# Study on the behaviors of rupture solutions for a class of elliptic MEMS
equations in $\mathbb{R}^{2}$

Qing Li, Yanyan Zhang School of Mathematical Sciences, East China Normal
University, Shanghai 200241, P.R. China. Email<EMAIL_ADDRESS>Corresponding author. School of Mathematical Sciences, Key Laboratory of
MEA(Ministry of Education) and Shanghai Key Laboratory of PMMP, East China
Normal University, Shanghai 200241, China. Email<EMAIL_ADDRESS>Y.
Zhang is sponsored by NSFC [No.12271505] and STCSM [No.22DZ2229014].

###### Abstract

This study examines nonnegative solutions to the problem

$\left\\{\begin{array}[]{lll}\Delta
u=\displaystyle\frac{\lambda|x|^{\alpha}}{u^{p}}&\hbox{ in}\,\
\mathbb{R}^{2}\setminus\\{0\\},\\\\[5.69054pt] u(0)=0\ \text{and}\ u>0&\hbox{
in}\,\ \mathbb{R}^{2}\setminus\\{0\\},\\\ \end{array}\right.$

where $\lambda>0,$ $\alpha>-2$, and $p>0$ are constants. The possible
asymptotic behaviors of $u(x)$ at $|x|=0$ and $|x|=\infty$ are classified
according to $(\alpha,p)$. In particular, the results show that for some
$(\alpha,p)$, $u(x)$ exhibits only “isotropic” behavior at $|x|=0$ and
$|x|=\infty$. However, in other cases, $u(x)$ may exhibit the “anisotropic”
behavior at $|x|=0$ or $|x|=\infty$. Furthermore, the relation between the
limit at $|x|=0$ and the limit at $|x|=\infty$ for a global solution is
investigated.

Keywords: MEMS; rupture solution; anisotropic; global solution; convergence

Mathematics Subject Classification (2020): 35B40, 35J75, 74F15,74K35

##  1 Introduction

In this paper, we consider the following nonlinear problem

$\left\\{\begin{array}[]{lll}\Delta
u=\displaystyle\frac{\lambda|x|^{\alpha}}{u^{p}}&\hbox{ in}\,\
\mathbb{R}^{2}\setminus\\{0\\},\\\\[5.69054pt] u(0)=0\ \text{and}\ u>0&\hbox{
in}\,\ \mathbb{R}^{2}\setminus\\{0\\},\\\ \end{array}\right.$ (1.1)

where $\lambda>0$, $\alpha>-2$, and $p>0$ are constants.

Equations of the form in (1.1) with a negative exponent arise in several
applied problems, such as in the study of steady states of thin films of
viscous fluids (see, for example, [24]), and in the modeling of electrostatic
micro-electromechanical systems (MEMS) (refer to [29] for physical derivations
and [27, 9] for mathematical analysis). In particular, $|x|^{\alpha}$
represents the power-law permittivity profile in MEMS modeling. Consequently,
the elliptic equation in (1.1) has been widely investigated over the past few
years (see [28, 23, 21, 20, 19, 8, 5, 7, 9, 13, 22, 16, 32, 12, 4, 18, 14] and
references therein), while most studies have focused on positive solutions.

Of special interest are solutions that give rise to singularities in the
equation, that is, such that $u=0$ in some regions. These solutions are called
rupture solutions because they represent rupture in the device in the physical
model. The parabolic version related to (1.1) was considered in [26, 10, 11,
15, 16, 31]. In such a setting, a solution $u$ that takes the value $0$ at
some point is called a touchdown solution.

In this study, for the elliptic equation in (1.1), we consider its rupture
solutions in the simplest case, where the rupture point is the origin. For
$\alpha=0,$ the rupture solutions for equation (1.1) in dimension $N$ were
investigated in [6, 21, 5, 20, 19, 28, 12]. In particular, the H$\ddot{o}$lder
continuity of rupture solutions and the Hausdorff dimensions of rupture sets
were studied in [5]; the existence of a single point rupture solution in some
$R^{N},N\geq 3$ was obtained in [28]; for (1.1) in $\mathbb{R}^{N}$ with
$N\geq 3,$ infinitely many non-radial rupture solutions were constructed in
[21] for some $p$. For general $\alpha$, the radial single point rupture
solution near the origin for (1.1) in dimension $N$ was recently considered in
[12, 4, 23].

In fact, the behaviors of the solutions near the origin were described in [17]
for problem (1.1) in a special case where $p=2$. The second author of the
present article and her collaborators conducted this study. More precisely, in
[17], the “isotropic” and “anisotropic” ruptures at the origin of the solution
for (1.1) were classified with respect to $\alpha$ for $p=2.$

In the present study, we extend the conclusions of [17] from $p=2$ to general
$p>0$ and expand the domain to the entire $\mathbb{R}^{2}.$ Compared with
[17], first, instead of only one $\alpha$, this study contains two parameters,
$\alpha$ and $p$, which yield richer results. Second, we will further study
global solutions on $\mathbb{R}^{2}\setminus\\{0\\}$, which is the main
concern of this paper and was not investigated in [17]. In addition, our
results show a fundamental difference for the case of $p=3$ compared with the
other cases.

Before describing the outcome of this study precisely, we rewrite equation in
(1.1) as

$u_{rr}+\frac{1}{r}u_{r}+\frac{1}{r^{2}}u_{\theta\theta}=\frac{\lambda
r^{\alpha}}{u^{p}},\quad(r,\theta)\in[0,+\infty)\times S^{1},$ (1.2)

where $u(x)=u(r,\theta)$ and $(r,\theta)$ is the polar coordinate.
Furthermore, we define $v(t,\theta)$ by

$v(t,\theta):=r^{-\frac{\alpha+2}{p+1}}u(r,\theta),\ \ \mbox{where}\,\ t=\ln
r\ \ \mbox{and}\ \ r=|x|.$ (1.3)

Denoting

$\beta:=\frac{\alpha+2}{p+1},$ (1.4)

$v(t,\theta)$ satisfies the following elliptic evolution problem

$-v_{tt}-2\beta
v_{t}=v_{\theta\theta}+\beta^{2}v-\displaystyle\frac{\lambda}{v^{p}},\quad(t,\theta)\in(-\infty,+\infty)\times
S^{1}.$ (1.5)

In association with the stationary problem of (1.5), we also define
$w(\theta)$ as a solution to

$w^{\prime\prime}+\beta^{2}w-\frac{\lambda}{w^{p}}=0\ \ \hbox{on}\,\ S^{1},$
(1.6)

and denote the set of all solutions as

$\mathfrak{S}=\big{\\{}w>0:\,w\ \mbox{is a solution of}\
\eqref{2.6}\big{\\}}.$ (1.7)

In this study, we focus on the rupture solutions $u$ of (1.1) satisfying

$0<C_{1}\leq r^{-\frac{\alpha+2}{p+1}}u(r,\theta)\leq C_{2}<\infty,$ (1.8)

which is equivalent to the solutions $v$ of (1.5) satisfying

$0<C_{1}\leq v\leq C_{2}<\infty,$ (1.9)

where $C_{1}$ and $C_{2}$ are constants independent of $(t,\theta)$ but
dependent on $v.$ Denote

$m_{0}:=\left(\frac{\lambda}{\beta^{2}}\right)^{\frac{1}{p+1}}=\left(\frac{\lambda(p+1)^{2}}{(\alpha+2)^{2}}\right)^{\frac{1}{p+1}}.$
(1.10)

Clearly, the trivial solution $v\equiv m_{0}$ for (1.5) satisfies the
requirement of (1.9). Nevertheless, it should be noted that some solutions for
equation (1.5) may not meet the requirements of equation (1.9). For instance,
following Guo and Wei [21], we can construct a solution
$v=[\frac{\lambda(p+1)^{2}}{2(1-p)}]^{\frac{1}{p+1}}(\sin\theta)^{\frac{2}{p+1}}$
for (1.5) with $\alpha=0,0<p<1$, which does not satisfy (1.9).

We are now in a position to state our first main result concerning the
structure of $\mathfrak{S}$ in terms of $\alpha$ and $p$, which plays a
fundamental role in the classification of solution behaviors.

Figure 1: Set
$\mathscr{M}:=\mathscr{M}_{1}\cup\mathscr{M}_{2}\cup\mathscr{M}_{3}\cup\mathscr{M}_{4}$

######  Theorem 1.1.

Consider the set

$\mathfrak{S}:=\Bigg{\\{}w(\theta)\in
C^{2}(S^{1}):\,w^{\prime\prime}+\bigg{(}\frac{\alpha+2}{p+1}\bigg{)}^{2}w-\frac{\lambda}{w^{p}}=0,\quad
w>0\Bigg{\\}}.$ (1.11)

as defined in (1.7), where $\alpha>-2,~{}p>0,~{}\lambda>0$ are given
constants. The results are as follows:

  1. $(1)$

If

$(\alpha,p)\in\mathscr{M}:=\mathscr{M}_{1}\cup\mathscr{M}_{2}\cup\mathscr{M}_{3}\cup\mathscr{M}_{4},$
(1.12)

then
$\mathfrak{S}=\big{\\{}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}\big{\\}}.$
Here (refer to Figure 1)

$\begin{split}\mathscr{M}_{1}&:=\bigg{\\{}(\alpha,p)|~{}\alpha>-2,\,\
0<p<1\\}\cap\\{(\alpha,p)|~{}\alpha\leq-1\\\ &\ \text{or}\
\exists~{}j\in\mathbb{N},\,\ j\geq 1,\,\ \operatorname{s.t.}\,\ \alpha\leq
j-1\ \text{and}\ p\leq\left(\frac{\alpha+2}{j}\right)^{2}-1\bigg{\\}},\\\
\mathscr{M}_{2}&:=\bigg{\\{}(\alpha,p)|~{}\alpha>-2,\,\ 1\leq
p<3\\}\cap\\{(\alpha,p)\big{|}~{}p\geq 2\alpha+3\\\ &\ \text{or}\
\exists~{}j\in\mathbb{N},\,\ j\geq 1,\,\ \operatorname{s.t.}\,\
\frac{2(\alpha+2)}{j+1}-1\leq
p\leq\left(\frac{\alpha+2}{j}\right)^{2}-1\bigg{\\}},\\\
\mathscr{M}_{3}&:=\bigg{\\{}(\alpha,3)|~{}\alpha>-2,\,\ \text{and}\,\
\alpha\notin 2\mathbb{N}\bigg{\\}},\\\
\mathscr{M}_{4}&:=\bigg{\\{}(\alpha,p)|~{}\alpha>-2,\,\
p>3\\}\cap\\{(\alpha,p)\big{|}~{}p\geq(\alpha+2)^{2}-1\\\ &\ \text{or}\
\exists~{}j\in\mathbb{N},\,\ j\geq 1,\,\ \operatorname{s.t.}\,\
\left(\frac{\alpha+2}{j+1}\right)^{2}-1\leq
p\leq\frac{2(\alpha+2)}{j}-1\bigg{\\}}.\end{split}$

  2. $(2)$

If $(\alpha,p)\in\\{(\alpha,p)|~{}\alpha>-2,\,\ p>0,p\neq
3\\}\backslash{\mathscr{M}}$, then $\mathfrak{S}$ contains exactly
$1+N_{0}(\alpha,p)$ connected components
$\mathfrak{S}_{0}=\big{\\{}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}\big{\\}}$,
$\mathfrak{S}_{1},\cdots,\mathfrak{S}_{i},$
$\cdots,\mathfrak{S}_{N_{0}(\alpha,p)}$. Here

$\mathfrak{S}_{i}=\big{\\{}w_{j_{i}}(\cdot+a);0\leq a<2\pi\big{\\}},\,\
i=1,2,\cdots,N_{0}(\alpha,p),$ (1.13)

$w^{j_{i}}(\theta)$ is the $\frac{2\pi}{{j_{i}}}-$periodic positive solution
of (1.6), $j_{i}=[b_{\alpha,p}]+i,$, $1\leq N_{0}(\alpha,p)<+\infty$ denotes
the number of integers in

$J_{\alpha,p}=\begin{cases}\big{(}\frac{\alpha+2}{\sqrt{p+1}},~{}\alpha+2\big{)}\
&0<p<1,\\\
\big{(}\frac{\alpha+2}{\sqrt{p+1}},~{}\frac{2(\alpha+2)}{p+1}\big{)}\ &1\leq
p<3,\\\ \big{(}\frac{2(\alpha+2)}{p+1},~{}\frac{\alpha+2}{\sqrt{p+1}}\big{)}\
&p>3.\end{cases}$

and $b_{\alpha,p}$ is the left endpoint of $J_{\alpha,p}$.

  3. $(3)$

If $p=3$ and $\alpha\in 2\mathbb{N},$ then $\mathfrak{S}$ contains exactly one
connected component. More precisely, it holds
$\mathfrak{S}=\\{w_{\epsilon,a}(\theta);(\epsilon,a)\in(0,1]\times[0,\pi/\beta)\\}$
with $\beta=(\alpha+2)/4$ and

$w_{\epsilon,a}(\theta)=\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{4}}\left[\epsilon\cos^{2}(\beta(\theta+a))+\epsilon^{-1}\sin^{2}(\beta(\theta+a))\right]^{\frac{1}{2}}.$
(1.14)

We point out that the minimum half-period of $w_{\epsilon,a}(\theta)$ is
$\frac{2}{\alpha+2}\pi$ for all $(\epsilon,a)\in(0,1]\times[0,\pi/\beta)$.
Unlike the case $p\neq 3$, this connected component contains the trivial
solution $w_{1,a}\equiv\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{4}}$.

######  Remark 1.1.

The set $\mathscr{M}$ depends only on $\alpha$ and $p,$ and is not affected by
$\lambda.$

######  Remark 1.2.

Because $\mathscr{M}$ contains an infinite number of parts, only a finite
number is shown in Figure 1.

######  Remark 1.3.

For the particular case $\alpha=0$(which is often discussed in the literature,
e.g., [5, 9, 20, 21]), equations (1.1) and (1.6) can be rewritten as

$\left\\{\begin{array}[]{lll}\Delta
u=\displaystyle\frac{\lambda}{u^{p}}&\hbox{ in}\,\
\mathbb{R}^{2}\setminus\\{0\\},\\\\[5.69054pt] u(0)=0\ \text{and}\ u>0&\hbox{
in}\,\ \mathbb{R}^{2}\setminus\\{0\\},\\\ \end{array}\right.$ (1.15)

and

$w^{\prime\prime}+\left(\frac{2}{p+1}\right)^{2}w-\frac{\lambda}{w^{p}}=0\ \
\hbox{on}\,\ S^{1},$ (1.16)

respectively. According to Theorem 1.1(1), now

${\mathscr{M}}=\\{(0,p)|p\in(0,3)\cup(3,+\infty)\\}.$

Therefore, for all $p\neq 3$, it holds
$\mathfrak{S}=\big{\\{}\big{(}\frac{\lambda(p+1)^{2}}{4}\big{)}^{\frac{1}{p+1}}\big{\\}},$
that is, equation (1.16) has only trivial solution. For $p=3$, according to
Theorem 1.1(3), (1.16) has a family of solutions:

$w_{\epsilon,a}(\theta)=\sqrt{2}\lambda^{\frac{1}{4}}\left[\epsilon\left(\cos{\frac{\theta+a}{2}}\right)^{2}+\epsilon^{-1}\left(\sin{\frac{\theta+a}{2}}\right)^{2}\right]^{\frac{1}{2}},\
(\epsilon,a)\in(0,1]\times[0,2\pi).$

This implies that there exists a family of non-radial rupture solutions for
(1.15) $u(r,\theta)=\sqrt{r}w_{\epsilon,a}(\theta),$ which is consistent with
the results presented in [21].

######  Remark 1.4.

For the particular case $p=2$ (where the equation in (1.1) is the standard
MEMS equation), according to Theorem 1.1, it holds that

${\mathscr{M}}=\displaystyle\bigg{\\{}(\alpha,2)\bigg{|}\alpha\in\big{(}-2,-{1\over
2}\big{]}\cup\displaystyle\bigcup^{7}_{k=2}\bigg{[}(k-1)\sqrt{3}-2,\frac{3k-4}{2}\bigg{]}\bigg{\\}},$

consistent with the result presented in [17, Theorem 1.2].

######  Remark 1.5.

For a better understanding of Theorem 1.1(2), we present the following
examples. If $p=2$ and $\alpha\in(1,2\sqrt{3}-2)$, then
$\big{[}\frac{\sqrt{3}(2+\alpha)}{3}\big{]}=1$, $N_{0}(\alpha)=1,$ $j_{1}=2,$
$\mathfrak{S}_{1}=\big{\\{}w_{2}(\cdot+a):\,a\in S^{1}\big{\\}}$, and
$\mathfrak{S}=\mathfrak{S}_{0}\cup\mathfrak{S}_{1}=\big{\\{}\big{(}\frac{9\lambda}{(2+\alpha)^{2}}\big{)}^{\frac{1}{3}}\big{\\}}\cup\big{\\{}w_{2}(\cdot+a):\,a\in
S^{1}\big{\\}}$; if $p=2$ and $\alpha\in(\frac{5}{2},3\sqrt{3}-2)$, then
$\big{[}\frac{\sqrt{3}(2+\alpha)}{3}\big{]}=2$, $N_{0}(\alpha)=1,$ $j_{1}=3,$
$\mathfrak{S}_{1}=\big{\\{}w_{3}(\cdot+a):\,a\in S^{1}\big{\\}}$, and
$\mathfrak{S}=\mathfrak{S}_{0}\cup\mathfrak{S}_{1}=\big{\\{}\big{(}\frac{9\lambda}{(2+\alpha)^{2}}\big{)}^{\frac{1}{3}}\big{\\}}\cup\big{\\{}w_{3}(\cdot+a):\,a\in
S^{1}\big{\\}}$.

######  Remark 1.6.

Equation (1.6) also arises from the study of the generalized curve-shortening
problem. In particular, for $p=3,$ equation (1.6) with $\beta=1$ is called the
affine curve-shortening problem (see [1, 25]).

Next, we investigate the asymptotic behavior of the solution $u$ to (1.1) at
the origin and infinity.

######  Theorem 1.2.

If $u(x)=u(r,\theta)$ is a solution of (1.1) and satisfies condition (1.8),
then there exist $w_{0},w_{\infty}\in\mathfrak{S}$ such that

$\displaystyle\|r^{-\frac{\alpha+2}{p+1}}u(r,\theta)-w_{0}(\theta)\|_{C^{2}(S^{1})}$
$\displaystyle\leq C_{1}(1-lnr)^{-\frac{s_{1}}{7(1-2s_{1})}}\quad\text{as}\
r\rightarrow 0^{+},$ (1.17)
$\displaystyle\|r^{-\frac{\alpha+2}{p+1}}u(r,\theta)-w_{\infty}(\theta)\|_{C^{2}(S^{1})}$
$\displaystyle\leq C_{2}(1+lnr)^{-\frac{s_{2}}{7(1-2s_{2})}}\quad\text{as}\
r\rightarrow+\infty,$ (1.18)

for some $s_{1},s_{2}\in(0,\frac{1}{2})$ depending on $w_{0},w_{\infty}$,
respectively. Here $C_{1},C_{2}$ are constants.

By combining Theorem 1.1 and Theorem 1.2, the following conclusion is
immediately obtained.

######  Corollary 1.3.

Suppose $u(x)=u(r,\theta)$ is a solution to (1.1) and satisfies condition
(1.8).

  1. $(1)$

If $(\alpha,p)\in{\mathscr{M}},$ then

$\lim_{r\to
0^{+}}r^{-\frac{2+\alpha}{p+1}}u(r,\theta)=\lim_{r\to+\infty}r^{-\frac{2+\alpha}{p+1}}u(r,\theta)=\left(\frac{\lambda(p+1)^{2}}{(\alpha+2)^{2}}\right)^{\frac{1}{p+1}}\quad\text{in}\
\ C^{2}(S^{1}).$ (1.19)

  2. $(2)$

If $(\alpha,p)\in\\{(\alpha,p)|~{}\alpha>-2,\,\ p>0,p\neq
3\\}\backslash{\mathscr{M}}$, then either (1.19) holds or there exist
$w_{j_{i}}(\theta+a_{0})$ and $w_{j_{k}}(\theta+a_{\infty}),$$1\leq i,k\leq
N_{0}(\alpha,p),$ $a_{0},a_{\infty}\in[0,2\pi)$ such that

$\lim_{r\to
0^{+}}r^{-\frac{2+\alpha}{p+1}}u(r,\theta)=w_{j_{i}}(\theta+a_{0})\
\text{and}\
\lim_{r\to+\infty}r^{-\frac{2+\alpha}{p+1}}u(r,\theta)=w_{j_{k}}(\theta+a_{\infty})$
(1.20)

in $C^{2}(S^{1}).$

  3. $(3)$

If $p=3$ and $\alpha\in 2\mathbb{N},$ then there exist
$(\epsilon_{0},a_{0}),(\epsilon_{\infty},a_{\infty})\in(0,1]\times[0,2\pi)$
such that

$\lim_{r\to
0^{+}}r^{-\frac{2+\alpha}{p+1}}u(r,\theta)=w_{\epsilon_{0},a_{0}}(\theta)\
\text{and}\ \lim_{r\to
0^{+}}r^{-\frac{2+\alpha}{p+1}}u(r,\theta)=w_{\epsilon_{\infty},a_{\infty}}(\theta)$
(1.21)

in $C^{2}(S^{1}).$

Based on previous preparations, we focus on the global rupture solutions, that
is, functions $u\in C^{2}(\mathbb{R}^{2}\setminus\\{0\\})$ satisfying equation
(1.1) and condition (1.8). This is the main concern of this study. According
to Theorem 1.2, the limit functions

$\begin{split}w_{0}(\theta)&=\lim\limits_{r\to
0}r^{-\frac{\alpha+2}{p+1}}u(r,\theta)=\lim\limits_{t\to-\infty}v(t,\theta),\\\
w_{\infty}(\theta)&=\lim\limits_{r\to+\infty}r^{-\frac{\alpha+2}{p+1}}u(r,\theta)=\lim\limits_{t\to+\infty}v(t,\theta),\end{split}$
(1.22)

both belong to $\mathfrak{S},$ which is defined in (1.7). Here,
$v=r^{-\frac{\alpha+2}{p+1}}u$ is a solution of (1.5). Our main concern is the
relation between $w_{0}(\theta)$ and $w_{\infty}(\theta)$. In particular, we
want to know that if there exist global solutions satisfying (1.22), then what
necessary conditions the pair of elements $w_{0}(\theta),w_{\infty}(\theta)$
must satisfy. In other words, we want to investigate for which pairs of
elements $w_{0}(\theta),w_{\infty}(\theta)$ there may exist global solutions
satisfying (1.22), and for which pairs there is no global solution satisfying
(1.22).

Inspired by [3], a key observation is derived from an estimate of energy: the
energy at the origin is always less than or equal to that at infinity. More
precisely, the energy functional $E$ is defined as

$E(\psi)=\int_{S^{1}}~{}\frac{1}{2}(\psi^{\prime})^{2}-\frac{1}{2}\beta^{2}\psi^{2}-h(\psi)~{}d\theta,$
(1.23)

where $h$ is defined by (2.1). If $w_{0}(\theta),w_{\infty}(\theta)$ are
defined as in (1.22), then we have (Proposition 4.1)

$E(w_{0})\leq E(w_{\infty}).$ (1.24)

Here, the equality sign holds if and only if
$w_{0}(\theta)=w_{\infty}(\theta)$ and $u$ is of the form
$u(r,\theta)=r^{\frac{\alpha+2}{p+1}}w(\theta),$ $w(\theta)\in\mathfrak{S}$.
By Theorem 1.1, this implies that if $(\alpha,p)$ satisfies (1.12), then
$u(r,\theta)=\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}r^{\frac{\alpha+2}{p+1}}$
is the unique global solution to (1.1) that satisfies (1.8) (Corollary 4.3).

For the special case $p=3$, it holds that $E(w)\equiv C$ on $\mathfrak{S},$
where $C$ is a constant independent of $w$ (Proposition 4.4). Consequently,
$u$ has the form $u(r,\theta)=r^{\frac{\alpha+2}{p+1}}w(\theta),$
$w(\theta)\in\mathfrak{S}$ (Corollary 4.5). To be more precise, there holds
$u(r,\theta)=r^{\frac{\alpha+2}{p+1}}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}$
for $\alpha\notin 2\mathbb{N}$ and
$u(r,\theta)=r^{\frac{\alpha+2}{p+1}}w_{\epsilon,a}(\theta)$ for $\alpha\in
2\mathbb{N}$, where $w_{\epsilon,a}(\theta)$ is as in (1.14).

For the rest of the cases, it is noted that the energy $E$ is constant on each
connected component of $\mathfrak{S}.$ Furthermore, the analytical results in
Section 4.2 and the numerical monotonicity results in Section 4.4 for $E$
indicate

$\begin{array}[]{lll}E(\mathfrak{S}_{0})<E(\mathfrak{S}_{1})<E(\mathfrak{S}_{2})<\cdots<E(\mathfrak{S}_{j})<\cdots<E(\mathfrak{S}_{N_{0}}),\quad\text{for}\
0<p<3,\\\
E(\mathfrak{S}_{1})<E(\mathfrak{S}_{2})<\cdots<E(\mathfrak{S}_{j})<\cdots<E(\mathfrak{S}_{N_{0}})<E(\mathfrak{S}_{0}),\quad\text{for}\
p>3,\end{array}$ (1.25)

where $\mathfrak{S}_{k},k=1,2,\ldots,N_{0}$ are defined in Theorem 1.1. From
(1.24), inequality (1.25) can be interpreted as follows: the asymptotic
profile of a global solution at the origin has a lower (or equal) frequency
than that at infinity.

In conclusion, we have the following two main theorems:

######  Theorem 1.4.

Let $\mathscr{M},w_{\epsilon,a}(\theta)$ be defined as in Theorem 1.1.

  1. $(1)$

If $(\alpha,p)\in\mathscr{M}$, then
$u(r,\theta)=\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}r^{\frac{\alpha+2}{p+1}}$
is the unique global solution to (1.1) that satisfies (1.8).

  2. $(2)$

If $p=3$ and $\alpha\in 2\mathbb{N}$, then $u(r,\theta)$ is a global solution
to (1.1) that satisfies (1.8) if and only if
$u(r,\theta)=r^{\frac{\alpha+2}{p+1}}w_{\epsilon,a}(\theta)$ for some
$(\epsilon,a)\in(0,1]\times[0,\pi/\beta).$

######  Remark 1.7.

The conclusion of Theorem 1.4 is stronger than Corollary 1.3(1)(3).

Next, for the remaining cases, we provide a necessary condition that
$(w_{0}(\theta),w_{\infty}(\theta)$ must satisfy in order to have a global
solution that satisfies $\eqref{120}.$

######  Theorem 1.5.

Let $(\alpha,p)\notin\mathscr{M},$ $p>0,p\neq 3,$
$\mathfrak{S}_{k},k=1,2,\ldots,N_{0}$ be defined as in Theorem 1.1. Assume
that $u(x)=u(r,\theta)$ is a global rupture solution of (1.1), satisfying
(1.8) and (1.22) with
$w_{0}(\theta)\in\mathfrak{S}_{m}$,$w_{\infty}(\theta)\in\mathfrak{S}_{n}$. If
(1.25) holds, then

  1. $(1)$

for $1<p<3$, it holds $0\leq m\leq n$;

  2. $(2)$

for $p>3$, either $0<m\leq n$, or $n=0,$ $0\leq m\leq N_{0}$.

######  Remark 1.8.

Each of the above theorems and its corollary show that $p=3$ is fundamentally
different from the other cases.

Before concluding this section, we want to stress some new features of our
study.

(1) For solutions that satisfy (1.1) and (1.8), Figure 1 completes the
classification of all possible rupture behaviors concerning $(\alpha,p)$ at
$|x|=0$ and $+\infty$. Specifically, it only admits the “isotropic” rupture
behavior in the sense of (1.19) for $(\alpha,p)$ in the colored region.
Otherwise, it also allows for the “anisotropic” rupture behavior in the sense
of (1.20) or (1.21).

(2) Our research reveals a remarkable discovery: when $p=3$, it exhibits
distinctive characteristics compared with the other cases. First, it should be
noted from Theorem 1.1 that when $p=3$, $\mathfrak{S}$ remains connected for
all $\alpha>-2$, unlike when $0<p\neq 3$, $\mathfrak{S}$ is not connected for
most $\alpha>-2$. Second, also by Theorem 1.1, for $p=3$, all nontrivial
functions in $\mathfrak{S}$ have the same frequency for all $\alpha>-2$, which
is not true for $0<p\neq 3$ . Finally, but most importantly, according to
Theorem 1.4, the global solutions for $p=3$ behave in a simpler manner than
the other cases.

(3) It is worth noting that elliptic integrals were employed in our study. In
fact, we were unable to determine the asymptotic behavior of $E$ for
$p\in(1,3)$ until we established a connection with elliptic integrals.

The remainder of this paper is organized as follows. Section 2 analyzes the
structure of set $\mathfrak{S}$ and presents the proof of Theorem 1.1. Then,
in Section 3, we examine the asymptotic behaviors of the solution $u$ near the
origin and infinity, and provide a proof for Theorem 1.2. The final section,
Section 4, focuses on studying the global solutions and presenting proof for
Theorems 1.4 and 1.5.

##  2 The structure of set $\mathfrak{S}$

In this section, we examine the set $\mathfrak{S}$ defined in (1.11) and apply
the phase-plane method to prove Theorem 1.1. For convenience, we define:

$h(x)=\left\\{\begin{array}[]{lll}\frac{\lambda}{(p-1)x^{p-1}},&\text{for}\
p\neq 1,\\\ -\lambda\ln x,&\text{for}\ p=1.\end{array}\right.$ (2.1)

Multiplying (1.6) by $w^{\prime}$ and then integrating over $S^{1}$ yields

$(w^{\prime})^{2}+{\beta}^{2}w^{2}+2h(w)=E,$ (2.2)

where $E$ is a constant. Define

$g(w)={\beta}^{2}w^{2}+2h(w),\quad w>0.$ (2.3)

Then (2.2) can be rewritten as

$(w^{\prime})^{2}=E-g(w).$

Note that $w_{0}=\big{(}\frac{\lambda}{{\beta}^{2}}\big{)}^{1/(p+1)}$ is the
unique critical point of $g$ and

$g(w_{0}):=E_{0}=\left\\{\begin{array}[]{lll}\frac{p+1}{p-1}\lambda^{\frac{2}{p+1}}{\beta}^{\frac{2(p-1)}{p+1}},&\text{for}\
p\neq 1\\\ \lambda(1-\ln{\lambda\over{\beta^{2}}}),&\text{for}\
p=1\end{array}\right..$

Furthermore, $g(w)$ decreases monotonically on $(0,w_{0})$ and increases
monotonically on $(w_{0},+\infty)$. Consequently, for $p>0$, (1.6) has a
nontrivial solution if and only if $E>g(w_{0})$. In addition, any nontrivial
solution of problem (1.6) has the following two properties: (i) it is
periodic; (ii) if $w(\theta)$ is a solution of (1.6), then $w(\theta+a)$ is
also a solution of (1.6) for any $a\in\mathbb{R}.$

Suppose that $w(\theta)$ is a nontrivial positive solution to (1.6). Denote
$w_{1}$ (resp. $w_{2}$) the minimum (resp. maximum) value of $w(\theta).$
Then, $w_{1}$ and $w_{2}$ are the two roots of

$g(w)=E\quad\text{for}\,\ E>g(w_{0}).$

Therefore, by setting $\tau=\frac{w_{2}}{w_{1}}$, we conclude from the above
that

$\left\\{\begin{array}[]{lll}w_{1}^{p+1}=\frac{2\lambda(1-{\tau}^{1-p})}{{\beta}^{2}(p-1){(\tau^{2}-1)}},&\text{for}\
p\neq 1,\\\
w_{1}^{2}=\frac{2\lambda\ln\tau}{\beta^{2}(\tau^{2}-1)},&\text{for}\
p=1.\end{array}\right.$ (2.4)

We point out that by ODE theory, for any given $\tau\in[1,+\infty),$ there
exists a unique (up to shift) solution $w(\theta)$ to (1.6) such that
$w_{2}/w_{1}=\tau.$

Without loss of generality, we can assume that $\theta=0$ is a minimum point
of $w(\theta)$ and $\theta=L>0$ is a maximum point of $w(\theta)$, such that
$w^{\prime}(\theta)>0$ for any $\theta\in(0,L).$ Therefore,
$w^{\prime}(0)=w^{\prime}(L)=0,$ where $L>0$ is the minimum half-period of
$w.$ Note also from (2.2) that

$d\theta=\frac{dw}{\sqrt{E-g(w)}},$ (2.5)

which implies that

$L=\int_{w_{1}}^{w_{2}}\frac{dw}{\sqrt{E-g(w)}}.$ (2.6)

By setting $y=\frac{w}{w_{1}}$, (2.6) can be rewritten as

$L(\tau)=\int_{1}^{\tau}\frac{dy}{\sqrt{\frac{E}{w_{1}^{2}}-{g(w_{1}y)\over{w_{1}^{2}}}}}=\left\\{\begin{array}[]{lll}\displaystyle\frac{1}{\beta}\int_{1}^{\tau}\frac{dy}{\sqrt{1+\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}-y^{2}-\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}{1\over{y^{p-1}}}}},&p\neq
1,\\\
\displaystyle\frac{1}{\beta}\int_{1}^{\tau}\frac{dy}{\sqrt{1-y^{2}+(\tau^{2}-1)\frac{\ln
y}{\ln\tau}}},&p=1,\end{array}\right.$ (2.7)

by (2.1), (2.3) and (2.4). Next, we analyze the range of $L(\tau)$.

######  Lemma 2.1.

$L(\tau)$ is continuous on $[1,+\infty)$ and satisfies

$\begin{split}\lim_{\tau\rightarrow
1}L(\tau)&=\frac{\pi}{\sqrt{p+1}\beta}=\frac{\sqrt{p+1}}{\alpha+2}\pi,\\\
\lim_{\tau\rightarrow+\infty}L(\tau)&=\frac{\pi}{\min\\{p+1,2\\}\beta}=\frac{p+1}{\min\\{p+1,2\\}(\alpha+2)}\pi.\\\
\end{split}$ (2.8)

Moreover, $L(\tau)$ decreases strictly for $p\in(0,3)$, increases strictly for
$p\in(3,+\infty)$, and $L(\tau)\equiv\frac{2\pi}{\alpha+2}$ for $p=3$.

Proof: We denote $Q(w)=-\beta^{2}w+\frac{\lambda}{w^{p}}$, and let
$w_{0}=\left(\frac{\lambda}{{\beta}^{2}}\right)^{\frac{1}{p+1}}$ be the unique
root of $Q(w).$ From [30, Lemma 3.2], we have

$L(E)\xlongrightarrow{E\rightarrow
E_{0}}\frac{\pi}{\sqrt{-Q^{\prime}(w_{0})}}=\frac{\pi}{\sqrt{p+1}\beta},\
\text{for}\ p\in(0,+\infty).$ (2.9)

Thus, the first equation of (2.8) follows directly from (2.9) because
$\tau\rightarrow 1$ is equivalent to $E\rightarrow E_{0}$. By setting
$\xi=\frac{y-1}{\tau-1}$, we rewrite (2.7) as

$L(\tau)=\frac{1}{\beta}\int_{0}^{1}\frac{\tau-1}{\sqrt{1+\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}{1\over{(\xi(\tau-1)+1)^{p-1}}}}}d\xi$
(2.10)

for $p\neq 1,$ and

$L(\tau)=\frac{1}{\beta}\int_{0}^{1}\frac{\tau-1}{\sqrt{1-(\xi(\tau-1)+1)^{2}+(\tau^{2}-1)\frac{\ln{(\xi(\tau-1)+1)}}{\ln\tau}}}d\xi$

for $p=1.$ We then obtain

$\lim_{\tau\rightarrow+\infty}L(\tau)=\left\\{\begin{array}[]{lll}\frac{1}{\beta}\int_{0}^{1}\frac{d\xi}{\sqrt{1-\xi^{2}}}=\frac{\pi}{2\beta},&\text{for}\
p\in[1,+\infty),\\\
\frac{1}{\beta}\int_{0}^{1}\frac{d\xi}{\sqrt{\xi^{1-p}-\xi^{2}}}=\frac{\pi}{(p+1)\beta},&\text{for}\
p\in(0,1].\\\ \end{array}\right.$ (2.11)

This indicates the validity of the second equation in (2.8). For $p=3,$ it can
be calculated that

$\begin{split}L(\tau)=&\frac{1}{\beta}\int_{0}^{1}~{}\frac{\tau-1}{\sqrt{1+\tau^{2}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{2}}{(\xi(\tau-1)+1)^{2}}}}~{}d\xi\\\
\xlongequal{x=(\xi(\tau-1)+1)^{2}}&\frac{1}{\beta}\int_{1}^{\tau^{2}}\frac{1}{2}\frac{dx}{\sqrt{(\frac{\tau^{2}-1}{2})^{2}-(x-\frac{\tau^{2}+1}{2})^{2}}}\\\
\equiv&\frac{\pi}{2\beta}.\end{split}$ (2.12)

Finally, for $p\neq 3$ the monotonicity of $L$ in $\tau$ can be easily deduced
from [1, Corollary 5.6]. The proof is complete.

$\Box$

Based on Lemma 2.1, we present the proof of Theorem 1.1 as follows.

Proof of Theorem 1.1. We provide only proofs for two specific cases:
$p\in(0,1)$ and $p=3.$ This is because the remaining cases, $1\leq p<3$ and
$p>3$ can be proven in the same manner as in the case $p\in(0,1).$

  * •

Case $p\in(0,1)$
According to Lemma 2.1, the range of the minimum half-period $L$ is
$I:=\left(\frac{\pi}{\alpha+2},\frac{\sqrt{p+1}}{\alpha+2}\pi\right).$ Thus,
(1.6) has no nontrivial solution if and only if the interval $I$ does not
contain $\frac{\pi}{j}$ for any integer $j\geq 1,$ which implies that either
$\alpha\leq-1$ or

$\alpha\leq j-1\quad\text{and}\quad
p\leq\left(\frac{\alpha+2}{j}\right)^{2}-1\quad\text{for some }\ j\geq 1.$
(2.13)

In other words, for $p\in(0,1),$ (1.6) has no nontrivial solution if and only
if $(\alpha,p)\in\mathscr{M}_{1}$. On the other hand, for
$(\alpha,p)\in(-2,+\infty)\times(0,1)\setminus\mathscr{M}_{1}$, in addition to
$\mathfrak{S}_{0}=\left\\{\left(\frac{\lambda(p+1)^{2}}{(\alpha+2)^{2}}\right)^{\frac{1}{p+1}}\right\\}\subset\mathfrak{S}$,
$\mathfrak{S}$ also contains nontrivial solutions since the interval $I$
contains $\frac{\pi}{j}$ for some integer $j\geq 1$. In other words, there
exists $j\geq 1$ such that
$\frac{\pi}{\alpha+2}<\frac{\pi}{j}<\frac{\sqrt{p+1}}{\alpha+2}\pi,$ i.e.,
$\frac{\alpha+2}{\sqrt{p+1}}<j<\alpha+2.$ If we denote the number of integers
in $(\frac{\alpha+2}{\sqrt{p+1}},\alpha+2)$ by $N_{0}(\alpha,p),$ then $I$
contains
$\\{\frac{\pi}{j_{i}}|j_{i}=[\frac{\alpha+2}{\sqrt{p+1}}]+i,i=1,2,\cdots,N_{0}(\alpha,p)\\}.$
This implies that (1.6) has $N_{0}(\alpha,p)$ periodic solutions
$w_{j_{1}}(\theta),w_{j_{2}}(\theta),\cdots,w_{j_{N_{0}(\alpha,p)}}(\theta),$
where the period of $w_{j_{i}}(\theta)$ is $\frac{2\pi}{j_{i}}$ and
$\min\limits_{\theta}w_{j_{i}}(\theta)=w_{j_{i}}(0).$ In conclusion, if
$(\alpha,p)\in(-2,+\infty)\times(0,1)\setminus\mathscr{M}_{1}$, then
$\mathfrak{S}$ contains precisely $1+N_{0}(\alpha,p)$ connected components
$\mathfrak{S}_{0}=\left\\{\left(\frac{\lambda}{\beta^{2}}\right)^{\frac{1}{p+1}}\right\\}$,
$\mathfrak{S}_{1},\cdots,\mathfrak{S}_{i},$
$\cdots,\mathfrak{S}_{N_{0}(\alpha,p)}$, where $\mathfrak{S}_{i}$ is defined
by (1.13).

  * •

Case $p=3$

In this case, $L(\tau)\equiv\frac{2}{\alpha+2}\pi.$ Hence, if
${{\alpha+2}\over 2}\notin\mathbb{N}^{+},$ i.e., $\alpha\neq 0,2,4,6,\cdots,$
then
$\mathfrak{S}=\left\\{\frac{2\lambda^{\frac{1}{4}}}{\sqrt{\alpha+2}}\right\\}.$
Otherwise, similar to [25], one can show that all solutions of (1.6) are given
by a 2-parameter family of functions (1.14).

The proof is complete. $\square$

##  3 The asymptotic behaviors as $|x|\rightarrow 0$ and
$|x|\rightarrow\infty$

This section focuses on the asymptotic behavior of the solution $u(r,\theta)$
as $r\rightarrow 0$ and $r\rightarrow\infty$. The goal is to prove Theorem
1.2. Note (1.3), it is reduced to study the behaviors of $v(t,\theta)$ as
$t\rightarrow-\infty$ and $t\rightarrow+\infty$.

Because inequality (1.17) in Theorem 1.2 can be obtained in a similar manner
as in [17], we only provide the proof of (1.18). In other words, we study only
the behavior of $v(t,\theta)$ as $t\rightarrow+\infty$. We begin with the
following proposition:

######  Proposition 3.1.

Let $v$ be a solution of the evolution equation

$-v_{tt}-2\beta
v_{t}=v_{\theta\theta}+\beta^{2}v-\displaystyle\frac{\lambda}{v^{p}},\quad(t,\theta)\in\\{(t_{0},+\infty)\times
S^{1}\\}$ (3.1)

and satisfy (1.9), where $\beta,\lambda,p$ are positive constants. Then there
exists $w_{\infty}\in\mathfrak{S}$ such that

$\begin{split}\|v(t,\cdot)-w_{\infty}(\cdot)\|_{C^{2}(S^{1})}\leq&C(1+t)^{-\frac{s_{2}}{7(1-2s_{2})}},\quad\text{as}\
t\rightarrow+\infty,\end{split}$ (3.2)

where $s_{2}\in(0,\frac{1}{2})$ is a constant depending on $w_{\infty}$. Here
$C$ is a positive constant.

Proof:  In contrast to [17, Proposition 3.5], we need to modify the definition
of functional $\mathscr{H}(v)$. For any $\varepsilon>0$ and $t\geq t_{0}$, we
define

$\mathscr{H}(v)={1\over
2}\int_{S^{1}}|v_{t}|^{2}d\theta-(1+2\beta\varepsilon)E(v)+\varepsilon(v_{\theta\theta}+j(v),v_{t})$
(3.3)

with

$E(v)=\int_{S^{1}}\frac{1}{2}(v_{\theta}^{2}-g(v))~{}d\theta,\,\
g(v)=\beta^{2}v^{2}+2h(v),$ $j(v)=\beta^{2}v-{\lambda\over v^{p}},$ (3.4)
$(v_{\theta\theta}+j(v),v_{t})=\int_{S^{1}}(v_{\theta\theta}+j(v))v_{t}~{}d\theta.$

Then, Proposition 3.1 can be proved in a similar way as in [17, Proposition
3.5] by applying dynamical system theory and the Lojasiewicz-Simon method.
$\square$ Proof of Theorem 1.2. Inequality (1.18) follows directly from
Proposition 3.1 by (1.3), and inequality (1.17) can be obtained in a similar
manner as in [17]. $\square$

##  4 Global Solutions

In this section, we study the global rupture solutions, that is, the functions
$u\in C^{2}(\mathbb{R}^{2}\setminus\\{0\\})$ that satisfy (1.1) and (1.8). Our
goal is to prove Theorem 1.4 and 1.5.

Recall the energy $E$ defined by (1.23) and
$(w_{0}(\theta),w_{\infty}(\theta))$ defined by (1.22). First, we prove that
the energy $E$ at the origin is always less than or equal to that at infinity.
More precisely, we have

######  Proposition 4.1.

It holds that $E(w_{0})\leq E(w_{\infty})$. Furthermore, if
$E(w_{0})=E(w_{\infty})$, then $v(t,\theta)\equiv
w_{0}(\theta)=w_{\infty}(\theta)$ for all $t\in\mathbb{R}$, that is,
$u(r,\theta)=r^{\frac{\alpha+2}{p+1}}w_{0}(\theta)$ for all
$(r,\theta)\in(0,+\infty)\times S^{1}$.

Proof: Multiplying (1.5) by $v_{t}$ and integrating over $[-T,T]\times S^{1}$
yields

$\int_{-T}^{T}\int_{S^{1}}~{}\left(-v_{tt}v_{t}-2\beta{v_{t}}^{2}\right)~{}d\theta
dt=\int_{-T}^{T}\int_{S^{1}}~{}\left(v_{\theta\theta}v_{t}+\beta^{2}v_{t}v-\frac{\lambda}{v^{p}}v_{t}\right)~{}d\theta
dt.$ (4.1)

Note that similar to [17], $v_{t}(t,\cdot)$ tends to $0$ in $C^{0}(S^{1})$ as
$t\to\pm\infty$. Let $T\to+\infty$, we obtain

$\begin{split}0\leq&\int_{-\infty}^{+\infty}\int_{S^{1}}2\beta{v_{t}}^{2}~{}d\theta
dt\\\
=&\int_{S^{1}}\frac{1}{2}\big{(}(w^{\prime}_{\infty})^{2}-(w^{\prime}_{0})^{2}\big{)}-\frac{\beta^{2}}{2}\big{(}w_{\infty}^{2}-w_{0}^{2}\big{)}-\big{(}h(w_{\infty})-h(w_{0})\big{)}~{}d\theta\\\
=&E(w_{\infty})-E(w_{0})\end{split}$

and $E(w_{0})\leq E(w_{\infty})$ follows. Moreover, if
$E(w_{0})=E(w_{\infty})$, then $v_{t}(t,\theta)\equiv 0,$ which implies that
$v$ is independent of $t$ and $v(t,\theta)\equiv
w_{0}(\theta)=w_{\infty}(\theta).$ The proof is complete. $\Box$

######  Corollary 4.2.

If $\mathfrak{S}$ contains only the trivial solution
$w(\theta)\equiv\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}},$ then
$v(t,\theta)\equiv\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}$ is
the unique solution of (1.5) satisfying (1.9), which means
$u(r,\theta)=\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}r^{\frac{\alpha+2}{p+1}}$
is the unique global solution to (1.1) satisfying (1.8).

######  Corollary 4.3.

By Corollary 4.2 and Theorem 1.1, if $(\alpha,p)$ satisfies (1.12), then
$u(r,\theta)=\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}r^{\frac{\alpha+2}{p+1}}$
is the unique global solution to (1.1) satisfying (1.8).

Next, assume that there exists a function $u(x)=u(r,\theta)$ which is a global
solution to (1.1), satisfying (1.8) and (1.22) with
$w_{0}(\theta),w_{\infty}(\theta)\in\mathfrak{S}.$ Our goal is to obtain the
necessary conditions that the pair of elements
$w_{0}(\theta),w_{\infty}(\theta)$ must satisfy. To do this, note that
$E(w_{0})\leq E(w_{\infty})$, we will study how $E(w)$ depends on
$w\in\mathfrak{S}.$ Note that the value of $E(w)$ does not change with the
shift of $w.$ Recall that $w_{1}$ (or $w_{2}$ ) denotes the minimum (or
maximum) value of $w(\theta)$, $\tau=\frac{w_{2}}{w_{1}}$ and for any given
$\tau\in[1,+\infty)$, there exists a unique (up to shift) solution $w(\theta)$
to (1.6) such that $w_{2}/w_{1}=\tau.$ We aim to prove the monotonicity of
$E(w)$ with respect to $\tau$. In fact, by Lemma 2.1, if $p\neq 3$,
monotonicity with respect to $\tau$ means monotonicity with respect to $L$
(the minimum half-period of $w$); if $p=3$, monotonicity with respect to
$\tau$ means monotonicity with respect to $\epsilon$ in Theorem 1.1.

Before continuing, let us give another form of $E(w)$ for $w\in\mathfrak{S}$.
Introduce

$F(w):=\frac{1}{2\pi}\bigg{(}\frac{\lambda}{\beta^{2}}\bigg{)}^{\frac{p-1}{p+1}}\int_{S^{1}}~{}\frac{1}{w^{p-1}}~{}d\theta,\
p\neq 1,\quad\text{and}\quad F_{1}(w):=\frac{1}{2\pi}\int_{S^{1}}~{}\ln
w~{}d\theta.$ (4.2)

If we multiply (1.6) by $w$ and integrate on $S^{1},$ then $E(w)$ on
$\mathfrak{S}$ can be rewritten as

$E(w)=\left\\{\begin{array}[]{lll}\int_{S^{1}}-\left(\frac{1}{2}+\frac{1}{p-1}\right)\frac{\lambda}{w^{p-1}}d\theta=\pi\lambda\frac{1+p}{1-p}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1-p}{p+1}}F(w),&\text{for}\
p\neq 1,\\\ \int_{S^{1}}(-\frac{\lambda}{2}+\lambda\ln
w)d\theta=-\pi\lambda+2\pi\lambda\ln\sqrt{\frac{\lambda}{\beta^{2}}}+2\pi\lambda
F_{1}(w),&\text{for}\ p=1.\end{array}\right.$ (4.3)

The monotonicity of $E(w)$ with respect to $\tau$ is reduced to obtain the
monotonicity of $F(w)$ and $F_{1}(w)$ with respect to $\tau$. For convenience,
we sometimes denote $E(w)$ as $E(\tau)$, as discussed previously. The same
applies to $F(\tau)$ and $F_{1}(\tau).$

If we rewrite

$\begin{split}F(w)=&\bigg{(}\frac{2\lambda}{\beta^{2}}\bigg{)}^{\frac{p-1}{p+1}}\frac{1}{L(\tau)}\int_{w_{1}}^{w_{2}}~{}\frac{dw}{w^{p-1}\sqrt{E-g(w)}},\\\
F_{1}(w)=&\frac{1}{L(\tau)}\int_{w_{1}}^{w_{2}}~{}\frac{\ln
wdw}{\sqrt{E-g(w)}},\end{split}$ (4.4)

by (2.5), it should be noted that both $F$ and $F_{1}$ depend only on $\tau$
and $p,$ not on $\beta$ and $\lambda.$ In fact, as in (2.7) and (2.10), by
setting $y=\frac{w}{w_{1}}$ and $\xi=\frac{y-1}{\tau-1}$, $F(w)$ and
$F_{1}(w)$ can be rewritten as

$F(\tau)=\frac{\displaystyle{\int_{0}^{1}~{}\left(\frac{2(1-\tau^{1-p})}{(p-1)(\tau^{2}-1)}\right)^{\frac{1-p}{p+1}}\frac{(\xi(\tau-1)+1)^{1-p}(\tau-1)}{\sqrt{1+\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}{1\over{(\xi(\tau-1)+1)^{p-1}}}}}~{}d\xi}}{\displaystyle{\int_{0}^{1}\frac{\tau-1}{\sqrt{1+\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}{1\over{(\xi(\tau-1)+1)^{p-1}}}}}~{}d\xi}}$
(4.5)

and

$F_{1}(\tau)=\frac{\displaystyle{\int_{0}^{1}~{}\ln\left[\left(\frac{2\ln\tau}{\tau^{2}-1}\right)^{\frac{1}{2}}(\xi(\tau-1)+1)\right]\frac{\tau-1}{\sqrt{1-(\xi(\tau-1)+1)^{2}+(\tau^{2}-1)\frac{\ln{(\xi(\tau-1)+1)}}{\ln\tau}}}~{}d\xi}}{\displaystyle{\int_{0}^{1}~{}\frac{\tau-1}{\sqrt{1-(\xi(\tau-1)+1)^{2}+(\tau^{2}-1)\frac{\ln{(\xi(\tau-1)+1)}}{\ln\tau}}}~{}d\xi}}.$
(4.6)

In the remainder of this section, for convenience, we denote the numerator in
formula (4.5) as

$\begin{split}H(\tau):=&\int_{0}^{1}~{}\left(\frac{2(1-\tau^{1-p})}{(p-1)(\tau^{2}-1)}\right)^{\frac{1-p}{p+1}}\frac{(\xi(\tau-1)+1)^{1-p}(\tau-1)}{\sqrt{1+\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}{1\over{(\xi(\tau-1)+1)^{p-1}}}}}~{}\mathrm{d}\xi\\\
\overset{\triangle}{=}&\int_{0}^{1}G(\xi,\tau,p)\mathrm{d}\xi.\end{split}$
(4.7)

Then by (2.10), $F(\tau)$ can also be rewritten as

$F(\tau)=\frac{H(\tau)}{\beta L(\tau)}.$ (4.8)

In the following, the cases $p=3,$ $p\in(0,+\infty)\setminus\\{1,3\\}$ and
$p=1$ are considered separately because of their differences. More precisely,
in Section 4.1, we prove that $E(w)$ is a constant on $\mathfrak{S}$ for a
particular case $p=3.$ For cases $p\in(0,+\infty)\setminus\\{1,3\\}$ and
$p=1$, we study the asymptotic behavior of $E(w)$ with respect to $\tau$ in
Sections 4.2 and 4.3, respectively. In Section 4.4, we discuss the
monotonicity of $E(\tau)$ with respect to $\tau$. Finally, we provide the
proofs of Theorems 1.4 and 1.5 in Section 4.5.

###  4.1 The particular case $p=3$

In this subsection, we show that, for the particular case $p=3$, $F(\tau)$ is
a constant independent of $\tau$, which implies by (4.3) that $E(w)$ is a
constant on $\mathfrak{S}$. More precisely, we have

######  Proposition 4.4.

For $p=3$, $F(\tau)\equiv 1$ holds for all $\tau\in[1,+\infty),$ which implies
by (4.3) that $E(w)\equiv-\pi\sqrt{\lambda}(\alpha+2)/2$ on $\mathfrak{S}$.

Proof:  Using (4.5) and (2.10), and recalling that
$L(\tau)\equiv\frac{\pi}{2\beta}$ for $p=3,$ by (4.8) we rewrite
$F(\tau)=\frac{H(\tau)}{\beta L(\tau)}=\frac{2}{\pi}H(\tau).$ Therefore, it is
reduced to prove $H(\tau)\equiv\pi/2.$ In fact,

$\begin{split}H(\tau)=&\int_{0}^{1}~{}\left(\frac{2(1-\tau^{-2})}{2(\tau^{2}-1)}\right)^{-\frac{1}{2}}\frac{(\xi(\tau-1)+1)^{-2}(\tau-1)}{\sqrt{1+\tau^{2}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{2}}{(\xi(\tau-1)+1)^{2}}}}~{}d\xi\\\
=&\tau\int_{0}^{1}~{}\frac{(\tau-1)(\xi(\tau-1)+1)^{-1}}{\sqrt{(1+\tau^{2})(\xi(\tau-1)+1)^{2}-(\xi(\tau-1)+1)^{4}-\tau^{2}}}~{}d\xi\\\
\xlongequal{x=(\xi(\tau-1)+1)^{2}}&\frac{\tau}{2}\int_{1}^{\tau^{2}}\frac{dx}{x\sqrt{(\frac{\tau^{2}-1}{2})^{2}-(x-\frac{\tau^{2}+1}{2})^{2}}}\\\
\xlongequal{x-\frac{\tau^{2}+1}{2}=\frac{\tau^{2}-1}{2}\sin
X}&\frac{\tau}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{dX}{\frac{\tau^{2}+1}{2}+\frac{\tau^{2}-1}{2}\sin
X}\\\
\xlongequal{Y=\tan\frac{X}{2}}&\frac{\tau^{2}+1}{2\tau}\int_{-1}^{1}\frac{dY}{1+\left({\frac{\tau^{2}+1}{2\tau}}(Y+\frac{\tau^{2}-1}{\tau^{2}+1})\right)^{2}}\\\
=&[\arctan{\tau}-\arctan(-\frac{1}{\tau})]\\\ \equiv&\pi/2.\end{split}$ (4.9)

So the proof is complete. $\Box$

If we combine Proposition 4.1, Proposition 4.4 and Theorem 1.1, we obtain

######  Corollary 4.5.

For $p=3,$ $u(r,\theta)$ is a global solution to (1.1) satisfying (1.8) if and
only if there exists $w_{\epsilon,a}(\theta)\in\mathfrak{S}$ such that
$u(r,\theta)=w_{\epsilon,a}(\theta)r^{\frac{\alpha+2}{p+1}}.$ In particular,
for $p=3$ and $\alpha\notin 2\mathbb{N},$
$u(r,\theta)=\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1}{p+1}}r^{\frac{\alpha+2}{p+1}}$
is the unique global solution to (1.1) satisfying (1.8).

###  4.2 The asymptotic behavior of $E(\tau)$ for
$p\in(0,+\infty)\setminus\\{1,3\\}$

Unlike the case $p=3,$ for cases $p\neq 3$, it is challenging to confirm
whether $E(\tau)$ is monotonous with respect to $\tau$, even though we believe
it is. To observe monotonicity, we first study the asymptotic behavior of
$E(\tau)$ as $\tau\to 1$ and $\tau\to+\infty.$ Using (4.3), this is reduced to
the study of asymptotic behavior of $F(\tau)$ for $p\neq 1$ and $F_{1}(\tau)$
for $p=1.$ In this subsection, we focus on the asymptotic behavior of
$F(\tau)$, and the asymptotic behavior of $F_{1}(\tau)$ is considered in
subsection 4.3.

####  4.2.1 The asymptotic behavior of $E(\tau)$ as $\tau\to 1$

######  Proposition 4.6.

It holds $\lim_{\tau\to 1}F(\tau)=1$ for all $p>0$ and $p\neq 1,$ which
implies $\lim_{\tau\to
1}E(\tau)=\pi\lambda\frac{1+p}{1-p}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1-p}{p+1}}$
by (4.3).

Proof: Note (4.8), and the asymptotic behavior of $L(\tau)$ is given in Lemma
2.1. Therefore, it is reduced to study the behavior of $H(\tau)$ as $\tau\to
1.$ Denote

$\begin{split}f_{1}(\xi,\tau):=&(\xi(\tau-1)+1)^{1-p}\left(\frac{2(1-\tau^{1-p})}{(p-1)(\tau^{2}-1)}\right)^{\frac{1-p}{p+1}},\\\
f_{2}(\xi,\tau):=&\frac{\tau-1}{\sqrt{1+\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}-(\xi(\tau-1)+1)^{2}-\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}{1\over{(\xi(\tau-1)+1)^{p-1}}}}}.\end{split}$
(4.10)

Then $H(\tau)=\int_{0}^{1}~{}f_{1}(\xi,\tau)f_{2}(\xi,\tau)~{}d\xi$, $\beta
L(\tau)=\int_{0}^{1}~{}f_{2}(\xi,\tau)~{}d\xi$. We claim that
$H(\tau)=\int_{0}^{1}~{}f_{1}(\xi,\tau)f_{2}(\xi,\tau)~{}d\xi$ converges
uniformly with respect to $\tau\in[1,2]$. In fact, for $p>1$, note that the
improper integral $L(\tau)$ converges uniformly on $[1,2]$, $f_{1}(\xi,\tau)$
monotonically decreases in $\xi$ and is uniformly bounded on $[1,2]$. The
uniform convergence of $H(\tau)$ follows by the Abel test. For $p<1$, note
that

$H(\tau)\leq\tau^{1-p}\left(\frac{2(1-\tau^{1-p})}{(p-1)(\tau^{2}-1)}\right)^{\frac{1-p}{p+1}}\beta
L(\tau),$

the uniform convergence of $H(\tau)$ on $[1,2]$ also holds.

Thus, by (4.8) and $\lim_{\tau\to 1}~{}f_{1}(\xi,\tau)=1$ we have

$\begin{split}\lim_{\tau\to 1}F(\tau)=\lim_{\tau\to
1}\frac{\int_{0}^{1}~{}f_{1}(\xi,\tau)f_{2}(\xi,\tau)~{}d\xi}{\int_{0}^{1}~{}f_{2}(\xi,\tau)~{}d\xi}=\frac{\int_{0}^{1}\lim_{\tau\to
1}~{}f_{1}(\xi,\tau)f_{2}(\xi,\tau)~{}d\xi}{\int_{0}^{1}\lim_{\tau\to
1}~{}f_{2}(\xi,\tau)~{}d\xi}=1.\end{split}$

The proof is complete. $\square$

####  4.2.2 The asymptotic behavior of $E(\tau)$ as $\tau\to\infty$

The cases $p\in(0,1),$ $p\in(3,+\infty)$ and $p\in(1,3)$ are considered in
propositions 4.7, 4.8, and 4.12, respectively. Note that the study of case
$p\in(1,3)$ is more complicated than that of the other two cases. In fact,
these involve elliptic integrals.

######  Proposition 4.7.

Let $p\in(0,1).$ Suppose that $H(\tau)$ is defined in (4.7). Then we have

$\lim_{\tau\to\infty}H(\tau)\overset{\triangle}{=}H(+\infty)=\left(\frac{2}{1-p}\right)^{\frac{1-p}{p+1}}\int_{0}^{1}\frac{\xi^{1-p}}{\sqrt{\xi^{1-p}-\xi^{2}}}d\xi.$
(4.11)

By Lemma 2.1 and (4.3) this implies that

$\lim_{\tau\to\infty}F(\tau)\overset{\triangle}{=}C(p)=\frac{p+1}{\pi}H(+\infty)$
(4.12)

and

$\lim_{\tau\to\infty}E(\tau)=\pi\lambda\frac{1+p}{1-p}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1-p}{p+1}}C(p).$
(4.13)

Proof:  As in the proof of Proposition 4.6, it can be seen that $H(\tau)$
converges uniformly on $[1,+\infty)$. Hence, we have

$\lim_{\tau\to\infty}H(\tau)=\int_{0}^{1}\lim_{\tau\to\infty}G(\xi,\tau,p)\mathrm{d}\xi=\left(\frac{2}{1-p}\right)^{\frac{1-p}{p+1}}\int_{0}^{1}\frac{\xi^{1-p}}{\sqrt{\xi^{1-p}-\xi^{2}}}d\xi$

and the proof is complete. $\square$

######  Remark 4.1.

We point out that $F(+\infty)>1$ for all $p\in(0,1).$

Next, in preparation for what follows, rewrite $H(\tau)$ in (4.7) as

$H(\tau)=A(\tau)\int_{1}^{\tau}\frac{dy}{y^{p-1}\sqrt{I(y,\tau)}}$ (4.14)

with
$A(\tau)=\left(\frac{2(1-\tau^{1-p})}{(p-1)(\tau^{2}-1)}\right)^{\frac{1-p}{p+1}},$
$I(y,\tau)=1+z(\tau)-y^{2}-z(\tau)y^{1-p}$ and
$z(\tau)=\frac{\tau^{p-1}(\tau^{2}-1)}{\tau^{p-1}-1}.$

For the case $p>3,$ the asymptotic behavior of $E(\tau)$ as $\tau\to\infty$
can be obtained as follows:

######  Proposition 4.8.

Let $p>3.$ Then it holds

$\lim_{\tau\to\infty}F(\tau)=+\infty,$ (4.15)

which implies $\lim_{\tau\to\infty}E(\tau)=+\infty.$

Proof:  Note that $F(\tau)=H(\tau)/\beta L(\tau)$ and
$\lim_{\tau\rightarrow+\infty}L(\tau)=\pi/2$ by Lemma 2.1. It is reduced to
proving

$\lim_{\tau\to\infty}H(\tau)=+\infty.$ (4.16)

Recall (4.14). By noting that

$I(y,\tau)\leq I_{y}(1,\tau)(y-1),\quad\forall\ y\in(1,\tau)$ (4.17)

due to the concavity of $I$ with respect to $y$ and $I(1,\tau)=0$. It holds
that

$\begin{split}H(\tau)=&A(\tau)\int_{1}^{\tau}\frac{dy}{y^{p-1}\sqrt{I(y,\tau)}}\\\
\geq&A(\tau)\int_{1}^{2}\frac{dy}{y^{p-1}\sqrt{I_{y}(1,\tau)(y-1)}}\\\
\geq&A(\tau)\frac{1}{2^{p-1}\sqrt{-2+z(\tau)(p-1)}}\int_{1}^{2}\frac{dy}{\sqrt{y-1}}\\\
\geq&C(p)\tau^{\frac{p-3}{p+1}}\end{split}$ (4.18)

for $\tau$ large, where $C(p)$ is a positive constant that depends only on
$p.$ Therefore, (4.16) holds and the proof is complete. $\square$

In the remainder of this subsection, we focus on studying the asymptotic
behavior of $E(\tau)$ as $\tau\rightarrow+\infty$ for the case $1<p<3,$ which
is more complicated than the previous two cases. In fact, these involve
elliptic integrals. In preparation, we provide the following lemmas:

######  Lemma 4.9.

It holds

$K(k^{2})\overset{\triangle}{=}\int_{0}^{\pi\over
2}\frac{d\theta}{\sqrt{1-k^{2}sin^{2}\theta}}=\frac{{\pi\over
2}}{AGM(1,\sqrt{1-k^{2}})}.$ (4.19)

where $K(k^{2})$ is the complete elliptic integral of the first kind and
$AGM(x,y)$ is the arithmetic-geometric mean of two positive real numbers $x$
and $y.$

This is a well-known result, and from which we obtain

######  Lemma 4.10.

It holds

$\int_{1}^{\tau}\frac{1}{\sqrt{y}\sqrt{y-1}\sqrt{\tau-y}}dy=\frac{2}{\sqrt{\tau}}K\left(1-{1\over\tau}\right)=\frac{2}{\sqrt{\tau}}\frac{{\pi/2}}{AGM({1\over{\sqrt{\tau}}},1)}.$
(4.20)

Proof:  By calculation, we have

$\begin{split}&\int_{1}^{\tau}\frac{1}{\sqrt{y}\sqrt{y-1}\sqrt{\tau-y}}dy\\\
=&\int_{1}^{\tau}\frac{1}{\sqrt{y}\sqrt{\frac{(\tau-1)^{2}}{4}-\left(y-\frac{\tau+1}{2}\right)^{2}}}dy\\\
\xlongequal{y-\frac{\tau+1}{2}=\frac{\tau-1}{2}\sin
X}=&\int_{-\pi/2}^{\pi/2}\frac{dX}{\sqrt{\frac{\tau+1}{2}+\frac{\tau-1}{2}\sin
X}}\\\ =&\int_{0}^{\pi}\frac{dX}{\sqrt{\frac{\tau+1}{2}+\frac{\tau-1}{2}\cos
X}}\\\
\xlongequal{X=2\theta}&\frac{2}{\sqrt{\tau}}\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-(1-\frac{1}{\tau})\sin^{2}\theta}}\\\
=&\frac{2}{\sqrt{\tau}}K\left(1-{1\over\tau}\right).\end{split}$ (4.21)

Thus the proof is complete by Lemma 4.9. $\square$

Second, we present another result (see,e.g. [2, Theorem 2.5]) as follows:

######  Lemma 4.11.

$\frac{{\pi/2}}{AGM(k,1)}\sim\ln(\frac{4}{k})$ as $k\rightarrow 0^{+}.$

Based on these preparations, we can derive the asymptotic behavior of
$F(\tau)$ as $\tau\rightarrow+\infty$ for $1<p<3.$

######  Proposition 4.12.

Let $1<p<3.$ Then it holds

$\lim_{\tau\to\infty}E(\tau)=\lim_{\tau\to\infty}F(\tau)=0.$ (4.22)

Proof:  Note (4.3), $F(\tau)=H(\tau)/\beta L(\tau)$ and
$\lim_{\tau\rightarrow+\infty}L(\tau)=\pi/2$ by Lemma 2.1. It is reduced to
prove

$\lim_{\tau\to\infty}H(\tau)=0\quad\text{for}\quad p\in(1,3).$ (4.23)

By calculation, $I(y,\tau)$ in (4.14) can be rewritten as

$I(y,\tau)=\frac{z(\tau)-z(y)}{\tau-y}\frac{y^{p-1}-1}{y^{p-1}(y-1)}(y-1)(\tau-y)$
(4.24)

and we have

$I(y,\tau)\geq\frac{z(\tau)-z(1)}{\tau-1}\frac{y^{p-1}-1}{y^{p-1}(y-1)}(y-1)(\tau-y)\quad\text{for}\
y\in(1,\tau),$ (4.25)

by using the convexity of $z(y)$ with respect to $y.$ Then from (4.14), Lemmas
4.10, and 4.11, we have

$\begin{split}H(\tau)&\leq
A(\tau)\left(\frac{z(\tau)-z(1)}{\tau-1}\right)^{-1/2}\int_{1}^{\tau}\left(y^{p-1}\sqrt{\frac{y^{p-1}-1}{y^{p-1}(y-1)}}\sqrt{y-1}\sqrt{\tau-y}\right)^{-1}dy\\\
&\leq
A(\tau)\left(\frac{z(\tau)-z(1)}{\tau-1}\right)^{-1/2}\int_{1}^{\tau}\left(y^{p-\frac{3}{2}}\sqrt{\frac{y^{p-1}-1}{y^{p-1}(y-1)}}\sqrt{y}\sqrt{y-1}\sqrt{\tau-y}\right)^{-1}dy\\\
&\leq C(p)\tau^{\frac{2(p-1)}{p+1}-{1\over
2}}\int_{1}^{\tau}y^{2-p}\left(\sqrt{y}\sqrt{y-1}\sqrt{\tau-y}\right)^{-1}dy\\\
&\leq\left\\{\begin{array}[]{lll}C(p)\tau^{\frac{2(p-1)}{p+1}-{1\over
2}}\tau^{2-p}\int_{1}^{\tau}\left(\sqrt{y}\sqrt{y-1}\sqrt{\tau-y}\right)^{-1}dy,&\text{for}\
p\in(1,2)\\\ C(p)\tau^{\frac{2(p-1)}{p+1}-{1\over
2}}\int_{1}^{\tau}\left(\sqrt{y}\sqrt{y-1}\sqrt{\tau-y}\right)^{-1}dy,&\text{for}\
p\in[2,+\infty)\\\ \end{array}\right.\\\
&=\left\\{\begin{array}[]{lll}C(p)\tau^{\frac{2(p-1)}{p+1}-{1\over
2}}\tau^{2-p}\frac{2}{\sqrt{\tau}}\frac{{\pi/2}}{AGM({1\over{\sqrt{\tau}}},1)},&\text{for}\
p\in(1,2)\\\ C(p)\tau^{\frac{2(p-1)}{p+1}-{1\over
2}}\frac{2}{\sqrt{\tau}}\frac{{\pi/2}}{AGM({1\over{\sqrt{\tau}}},1)},&\text{for}\
p\in[2,+\infty)\\\ \end{array}\right.\\\
&\leq\left\\{\begin{array}[]{lll}C(p)\tau^{\frac{-(p-1)^{2}}{p+1}}\ln(4\sqrt{\tau}),&\text{for}\
p\in(1,2)\\\ C(p)\tau^{-\frac{3-p}{p+1}}\ln(4\sqrt{\tau}),&\text{for}\
p\in[2,+\infty),\\\ \end{array}\right.\\\ \end{split}$ (4.26)

which implies (4.23). The proof is complete. $\square$

######  Remark 4.2.

Finally, we summarize the results obtained in this subsection 4.2.2 as
follows:

$\lim_{\tau\to\infty}F(\tau)=\left\\{\begin{array}[]{lll}C(p)<+\infty,&\
0<p<1,\\\ 0,&\ 1<p<3,\\\ +\infty,&\ p>3.\\\ \end{array}\right.\\\ $ (4.27)

###  4.3 The asymptotic behavior of $E(\tau)$ for $p=1$

In this subsection, we will focus on the asymptotic behavior of $F_{1}(\tau).$

######  Proposition 4.13.

Let $p=1$. Then, it holds $\lim_{\tau\to 1}F_{1}(\tau)=0$, which implies by
(4.3) that $\lim_{\tau\to
1}E(\tau)=-\pi\lambda+2\pi\lambda\ln\frac{2\sqrt{\lambda}}{\alpha+2}$.

Proof:  Note that

$\lim_{\tau\to
1}\ln\left[\left(\frac{2\ln\tau}{\tau^{2}-1}\right)^{\frac{1}{2}}(\xi(\tau-1)+1)\right]=0,\
\forall\xi\in(0,1).$ (4.28)

This proposition can be proven in the same manner as Proposition 4.6.
$\square$

######  Proposition 4.14.

Let $p=1$. Then, it holds $\lim_{\tau\to+\infty}F_{1}(\tau)=+\infty$, which
implies by (4.3) that $\lim_{\tau\to 1}E(\tau)=+\infty$. More precisely,
$F_{1}(\tau)=\ln\sqrt{\ln\tau}+o(\ln\tau)$ as $\tau\to+\infty.$

Proof:  Rewrite $F_{1}$ in (4.6) as

$F_{1}(\tau)=I_{1}(\tau)+I_{2}(\tau),$

where $I_{1}(\tau)=\frac{1}{2}\ln\frac{2\ln\tau}{\tau^{2}-1}$ and

$I_{2}(\tau)=\frac{\displaystyle{\int_{0}^{1}~{}\ln\left[(\xi(\tau-1)+1)\right]\frac{\tau-1}{\sqrt{1-(\xi(\tau-1)+1)^{2}+(\tau^{2}-1)\frac{\ln{(\xi(\tau-1)+1)}}{\ln\tau}}}~{}d\xi}}{\displaystyle{\int_{0}^{1}~{}\frac{\tau-1}{\sqrt{1-(\xi(\tau-1)+1)^{2}+(\tau^{2}-1)\frac{\ln{(\xi(\tau-1)+1)}}{\ln\tau}}}~{}d\xi}}.$
(4.29)

Note that $I_{2}(\tau)=\ln\tau+o(\ln\tau)$ as $\tau\to\infty$ by Abel’s Lemma
and

$\lim_{\tau\to\infty}\frac{\ln\left[(\xi(\tau-1)+1)\right]}{\ln\tau}=1$

for any given $\xi\in(0,1)$. Therefore, we can conclude that
$F_{1}(\tau)=\ln\sqrt{\ln\tau}+o(\ln\tau),$ and this proposition follows.

$\square$

###  4.4 The monotonicity of $E(\tau)$

In this subsection, we examine the monotonicity of $E(\tau)$ with respect to
$\tau.$ It was previously established in Subsection 4.1 that $E(\tau)$ is a
constant independent of $\tau$ for the particular case $p=3.$ However, for
$p\in(0,+\infty)\setminus\\{3\\}$, the monotonicity remains uncertain.

As an initial step towards studying the monotonicity of $E(\tau),$ we reduce
it to examine the monotonicity of $F(\tau)$ and $F_{1}(\tau),$ as per (4.3).
For easy reference, Table 1 provides a summary of the asymptotic behaviors of
$F(\tau)$ and $F_{1}(\tau)$ obtained in Subsections 4.1-4.3. From Table 1, we
can infer that if $F(\tau)$ and $F_{1}(\tau)$ are monotonic with respect to
$\tau$, then they can only be expressed as follows:

$\begin{split}&(1)\ F(\tau)\ \text{decreases monotonically with respect to}\
\tau\ \text{for}\ p\in(1,3);\\\ &(2)\ F(\tau)\ \text{increases monotonically
with respect to}\ \tau\ \text{for}\ p\in(0,1)\cup(3,+\infty);\\\ &(3)\
F_{1}(\tau)\ \text{increases monotonically with respect to}\ \tau\ \text{for}\
p=1.\end{split}$ (4.30)

However, providing a mathematical proof for (4.30) is challenging. Therefore,
we present the numerical results (Figure 2).

$F(\tau)$ | $\tau\to 1$ | $\tau\to\infty$
---|---|---
$p<1$ | $1$ | $C(p)(>1)$
$1<p<3$ | $1$ | $0$
$p=3$ | $1$ | $1$
$p>3$ | $1$ | $+\infty$
$F_{1}(\tau)$ | $\tau\to 1$ | $\tau\to\infty$
$p=1$ | $0$ | $+\infty$
$E(\tau)$ | $\tau\to 1$ | $\tau\to\infty$
$p<1$ | $\pi\lambda\frac{1+p}{1-p}\big{[}\frac{(p+1)^{2}}{(\alpha+2)^{2}}\lambda\big{]}^{\frac{1-p}{p+1}}$ | $\pi\lambda\frac{1+p}{1-p}\big{(}\frac{\lambda}{\beta^{2}}\big{)}^{\frac{1-p}{p+1}}C(p)$
$p=1$ | $-\pi\lambda+2\pi\lambda\ln\frac{2\sqrt{\lambda}}{\alpha+2}$ | $+\infty$
$1<p<3$ | $\pi\lambda\frac{1+p}{1-p}\big{[}\frac{(p+1)^{2}}{(\alpha+2)^{2}}\lambda\big{]}^{\frac{1-p}{p+1}}$ | $0$
$p=3$ | $-\pi\sqrt{\lambda}(\alpha+2)/2$ | $-\pi\sqrt{\lambda}(\alpha+2)/2$
$p>3$ | $\pi\lambda\frac{1+p}{1-p}\big{[}\frac{(p+1)^{2}}{(\alpha+2)^{2}}\lambda\big{]}^{\frac{1-p}{p+1}}$ | $+\infty$
Table 1: Asymptotic behaviors of $F(\tau),$ $F_{1}(\tau)$ and $E(\tau)$

(a) $p\in(0,1)\cup(1,3.5)$

(b) $p\in(3,50)$

(c) $p=0.5,2,3,4$

(d) $p=20$

(e) $p=1$

Figure 2: $F$ and $F_{1}$

###  4.5 The proof of Theorems 1.4 and 1.5

Proof of Theorem 1.4. This theorem can be obtained directly from Theorem 1.1,
Corollary 4.3 and 4.5.

Proof of Theorem 1.5. Combining (4.30) with Lemma 2.1 and (4.3), and recalling
that $w\in\mathfrak{S}_{0}\Leftrightarrow\tau=1,$ leads to the presentation of
Table 2.

|  $j=\frac{\pi}{L(\tau)}$ |  $L(\tau)$ |  $\tau$ |  $F(\tau)$ |  $E(\tau)$ |  $E(\mathfrak{S}_{0})$
---|---|---|---|---|---|---
$p<1$ |  $\nearrow$ |  $\searrow$ |  $\nearrow$ |  $\nearrow$ |  $\nearrow$ |  minimum
$p=1$ |  $\nearrow$ |  $\searrow$ |  $\nearrow$ |  $\nearrow$ |  $\nearrow$ |  minimum
$1<p<3$ |  $\nearrow$ |  $\searrow$ |  $\nearrow$ |  $\searrow$ |  $\nearrow$ |  minimum
$p>3$ |  $\nearrow$ |  $\searrow$ |  $\searrow$ |  $\searrow$ |  $\nearrow$ |  maximum
Table 2: Monotonicity of $E(w)$ with respect to the frequency $j$ of $w$

By Table 2, we can conclude the subsequent energy inequalities:

$\begin{array}[]{lll}E(\mathfrak{S}_{0})<E(\mathfrak{S}_{1})<E(\mathfrak{S}_{2})<\cdots<E(\mathfrak{S}_{j})<\cdots<E(\mathfrak{S}_{N_{0}}),\quad\text{for}\
0<p<3,\\\
E(\mathfrak{S}_{1})<E(\mathfrak{S}_{2})<\cdots<E(\mathfrak{S}_{j})<\cdots<E(\mathfrak{S}_{N_{0}})<E(\mathfrak{S}_{0}),\quad\text{for}\
p>3,\end{array}$ (4.31)

where $\mathfrak{S}_{k},k=1,2,\ldots,N_{0}$ are as stated in Theorem 1.1.
Recall that $E(w_{0})\leq E(w_{\infty})$ from Proposition 4.1. Theorem 1.5
follows and the proof is complete.

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# A Physics-based and Data-driven Approach for Localized Statistical Channel
Modeling

Shutao Zhang12, Xinzhi Ning34, Xi Zheng 5, Qingjiang Shi32, Tsung-Hui Chang12,
and Zhi-Quan Luo124 Email<EMAIL_ADDRESS>The work is supported
in part by the Shenzhen Science and Technology Program under Grant No.
RCJC20210609104448114, and in part by the NSFC, China, under Grant No.
61731018 and 62071409, and by Guangdong Provincial Key Laboratory of Big Data
Computing. 1School of Science and Engineering, The Chinese University of Hong
Kong, Shenzhen, China  2Shenzhen Research Institute of Big Data, Shenzhen,
China  3School of Software Engineering, Tongji University, Shanghai, China
4Pengcheng Lab, Shenzhen, China  5Networking and User Experience Lab, Huawei
Technologies, Shenzhen, China

###### Abstract

Localized channel modeling is crucial for offline performance optimization of
5G cellular networks, but the existing channel models are for general
scenarios and do not capture local geographical structures. In this paper, we
propose a novel physics-based and data-driven localized statistical channel
modeling (LSCM), which is capable of sensing the physical geographical
structures of the targeted cellular environment. The proposed channel modeling
solely relies on the reference signal receiving power (RSRP) of the user
equipment, unlike the traditional methods which use full channel impulse
response matrices. The key is to build the relationship between the RSRP and
the channel’s angular power spectrum. Based on it, we formulate the task of
channel modeling as a sparse recovery problem where the non-zero entries of
the sparse vector indicate the channel paths’ powers and angles of departure.
A computationally efficient weighted non-negative orthogonal matching pursuit
(WNOMP) algorithm is devised for solving the formulated problem. Finally,
experiments based on synthetic and real RSRP measurements are presented to
examine the performance of the proposed method.

##  I Introduction

With the rapid development of the fifth generation (5G) mobile communications,
cellular network optimization becomes the fundamental means to improve the
quality of experience (QoE) of the users in wireless communication [1, 2, 3].
Since network optimization must be performed in the offline fashion, one of
the key challenges is how to characterize the large-scale behaviors of the
wireless propagation channels between the base station and the user equipment
under the network coverage. Thus, a channel model that can capture the
localized physical geographical structures of the targeted environment is
paramount for the network optimization of the 5G systems.

Unfortunately, the existing channel models commonly used in the 5G systems are
not applicable to cellular network optimization. For example, the geometry-
based stochastic models (GBSM) have been intensively studied [4, 5, 6]. They
are employed for some typical scenarios, such as indoor and outdoor, rural and
urban scenarios, but lack the ability to describe a specific and localized
environment. As for deterministic channel modeling methods like ray tracing
[7], they can describe the localized environment, but the complexity of using
Maxwell’s equations is unaffordable for cellular network optimization.
Additionally, ray tracing is based on map information, which may not always be
available [8].

In this paper, we propose a physics-based and data-driven localized
statistical channel modeling (LSCM) method, which is capable of sensing the
physical propagation environment for the target region. The proposed LSCM can
efficiently provide the statistical information of channel angular power
spectrum (APS), thus revealing the localized multi-path structure and
topography. In contrast to the existing GBSM which relies on channel impulse
response (CIR) measurements, our proposed LSCM considers the measurements of
reference signal receiving power (RSRP) [9] from multiple transmit beams only.
Compared with the high-dimensional CIR, the benefit of using RSRP is that it
can save a lot of storage space during data collection, and enable
computationally efficient approaches. However, how to extract the channel
statistics from the RSRP data has never been studied before and is a
challenging task.

To this end, we first build the relationship between the RSRP and the APS. We
show later that extracting the APS can be achieved through solving a sparse
recovery problem, where the non-zero entries of the sparse vector indicate the
channel paths’ powers and angles of departure. While the classical compressive
sensing (CS) theory suggests that orthogonal matching pursuit (OMP) is a good
solver [10], its convergence is guaranteed for some well-conditioned matrices,
which in general does not hold. On the contrary, the column magnitudes of the
coefficient matrix in the formulated problem can vary dramatically, which will
significantly deteriorate the channel modeling performance. To solve this
intractable sparse recovery problem, we design the weighted non-negative OMP
(WNOMP) algorithm. Simulation results based on synthetic data and real-world
RSRP measurements demonstrate that the proposed channel modeling method is
effective and the WNOMP outperforms the classical methods.

Figure 1: Angular discretization of the channel model in the downlink.

##  II MIMO Downlink System and RSRP

In this section, we consider the channel modeling of the massive multiple
input multiple output (MIMO) downlink communication system in the target grid,
followed by the RSRP measured from multiple beams.

###  II-A Angular discretization channel modeling

The technique of beamforming is exploited in the massive MIMO of 5G systems,
yielding high-resolution multi-path channels in the angular domain. Thus, we
consider the channel modeling of the massive MIMO downlink communication
system with beamforming, where the signal is transmitted from a base station
to the user equipment through several propagation paths, as shown in Fig. 1.

Similar to the 3GPP’s technical report [11], we focus on the tilt angle of
departure (AoD), azimuth AoD and channel gain from the base station to the
user equipment with one receive antenna, ignoring the arrival angle and delay.
Suppose the uniform rectangular array of the base station contains
$N_{T}=N_{x}\times N_{y}$ antennas. The CIR of antenna $(x,y)$ from the base
station to the user equipment is given by

$\displaystyle h_{x,y}(t)=\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}$
$\displaystyle\sqrt{\alpha_{i,j}(t)}\times g_{i,j}\times
e^{-j2\pi\frac{d_{x}x}{\lambda}\cos\theta_{i}\sin\varphi_{j}}$
$\displaystyle\times e^{-j2\pi\frac{d_{y}y}{\lambda}\sin\theta_{i}}\times
e^{-j\omega_{i,j}(t)-j\omega_{x,y}(t)}.$ (1)

TABLE I: Summary of notation

The definition of notation in (II-A) is summarized in Table I. The tilt AoD
and azimuth AoD over the free space are discretized into $N_{V}$ angles and
$N_{H}$ angles, respectively. If there does not exist a path in the angles
$\left(\theta_{i},\varphi_{j}\right)$, the corresponding channel gain
$\alpha_{i,j}(t)$ is zero, otherwise $\alpha_{i,j}(t)>0$ for the path
$\left(\theta_{i},\varphi_{j}\right)$ in the channel. It is readily known that
the channel gain ${\alpha_{i,j}}(t)$ is sparse, and contains multiple
propagation paths in the angular domain.

The channel gain ${\alpha_{i,j}}(t)$ consists of path loss and the shadowing
effect. Path loss is determined by the physical environment (distance, carrier
frequency, buildings) which is assumed to be relatively static, while
shadowing is caused by the obstacles and is usually modeled as the log-normal
distribution. Thus, we assume ${\alpha_{i,j}}(t)$ follows the log-normal
distribution, with its mean representing path loss and its covariance
representing the shadowing effect. $\omega_{i,j}(t)$ is the random phase error
between different angles caused by the reflection, diffraction and scattering
effect of electromagnetic waves, while $\omega_{x,y}(t)$ is the random phase
error of different antennas caused by the imperfect hardware of the antenna
array. Inspired by [11], we assume $\omega_{i,j}(t)$ follows the uniform
distribution between $-\pi$ and $\pi$, and $\omega_{x,y}(t)$ follows the
Gaussian distribution with zero mean and covariance $\sigma^{2}$.

(a) RSRP = -75.92 dB

(b) RSRP = -73.29 dB

(c) RSRP = -64.12 dB

(d) RSRP = -59.61 dB

(e) RSRP = -70.73 dB

Figure 2: An example of channel measurements of the RSRP from five beams.
There exists one propagation path between the base station and target grid,
with angle of departure
$\left[\widehat{\theta},\widehat{\varphi}\right]=[-2.84,23.82]$. The
directions of main lobe of different beams are: (a)[0, -15], (b)[0, 0], (c)[0,
15], (d)[0, 30], (e)[0, 45].

###  II-B RSRP measurements

The channel measurement of our proposed LSCM is the RSRP measured from
multiple beams. In the downlink of 5G cellular systems, the RSRP can be
measured from synchronization signals block (SSB) beams or channel state
information-reference signal (CSI-RS) beams. Denote the precoding matrix of
the $m_{\rm th}$ beam as $\bm{W}^{(m)}\in\mathbb{C}^{N_{x}\times N_{y}}$,
whose size is the same as that of the antenna array. The entry of
$\bm{W}^{(m)}$ is
$\left(w_{x,y}^{(m)}\right)=\left(e^{j\phi_{x,y}^{(m)}}\right)$. Denote the
CIR matrix from the base station to the user equipment as
$\bm{H}\in\mathbb{C}^{N_{x}\times N_{y}}$, where the entry of $\bm{H}$ is
$h_{x,y}(t)$ in (II-A). The RSRP of the $m_{\rm th}$ beam at time $t$ is
defined as

${rsrp}_{m}(t)=P\left|\operatorname{tr}\left(\bm{H}^{{T}}\bm{W}^{(m)}\right)\right|^{2}=P\left|\sum_{x,y}h_{x,y}(t)w_{x,y}^{(m)}\right|^{2},$
(2)

where $P$ denotes the transmit power. The quality of the channel can be
represented through ${rsrp}_{m}(t)$, and a larger value of ${rsrp}_{m}(t)$
reflects better channel quality.

Figure 2 shows an example of channel measurement of RSRP measured from five
beams. These five beam patterns are generated according to different
beamforming weights, and the directions of main lobe of these five beams are
different. The receiver at the target grid is highlighted as a blue point. The
target grid can receive five RSRP measurements, corresponding to the received
power of five beams. To be specific, the strongest value of RSRP measurements
is -59.61 dB in Fig. 2(d), because its main lobe’s direction is
$\left[\bar{\theta},\bar{\varphi}\right]=[0,30]$, which is close to the angle
of departure of the propagation path
$\left[\widehat{\theta},\widehat{\varphi}\right]=[-2.84,23.82]$.

##  III Physics-based and Data-driven LSCM

In this section, the relation between the first-order statistics of the RSRP
and channel gain is established, which is the key of the proposed LSCM. Then,
the WNOMP is designed for extracting the channel statistics of the APS from
the RSRP.

###  III-A The first-order statistics of the RSRP and channel gain

As for ${rsrp}_{m}(t)$, it is a random variable inferred from (2), since
$h_{x,y}(t)$ contains three independent random variables, i.e.,
$\omega_{x,y}(t)$, $\omega_{i,j}(t)$ and ${\alpha_{i,j}}(t)$. Theorem 1 shows
the relationship between the first-order statistics of ${rsrp}_{m}(t)$ and
${\alpha_{i,j}}(t)$, i.e., their expectation
$\mathbb{E}\left({rsrp}_{m}(t)\right)$ and
$\mathbb{E}\left({\alpha_{i,j}}(t)\right)$, implying the statistical
relationship of the RSRP and angular information.

######  Theorem 1.

(The relationship between the first-order statistics of RSRP and channel gain)
Suppose the expectation of ${rsrp}_{m}(t)$ is
${\rm{RSRP}}_{m}\triangleq\mathbb{E}\left({rsrp}_{m}(t)\right)$, and the
expectation of channel gain is
$X_{i,j}\triangleq\mathbb{E}\left({\alpha_{i,j}}(t)\right)$, then we have

$\displaystyle{\rm RSRP}_{m}=\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}{\rm
A}_{i,j}^{(m)}X_{i,j},$ (3)

where

$\displaystyle{\rm A}_{i,j}^{(m)}$ $\displaystyle\triangleq
Pg_{i,j}^{2}\Bigg{(}\\!N_{x}N_{y}\left(1-e^{-\sigma^{2}}\right)$
$\displaystyle\quad\
+e^{-\sigma^{2}}\sum_{x,y}\sum_{x^{\prime},y^{\prime}}\cos\left(\psi_{i,j,x,y}^{(m)}-\psi_{i,j,x^{\prime},y^{\prime}}^{(m)}\\!\right)\\!\Bigg{)},$
(4) $\displaystyle\psi^{(m)}_{i,j,x,y}$
$\displaystyle=2\pi{d_{x}x\over\lambda}\cos\theta_{i}\sin\varphi_{j}+2\pi{d_{y}y\over\lambda}\sin\theta_{i}-\phi_{x,y}^{(m)}.$
(5)

###### Proof.

Please refer to Appendix A. ∎

The statistical relationship between the expectation of RSRP measurements
${\rm{RSRP}}_{m}$ and the first-order statistics of channel gain $X_{i,j}$ is
connected by the coefficient ${\rm A}_{i,j}^{(m)}$, as revealed by Theorem 1.
According to this finding, it is possible to extract the angular information
from RSRP measurements if we can calculate $X_{i,j}$ from ${\rm{RSRP}}_{m}$.
The APS can be expressed by $X_{i,j}$, where the subscripts $i$ and $j$
indicate the tilt angle $\theta_{i}$ and azimuth angle $\varphi_{j}$,
respectively.

Based on (3), we reshape $X_{i,j}$ with the subscripts
$i=1,\cdots,N_{V},j=1,\cdots,N_{H}$ as a vector, i.e.,

$\displaystyle{\bf
x}=\left[X_{1,1},X_{1,2},\cdots,X_{1,N_{H}},X_{2,1},\cdots,X_{N_{V},N_{H}}\right]^{T},$
(6)

where the length of ${\bf x}$ is $N=N_{V}\times N_{H}$. If the path
$\left(\theta_{i},\varphi_{j}\right)$ does not exist, then
$\mathbb{E}\left({\alpha_{i,j}}(t)\right)=0$. As there are only several
propagation paths from the base station to the user equipment, the channel
vector ${\bf x}$ is sparse.

Suppose there are $M$ measured beams in total, and define ${\bf
y}\in\mathbb{R}^{M\times 1}$ as the vector of the expectation of RSRP
measurements from different beams, i.e.,

$\displaystyle{\bf y}=\left[{\rm RSRP}_{1},{\rm RSRP}_{2},\cdots,{\rm
RSRP}_{m},\cdots,{\rm RSRP}_{M}\right]^{T}.$ (7)

Define the vector ${\bf a}^{(m)}\in\mathbb{R}^{N\times 1}$ as

$\displaystyle{\bf a}^{(m)}=\left[{\rm A}_{1,1}^{(m)},{\rm
A}_{1,2}^{(m)},\cdots,{\rm A}_{1,N_{H}}^{(m)},{\rm A}_{2,1}^{(m)},\cdots,{\rm
A}_{N_{V},N_{H}}^{(m)}\right]^{T},$ (8)

and the coefficient matrix ${\bf A}\in\mathbb{R}^{M\times N}$ is

$\displaystyle{\bf A}=\left[{\bf a}^{(1)},{\bf a}^{(2)},\cdots,{\bf
a}^{(m)},\cdots,{\bf a}^{(M)}\right]^{T}.$ (9)

Then we can recast (3) as

$\displaystyle{\bf y}={\bf A}{\bf x}.$ (10)

###  III-B The designed WNOMP algorithm

To construct the localized statistical channel model, we want to recover the
sparse channel vector $\bf x$ from the expectation of RSRP measurements $\bf
y$ and the coefficient matrix $\bf A$. According to (10), as the expectation
of channel gain in terms of different angles ${x}_{n}\geq 0,\ n=1,\cdots,N$,
is non-negative, we can formulate the optimization problem as

$\displaystyle\ \min_{\bf x}$ $\displaystyle\ \|{\bf{Ax-y}}\|_{2}^{2}$ (11)
s.t. $\displaystyle\ \|{\bf x}\|_{0}\leq K,$ $\displaystyle\ {x}_{n}\geq 0,\
n=1,\cdots,N,$

where $K$ is the maximum number of non-zero entries, representing the maximum
number of propagation paths.

The OMP is an iterative algorithm, which selects one column into the set of
indexes of non-zero entries $\mathcal{S}$ at each iteration. The selection
criteria is to select the column that is best correlated with the residual of
current iteration. Problem (11) can be solved by the non-negative OMP (NNOMP),
as shown in Algorithm 1. The biggest difference between NNOMP and OMP lies in
Step 5 of Algorithm 1, where the subproblem of non-negative least squares
(NNLS) is solved instead of the unconstrained least squares (ULS). After
solving the NNLS subproblem, the entries of $\bf x$ in the complement set
$\mathcal{S}^{c}$ are forced zero to ensure the sparsity. A fast
implementation of NNOMP is proposed in [12] based on the active-set algorithm.
The role of $\max\left({\bf A}^{T}{\bf r}_{k}\right)<0$ is also explained in
[12] by using the Karush-Kuhn-Tucker (KKT) condition. The condition ${\bf
a}_{n}^{T}{\bf r}_{k}>0$ means that the $n_{\rm th}$ column is the descending
column, which can be added into the set $\mathcal{S}$ and reduce the residual.

Algorithm 1 Non-negative OMP (NNOMP)

0: $K,\ {\bf A}\triangleq\left\\{{\bf a}_{1},{\bf a}_{2},\cdots,{\bf
a}_{n},\cdots,{\bf a}_{N}\right\\},\ {\bf y}$

1: Initialize  $k=0,\ \mathcal{S}=\varnothing$, $\bf x=0$, ${\bf r}_{0}=\bf y$

2: repeat

3: $i=\arg\max_{n}$ ${\bf a}_{n}^{T}{\bf r}_{k}$

4: $\mathcal{S}=\mathcal{S}\cup\left\\{i\right\\}$

5: ${\bf x}_{\mathcal{S}}\leftarrow\arg\min_{{\bf z}\geq 0}\left\|{\bf y}-{\bf
A}_{\mathcal{S}}{\bf z}\right\|_{2}$, ${\bf x}_{\mathcal{S}^{\rm
c}}\leftarrow\bf 0$

6: ${\bf r}_{k+1}={\bf y}-{\bf Ax}$

7: $k=k+1$

8: until $\max\left({\bf A}^{T}{\bf r}_{k}\right)<0$ or $|\mathcal{S}|$ equals
to $K$

8: ${\bf x}$

However, in our case, because our coefficient matrix $\bf A$ is not column-
normalized, this sparse recovery problem is intractable. The magnitude of each
column varies significantly, which influences the accuracy of finding the
correct column that is best correlated with the residual. The NNOMP algorithm
prefers the large-magnitude columns of coefficient matrix $\bf A$, resulting
in a low probability to select the correct index of non-zero entries. For
example, although there exists one column that is parallel to the residual,
the NNOMP algorithm will still select the wrong column with large magnitude.

To deal with this problem, we design the WNOMP algorithm, which is shown in
Algorithm 2. The WNOMP algorithm can strike a good balance between the effect
of column magnitude and correlation with the residual by setting the dynamic
weight for each selection. The weight of magnitude should be smaller than that
of the correlation. If we define the column-wise normalized coefficient matrix
as

$\displaystyle\widehat{\bf A}=\left[{{\bf a}_{1}\over\|{\bf
a}_{1}\|_{2}},{{\bf a}_{2}\over\|{\bf a}_{2}\|_{2}},\cdots,{{\bf
a}_{n}\over\|{\bf a}_{n}\|_{2}},\cdots,{{\bf a}_{N}\over\|{\bf
a}_{N}\|_{2}}\right],$ (12)

where ${\bf a}_{n}\in\mathbb{R}^{M\times 1}$ is the $n_{\rm th}$ column vector
of coefficient matrix $\bf A$. The dynamic weight $\lambda_{k}$ can be
calculated as

$\displaystyle\lambda_{k}={\|\widehat{\bf A}^{T}{\bf
r}_{k}\|_{2}\over\sum_{n=1}^{N}\|{\bf a}_{n}\|_{2}}.$ (13)

The dynamic weight $\lambda_{k}$ plays an important role in the selection of
Step 4 of Algorithm 2. An intuitive idea to reduce the influence of column is
to use the column-normalized matrix $\widehat{\bf A}$, i.e., the selection in
Step 4 of Algorithm 2 is modified as $i=\arg\max_{n}\left(\left({{\bf
a}_{n}/\|{\bf a}_{n}\|_{2}}\right)^{\rm T}{\bf r}_{k}\right)$. However, this
modification is not good, because the propagation paths are more likely to
exist in the angles with large antenna gain of the transmit antenna array in
the base station, which means the large-magnitude column corresponds to the
non-zero entries with a large probability. As for the selection
$i=\arg\max_{n}\left({{\bf a}_{n}}^{\rm T}{\bf r}_{k}\right)$, it can be
viewed as the multiplication of the modification case and $\|{\bf
a}_{n}\|_{2}$, where the correlation with residual contributes little to the
selection because the value of $\|{\bf a}_{n}\|_{2}$ is much larger than
$\left({{\bf a}_{n}/\|{\bf a}_{n}\|_{2}}\right)^{\rm T}{\bf r}_{k}$. As a
result, the WNOMP considers these two terms separately, and use $\lambda_{k}$
to balance them. The Step 7 of Algorithm 2 compresses the set $\mathcal{S}$ by
updating it with the support of $\bf x$, because some components of
$\mathcal{S}$ may vanish due to the activation of the non-negative
constraints.

Algorithm 2 Weighted non-negative OMP (WNOMP)

0: $K,\ {\bf A}\triangleq\left\\{{\bf a}_{1},{\bf a}_{2},\cdots,{\bf
a}_{n},\cdots,{\bf a}_{N}\right\\},\ \widehat{\bf A},\ {\bf y}$

1: Initialize  $k=0,\ \mathcal{S}=\varnothing$, $\bf x=0$, ${\bf r}_{0}=\bf y$

2: repeat

3: $\lambda_{k}={\left(\|\widehat{\bf A}^{T}{\bf
r}_{k}\|_{2}\right)/\left(\sum_{n=1}^{N}\|{\bf a}_{n}\|_{2}\right)}$

4: $i=\arg\max_{n}$ $\left(\left({{\bf a}_{n}/\|{\bf a}_{n}\|_{2}}\right)^{\rm
T}{\bf r}_{k}+\lambda_{k}\|{\bf a}_{n}\|_{2}\right)$

5: $\mathcal{S}=\mathcal{S}\cup\left\\{i\right\\}$

6: ${\bf x}_{\mathcal{S}}\leftarrow\arg\min_{{\bf z}\geq 0}\left\|{\bf y}-{\bf
A}_{\mathcal{S}}{\bf z}\right\|_{2}$, ${\bf x}_{\mathcal{S}^{\rm
c}}\leftarrow\bf 0$

7: $\mathcal{S}=\operatorname{supp}\left(\bf x\right)$

8: ${\bf r}_{k+1}={\bf y}-{\bf Ax}$

9: $k=k+1$

10: until $\max\left({\bf A}^{T}{\bf r}_{k}\right)<0$ or $|\mathcal{S}|$
equals to $K$

10: ${\bf x}$

Figure 3: The magnitude of each column of coefficient matrix $\bf A$.

(a) Accuracy versus # of angular discretization $N$

(b) Accuracy versus # of measured beams $M$

(c) Accuracy versus # of non-zero entries $K$

Figure 4: Simulation results of synthetic data with respect to the performance
of accuracy.

##  IV Simulation Results and Discussions

In the following, we show numerical results to analyze the performance of the
algorithms proposed in this paper. To generate $\bf A$, we set the interval of
the tilt angle to be two degrees, and the interval of the azimuth angle to be
five degrees, as shown in Fig. 3. Due to the antenna gain $g_{i,j}$, the
magnitude (i.e., $l_{2}$ norm) of each column in $\bf A$ varies significantly,
making problem (11) intractable. Both synthetic and real-world measurement
data are considered.

For synthetic data, the location of non-zero entries of synthetic data are
generated following the uniform distribution in the interval $\left[1,\
N\right]$. The accuracy is defined as the ratio of the number of correctly
recovered non-zero entries to the total number of non-zero entries, i.e., the
success of support recovery. Besides the NNOMP and the proposed WNOMP for the
$l_{0}$ norm constraint, we also consider the least absolute shrinkage and
selection operator (LASSO) for sparse recovery, which uses $l_{1}$
regularization and can be solved by proximal gradient descent. The performance
of the proposed algorithm is related to the problem size, i.e., $N$, $M$ and
$K$. As shown in Fig. 4(a), we fix $M$ = 32 and $K$ = 5, the accuracy
decreases when $N$ increases because the algorithms may select the wrong
column with more choices. Here, we change the value of $N$ by picking the
columns with the top $N$ highest magnitude. The NNOMP performs worst, while
WNOMP performs better than LASSO. As shown in Fig. 4(b), the accuracy
increases when $M$ increases for $N$ = 400 and $K$ = 5, which indicates that
the channel model is more accurate with more beam measurements. As shown in
Fig. 4(c), the accuracy decreases when $K$ increases for $N=400$ and $M=32$.

Figure 5: RSRP distribution in real-world drive testing.

As for the real-world measurement data, they are collected by using the drive-
test in street of the Chengdu city, China, as shown in Fig. 5. The carrier
frequency is $2.6$ GHz with $100$ MHz bandwidth. There are $N_{T}=32$ transmit
antennas at the base station. Depending on the velocity of the driving car,
dozens of samples are collected when passing through a $10\times 10$ meters
grid in the interval of several seconds. For each grid, the user equipment can
measure the RSRP of the CSI-RS beams (32 beams in total) and SSB beams (8
beams in total) transmitted from its serving cell and other neighborhood
cells. In general, the number of the measured beams in the serving cell is
more than the number of the measured beams in the neighborhood cells. Based on
${rsrp}_{m}(t)$, we obtain the RSRP by averaging the samples, then use them
and the coefficient matrix $\bf A$ to formulate and solve the problem (11).

TABLE II: MAE performance of predicting the SSB beams
(LASSO $\rightarrow$ NNOMP $\rightarrow$ WNOMP) |  # of grids |  MAE (dB)
---|---|---
Serving cell  |  718  |  6.14 $\rightarrow$ 4.45 $\rightarrow$ 4.93
Neighborhood  |  1566  |  7.32 $\rightarrow$ 5.85 $\rightarrow$ 5.58
All grids  |  2284  |  6.95 $\rightarrow$ 5.41 $\rightarrow$ 5.38
TABLE III: MAE performance of predicting the CSI-RS beams
(LASSO $\rightarrow$ NNOMP $\rightarrow$ WNOMP) |  # of grids |  MAE (dB)
---|---|---
Serving cell  |  631  |  6.07 $\rightarrow$ 5.37 $\rightarrow$ 4.89
Neighborhood  |  1055  |  11.5 $\rightarrow$ 10.6 $\rightarrow$ 8.96
All grids  |  1686  |  9.44 $\rightarrow$ 8.68 $\rightarrow$ 7.44

In particular, to evaluate the proposed channel modeling method in practical
systems, we collect the RSRP measurements both before and after rotating the
antenna array for certain degrees. The former RSRP data is used for channel
modeling by solving (11) and the solution denoted by $\widehat{\bf x}$. Then
we construct the coefficient matrix for the array after rotation, denoted by
$\widehat{\bf A}$. The estimated RSRP after array rotation is thus given by
$\widehat{\bf y}=\widehat{\bf A}\widehat{\bf x}$. Suppose the RSRP after
rotating as $\widetilde{\bf y}$, we then use the following mean absolute error
(MAE)

$\displaystyle{\rm MAE}={1\over M}\|\widehat{\bf y}-\widetilde{\bf y}\|_{1}$
(14)

as the performance measure.

As shown in Table II of predicting SSB beams, we consider 2284 grids in total
and calculate the average MAE. The MAE of serving cell is better than that of
neighborhood cells due to more beam measurements in serving cell. The MAE from
1686 grids of predicting CSI-RS beam is shown in Table III, which is worse
than that of the prediction of SSB beam, because the beam pattern of CSI-RS is
more complex. Although LASSO performs better than the NNOMP in the synthetic
data, its performance deteriorates in the real-world measurement data, where
the paths does not follow the uniform distribution. The MAE performance of the
proposed WNOMP is better than those of LASSO and NNOMP algorithms, which means
that the channel model established by WNOMP is more accurate than the channel
models established by LASSO and NNOMP.

##  V Conclusions

In this paper, we propose the LSCM to sense the physical environment that is
localized for the specific geographical location. The proposed LSCM can
generate a high-resolution multi-path channel in the angular domain by
exploiting the first-order statistics of RSRP measured from multiple beams.
The statistical distribution of the APS in the LSCM is consistent with that of
true wireless propagation environment. To construct the LSCM, we design the
WNOMP algorithm to deal with the coefficient matrix without column-wise
normalization at a low computational complexity. Simulation results verify the
WNOMP algorithm outperforms the classic LASSO and NNOMP algorithms in terms of
accuracy and MAE. Our future work will focus on the joint estimation of the
APS by taking advantage of the spatial consistency in multiple grids [13].

##  Appendix A Proof of Theorem 1

We want to establish the relationship between
$\mathbb{E}\left({rsrp}_{m}(t)\right)$ and
$\mathbb{E}\left({\alpha_{i,j}}(t)\right)$. We have

$\displaystyle\mathbb{E}\left({rsrp}_{m}(t)\right)$
$\displaystyle=\mathbb{E}_{\alpha_{i,j}(t),\omega_{i,j}(t),\omega_{x,y}(t)}\left({rsrp}_{m}(t)\right)$
$\displaystyle=\mathbb{E}_{\alpha_{i,j}(t)}\left(\mathbb{E}_{\omega_{i,j}(t),\omega_{x,y}(t)}\left({rsrp}_{m}(t)\right)\right)$
$\displaystyle=\mathbb{E}_{\alpha_{i,j}(t)}\left({RSRP}_{m}(t)\right),$ (15)

where ${RSRP}_{m}(t)$ is the expectation of ${rsrp}_{m}(t)$ w.r.t. both
$\omega_{x,y}(t)$ and $\omega_{i,j}(t)$. Then, we need to show how
${RSRP}_{m}(t)$ is expressed by $\alpha_{i,j}(t)$. According to the definition
in (2),

$\displaystyle{rsrp}_{m}(t)=P\left|\sum_{x,y}h_{x,y}(t)w_{x,y}^{(m)}\right|^{2}$
$\displaystyle=P\\!\left(\\!\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}\sqrt{\alpha_{i,j}(t)}g_{i,j}\sum_{x,y}\cos\left(\\!\psi^{(m)}_{i,j,x,y}+\left(\\!{\omega_{i,j}(t)}+{\omega_{x,y}(t)}\\!\right)\\!\right)\\!\right)^{2}$
$\displaystyle\
\quad+P\left(\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}\sqrt{\alpha_{i,j}(t)}g_{i,j}\sum_{x,y}\sin\left(\psi^{(m)}_{i,j,x,y}+\left({\omega_{i,j}(t)}+{\omega_{x,y}(t)}\\!\right)\\!\right)\\!\right)^{2},$
(16)

where
$\psi^{(m)}_{i,j,x,y}=2\pi{d_{x}x\over\lambda}\cos\theta_{i}\sin\varphi_{j}+2\pi{d_{y}y\over\lambda}\sin\theta_{i}-\phi_{x,y}^{(m)}$.

The expectation of first term and second term in (16) w.r.t.
${\omega_{i,j}(t)}$ and ${\omega_{x,y}(t)}$ are equal. Then, we have

$\displaystyle{RSRP}_{m}(t)$ $\displaystyle=\mathbb{E}_{\omega_{i,j}(t),\
\omega_{x,y}(t)}\left({rsrp}_{m}(t)\right)$ (17)
$\displaystyle=2P\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}\sum_{i^{\prime}=1}^{N_{V}}\sum_{j^{\prime}=1}^{N_{H}}\sqrt{\alpha_{i,j}(t)}\sqrt{\alpha_{i^{\prime}},j^{\prime}(t)}g_{i,j}g_{i^{\prime},j^{\prime}}$
$\displaystyle\qquad\qquad\times\mathbb{E}\Bigg{[}\sum_{x,y}\cos\left(\psi_{i,j,x,y}^{(m)}+\omega_{i,j}+\omega_{x,y}\right)$
$\displaystyle\qquad\qquad\qquad\
\sum_{x^{\prime},y^{\prime}}\cos\left(\psi_{i^{\prime},j^{\prime},x^{\prime},y^{\prime}}^{(m)}+\omega_{i^{\prime},j^{\prime}}+\omega_{x^{\prime},y^{\prime}}\right)\Bigg{]}.$
(18)

It is inferred from the distribution of $\omega_{i,j}$ that

$\displaystyle\mathbb{E}\Bigg{[}\sum_{x,y}\cos\left(\psi_{i,j,x,y}^{(m)}+\left(\omega_{i,j}+\omega_{x,y}\right)\right)$
$\displaystyle\quad\sum_{x^{\prime},y^{\prime}}\cos\left(\psi_{i^{\prime},j^{\prime},x^{\prime},y^{\prime}}^{(m)}+\left(\omega_{i^{\prime},j^{\prime}}+\omega_{x^{\prime},y^{\prime}}\right)\right)\Bigg{]}=0,$
(19)

when $(i,j)\neq\left(i^{\prime},j^{\prime}\right)$. As a result,

$\displaystyle{RSRP}_{m}(t)$
$\displaystyle=2P\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}\alpha_{i,j}(t)g_{i,j}^{2}\mathbb{E}\left[\left(\sum_{x,y}\cos\left(\psi_{i,j,x,y}^{(m)}+\omega_{i,j}+\omega_{x,y}\\!\right)\\!\right)^{2}\\!\right]$
(20)
$\displaystyle=P\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}\alpha_{i,j}(t)g_{i,j}^{2}$
$\displaystyle\qquad\times\sum_{x,y}\sum_{x^{\prime},y^{\prime}}\mathbb{E}\left[\cos\left(\psi_{i,j,x,y}^{(m)}+\omega_{x,y}-\psi_{i,j,x^{\prime},y^{\prime}}^{(m)}-\omega_{x^{\prime},y^{\prime}}\right)\right].$
(21)

At last, we can obtain

$\displaystyle{RSRP}_{m}(t)$
$\displaystyle=P\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}{\alpha_{i,j}}(t)g_{i,j}^{2}\Bigg{(}N_{x}N_{y}\left(1-e^{-\sigma^{2}}\right)$
$\displaystyle\qquad\
+e^{-\sigma^{2}}\sum_{x,y}\sum_{x^{\prime},y^{\prime}}\cos\left(\psi_{i,j,x,y}^{(m)}-\psi_{i,j,x^{\prime},y^{\prime}}^{(m)}\right)\Bigg{)}$
(22) $\displaystyle=\sum_{i=1}^{N_{V}}\sum_{j=1}^{N_{H}}{\rm
A}_{i,j}^{(m)}{\alpha_{i,j}}(t),$ (23)

where

$\displaystyle{\rm A}_{i,j}^{(m)}$ $\displaystyle\triangleq
Pg_{i,j}^{2}\Bigg{(}N_{x}N_{y}\left(1-e^{-\sigma^{2}}\right)$
$\displaystyle\qquad\qquad\
+e^{-\sigma^{2}}\sum_{x,y}\sum_{x^{\prime},y^{\prime}}\cos\left(\psi_{i,j,x,y}^{(m)}-\psi_{i,j,x^{\prime},y^{\prime}}^{(m)}\right)\Bigg{)}.$
(24)

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