# Liouville type theorems for subelliptic systems on the Heisenberg group with
general nonlinearity

Rong Zhang, Vishvesh Kumar and Michael Ruzhansky  School of Mathematical
Sciences, Nanjing Normal University, Nanjing, 210023, China
and
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent
University, Ghent, Belgium<EMAIL_ADDRESS>Department of Mathematics:
Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
<EMAIL_ADDRESS>/<EMAIL_ADDRESS>Department of Mathematics:
Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
and
School of Mathematical Sciences, Queen Mary University of London, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish Liouville type results for semilinear subelliptic
systems associated with the sub-Laplacian on the Heisenberg group
$\mathbb{H}^{n}$ involving two different kinds of general nonlinearities. The
main technique of the proof is the method of moving planes combined with some
integral inequalities replacing the role of maximum principles. As a special
case, we obtain the Liouville theorem for the Lane-Emden system on the
Heisenberg group $\mathbb{H}^{n}$, which also appears to be a new result in
the literature.

###### Key words and phrases:

Liouville-type theorem; Lane-Emden system; Method of moving plane; Heisenberg
group; Semilinear subelliptic systems, Integral inequalities.

###### 1991 Mathematics Subject Classification:

35R03; 35A01; 35D99; 35B53

##  1\. Introduction

The result related to the nonexistence of solutions, commonly known as
Liouville type theorem, serves as a powerful tool to obtain a priori estimates
for a system of equations on bounded domains in the Euclidean space
$\mathbb{R}^{n}$ or in the Heisenberg group $\mathbb{H}^{n}$. Indeed, in the
Euclidean setting, one uses the “blow up” method, also called the rescaling
method, developed by Gidas and Spruck [32]. After the blowing up, an equation
in a bounded domain becomes an equation in the whole Euclidean space. Now, a
Liouville type theorem on $\mathbb{R}^{n}$ followed by a contradiction
argument immediately provides a priori bounds. Following this technique,
Birindelli et al. [6] have developed the blow up method for the Heisenberg
group and proved a priori bounds for the supremum of the solution to the
nonlinear Dirichlet subelliptic problem on a bounded domain in
$\mathbb{H}^{n}$ satisfying an intrinsic cone condition. The idea was to
reduce the problem of establishing the desired a priori estimates on bounded
domains to a problem of Liouville type on the full Heisenberg group. The
Liouville theorem for sub-Laplacian on the Heisenberg group is well known, see
Folland [24, 25, 26], and Geller [30] and also [22, Section 3.2.8].

The main aim of this paper is to study the nonexistence of positive solutions
to the following semilinear subelliptic system on the Heisenberg group
$\mathbb{H}^{n}$:

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}u(\xi)=f(v(\xi)),&\xi\in\mathbb{H}^{n},\\\ \
-\Delta_{\mathbb{H}^{n}}v(\xi)=g(u(\xi)),&\xi\in\mathbb{H}^{n},\end{cases}$
(1.1)

and

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}u(\xi)=f(u(\xi),v(\xi)),&\xi\in\mathbb{H}^{n},\\\ \
-\Delta_{\mathbb{H}^{n}}v(\xi)=g(u(\xi),v(\xi)),&\xi\in\mathbb{H}^{n},\end{cases}$
(1.2)

where $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian on $\mathbb{H}^{n},n\geq
2$ and $f,g$ are two continuous functions. It is worth noting that a
particular instance of (1.1) is the famous Lane-Emden system. In the Euclidean
framework, the Lane-Emden system was investigated by several prominent
mathematicians. We will briefly discuss this development as follows. Recall
that the semi-linear Lane-Emden system in $\mathbb{R}^{n}$ takes the form

$\begin{cases}-\Delta u(x)=v^{p}(x),\\\ -\Delta v(x)=u^{q}(x).\end{cases}$
(1.3)

It is known that if $(p,q)$ are in critical or supercritical case (i.e.
$\frac{1}{p+1}+\frac{1}{q+1}\leq\frac{n-2}{n}$), then system (1.3) admits some
positive classical solutions on $\mathbb{R}^{n}$ (see [52]). The well-known
Lane-Emden conjecture states that in the subcritical case (i.e.
$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2}{n}$), system (1.3) does not have a
positive classical solution. The conjecture is known to be true for radial
solutions in all dimensions (see [44, 46]). For non-radial solutions, and
$n\leq 2$, the conjecture is a consequence of known results (see [45, 46,
55]). For $n\geq 3$, when $p$ and $q$ are both subcritical case (i.e.
$p,q\leq\frac{n+2}{n-2}$, but are not both equal to $\frac{n+2}{n-2}$), the
conjecture was also proved (see [28, 49]). For $n=3$, it was also proved by
Serrin and Zou [51] in the subcritical case (i.e.
$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2}{n}$), but under the additional
assumption that $(u,v)$ have at most polynomial growth at $\infty$. This
assumption was removed by Polacik, Quittner, and Souplet (see [48, 54]), and
therefore, the conjecture was solved for $n=3$.

In the last two decades, Liouville type results for nonlinear subelliptic
equations on the Heisenberg group $\mathbb{H}^{n}$ attracted a lot of
attention. In particular, the subelliptic Lane-Emden inequality

$\Delta_{\mathbb{H}^{n}}u+u^{p}\leq 0,$ (1.4)

on $\mathbb{H}^{n}$ was investigated by Birindelli et al. [5] and they proved
that, for $1<p\leq\frac{Q}{Q-2},$ the above inequality (1.4) does not possess
any positive classical solution. Here $Q=2n+2$ is the homogeneous dimension of
the Heisenberg group $\mathbb{H}^{n}$. Indeed, they went on to show that the
exponent $\frac{Q}{Q-2}$ is optimal in the sense that, for $p>\frac{Q}{Q-2},$
the subelliptic inequality (1.4) have a nontrivial positive solution. Their
results were applied to obtain a priori bounds for subelliptic Dirichlet
problems on bounded domains in $\mathbb{H}^{n}$ (see [6]). The results of [5]
were extended to the general stratified Lie groups in [13]. After these
seminal works, there are plenty of works devoted to the study of Liouville
type results for the subelliptic problem on $\mathbb{H}^{n}:$

$-\Delta_{\mathbb{H}^{n}}u=u^{p}.$ (1.5)

The Euclidean and elliptic counterpart of (1.5) was studied by Gidas and
Spruck [31]. They showed the nonexistence of a positive solution for the range
$0<p<\frac{n+2}{n-2}.$ Later, Chen and Li [14] simplified the proof of these
results using the so-called “the method of moving planes” and “the Kelvin
transform”. The method of moving planes has become a powerful tool in studying
qualitative properties for solutions of elliptic equations and systems. The
method of moving planes, which goes back to Alexandrov [1, 2, 3] and J. Serrin
[53], has been developed by many researchers later, (see [29, 64, 40, 20, 17,
15, 21, 4, 16, 42]).

Inspired by the work of Chen and Li [14], the method of moving planes was
introduced by Birindelli and Prajapat [9] in the setting of the Heisenberg
group $\mathbb{H}^{n},$ to study Liouville theorems for positive cylindrical
solutions to the semilinear subelliptic equation (1.5) on $\mathbb{H}^{n}$ by
suitably modifying the method of moving planes developed by Chen and Li [14].

We recall here that function $u$ defined on $\mathbb{H}^{n}$ is called
cylindrical if, for any $(x,y,t)\in\mathbb{H}^{n}$, where
$(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ and $t\in\mathbb{R}$ is the
anisotropic direction, we have $u(x,y,t)=u(r,t)$ with
$r=\sqrt{|x|^{2}+|y|^{2}}$.

Birindelli and Prajapat [9] established Gidas and Spruck [31] type results for
(1.5) and proved that (1.5) does not have any positive cylindrical solution
for $0<p<\frac{Q+2}{Q-2}.$ The sub-Laplacian $\mathbb{H}^{n}$ satisfies Bony’s
maximal principle [11] but it is not invariant under the usual hyperplane
reflection as in the Euclidean case. This made authors [9] define a new kind
of reflection called the “$H$-reflection” on the Heisenberg group with respect
to the plane $T_{\lambda}:=\\{(x,y,t)\in\mathbb{H}^{n}:t=\lambda\\}.$ The main
reason for working only with cylindrical functions in [9] is due to the fact
that the $H$-reflection with respect to the plane $T_{\lambda}$ leaves the
plane invariant but not fixed. Now, in order to settle the Gidas and Spruck
conjecture, it is enough to show that any positive solution to (1.4) is
cylindrical. Therefore, several attempts have been made in this direction; we
refer to [7, 10, 35] and references therein for more details. In the
aforementioned papers dealing with nonexistence results, the maximum principle
is exploited along with the method of moving planes and the Kelvin transform
or CR transform. We would like to also mention that many of the above results
are obtained for general nonlinearity $f(u)$ with some suitable conditions on
$f$ such as the Lipschitz continuity although we have discussed here some
particular instances. Finally, we mention a recent advance made by Ma and Ou
[43], in which the authors established the Liouville theorem for the classical
solution of (1.5) for the subcritical case $1<p<\frac{Q+2}{Q-2}.$ They used a
suitable generalization of the Jerison-Lee’s divergence identity and then an a
priori integral estimate.

Another very useful method to obtain Liouville type results is the combination
of integral inequalities along with the method of moving planes and the Kelvin
transform (see [18]). These integral inequalities substitute the role of
maximal principles in a differential form. The attribution of these integral
inequalities goes to the work of Terracini [56, 57]. The main advantage of
this method is that one can handle the subcritical, critical, and
supercritical cases of the Lane-Emden equation simultaneously [56]. After
these works, the method of moving planes combined with integral inequalities
was widely used to obtain Liouville type results for elliptic equations with
general nonlinearities [18, 60]. These methods were also adapted to study the
nonexistence of solutions to elliptic systems, we cite [12, 19, 36, 61, 37]
and references therein for more details.

Recently, several works have been devoted to the study of the classification
of cylindrical solutions to nonlinear subelliptic equations with general
nonlinearity on the Heisenberg group $\mathbb{H}^{n}$:

$-\Delta_{\mathbb{H}^{n}}u=f(u).$ (1.6)

Indeed, Yu [62] extended the method of integral inequalities [18, 56, 57] in
the setting of the Heisenberg group and combined it with the method of moving
planes on the Heisenberg group, to establish the Liouville type theorem for
(1.6). In [63], a similar type of problem was studied for a subellipitic
equation arising from the study of nonlocal equations on the Heisenberg group.
For Liouville type theorems obtained using different methods such as a vector
field method and test function method we refer to [59, 41] and references
therein.

Inspired by the above works, it is natural to study subelliptic systems (1.1)
and (1.2) on the Heisenberg group. Let us first define the notion of a weak
solution in this context.

We say that $(u,v)\in\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)\times\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)$ is a weak solution of system (1.2) if it
satisfies

$\int_{\mathbb{H}^{n}}\nabla_{\mathbb{H}^{n}}u\nabla_{\mathbb{H}^{n}}\varphi=\int_{\mathbb{H}^{n}}f(u,v)\varphi,\quad\varphi\in
C_{c}^{1}(\mathbb{H}^{n}),$ (1.7)
$\int_{\mathbb{H}^{n}}\nabla_{\mathbb{H}^{n}}v\nabla_{\mathbb{H}^{n}}\varphi=\int_{\mathbb{H}^{n}}g(u,v)\varphi,\quad\varphi\in
C_{c}^{1}(\mathbb{H}^{n}).$ (1.8)

In this paper, we are concerned with the nonexistence result of the
cylindrical weak solution to the semilinear subelliptic systems systems (1.1)
and (1.2) with general nonlinearity. The coupled nonlinearities in such
systems make it difficult to find a starting point to apply the method of
moving planes. To overcome this difficulty, based on CR transform [39]
together with the method of moving planes in $\mathbb{H}^{n},$ we use integral
inequalities of Terracini [56, 57], which earlier proved to be helpful in [62]
for the subellipic equation (1.6). Moreover, since the nonlinearities $f$ and
$g$ are only assumed to be continuous rather than Lipschitz continuous, the
classical Bony’s maximal principle [11] can not be applied. This is one of the
major reasons why we work with integral inequalities combined with the method
of moving planes.

In fact, by using the aforementioned tools, we establish the following results
in this direction.

######  Theorem 1.1.

Let $n\geq 2$ and let $(u,v)\in\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)\times\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)$ be a positive cylindrical solution to system
(1.1). Assume that $f,g:[0,+\infty)\rightarrow\mathbb{R}$ are two continuous
functions satisfying the following conditions:

$(i)$ $f(t)$ and $g(t)$ are nondecreasing in $(0,+\infty)$;

$(ii)$ $h(t):=\frac{f(t)}{t^{\frac{Q+2}{Q-2}}}$ and
$k(t):=\frac{g(t)}{t^{\frac{Q+2}{Q-2}}}$ are nonincreasing in $(0,+\infty)$;

$(iii)$ either $h$ or $k$ is not a constant function on
$(0,\sup_{\xi\in\mathbb{H}^{n}}v(\xi))$ and
$(0,\sup_{\xi\in\mathbb{H}^{n}}u(\xi)),$ respectively.

Then $(u,v)\equiv(C_{1},C_{2})$ for some constants $C_{1}$ and $C_{2}$ with
$f(C_{1})=0$ and $g(C_{2})=0.$

Based on Theorem 1.1, we state the following interesting consequence about the
nonexistence of positive cylindrical solutions to the Lane-Emden system on the
Heisenberg group $\mathbb{H}^{n}$ in the subcritical case for any $n\geq 1$
(cf. Remark 1.4).

######  Corollary 1.2.

Let $n\geq 1.$ Consider the following Lane-Emden system

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}u(\xi)=v^{p}(\xi),&\xi\in\mathbb{H}^{n},\\\ \
-\Delta_{\mathbb{H}^{n}}v(\xi)=u^{q}(\xi),&\xi\in\mathbb{H}^{n}.\end{cases}$
(1.9)

Then, there is no positive cylindrical solution

$(u,v)\in\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)\times\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)$

to (1.9) for $0<p,q<\frac{Q+2}{Q-2}$.

We note that Pohozaev and Véron [47] obtained a similar result for general
solution as of Corollary 1.2 in the case of inequality for the range
$1<p,q\leq\frac{Q}{Q-2}.$ They used test function method to established these
Liouville type theorems. These results were recently extended to Kirchhoff
type systems on $\mathbb{H}^{n}$ by the third author and his collaborators
[41] using the test function method.

The above Theorem 1.1 can be extended to a more general case given by the
system (1.2), that is both $f$ and $g$ depend on $(u,v)$. More precisely, we
have the following Liouville type theorem for the system (1.2).

######  Theorem 1.3.

Let $n\geq 2$ and let $(u,v)\in\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)\times\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)$ be a positive cylindrical solution to system
(1.2). Assume that $f,g:[0,+\infty)\times[0,+\infty)\rightarrow\mathbb{R}$ are
two continuous positive functions satisfying the following conditions:

$(i)$ $f(s,t)$ and $g(s,t)$ are nondecreasing in $t$ for fixed $s$ and,
$f(s,t)$ and $g(s,t)$ are nondecreasing in $s$ for fixed $t$;

$(ii)$ there exist $p_{1}\geq 0,q_{1}>0$ with $p_{1}+q_{1}=\frac{Q+2}{Q-2}$
such that $\frac{f(s,t)}{s^{p_{1}}t^{q_{1}}}$ is nonincreasing in $t$ for
fixed $s$ and $\frac{f(s,t)}{s^{p_{1}}t^{q_{1}}}$ is nonincreasing in $s$ for
fixed $t$;

$(iii)$ there exist $p_{2}>0,q_{2}\geq 0$ with $p_{2}+q_{2}=\frac{Q+2}{Q-2}$
such that $\frac{g(s,t)}{s^{p_{2}}t^{q_{2}}}$ is nonincreasing in $t$ for
fixed $s$ and $\frac{g(s,t)}{s^{p_{2}}t^{q_{2}}}$ is nonincreasing in $s$ for
fixed $t$;

$(iv)$ either $f$ or $g$ is not a constant multiple of $s^{p_{1}}t^{q_{1}}$ or
$s^{p_{2}}t^{q_{2}}$, respectively.

Then, we have $(u,v)\equiv(C_{1},C_{2})$ for some constants $C_{1}$ and
$C_{2}$ such that $f(C_{1},C_{2})=0$ and $g(C_{1},C_{2})=0$.

######  Remark 1.4.

We would like to discuss the condition $n\geq 2$ in our main results (Theorem
1.1 and Theorem 1.3). We need to impose this technical condition due to the
unavailability of corresponding results in the Euclidean space
$\mathbb{R}^{N}$ for $N=2$ (see [36]). We emphasise here that condition $n\geq
2$ is only used in the proof of main results because of the aforementioned
reason, although all the supporting Lemmata hold for all $n\geq 1.$

The paper is organized as follows: In Section 2 we collect some well-known
preliminaries related to the analysis of the Heisenberg group
$\mathbb{H}^{n}$. Section 3 is devoted to the proof of Theorem 1.1 by using
the method of moving planes along with newly developed integral inequalities.
In Section 4, we will present the proof of Theorem 1.3.

##  2\. Preliminaries: the Heisenberg group

In this section, we introduce some definitions, set up notation, and recall
some basic results concerning the Heisenberg group $\mathbb{H}^{n}$. We refer
to [23, 8, 58, 22, 50] for a complete overview of the material presented here.

The Heisenberg group $\mathbb{H}^{n}$ is $(\mathbb{R}^{2n+1},\circ),\,\,n\geq
1$, endowed with the group law $\circ$ defined by

$\xi\circ\bar{\xi}:=\bigg{(}x+\bar{x},y+\bar{y},t+\bar{t}+2\sum_{i=1}^{n}(x_{i}\bar{y}_{i}-y_{i}\bar{x}_{i})\bigg{)},$
(2.1)

where
$\xi:=(x,y,t)=(x_{1},\cdot\cdot\cdot,x_{n},y_{1},\cdot\cdot\cdot,y_{n},t)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}$
and $\bar{\xi}=(\bar{x},\bar{y},\bar{t})$.

Denote by $\delta_{\tau}$ the dilations on $\mathbb{H}^{n}$ defined as

$\delta_{\tau}(\xi)=(\tau x,\tau y,\tau^{2}t),\quad\tau>0,$ (2.2)

so that
$\delta_{\tau}(\bar{\xi}\circ\xi)=\delta_{\tau}(\bar{\xi})\circ\delta_{\tau}(\xi).$

The left invariant vector fields
$\\{X_{1},\cdot\cdot\cdot,X_{n},Y_{1},\cdot\cdot\cdot,Y_{n},T\\}$
corresponding to $\mathbb{H}^{n}$ are defined by

$X_{i}=\frac{\partial}{\partial x_{i}}+2y_{i}\frac{\partial}{\partial t},\quad
Y_{j}=\frac{\partial}{\partial y_{j}}-2x_{j}\frac{\partial}{\partial t},\quad
i,j=1,2,\ldots,n,\,\,\text{and}\,\,\,\,T=\frac{\partial}{\partial t},$

forming a basis for the Lie algebra of $\mathbb{H}^{n}$. It is easy to check
that

$[X_{i},Y_{j}]=-4T\delta_{ij},\ [X_{i},X_{j}]=[Y_{i},Y_{j}]=0,\quad
i,j=1,\cdot\cdot\cdot,n.$

The horizontal gradient on $\mathbb{H}^{n}$ of a suitable function $\phi$ is
defined as

$\nabla_{\mathbb{H}^{n}}\phi=(X_{1}\phi,\ldots,X_{n}\phi,Y_{1}\phi,\ldots,Y_{n}\phi).$
(2.3)

The sub-Laplacian $\Delta_{\mathbb{H}^{n}}$ on the Heisenberg group
$\mathbb{H}^{n}$ is defined as

$\displaystyle\Delta_{\mathbb{H}^{n}}$
$\displaystyle:=\sum_{i=1}^{n}(X_{i}^{2}+Y_{i}^{2})$ (2.4)
$\displaystyle=\sum_{i=1}^{n}\bigg{(}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\frac{\partial^{2}}{\partial
y_{i}^{2}}+4y_{i}\frac{\partial^{2}}{\partial x_{i}\partial
t}-4x_{i}(\frac{\partial^{2}}{\partial y_{i}\partial
t}+4x_{i}^{2}+y_{i}^{2})\frac{\partial^{2}}{\partial t^{2}}\bigg{)}.$

The family ${X_{1},\cdot\cdot\cdot,X_{n},Y_{1},\cdot\cdot\cdot,Y_{n},}$
satisfies the Hörmander’s rank condition (see [38]), which implies that
$\Delta_{\mathbb{H}^{n}}$ is hypoelliptic.

The even integer $Q=2n+2$ is called the homogeneous dimension of
$\mathbb{H}^{n}$. Denote by $|\xi|_{\mathbb{H}^{n}}$ the (Kaplan) distance
from $\xi$ to the zero (see [27]):

$|\xi|_{\mathbb{H}^{n}}:=\bigg{(}\sum_{i=1}^{n}(x_{i}^{2}+y_{i}^{2})^{2}+t^{2}\bigg{)}^{\frac{1}{4}}.$
(2.5)

The distance between two points $\xi$ and $\eta$ in $\mathbb{H}^{n}$ is
defined by

$d_{\mathbb{H}^{n}}(\xi,\eta)=|\eta^{-1}\circ\xi|_{\mathbb{H}^{n}},$

where $\eta^{-1}$ denotes the inverse of $\eta$ with respect to $\circ$,
namely $\eta^{-1}=-\eta$.

We say that a function $u$ on $\mathbb{H}^{n}$ is cylindrical  if for any
$(x,y,t)\in\mathbb{H}^{n}$, we have $u(x,y,t)=u(r,t)$ with
$r:=\sqrt{|x|^{2}+|y|^{2}}$.

It is easy to see that if $u$ is cylindrical then

$\Delta_{\mathbb{H}^{n}}u(r,t)=\frac{\partial^{2}u}{\partial
r^{2}}+\frac{2n-1}{r}\frac{\partial u}{\partial
r}+4r^{2}\frac{\partial^{2}u}{\partial t^{2}}.$

The open ball of radius $R>0$ centered at $\xi$ is the set

$B_{\mathbb{H}^{n}}(\xi,R):=\\{\eta\in\mathbb{H}^{n}\mid
d_{\mathbb{H}^{n}}(\xi,\eta)<R\\}.$

The map $\xi\mapsto|\xi|_{\mathbb{H}^{n}}$ is homogeneous of degree one with
respect to the dilations $\delta_{\tau}$ and consequently, we have

$|B_{\mathbb{H}^{n}}(\xi,R)|=|B_{\mathbb{H}^{n}}(0,R)|=R^{Q}|B_{\mathbb{H}^{n}}(0,1)|,$

where $|\cdot|$ denotes the Haar measure on $\mathbb{H}^{n}$.

##  3\. The Proof of Theorem 1.1

In this section, we outline the proof of Theorem 1.1 concerning the Liouville
type theorem for system (1.1). We begin with some statements required for the
proof.

Let $(u,v)$ be a pair of nonnegative continuous functions defined on
$\mathbb{H}^{n}$. We introduce the CR inversion of $u$ and $v$ centered at the
origin, denoted by $\bar{u}$ and $\bar{v}$, respectively as follows:

$\bar{u}(\xi)=\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q-2}}u(\tilde{\xi}),\,\,\bar{v}(\xi)=\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q-2}}v(\tilde{\xi}),\,\,\,\
\xi\in\mathbb{H}^{n}\backslash\\{0\\},$

where $\xi=(x,y,t)\in\mathbb{H}^{n}\backslash\\{0\\}$,
$\tilde{\xi}=(\tilde{x},\tilde{y},\tilde{t})\in\mathbb{H}^{n}$, and

$\tilde{x}_{i}=\frac{x_{i}t+y_{i}r^{2}}{{|\xi|_{\mathbb{H}}^{4}}},\quad\tilde{y}_{i}=\frac{y_{i}t-x_{i}r^{2}}{{|\xi|_{\mathbb{H}}^{4}}},\quad\tilde{t}=-\frac{t}{{|\xi|_{\mathbb{H}}^{4}}},$

where $r^{2}:=(|x|^{2}+|y|^{2}).$ The CR inversion on $\mathbb{H}^{n}$ was
first defined by Jerison and Lee [39] and serves as a suitable replacement of
the Kelvin transform on $\mathbb{R}^{n}.$ Obviously, $\bar{u}$ and $\bar{v}$
are continuous and nonnegative on $\mathbb{H}^{n}\backslash\\{0\\}$. A direct
computation yields (see [9, 62] for the proof) that:

######  Lemma 3.1.

Let $(u,v)\in\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)\times\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)$ be a positive weak cylindrical solution of
system (1.1). Then $(\bar{u},\bar{v})$ satisfy the following system:

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}\bar{u}(\xi)=\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)),&\xi\in\mathbb{H}^{n}\backslash\\{0\\},\\\
\
-\Delta_{\mathbb{H}^{n}}\bar{v}(\xi)=\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}g(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi)),&\xi\in\mathbb{H}^{n}\backslash\\{0\\}.\end{cases}$
(3.1)

Moreover, $(\bar{u},\bar{v})$ satisfy

$\lim_{|\xi|_{\mathbb{H}^{n}}\rightarrow\infty}|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi)=u(0),\
\lim_{|\xi|_{\mathbb{H}^{n}}\rightarrow\infty}|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)=v(0),$
(3.2)

and therefore, $\bar{u},\bar{v}\in L^{\tau+1}(\mathbb{H}^{n}\backslash
B_{r}(0))\cap L^{\infty}(\mathbb{H}^{n}\backslash B_{r}(0))$ for any $r>0$,
where $\tau=\frac{Q+2}{Q-2}$.

Define

$h(t):=\frac{f(t)}{t^{\frac{Q+2}{Q-2}}},\quad
k(t):=\frac{g(t)}{t^{\frac{Q+2}{Q-2}}},$ (3.3)

and substitute it in (3.1) to obtain the following system:

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}\bar{u}(\xi)=h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))\bar{v}^{\frac{Q+2}{Q-2}}(\xi),&\xi\in\mathbb{H}^{n}\backslash\\{0\\},\\\
\
-\Delta_{\mathbb{H}^{n}}\bar{v}(\xi)=k(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi))\bar{u}^{\frac{Q+2}{Q-2}}(\xi),&\xi\in\mathbb{H}^{n}\backslash\\{0\\}.\end{cases}$
(3.4)

Now, we are in a position to apply the moving planes method. Before that let
us recall the necessary notation.

We define the set

$\Sigma_{\lambda}:=\\{\xi:=(x,y,t)\in\mathbb{H}^{n}\mid t>\lambda\\},$

and the plane

$T_{\lambda}:=\\{\xi=(x,y,t)\in\mathbb{H}^{n}\mid t=\lambda\\}.$

For $\xi=(x,y,t)\in\mathbb{H}^{n}$, the $H$-reflection of $\xi$ with respect
to the plane $T_{\lambda}$ is defined by

$\xi_{\lambda}:=(y,x,2\lambda-t).$

It is well-known that $-\Delta_{\mathbb{H}^{n}}$ is invariant under the action
of $H$-reflection. This means that, if $-\Delta_{\mathbb{H}^{n}}u(\xi)=f(\xi)$
then $-\Delta_{\mathbb{H}^{n}}u(\xi_{\lambda})=f(\xi_{\lambda}).$

We define the $H$-reflection $\bar{u}_{\lambda}$ of a cylindrical function
$\bar{u}$ with respect to $T_{\lambda}$ by

$\bar{u}_{\lambda}(x,y,t)=\bar{u}_{\lambda}(r,t):=\bar{u}(r,2\lambda-t)=\bar{u}(y,x,2\lambda-t).$

Similarly, we have

$\bar{v}_{\lambda}(x,y,t)=\bar{v}_{\lambda}(r,t):=\bar{v}(r,2\lambda-t)=\bar{v}(y,x,2\lambda-t).$

Let $0_{\lambda}=(0,0,2\lambda)$, then it follows from (3.4) and the
invariance with respect to the $H$-reflection that
$(\bar{u}_{\lambda},\bar{v}_{\lambda})$ satisfies

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}\bar{u}_{\lambda}(\xi)=h(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}}(\xi),&\xi\in\mathbb{H}^{n}\backslash\\{0_{\lambda}\\},\\\
\
-\Delta_{\mathbb{H}^{n}}\bar{v}_{\lambda}(\xi)=k(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi))\bar{u}_{\lambda}^{\frac{Q+2}{Q-2}}(\xi),&\xi\in\mathbb{H}^{n}\backslash\\{0_{\lambda}\\}.\end{cases}$
(3.5)

In order to compare the value of $\bar{u}_{\lambda}$ and $\bar{v}_{\lambda}$
with $\bar{u}$ and $\bar{v}$, respectively, we define

$U_{\lambda}(\xi):=\bar{u}(\xi)-\bar{u}_{\lambda}(\xi),\quad
V_{\lambda}(\xi):=\bar{v}(\xi)-\bar{v}_{\lambda}(\xi).$

It is evident from the definition of $U_{\lambda}$ and $V_{\lambda}$ and (3.2)
that

$\lim_{|\xi|_{\mathbb{H}^{n}}\rightarrow\infty}U_{\lambda}(\xi)=0,\
\lim_{|\xi|_{\mathbb{H}^{n}}\rightarrow\infty}V_{\lambda}(\xi)=0.$ (3.6)

In order to use the method of moving planes, the first step is to show that we
can start the process. Namely, we first prove the following lemma.

######  Lemma 3.2.

For any fixed $\lambda>0$, we have $\bar{u},\bar{v}\in
L^{\tau+1}(\Sigma_{\lambda})\cap L^{\infty}(\Sigma_{\lambda})$,
$U_{\lambda}^{+}(\xi),V_{\lambda}^{+}(\xi)\in L^{\tau+1}(\Sigma_{\lambda})\cap
L^{\infty}(\Sigma_{\lambda})\cap H^{1}(\Sigma_{\lambda})$ with
$\tau=\frac{Q+2}{Q-2}$, where $U_{\lambda}^{+}=\max(U_{\lambda},0)$,
$V_{\lambda}^{+}=\max(V_{\lambda},0)$. Moreover, there exist $C_{\lambda}>0$
nonincreasing in $\lambda$, such that

$\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}\leq
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}},$
(3.7)
$\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}\leq
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}},$
(3.8)

where
$\Sigma_{\lambda}^{u}=\\{\xi\in\Sigma_{\lambda}\backslash\\{0_{\lambda}\\}\mid
h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u})>0,U_{\lambda}(\xi)>0\\}$ and
$\Sigma_{\lambda}^{v}=\\{\xi\in\Sigma_{\lambda}\backslash\\{0_{\lambda}\\}\mid
h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})>0,V_{\lambda}(\xi)>0\\}$.

###### Proof.

We only prove (3.7), the proof of (3.8) is similar. For any fixed $\lambda>0$,
there exists $r>0$ such that $\Sigma_{\lambda}\subset\mathbb{H}^{n}\backslash
B_{r}(0)$, then $\bar{u}$ and $U_{\lambda}^{+}\leq\bar{u}\in
L^{\tau+1}(\Sigma_{\lambda})\cap L^{\infty}(\Sigma_{\lambda})$ (see Lemma
3.1), and $\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}$ is integrable in
$\Sigma_{\lambda}$.

We choose a cylindrical symmetric cut-off function $0\leq\eta_{\epsilon}\leq
1$ on $\mathbb{H}^{n}$ such that

$\eta_{\epsilon}(\xi)=\begin{cases}1,&2\epsilon\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\frac{1}{\epsilon},\\\
0,&|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\epsilon\ or\
|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\geq\frac{2}{\epsilon},\end{cases}$

and

$|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|\leq\begin{cases}\frac{2}{\epsilon},&\epsilon\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq
2\epsilon,\\\
2\epsilon,&\frac{1}{\epsilon}\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\frac{2}{\epsilon}.\end{cases}$

Now, we choose $\phi_{\epsilon}=\eta_{\epsilon}^{2}U_{\lambda}^{+}$ as a test
function, then it follows from (3.4) and (3.5) using the simple calculation
$\nabla_{\mathbb{H}^{n}}\phi_{\epsilon}=2(\nabla_{\mathbb{H}^{n}}\eta_{\epsilon})\eta_{\epsilon}U_{\lambda}^{+}+\eta_{\xi}^{2}\nabla_{\mathbb{H}^{n}}U_{\lambda}^{+}$
that

$\displaystyle\int_{\Sigma_{\lambda}\cap\left[2\epsilon\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\frac{1}{\epsilon}\right]}$
$\displaystyle|\nabla_{\mathbb{H}^{n}}(U_{\lambda}^{+})|^{2}d\xi\leq\int_{\Sigma_{\lambda}}|\nabla_{\mathbb{H}^{n}}(U_{\lambda}^{+}\eta_{\epsilon})|^{2}d\xi$
(3.9)
$\displaystyle=\int_{\Sigma_{\lambda}}2\eta_{\epsilon}U_{\lambda}^{+}\nabla_{\mathbb{H}^{n}}U_{\lambda}^{+}\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}d\xi+\int_{\Sigma_{\lambda}}\eta_{\xi}^{2}|\nabla_{\mathbb{H}^{n}}U_{\lambda}^{+}|^{2}d\xi$
$\displaystyle\quad\quad\quad+\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi$
$\displaystyle=\int_{\Sigma_{\lambda}}(\nabla_{\mathbb{H}^{n}}U_{\lambda})2\eta_{\epsilon}U_{\xi}^{+}\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}d\xi$
$\displaystyle\quad\quad\quad+\int_{\Sigma_{\lambda}}\eta_{\xi}^{2}|\nabla_{\mathbb{H}^{n}}U_{\xi}^{+}|^{2}d\xi+\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi$
$\displaystyle=\int_{\Sigma_{\lambda}}\nabla_{\mathbb{H}^{n}}U_{\lambda}\nabla_{\mathbb{H}^{n}}\phi_{\epsilon}d\xi+\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi$
$\displaystyle=\int_{\Sigma_{\lambda}}-\Delta_{\mathbb{H}^{n}}U_{\lambda}\phi_{\epsilon}d\xi+\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi$
$\displaystyle=\int_{\Sigma_{\lambda}}\bigg{(}h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})\bar{v}^{\frac{Q+2}{Q-2}}-h(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda})\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}}\bigg{)}\phi_{\epsilon}d\xi+I_{\epsilon},$

where
$I_{\epsilon}=\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi$.

Since $h$ is a nonincreasing function,
$|\xi|_{\mathbb{H}^{n}}\geq|\xi_{\lambda}|_{\mathbb{H}^{n}}$ for
$\xi\in\Sigma_{\lambda}$ with $\lambda>0$ we conclude that, for
$\bar{v}(\xi)\geq\bar{v}_{\lambda}(\xi)\geq 0$, we have

$h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})\leq
h(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}).$ (3.10)

If $0\leq\bar{v}(\xi)\leq\bar{v}_{\lambda}(\xi)$, and since $f$ is
nondecreasing and $h$ is nononcreasing, we get

$\displaystyle
h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})\bar{v}^{\frac{Q+2}{Q-2}}$
$\displaystyle=\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\leq\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}$
(3.11)
$\displaystyle=h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda})\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}}\leq
h(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda})\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}}.$

Therefore, using (3.10) and (3.11) in (3.9) we obtain

$\displaystyle\int_{\Sigma_{\lambda}\cap\left[2\epsilon\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\frac{1}{\epsilon}\right]}|\nabla_{\mathbb{H}^{n}}(U_{\lambda}^{+})|^{2}d\xi$
$\displaystyle\leq\int_{\Sigma_{\lambda}}h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})(\bar{v}^{\frac{Q+2}{Q-2}}-\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}})\phi_{\epsilon}d\xi+I_{\epsilon}$
(3.12)
$\displaystyle=\int_{\Sigma_{\lambda}^{v}}h^{+}(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})(\bar{v}^{\frac{Q+2}{Q-2}}-\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}})\phi_{\epsilon}d\xi+I_{\epsilon}.$

Since $v$ is positive and locally bounded, there are constants
$0<C_{\lambda}^{{}^{\prime}}\leq C_{\lambda}^{{}^{\prime\prime}}$ such that

$0<C_{\lambda}^{{}^{\prime}}:=\inf_{\xi\in\Sigma_{\lambda}}|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)\leq|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)\leq
C_{\lambda}^{{}^{\prime\prime}},\quad\forall\ \xi\in\Sigma_{\lambda},$ (3.13)

and consequently, we have

$0\leq h^{+}(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})\leq
h^{+}(C_{\lambda}^{{}^{\prime}}):=C_{\lambda},\quad\forall\
\xi\in\Sigma_{\lambda}.$ (3.14)

This shows that $C_{\lambda}$ is nonincreasing in $\lambda$ since $h$ is a
nonincreasing function and for $\lambda_{1}\leq\lambda_{2},$ we can easily
deduce from (3.13) that $C_{\lambda_{1}}^{\prime}\leq
C_{\lambda_{2}}^{\prime}$. Indeed, it follows from (3.13) that
$C_{\lambda_{1}}^{{}^{\prime}}\leq\inf_{\xi\in\Sigma_{\lambda_{1}}}|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)$
and the condition $\lambda_{1}\leq\lambda_{2}$ then yields that
$C_{\lambda_{1}}^{{}^{\prime}}\leq\inf_{\xi\in\Sigma_{\lambda_{1}}}|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)\leq\inf_{\xi\in\Sigma_{\lambda_{2}}}|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)=C_{2}^{\prime}$
as $\Sigma_{\lambda_{1}}\subseteq\Sigma_{\lambda_{2}}.$ Moreover, for
$0\leq\bar{v}_{\lambda}\leq\bar{v}$ as $\bar{v}\in
L^{\infty}(\Sigma_{\lambda})$ for $\lambda>0$, we have using mean value
theorem that

$(\bar{v}^{\frac{Q+2}{Q-2}}-\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}})\leq\eta^{\frac{Q+2}{Q-2}-1}\frac{Q+2}{Q-2}V_{\lambda}^{+}\leq\bar{v}^{\frac{4}{Q-2}}\frac{Q+2}{Q-2}V_{\lambda}^{+}\leq\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}V_{\lambda}^{+},$
(3.15)

where $\eta$ lies between $(\bar{v}_{\lambda},\bar{v})$ and the last
inequality follows from the fact that $\bar{v}$ decays at infinity as
$\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q-2}}.$ Here and in the following of the
paper, we always use the same $C_{\lambda}$ to stand for different constants.

Combining the Hölder inequality with (3.14) and (3.15), we obtain, by setting
$\tau=\frac{Q+2}{Q-2},$

$\displaystyle\int_{\Sigma_{\lambda}\cap\left[2\epsilon\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\frac{1}{\epsilon}\right]}$
$\displaystyle|\nabla_{\mathbb{H}^{n}}U_{\lambda}^{+}|^{2}d\xi\leq\int_{\Sigma_{\lambda}^{v}}h^{+}(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})(\bar{v}^{\frac{Q+2}{Q-2}}-\bar{v}_{\lambda}^{\frac{Q+2}{Q-2}})\phi_{\epsilon}d\xi+I_{\epsilon}$
(3.16) $\displaystyle\leq
C_{\lambda}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{4}}\eta_{\epsilon}^{2}U_{\lambda}^{+}V_{\lambda}^{+}d\xi+I_{\epsilon}$
$\displaystyle\leq
C_{\lambda}\left(\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\right)^{\frac{2}{Q}}\left(\int_{\Sigma_{\lambda}}(V_{\lambda}^{+}\eta_{\epsilon})^{1+\tau}\right)^{\frac{1}{1+\tau}}\left(\int_{\Sigma_{\lambda}}(U_{\lambda}^{+}\eta_{\epsilon})^{1+\tau}\right)^{\frac{1}{1+\tau}}+I_{\epsilon}.$

Now, we claim that $I_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$. To
show this we define the following set

$D_{\epsilon}=\bigg{\\{}\xi\in\Sigma_{\lambda}:\epsilon\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq
2\epsilon\ or\
\frac{1}{\epsilon}\leq|0_{\lambda}^{-1}\circ\xi|_{\mathbb{H}^{n}}\leq\frac{2}{\epsilon}\bigg{\\}}.$

Then, it is clear from the definition of $\eta_{\epsilon}$ that

$\int_{D_{\epsilon}}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{Q}d\xi\leq C.$

Thus, a simple use of Hölder’s inequality yields

$\displaystyle I_{\epsilon}$
$\displaystyle=\int_{\Sigma_{\lambda}^{u}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi\leq\bigg{(}\int_{D_{\epsilon}}(U_{\lambda}^{+})^{\tau+1}d\xi\bigg{)}^{\frac{2}{\tau+1}}\bigg{(}\int_{D_{\epsilon}}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{Q}d\xi\bigg{)}^{\frac{2}{Q}}$
(3.17) $\displaystyle\leq
C\bigg{(}\int_{D_{\epsilon}}(U_{\lambda}^{+})^{\tau+1}d\xi\bigg{)}^{\frac{2}{\tau+1}}\rightarrow
0,\,\,\text{as}\,\,\epsilon\rightarrow 0,$

provided that $U_{\lambda}^{+}\in L^{\tau+1}(\Sigma_{\lambda}).$ This combined
with (3.16) implies that $U^{+}\in H^{1}(\Sigma_{\lambda})$ as
$U_{\lambda}^{+}\in L^{\tau+1}(\Sigma_{\lambda})\cap
L^{\infty}(\Sigma_{\lambda})$ and $\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}$ is
integrable in $\Sigma_{\lambda}$

On the other hand, by Sobolev inequality (see [34, 33]), we have

$\int_{\Sigma_{\lambda}}|\nabla_{\mathbb{H}^{n}}U_{\lambda}^{+}\eta_{\epsilon}|^{2}d\xi\geq
C\left(\int_{\Sigma_{\lambda}}(U_{\lambda}^{+}\eta_{\epsilon})^{\frac{2Q}{Q-2}}d\xi\right)^{\frac{Q-2}{Q}}.$
(3.18)

Applying monotone and dominated convergence theorem along with (3.18) by
letting $\epsilon\rightarrow 0$ in (3.16), we obtain

$\displaystyle\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}\leq
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}.$

This completes the proof of Lemma 3.2. ∎

######  Lemma 3.3.

There exists $\lambda_{o}>0$ such that for all $\lambda\geq\lambda_{o}$,
$U_{\lambda}(\xi)\leq 0$ and $V_{\lambda}(\xi)\leq 0$ for all
$\xi\in\Sigma_{\lambda}\backslash\\{0_{\lambda}\\}$.

###### Proof.

Since $\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}$ is integrable in
$\mathbb{H}^{n}\backslash B_{r}(0)$ for any $r>0$, we have

$\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\leq\int_{\Sigma_{\lambda}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\rightarrow
0,\quad as\ \lambda\rightarrow+\infty.$

It follows that there exists $\lambda_{o}>0$ such that, for all
$\lambda\geq\lambda_{o}$, we get

$C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{1}{Q}}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{1}{Q}}\leq\frac{1}{2}.$

By Lemma 3.2, we obtain that

$\int_{\Sigma_{\lambda}}|U_{\lambda}^{+}|^{2}d\xi=0,\ \text{and}\
\int_{\Sigma_{\lambda}}|V_{\lambda}^{+}|^{2}d\xi=0,$

for all $\lambda\geq\lambda_{o}$, this implies that $U_{\lambda}(\xi)\leq 0$
and $V_{\lambda}(\xi)\leq 0$ for all
$\xi\in\Sigma_{\lambda}\backslash\\{0_{\lambda}\\}$ and
$\lambda\geq\lambda_{o}$.

This completes the proof of Lemma 3.3. ∎

Next, we can move the plane from the right to the left. More precisely, we
define

$\lambda_{1}:=\inf\\{\lambda>0\mid U_{\mu}(\xi)\leq 0,\ V_{\mu}(\xi)\leq 0,\
\forall\,\xi\in\Sigma_{\mu}\backslash\\{0_{\mu}\\},\forall\,\mu\geq\lambda\\}.$

This is well-defined by Lemma 3.3. Now we have the following result.

######  Lemma 3.4.

If $\lambda_{1}>0$, then $U_{\lambda_{1}}(\xi)\equiv 0$,
$V_{\lambda_{1}}(\xi)\equiv 0$ for all
$\xi\in\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$.

###### Proof.

We deduce from Lemma 3.3, by using continuity of $U_{\lambda}$ and
$V_{\lambda}$, that $U_{\lambda_{1}}(\xi)\leq 0$, $V_{\lambda_{1}}(\xi)\leq 0$
for $\xi\in\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$.

By (3.11), for $\xi\in\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$, we
have $h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v})\bar{v}^{\frac{Q+2}{Q-2}}\leq
h(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}})\bar{v}_{\lambda_{1}}^{\frac{Q+2}{Q-2}}$,
providing that $V_{\lambda_{1}}(\xi)\leq 0$. Therefore, by (3.4) and (3.5), we
get

$\displaystyle-\Delta_{\mathbb{H}^{n}}\bar{u}\leq-\Delta_{\mathbb{H}^{n}}\bar{u}_{\lambda_{1}},$

which implies that $-\Delta_{\mathbb{H}^{n}}U_{\lambda_{1}}\leq 0$. Since
$U_{\lambda_{1}}\leq 0$, by the maximum principle, either
$U_{\lambda_{1}}\equiv 0$ or $U_{\lambda_{1}}<0$ in
$\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$.

Now, let us suppose that $U_{\lambda_{1}}<0$ in
$\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$. We note that
$\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\chi_{\Sigma_{\lambda}^{v}}\rightarrow
0$ pointwise as $\lambda\rightarrow\lambda_{1}$ in
$\mathbb{H}^{n}\backslash(T_{\lambda_{1}}\cup\\{0_{\lambda_{1}}\\})$, where
$\chi_{A}$ is the characteristic function of the set $A$. Also, for
$\lambda\in(\lambda_{1}-\delta,\lambda_{1}]$, we have
$\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\chi_{\Sigma_{\lambda}^{v}}\leq\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\chi_{\Sigma_{\lambda-\delta}}\in
L^{1}(\Sigma_{\lambda})$. Therefore, by dominated convergence theorem, we
obtain

$\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\rightarrow
0,\ as\ \lambda\rightarrow\lambda_{1},$

and consequently, for $\lambda\in(\lambda_{1}-\delta,\lambda_{1})$,

$C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{1}{Q}}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{1}{Q}}\leq\frac{1}{2}.$

Following the similar argument as in the proof of Lemma 3.3, we conclude that
$U_{\lambda}(\xi)\leq 0$ and $V_{\lambda}(\xi)\leq 0$ in
$\Sigma_{\lambda}\backslash\\{0_{\lambda}\\}$ for
$\lambda\in(\lambda_{1}-\delta,\lambda_{1}]$, which contradicts with the
definition of $\lambda_{1}$.

This completes the proof of Lemma 3.4. ∎

######  Lemma 3.5.

Assume that $u,v,f$ and $g$ are as in the statement of Theorem 1.1, and
suppose that $(u,v)$ is positive cylindrical on $\mathbb{H}^{n}$. Then
$\bar{u},\bar{v}$ are symmetric with respect to $T_{0}$, that is, $\bar{u}$
and $\bar{v}$ are even functions in $t$-variable.

###### Proof.

To prove that $\bar{u},\bar{v}$ are symmetric, we still use the method of
moving plane and prove the symmetry. We can carry out the procedure as above.
If $\lambda_{1}>0$, then it follows from Lemma 3.4 that $\bar{u},\bar{v}$ are
symmetric with respect to $T_{\lambda_{1}}$. This means that
$\bar{u}=\bar{u}_{\lambda_{1}}$ and $\bar{v}=\bar{v}_{\lambda_{1}},$ which
implies that
$-\Delta_{\mathbb{H}^{n}}\bar{u}=-\Delta_{\mathbb{H}^{n}}\bar{u}_{\lambda_{1}}.$
This combined with $\bar{v}=\bar{v}_{\lambda_{1}},$ (3.4) and (3.5) show that

$h(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))=h(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)).$
(3.19)

By the assumption that $h$ is nonincreasing and the fact that
$|\xi_{\lambda_{1}}|<|\xi|$ for $\xi\in\Sigma_{\lambda_{1}},$ we conclude from
(3.19) that $h(s)$ is constant in a left neighbourhood of
$s=|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)=v\left(\frac{(\tilde{x},\tilde{y},-t)}{|\xi|_{\mathbb{H}^{n}}^{4}}\right),\
t>\lambda_{1}$. In a similar manner, we can show that $h$ is constant in a
right neighbourhood of
$s=|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)=v\left(\frac{(\tilde{x},\tilde{y},-t)}{|\xi|_{\mathbb{H}^{n}}^{4}}\right),\
t<\lambda_{1}$. In particular, this holds for $s$ close to $0$ because
$s=v\left(\frac{(\tilde{x},\tilde{y},-t)}{|\xi|_{\mathbb{H}^{n}}^{4}}\right)$
converges to $0$ at infinity by (3.2). Therefore we conclude that if
$\lambda_{1}>0,$ then $h$ is constant on
$(0,\sup_{\xi\in\mathbb{H}^{n}}v(\xi)),$ which is a contradiction to our
assumption $(iii)$. We also derive the same contradiction if $k$ is assumed
not be constant on $(0,\sup_{\xi\in\mathbb{H}^{n}}u(\xi)).$

If $\lambda_{1}=0$, then we conclude by continuity that
$\bar{u}(\xi)\leq\bar{u}_{0}(\xi)$ and $\bar{v}(\xi)\leq\bar{v}_{0}(\xi)$ for
all $\xi\in\Sigma_{0}$. In this case, we can also perform the moving plane
procedure from the left and find a corresponding $\lambda^{\prime}_{1}$. If
$\lambda^{\prime}_{1}<0$, an analogue to Lemma 3.4 shows that
$\bar{u},\bar{v}$ are symmetric with respect to $T_{\lambda^{\prime}_{1}}$ and
we can obtain contradiction in this case as for $\lambda_{1}>0$ previously. If
$\lambda^{\prime}_{1}=0$, then we conclude by continuity that
$\bar{u}_{0}(\xi)\leq\bar{u}(\xi)$ and $\bar{v}_{0}(\xi)\leq\bar{v}(\xi)$ for
all $\xi\in\Sigma_{0}$. The fact and the above inequalities imply that
$\bar{u}_{0}(\xi)=\bar{u}(\xi)$ and $\bar{v}_{0}(\xi)=\bar{v}(\xi)$ and
$\lambda_{1}=\lambda_{1}^{\prime}=0.$ This shows that $\bar{u}$ and $\bar{v}$
are symmetric with respect to $T_{0}$. This completes the proof of Lemma 3.5.
∎

Next we are ready to present the proof of Theorem 1.1.

###### Proof.

By Lemma 3.5, we conclude that the CR inversion $\bar{u}$ and $\bar{v}$ of
functions $u$ and $v$ respectively are symmetric with respect to $T_{0},$ that
is, $\bar{u}$ and $\bar{v}$ of $u$ and $v$ are even in $t$-variable. Since the
choice of origin is arbitrary in $t$-axis, then we conclude that $u$ and $v$
are independent of $t$. However, this shows that $u$ and $v$ satisfy the
system

$\begin{cases}\ -\Delta u(\xi)=f(v(\xi)),&\xi\in\mathbb{R}^{2n},\\\ \ -\Delta
v(\xi)=g(u(\xi)),&\xi\in\mathbb{R}^{2n}.\end{cases}$ (3.20)

Since $f,g$ are nondecreasing in $(0,\infty)$, and

$\frac{f(t)}{t^{\frac{2n+2}{2n-2}}}=\frac{f(t)}{t^{\frac{Q+2}{Q-2}}}t^{\frac{Q+2}{Q-2}-\frac{2n+2}{2n-2}},$
$\frac{g(t)}{t^{\frac{2n+2}{2n-2}}}=\frac{g(t)}{t^{\frac{Q+2}{Q-2}}}t^{\frac{Q+2}{Q-2}-\frac{2n+2}{2n-2}},$

are nonincreasing in $t$, it follows immediately from [36, Theorem 1.1] as
$2n\geq 3$ along with the assumption $h$ or $k$ is not constant function, that
$(u,v)\equiv(C_{1},C_{2})$ for some constant $C_{1}$ and $C_{2}$ with
$f(C_{1})=0$ and $g(C_{2})=0.$

This completes the proof of Theorem 1.1. ∎

Next we provide a proof of Corollary 1.2.

Proof of Corollary 1.2: For $n\geq 2,$ the proof is an immediate consequence
of Theorem 1.1. For $n=1,$ we repeat the proof of Theorem 1.1 including all
supporting lemmata (see Remark 1.4) for the special case when $f(t)=t^{p}$ and
$g(t)=t^{q}$ and use the corresponding Euclidean result from [45, 51, 55] (see
also [54]) for the dimension two instead of [36, Theorem 1.1] to complete the
proof.

##  4\. Proof of Theorem 1.3

In this section, by using the similar methods as used in Section 3, we study
the positive cylindrical (weak) solutions to the semilinear systems in the
Heisenberg group with more general nonlinearity

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}u(\xi)=f(u(\xi),v(\xi)),&\xi\in\mathbb{H}^{n},\\\ \
-\Delta_{\mathbb{H}^{n}}v(\xi)=g(u(\xi),v(\xi)),&\xi\in\mathbb{H}^{n},\end{cases}$
(4.1)

where $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian on the Heisenberg group
$\mathbb{H}^{n}$.

We establish Liouville type result stated in Theorem 1.3 for the system (4.1).
The spirit of the proofs is still the moving plane method and will be
completed similarly to that of Theorem 1.1 with the help of some analogue of
Lemmas 3.2-3.4 in this more general context. Here, we carry over the notation
used in the previous Section.

Suppose that $(u,v)\in\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)\times\left(H_{loc}^{1}(\mathbb{H}^{n})\cap
C^{0}(\mathbb{H}^{n})\right)$ be a weak cylindrical solution of system (4.1).
If $(u,v)$ solves (4.1), then a direct calculation yields that

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}\bar{u}(\xi)=\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)),&\xi\in\mathbb{H}^{n}\backslash\\{0\\},\\\
\
-\Delta_{\mathbb{H}^{n}}\bar{v}(\xi)=\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}g(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)),&\xi\in\mathbb{H}^{n}\backslash\\{0\\},\end{cases}$
(4.2)

and

$\begin{cases}\
-\Delta_{\mathbb{H}^{n}}\bar{u}_{\lambda}(\xi)=\frac{1}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi)),&\xi\in\mathbb{H}^{n}\backslash\\{0_{\lambda}\\},\\\
\
-\Delta_{\mathbb{H}^{n}}\bar{v}_{\lambda}(\xi)=\frac{1}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}g(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi)),&\xi\in\mathbb{H}^{n}\backslash\\{0_{\lambda}\\}.\end{cases}$
(4.3)

######  Lemma 4.1.

Under the assumptions of Theorem 1.3, for any fixed $\lambda>0$, we have
$\bar{u},\bar{v}\in L^{\tau+1}(\Sigma_{\lambda})\cap
L^{\infty}(\Sigma_{\lambda})$, $U_{\lambda}^{+}(\xi),V_{\lambda}^{+}(\xi)\in
L^{\tau+1}(\Sigma_{\lambda})\cap L^{\infty}(\Sigma_{\lambda})\cap
H^{1}(\Sigma_{\lambda})$. Moreover, there exists $C_{\lambda}>0$, non-
increasing in $\lambda$, such that the followling estimates holds:

$\displaystyle\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}\leq$
$\displaystyle
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}$
$\displaystyle+C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}},$
(4.4)

and

$\displaystyle\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}\leq$
$\displaystyle
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}$
$\displaystyle+C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}.$
(4.5)

###### Proof.

We just prove (4.1), the proof of (4.1) is similar. For any fixed $\lambda>0$,
there exists $r>0$ such that $\Sigma_{\lambda}\subset\mathbb{H}^{n}\backslash
B_{r}(0)$, then $\bar{u}$ and $U_{\lambda}^{+}\leq\bar{u}\in
L^{\tau+1}(\Sigma_{\lambda})\cap L^{\infty}(\Sigma_{\lambda})$ and
$\frac{1}{|\xi|_{\mathbb{H}^{n}}}$ is integrable in $\Sigma_{\lambda}$.

Since $\lambda>0$, it follows that
$|\xi|_{\mathbb{H}^{n}}>|\xi_{\lambda}|_{\mathbb{H}^{n}}$ for all
$\xi\in\Sigma_{\lambda}$, we still choose a cylindrical symmetric cut-off
function $\eta_{\epsilon}$ as in Section 3.

(i) If $U_{\lambda}(\xi)\geq 0$ and $V_{\lambda}(\xi)\leq 0$, then by the
assumptions $(i)$ in Theorem 1.3, we have

$f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))\geq
f\big{(}|\xi^{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\big{)}.$
(4.6)

By the assumptions $(ii)$ in Theorem 1.3, we have

$\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)})}{[|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi)]^{p_{1}}[|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}]^{q_{1}}}\geq\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{[|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi)]^{p_{1}}[|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)]^{q_{1}}}.$
(4.7)

By using (4.6) in (4.7), we deduce that

$f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))\geq
f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))\left(\frac{|\xi_{\lambda}|_{\mathbb{H}^{n}}}{|\xi|_{\mathbb{H}^{n}}}\right)^{Q+2}\Big{(}\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\Big{)}^{\frac{Q+2}{Q-2}},$

which further implies that

$\displaystyle\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}$ $\displaystyle
f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))-\frac{1}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))$
$\displaystyle\leq\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))\bigg{(}1-\Big{(}\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\Big{)}^{\frac{Q+2}{Q-2}}\bigg{)}$
$\displaystyle\leq\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))\frac{Q+2}{Q-2}\bigg{(}1-\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\bigg{)}$
$\displaystyle\leq\frac{C}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{\bar{u}(\xi)}(\bar{u}(\xi)-\bar{u}_{\lambda}(\xi))$
$\displaystyle\leq\frac{C}{|\xi|_{\mathbb{H}^{n}}^{4}}\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi)}(\bar{u}(\xi)-\bar{u}_{\lambda}(\xi))$
$\displaystyle\leq\frac{C}{|\xi|_{\mathbb{H}^{n}}^{4}}\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi)(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))^{\frac{4}{Q-2}}}(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))^{\frac{4}{Q-2}}(\bar{u}(\xi)-\bar{u}_{\lambda}(\xi))$
$\displaystyle\leq
C\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi)(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))^{\frac{4}{Q-2}}}(\bar{v}(\xi))^{\frac{4}{Q-2}}(\bar{u}(\xi)-\bar{u}_{\lambda}(\xi))$
$\displaystyle\leq\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}(\bar{u}(\xi)-\bar{u}_{\lambda}(\xi)),$

where we have used the mean value theorem with the observation that
$\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\leq 1$ in the second inequality
and the fact that $f$ is a positive continuous function such that
$\frac{f(s,t)}{s\,\,t^{\frac{4}{Q-2}}}$ is nonincreasing in $s$ (assumption
$(ii)$) along with the $\bar{v}(\xi)$ decay as
$\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q-2}}$ to deduce the last inequality.

$(ii)$ If $U_{\lambda}(\xi)\geq 0$ and $V_{\lambda}(\xi)>0$, then by arguing
similar to $(i)$ we have

$\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}\bar{u}_{\lambda}^{p_{1}}(\xi)\bar{v}_{\lambda}^{q_{1}}(\xi)}\geq\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}\bar{u}^{p_{1}}(\xi)\bar{v}^{q_{1}}(\xi)},$

that is

$\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi^{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}\geq\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\bigg{(}\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\bigg{)}^{p_{1}}\bigg{(}\frac{\bar{v}_{\lambda}(\xi)}{\bar{v}(\xi)}\bigg{)}^{q_{1}}.$

So we have

$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}-$
$\displaystyle\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle\leq$
$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\bigg{(}1-\Big{(}\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\Big{)}^{p_{1}}\Big{(}\frac{\bar{v}_{\lambda}(\xi)}{\bar{v}(\xi)}\Big{)}^{q_{1}}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\bigg{(}1-\Big{(}\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\Big{)}^{\frac{Q+2}{Q-2}}\Big{(}\frac{\bar{v}_{\lambda}(\xi)}{\bar{v}(\xi)}\Big{)}^{\frac{Q+2}{Q-2}}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\frac{Q+2}{Q-2}\bigg{(}\Big{(}1-\frac{\bar{u}_{\lambda}(\xi)}{\bar{u}(\xi)}\Big{)}+\Big{(}1-\frac{\bar{v}_{\lambda}(\xi)}{\bar{v}(\xi)}\Big{)}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\frac{C}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}\bigg{(}\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{\bar{u}(\xi)}\big{(}\bar{u}(\xi)-\bar{u}_{\lambda}(\xi)\big{)}$
$\displaystyle+\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{\bar{v}(\xi)}\big{(}\bar{v}(\xi)-\bar{v}_{\lambda}(\xi)\big{)}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}\bigg{(}\big{(}\bar{u}(\xi)-\bar{u}_{\lambda}(\xi)\big{)}+\big{(}\bar{v}(\xi)-\bar{v}_{\lambda}(\xi)\big{)}\bigg{)}.$

$(iii)$ If $U_{\lambda}(\xi)<0$ and $V_{\lambda}(\xi)\geq 0$, then we have
change the role of $\bar{u},\bar{v}$ in case $(i)$ and obtain

$\displaystyle\frac{1}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))$
$\displaystyle-\frac{1}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi^{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))$
$\displaystyle\leq\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}(\bar{v}(\xi)-\bar{v}_{\lambda}(\xi)).$

$(iv)$ If $U_{\lambda}(\xi)<0$ and $V_{\lambda}(\xi)<0$, then we have

$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle\leq\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))}{|\xi^{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle=\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{[|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi)]^{p_{1}}[|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi)]^{q_{1}}}\bar{u}_{\lambda}^{p_{1}}(\xi)\bar{v}_{\lambda}^{q_{1}}(\xi)$
$\displaystyle\leq\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{[|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi)]^{p_{1}}[|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi)]^{q_{1}}}\bar{u}_{\lambda}^{p_{1}}(\xi)\bar{v}_{\lambda}^{q_{1}}(\xi)$
$\displaystyle\leq\frac{f(|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda}(\xi),|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda}(\xi))}{|\xi_{\lambda}|_{\mathbb{H}^{n}}^{Q+2}}.$

The rest of calculation follows exactly similar to case $(ii),$ therefore we
skip it.

Therefore, we deduce from (4.2) and (4.3) by using cases $(i)$-$(iv)$ that

$-\Delta_{\mathbb{H}^{n}}U_{\lambda}\leq\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}\bigg{(}\big{(}\bar{u}(\xi)-\bar{u}_{\lambda}(\xi)\big{)}^{+}+\big{(}\bar{v}(\xi)-\bar{v}_{\lambda}(\xi)\big{)}^{+}\bigg{)}.$
(4.8)

Similarly, we obtain

$-\Delta_{\mathbb{H}^{n}}V_{\lambda}\leq\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}\bigg{(}\big{(}\bar{u}(\xi)-\bar{u}_{\lambda}(\xi)\big{)}^{+}+\big{(}\bar{v}(\xi)-\bar{v}_{\lambda}(\xi)\big{)}^{+}\bigg{)}.$
(4.9)

Hence, after a calculation similar to that of (3.9), we conclude that

$\displaystyle\int_{\Sigma_{\lambda}}|\nabla_{\mathbb{H}^{n}}(U_{\lambda}^{+}\eta_{\epsilon})|^{2}d\xi=$
$\displaystyle\int_{\Sigma_{\lambda}}-\Delta_{\mathbb{H}^{n}}(U_{\lambda}^{+}\eta_{\epsilon}^{2}U_{\lambda}^{+})d\xi+I_{\epsilon}$
(4.10) $\displaystyle\leq$
$\displaystyle\int_{\Sigma_{\lambda}}\frac{C_{\lambda}}{|\xi|_{\mathbb{H}^{n}}^{4}}(U_{\lambda}^{+}+V_{\lambda}^{+})\eta_{\epsilon}^{2}U_{\lambda}^{+}d\xi+I_{\epsilon},$

where
$I_{\epsilon}=\int_{\Sigma_{\lambda}^{u}}(U_{\lambda}^{+})^{2}|\nabla_{\mathbb{H}^{n}}\eta_{\epsilon}|^{2}d\xi$.
We can also prove that $I_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$
as in (3.17).

By the Hölder inequality, we obtain

$\displaystyle\int_{\Sigma_{\lambda}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{4}}(U_{\lambda}^{+}+V_{\lambda}^{+})$
$\displaystyle\eta_{\epsilon}^{2}U_{\lambda}^{+}d\xi\leq
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+}\eta_{\epsilon})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}$
(4.11)
$\displaystyle+C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}(\int_{\Sigma_{\lambda}}(V_{\lambda}^{+}\eta_{\epsilon})^{\frac{2Q}{Q-2}}d\xi\big{)}^{\frac{Q-2}{2Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+}\eta_{\epsilon})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}.$

Now, combining (4.10) with (4.11) and letting $\epsilon\rightarrow 0$ in
(4.10), we get

$\displaystyle\int_{\Sigma_{\lambda}}|\nabla_{\mathbb{H}^{n}}U_{\lambda}^{+}|^{2}d\xi\leq$
$\displaystyle
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}$
(4.12)
$\displaystyle+C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}(\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\big{)}^{\frac{Q-2}{2Q}}\bigg{(}\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}.$

Similarly, we have

$\displaystyle\int_{\Sigma_{\lambda}}|\nabla_{\mathbb{H}^{n}}V_{\lambda}^{+}|^{2}d\xi\leq$
$\displaystyle
C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{Q}}$
(4.13)
$\displaystyle+C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{2}{Q}}(\int_{\Sigma_{\lambda}}(U_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\big{)}^{\frac{Q-2}{2Q}}\bigg{(}\int_{\Sigma_{\lambda}}(V_{\lambda}^{+})^{\frac{2Q}{Q-2}}d\xi\bigg{)}^{\frac{Q-2}{2Q}}.$

This completes the proof of Lemma 4.1. ∎

######  Lemma 4.2.

Under the assumptions of Theorem 1.3, there exists $\lambda_{o}>0$ such that
for all $\lambda\geq\lambda_{o}$, $U_{\lambda}\leq 0$ and $V_{\lambda}\leq 0$
for all $\xi\in\Sigma_{\lambda}\backslash\\{0_{\lambda}\\}$.

###### Proof.

This proof of Lemma 4.2 is similar to Lemma 3.3, therefore, we skip it. ∎

Now we can move the plane from the right to the left. More precisely, we
define

$\lambda_{1}=\inf\\{\lambda>0\mid U_{\mu}(\xi)\leq 0,\ V_{\mu}(\xi)\leq 0,\
\xi\in\Sigma_{\mu}\backslash\\{0_{\mu}\\},\ \mu\geq\lambda\\},$

then we have the following lemma.

######  Lemma 4.3.

If $\lambda_{1}>0$, then $U_{\lambda_{1}}(\xi)\equiv 0$ and
$V_{\lambda_{1}}(\xi)\equiv 0$ for all
$\xi\in\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$.

###### Proof.

By continuity, we deduce that $U_{\lambda_{1}}(\xi)\leq 0$,
$V_{\lambda_{1}}(\xi)\leq 0$ for
$\xi\in\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$. Next, we will
prove that if $U_{\lambda_{1}}=0$ at some point
$\hat{\xi}\in\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$, then in a
neighborhood of $\hat{\xi}$, $U_{\lambda_{1}}=0$, and hence by continuity,
$U_{\lambda_{1}}\equiv 0$ in
$\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$.

Since $U_{\lambda_{1}}(\hat{\xi})=0$, we have
$|\xi|_{\mathbb{H}^{n}}^{Q+2}\bar{u}(\hat{\xi})>|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q+2}\bar{u}_{\lambda_{1}}(\hat{\xi})$
for $\xi\in\Omega$ a neighborhood of $\hat{\xi}$. By using the same arguments
as we used in the proof of Lemma 4.1, we have

$\begin{cases}-\Delta_{\mathbb{H}^{n}}U_{\lambda_{1}}+CU_{\lambda_{1}}\leq
0,&in\ \Omega,\\\ U_{\lambda_{1}}\leq 0,\ U_{\lambda_{1}}(\hat{\xi})=0,&in\
\Omega,\end{cases}$ (4.14)

where $C>0$ is a constant depending on $\hat{\xi}$, $\Omega$ is a neighborhood
of $\hat{\xi}$. Hence, by the maximal principle, $U_{\lambda_{1}}\equiv 0$, in
$\Omega$.

Now we claim that $U_{\lambda_{1}}\equiv 0$ implies $V_{\lambda_{1}}\equiv 0$.
In fact, by (4.2) and (4.3), we obtain

$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}=$
$\displaystyle\frac{f(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda_{1}}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))}{|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle\geq$
$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))}{|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle>$
$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}.$

Since $f(s,t)$ is nondecreasing in $t$, we deduce from the above inequality
that

$|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi)>|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi).$
(4.15)

On the other hand,

$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}=$
$\displaystyle\frac{f(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}_{\lambda_{1}}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))}{|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle\geq$
$\displaystyle\frac{f(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q+2}}$
$\displaystyle\geq$
$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}.$

Hence

$\displaystyle\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi|_{\mathbb{H}^{n}}^{Q+2}}=\frac{f(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q+2}},$
(4.16)

therefore

$\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi))^{p_{1}}(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))^{q_{1}}}=\frac{f(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi))^{p_{1}}(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))^{q_{1}}}.$
(4.17)

By (4.15), (4.17) and assumption $(i)$ of Theorem 1.3, we obtain

$\frac{f(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))}{(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi))^{p_{1}}(|\xi|_{\mathbb{H}^{n}}^{Q-2}\bar{v}(\xi))^{q_{1}}}=\frac{f(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi),|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))}{(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{u}(\xi))^{p_{1}}(|\xi_{\lambda_{1}}|_{\mathbb{H}^{n}}^{Q-2}\bar{v}_{\lambda_{1}}(\xi))^{q_{1}}},$
(4.18)

and consequently, it follows from (4.16) and (4.18) that
$\bar{v}^{q_{1}}=\bar{v}_{\lambda_{1}}^{q_{1}}$ and therefore,
$V_{\lambda_{1}}\equiv 0$ as $q_{1}>0$.

Suppose that $U_{\lambda_{1}}\not\equiv 0$ and $V_{\lambda_{1}}\not\equiv 0$
in $\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$, then
$U_{\lambda_{1}}<0$, $V_{\lambda_{1}}<0$ in
$\Sigma_{\lambda_{1}}\backslash\\{0_{\lambda_{1}}\\}$. So
$\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\chi_{\Sigma_{\lambda}^{u}}$ converges
pointwise to zero as $\lambda\rightarrow\lambda_{1}$ in
$\mathbb{H}^{n}\backslash(T_{\lambda_{1}}\cup\\{0_{\lambda_{1}}\\})$. Then,
for $\lambda\in(\lambda_{1}-\delta,\lambda_{1})$, we have
$\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\chi_{\Sigma_{\lambda}^{u}}\leq\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\chi_{\Sigma_{\lambda_{1}-\delta}}\in
L^{1}(\Sigma_{\lambda})$. Next, the dominated convergence theorem allows us to
conclude that

$\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}\rightarrow
0,\ as\ \lambda\rightarrow\lambda_{1}.$

This eventually implies that

$C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{u}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{4}{Q}}<1.$

Similarly, we have

$C_{\lambda}\bigg{(}\int_{\Sigma_{\lambda}^{v}}\frac{1}{|\xi|_{\mathbb{H}^{n}}^{2Q}}d\xi\bigg{)}^{\frac{4}{Q}}<1.$

By Lemma 4.1, this implies that $U_{\lambda}\leq 0$, and $V_{\lambda}\leq 0$
for $\lambda\in(\lambda_{1}-\delta,\lambda_{1})$, which contradicts with the
definition of $\lambda_{1}$.

This completes the proof of Lemma 4.3. ∎

######  Lemma 4.4.

Let $u,v,f,g$ be as in Theorem 1.3 and suppose that $u,v$ are positive. Let
$\bar{u},\bar{v}$ be the $CR$ inversion of $u$ and $v$ respectively. Then
$\bar{u},\bar{v}$ are symmetric with respect to $T_{0}$, that is, $\bar{u}$
and $\bar{v}$ are even functions in $t$-variable..

###### Proof.

The proof of this lemma is verbatim to the proof of Lemma 3.5 by using Lemma
4.3 instead of Lemma 3.4, so we do not repeat it. ∎

Next we give the proof of Theorem 1.3.

###### Proof.

By Lemma 4.4, we conclude that functions $\bar{u}$ and $\bar{v}$ are symmetric
with respect to $T_{0},$ that is, $\bar{u}$ and $\bar{v}$ are even in
$t$-variable. Consequently, $u$ and $v$ are even in $t$-variable. Since the
choice of origin is arbitrary in $t$-axis, we conclude that $u$ and $v$ are
independent of $t$. Thus, $u$ and $v$ satisfy the system

$\begin{cases}\ -\Delta u(\xi)=f(u(\xi),v(\xi)),&\xi\in\mathbb{R}^{2n},\\\ \
-\Delta v(\xi)=g(u(\xi),v(\xi)),&\xi\in\mathbb{R}^{2n}.\end{cases}$ (4.19)

Since $f(s,t),g(s,t)$ are nondecreasing, and

$\frac{f(s,t)}{s^{a_{1}}t^{b_{1}}}=\frac{f(s,t)}{s^{p_{1}}t^{q_{1}}}s^{p_{1}-a_{1}}t^{q_{1}-b_{1}},$
$\frac{g(s,t)}{s^{a_{2}}t^{b_{2}}}=\frac{g(s,t)}{s^{p_{2}}t^{q_{2}}}s^{p_{2}-a_{2}}t^{q_{2}-b_{2}},$

where $a_{i}+b_{i}=\frac{2n+2}{2n-2}\geq\frac{Q+2}{Q-2}=p_{i}+q_{i}$,
$a_{i}\geq p_{i}$ and $b_{i}\geq q_{i},i=1,2$, then by using [36, Theorem 3.1]
as $2n\geq 3$ and and $h$ or $k$ is not constant function, we deduce that
$(u,v)=(C_{1},C_{2})$ for some constant $C_{1},C_{2}$ such that
$f(C_{1},C_{2})=g(C_{1},C_{2})=0$.

This completes the proof of Theorem 1.3. ∎

## Acknowledgments

RZ is supported by the National Natural Science Foundation of China (Grant No.
11871278) and the National Natural Science Foundation of China (Grant No.
11571093). VK and MR are supported by the FWO Odysseus 1 grant G.0H94.18N:
Analysis and Partial Differential Equations, the Methusalem programme of the
Ghent University Special Research Fund (BOF) (Grant number 01M01021) and by
FWO Senior Research Grant G011522N. MR is also supported by EPSRC grant
EP/R003025/2.

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