2p)+(2+2p)\alpha^{2}+(2q-2)\frac{(1-\alpha^{2})^{2}}{1+\alpha^{2}}}}.$

On the other hand, we choose $g(u,E)\equiv 1$ and
$f(u,E)=\left(qE+\frac{2^{q}q\gamma}{p}u^{2p}\right)^{\frac{1}{q}}-u^{2}-u^{-2}$
in Lemma 6.4, then $u_{0}=u_{\gamma}$ and $E_{0}=\bar{E}_{\gamma}^{*}$. It
follows from $f(u_{\gamma},\bar{E}_{\gamma}^{*})=\frac{\partial f}{\partial
u}(u_{\gamma},\bar{E}_{\gamma}^{*})=0$ that $u_{\gamma}$ satisfies (6.17).
Moreover, we calculate

$\displaystyle A=-\frac{1}{2}\frac{\partial^{2}f}{\partial
u^{2}}(u_{\gamma},\bar{E}_{\gamma}^{*})$ $\displaystyle=$ $\displaystyle
1+3u_{\gamma}^{-4}-\frac{2^{q}\gamma}{q}(2p-1)u_{\gamma}^{2p-2}\left(qE+\frac{2^{q}q\gamma}{p}u^{2p}\right)^{\frac{1-q}{q}}+\frac{2^{2q+1}(q-1)\gamma^{2}}{q^{2}}u_{\gamma}^{4p-2}\left(qE+\frac{2^{q}q\gamma}{p}u^{2p}\right)^{\frac{1-2q}{q}}$
$\displaystyle=$ $\displaystyle
1+3u_{\gamma}^{-4}+(1-2p)(1-u_{\gamma}^{-4})+(2q-2)\frac{(u_{\gamma}^{2}-u_{\gamma}^{-2})^{2}}{u_{\gamma}^{2}(u_{\gamma}^{2}+u_{\gamma}^{-2})},$

where we used the definition of $u_{\gamma}$. Using Lemma 6.4, we obtain

$\displaystyle\lim\limits_{E\to(\bar{E}_{\gamma}^{*})^{+}}\Theta_{p,q}(\gamma,E)=\frac{\pi}{\sqrt{A}}=\frac{\pi}{\sqrt{(2-2p)+(2+2p)u_{\gamma}^{-4}+(2q-2)\frac{(1-u_{\gamma}^{-4})^{2}}{1+u_{\gamma}^{-4}}}}.$

∎

######  Remark 6.5.

Since $r$ tends to 1 if and only if $E$ tends to $\bar{E}_{\gamma}^{*}$, and
at the same time $\alpha$ tends to $u_{\gamma}^{-2}$, then (6.12) and (6.16)
are essentially equivalent.

Since (6.8), (6.9) and $(r^{2}-1)(1-\alpha^{2})>0$, Lemma 4.1 implies the
monotonicity in $p$.

######  Theorem 6.6.

Let $p\leq-1$ and $q\geq 1$. $\Theta_{p,q}\\{\alpha,r\\}$ is monotone
increasing in $p$.

As for the monotonicity in $q$, we need a lemma. Given $A,B>1$ and
$0<\lambda<1$, denote

$K(q)=(\lambda A^{q}+(1-\lambda)B^{q})^{\frac{1}{q}}.$

######  Lemma 6.7.

$K(q)$ is monotone increasing in $q>0$.

###### Proof.

Taking the derivative of $K(q)$ yields

$\displaystyle\frac{K^{\prime}(q)}{K(q)}$
$\displaystyle=\frac{1}{q^{2}(\lambda A^{q}+(1-\lambda)B^{q})}\left[\lambda
A^{q}\log A^{q}+(1-\lambda)B^{q}\log B^{q}\right.$
$\displaystyle\quad\quad\left.-(\lambda A^{q}+(1-\lambda)B^{q})\log(\lambda
A^{q}+(1-\lambda)B^{q})\right].$

It follows from the convexity of $k(t)=t\log t$ that $K^{\prime}(q)>0$. The
proof is complete. ∎

Choosing $A=r^{2}+\alpha^{2},\ B=1+r^{2}\alpha^{2}$ and
$\lambda=\frac{s^{2p}-1}{r^{2p}-1}$, and using Lemma 6.7, we obtain

######  Theorem 6.8.

Let $p\leq-1$ and $q\geq 1$. $\Theta_{p,q}\\{\alpha,r\\}$ is monotone
decreasing in $q$.

Next, we prove the monotonicity in $\alpha$.

######  Lemma 6.9.

Let $p<-1$, $q\geq 1$ and $0<\alpha<1$. $F_{p,q}(s,\alpha,r)$ is monotone
decreasing in $\alpha$.

###### Proof.

Let $A(\alpha)=r^{2}+\alpha^{2},\ B(\alpha)=1+r^{2}\alpha^{2}$ and
$\lambda=\frac{s^{2p}-1}{r^{2p}-1}$. Since $0<\alpha<1<s<r$, by (4.5), we have
$A>B>1$ and

$\lambda r^{-2}+(1-\lambda)<s^{-2}.$ (6.18)

By (6.9), fixing $1<s<r$, we define

$L(\alpha)=F_{p,q}(s,\alpha,r)=(\lambda
A^{q}+(1-\lambda)B^{q})^{\frac{1}{q}}-s^{2}-\alpha^{2}r^{2}s^{-2}.$

Taking the derivative of $L(\alpha)$ yields

$\displaystyle L^{\prime}(\alpha)=2\alpha r^{2}\left[(\lambda
A^{q}+(1-\lambda)B^{q})^{\frac{1-q}{q}}(\lambda
r^{-2}A^{q-1}+(1-\lambda)B^{q-1})-s^{-2}\right].$

Since $q\geq 1$, $1<s<r$ and $A>B>1$, using Lemma 6.7 and (6.18), we have

$\displaystyle L^{\prime}(\alpha)$ $\displaystyle\leq 2\alpha
r^{2}\left[(\lambda A^{q-1}+(1-\lambda)B^{q-1})^{-1}(\lambda
r^{-2}A^{q-1}+(1-\lambda)B^{q-1})-s^{-2}\right]$ $\displaystyle=2\alpha r^{2}\
\frac{\lambda A^{q-1}(r^{-2}-s^{-2})+(1-\lambda)B^{q-1}(1-s^{-2})}{\lambda
A^{q-1}+(1-\lambda)B^{q-1}}$ $\displaystyle\leq 2\alpha r^{2}B^{q-1}\
\frac{\lambda(r^{-2}-s^{-2})+(1-\lambda)(1-s^{-2})}{\lambda
A^{q-1}+(1-\lambda)B^{q-1}}$ $\displaystyle<0.$

This completes the proof of the lemma. ∎

Therefore, we obtain

######  Theorem 6.10.

Let $p<-1$, $q\geq 1$ and $0<\alpha<1$. $\Theta_{p,q}\\{\alpha,r\\}$ is
monotone increasing in $\alpha$.

Finally, we derive the upper and lower bound for $\Theta_{p,q}$. Denote

$\hat{\Theta}_{p,q}(r):=\Theta_{p,q}\\{0,r\\}=\int_{1}^{r}\frac{ds}{\sqrt{\left(\frac{1-s^{2p}}{1-r^{2p}}r^{2q}+\frac{s^{2p}-r^{2p}}{1-r^{2p}}\right)^{\frac{1}{q}}-s^{2}}}.$

Making the change of variable
$s^{\beta}=x=\frac{r^{\beta}-1}{2}z+\frac{r^{\beta}+1}{2}$ with
$\beta=\frac{4q-2p}{3}$ yields

$\hat{\Theta}_{p,q}(r)=\int_{-1}^{1}\frac{dz}{\sqrt{\hat{J}_{p,q}(r,z)}},$

where

$\hat{J}_{p,q}(r,z)=\frac{4\beta^{2}}{(r^{\beta}-1)^{2}}\left[x^{\frac{2(\beta-1)}{\beta}}\left(\frac{r^{2q}-r^{2p}}{1-r^{2p}}+\frac{1-r^{2q}}{1-r^{2p}}x^{\frac{2p}{\beta}}\right)^{\frac{1}{q}}-x^{2}\right].$

Note that $\hat{\Theta}_{p,q}(r)$ is the period function ${\Theta}(2p,2q,r)$
given by our previous paper [LW22]. We have the following properties.

######  Proposition 6.11 ([LW22]).

Let $p\leq-1$ and $q\geq 1$. Then $\hat{\Theta}_{p,q}(r)$ is continuous on
$(1,\infty)$, and has the following limiting values:

$\displaystyle\lim\limits_{r\to\infty}\hat{\Theta}_{p,q}(r)=\frac{\pi}{2},$
(6.19) $\displaystyle\lim\limits_{r\to
1}\hat{\Theta}_{p,q}(r)=\frac{\pi}{\sqrt{2q-2p}}.$ (6.20)

######  Proposition 6.12 ([LW22]).

Let $p<-1$, $q\geq 1$ and $r>1$. Then $\frac{\partial}{\partial
r}\hat{J}_{p,q}(r,z)<0$ for any $z\in(-1,1)$. Moreover,
$\hat{\Theta}_{p,q}(r)$ is monotone increasing in $r$.

This together with Theorem 6.6, Theorem 6.8 and Theorem 6.10 gives

######  Corollary 6.13.

Let $p<-1$ and $q\geq 1$. Then for any $0<\alpha<1<r$

$\frac{\pi}{2}>\Theta_{p,q}\\{\alpha,r\\}>\hat{\Theta}_{p,q}(r)>\frac{\pi}{\sqrt{2q-2p}}.$
(6.21)

###  6.4. Proofs of Theorem 1.3 (ii) and Theorem 1.4

If $q\geq 1$ and $q-8\leq p<-1$, using Corollary 6.13, we have

$\displaystyle\frac{\pi}{2}>\Theta_{p,q}\\{\alpha,r\\}>\frac{\pi}{4}$

for any $0<\alpha<1<r$. Therefore, if $\gamma>0$ and $E>\bar{E}_{\gamma}^{*}$,
then

$\displaystyle\Theta_{p,q}(\gamma,E)=\Theta_{p,q}\\{\alpha(\gamma,E),r(\gamma,E)\\}\in\left(\frac{\pi}{4},\frac{\pi}{2}\right).$

It follows from Lemma 6.2 (i) that the solution to (6.1) must be constant.
Moreover, the solution to (6.1) is unique. We complete the proof of Theorem
1.3 (ii).

If $p<-1,\ q\geq 1$ and $\gamma>\gamma_{p,q,l}$, then it follows from (6.15)
and (6.16) that

$\lim\limits_{E\to+\infty}\Theta_{p,q}(\gamma,E)=\frac{\pi}{2}$

and

$\lim\limits_{E\to(\bar{E}_{\gamma}^{*})^{+}}\Theta_{p,q}(\gamma,E)<\frac{\pi}{2(l+1)}.$

Therefore, there exist at least $l$ values $\\{E_{j}\\}_{j=1}^{l}$ such that
$E_{j}>\bar{E}_{\gamma}^{*}$ and
$\Theta_{p,q}(\gamma,E_{j})=\frac{\pi}{2(j+1)}$ for $1\leq j\leq l$. It
follows from Lemma 6.2 (ii) that there exist at least $(l+1)$ solutions to
(6.1). We complete the proof of Theorem 1.4.

## References