# Symmetry for a quasilinear elliptic equation in Hyperbolic space

Ramya Dutta† and Kunnath Sandeep††

###### Abstract.

In this article we establish the sharp decay estimates and Hyperbolic symmetry
of solutions to the quasilinear elliptic equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{H}^{n}}u-\lambda|u|^{p-2}u=|u|^{q-2}u\text{
in }\mathbb{H}^{n}$ $\displaystyle\displaystyle\quad u\geq
0,\;u\in\mathcal{D}^{1,p}(\mathbb{H}^{n})$

in the $\displaystyle n$-dimensional real Hyperbolic space
$\displaystyle\mathbb{H}^{n}$ where,
$\displaystyle\lambda\in[0,\lambda_{\text{max}})$ with
$\displaystyle\lambda_{\text{max}}=\left(\frac{n-1}{p}\right)^{p}$ the best
constant of the Poincaré inequality in $\displaystyle\mathbb{H}^{n}$ and
$\displaystyle p<q\leq p^{\ast}$, $\displaystyle p^{\ast}=\frac{np}{n-p}$.

###### Key words and phrases:

Quasilinear Elliptic Equation, Symmetry, Hyperbolic space

###### 2020 Mathematics Subject Classification:

35B06,35B33,35B44,35B45,35J92

† Université Claude Bernard Lyon 1, 43 boulevard du 11 Novembre 1918, 69622
Villeurbanne cedex, France. E-mail<EMAIL_ADDRESS>
†† TIFR Centre for Applicable Mathematics, Post Bag No. 6503, Sharadanagar,
Yelahanka New Town, Bangalore 560065. Email<EMAIL_ADDRESS>
##  1\. Introduction

Understanding the sharp constants and extremals of Sobolev Inequalities is an
important problem due to its applicability in the study of many partial
differential equations. One of the important example in this direction is the
well known Yamabe problem from differential geometry where the knowledge of
extremals of the Euclidean Sobolev inequality or more generally the
classification of positive solutions of the corresponding Euler-Lagrange
equation is used in a very crucial way. In the case of Euclidean space
extremals of Sobolev inequalities have been classified by now and one of the
main development in the process has been establishing the radial symmetry of
solutions.

In this article we study the Hyperbolic symmetry of solutions of the following
quasilinear PDE in the $\displaystyle n$ dimensional real Hyperbolic space
$\displaystyle(\mathbb{H}^{n},g)$ of constant sectional curvature
$\displaystyle-1$:

$\displaystyle\displaystyle\begin{cases}-\Delta_{p}^{\mathbb{H}^{n}}u-\lambda|u|^{p-2}u=|u|^{q-2}u\text{
in }\mathbb{H}^{n}\\\ u\geq
0,\;u\in\mathcal{D}^{1,p}(\mathbb{H}^{n}).\end{cases}$ (1.1)

The equation (1.1) is the Euler-Lagrange equation satisfied by the extremals
of the Poincaré-Sobolev inequality in $\displaystyle(\mathbb{H}^{n},g)$ given
by

$\displaystyle\displaystyle\int_{\mathbb{H}^{n}}|\nabla_{g}u|_{g}^{p}-\lambda|u|^{p}\,dV_{g}\geq
S_{\lambda,q}\left(\int_{\mathbb{H}^{n}}|u|^{q}\,dV_{g}\right)^{p/q},\forall\,u\in
C_{c}^{1}(\mathbb{H}^{n})$ (1.2)

where, $\displaystyle\lambda\in\left[0,\lambda_{\text{max}}\right)$ and
$\displaystyle\displaystyle\lambda_{\text{max}}:=\inf_{u\in
C_{c}^{1}(\mathbb{H}^{n})\setminus
0}\frac{\int_{\mathbb{H}^{n}}|\nabla_{g}u|_{g}^{p}\,dV_{g}}{\int_{\mathbb{H}^{n}}|u|_{g}^{p}\,dV_{g}}=\left(\frac{n-1}{p}\right)^{p}$
denotes the best constant in the Poincaré inequality in Hyperbolic space,
$\displaystyle\Delta_{p}^{\mathbb{H}^{n}}u:=\operatorname{div}_{g}(|\nabla_{g}u|^{p-2}\nabla_{g}u)$
is the corresponding $\displaystyle p$-Laplace operator, $\displaystyle 1<p<n$
, $\displaystyle p<q\leq p^{\ast}$ , $\displaystyle p^{\ast}=\frac{np}{n-p}$
denotes the critical Sobolev exponent and
$\displaystyle\mathcal{D}^{1,p}(\mathbb{H}^{n})$ is the completion of
$\displaystyle C_{c}^{1}(\mathbb{H}^{n})$ with the norm
$\displaystyle\left(\int_{\mathbb{H}^{n}}|\nabla_{g}u|_{g}^{p}\,dV_{g}\right)^{1/p}$.
Note that when $\displaystyle n=2$ we are only considering $\displaystyle p<2$
and $\displaystyle q\in(2,p^{\ast}]$ where, $\displaystyle
p^{\ast}=\frac{2p}{2-p}$. A solution of (1.1) is a $\displaystyle
u\in\mathcal{D}^{1,p}(\mathbb{H}^{n})$ satisfying

$\displaystyle\int_{\mathbb{H}^{n}}\left[|\nabla_{g}u|_{g}^{p-2}g(\nabla_{g}u,\nabla_{g}\phi)-\lambda|u|^{p-2}u\phi\right]\,dV_{g}=\int_{\mathbb{H}^{n}}|u|^{q-2}u\phi\,dV_{g},\;\forall\,\phi\in\mathcal{D}^{1,p}(\mathbb{H}^{n}).$

By writing this equation in either the Poincaré ball model or in the upper
half-space model, it can be easily seen from the standard regularity theory
(see [35], [18]) that the solutions of (1.1) are indeed $\displaystyle
C^{1,\alpha}$ for some $\displaystyle\alpha\in(0,1)$.

Our main result in this article is the sharp asymptotics and Hyperbolic
symmetry of solutions of (1.1).

######  Theorem 1.1.

Let, $\displaystyle u$ be a solution of (1.1), then there is a point
$\displaystyle O\in\mathbb{H}^{n}$ and a positive strictly monotone decreasing
function $\displaystyle\Phi:[0,\infty)\to\mathbb{R}_{+}$ such that
$\displaystyle u(x)=\Phi(\operatorname{dist}_{\mathbb{H}^{n}}(O,x))$. The
function $\displaystyle\Phi(t)$ is $\displaystyle C^{1}$ at $\displaystyle
t=0$ with $\displaystyle\Phi^{\prime}(0)=0$ and smooth for all $\displaystyle
t>0$. It satisfies the pointwise bounds $\displaystyle
ce^{-\alpha_{\lambda}t}\leq\Phi(t)\leq Ce^{-\alpha_{\lambda}t}$ for all
$\displaystyle t\geq 0$, $\displaystyle
ce^{-\alpha_{\lambda}t}\leq-\Phi^{\prime}(t)\leq Ce^{-\alpha_{\lambda}t}$ for
all $\displaystyle t\geq 1$, where, $\displaystyle c,C>0$ are some positive
constants (which depend on $\displaystyle\Phi$) and
$\displaystyle\lim\limits_{t\to\infty}\frac{\Phi^{\prime}(t)}{\Phi(t)}=-\alpha_{\lambda}$,
where, $\displaystyle\alpha_{\lambda}$ is the unique positive root of
$\displaystyle|\alpha|^{p-2}\alpha(n-1-(p-1)\alpha)=\lambda$ such that
$\displaystyle\alpha_{\lambda}\in\left(\frac{n-1}{p},\frac{n-1}{p-1}\right]$.

In the case of Euclidean Sobolev inequality, the corresponding Euler-Lagrange
equation

$\displaystyle\displaystyle-\Delta_{p}u=|u|^{p^{\ast}-2}u\text{ in
}\mathbb{R}^{n},u\geq 0,u\in\mathcal{D}^{1,p}(\mathbb{R}^{n})$ (1.3)

has been studied in detail. When $\displaystyle p=2$, the symmetry of
solutions has been established in the seminal work of Gidas-Ni-Nirenberg [20]
using the moving plane method. However when $\displaystyle p\not=2$ there are
many obstructions for proving the symmetry of solutions for the $\displaystyle
p$-Laplace equations using the moving plane method mainly due to the lack of
strong comparison principle, lack of tools to get the moving plane argument
started from infinity (in the case of problems in unbounded domain) etc. Note
that the $\displaystyle p$-Laplace operator is elliptic on points where
$\displaystyle\nabla u\not=0$ and in the complement it is either degenerate or
singular depending on $\displaystyle p>2$ or $\displaystyle p<2$ respectively.
The first significant progress in applying the moving plane method for the
$\displaystyle p$-Laplace case was made by Damascelli in [10] where he proved
the radial symmetry for positive solutions of equations of the form
$\displaystyle-\Delta_{p}u=f(u)$ with Dirichlet boundary condition in bounded
domains in the Euclidean space. The results in the entire space
$\displaystyle\mathbb{R}^{n}$ was established in [12] for solutions of the
same PDE with proper decay/integrability, where $\displaystyle f$ was non
increasing in some interval $\displaystyle(0,s_{0})$. See also [32] for
related developments.

In the case of Sobolev extremals, the nonlinear term is strictly increasing
and hence the results of [12] can not be applied to obtain radial symmetry of
Sobolev extremals. It was shown in [13] that when $\displaystyle 1<p<2$ the
symmetry of (1.3) follows if the solutions have a decay namely $\displaystyle
u(x)\leq C|x|^{-m}$, $\displaystyle|\nabla u|\leq C|x|^{-(m+1)}$ for
$\displaystyle|x|$ large (additionally $\displaystyle u(x)\geq c|x|^{-m}$ for
$\displaystyle|x|$ large when $\displaystyle p^{\ast}<2$) where,
$\displaystyle m>\frac{n-p}{p}$. The symmetry result was proved in [11]
without any additional decay assumption when $\displaystyle\frac{2n}{n+2}\leq
p<2$ (i.e., $\displaystyle p<2$ and $\displaystyle p^{\ast}\geq 2$).

Tools for handling the issue of strong comparison in the general case
$\displaystyle p>2$ was developed by Damascelli and Sciunzi in [14], [15].
Again these informations are not enough to obtain symmetry results for (1.3)
in the full range of $\displaystyle 1<p<n$. The first break through in this
direction was obtained by Vétois [37] where he proved that the solutions of
(1.3) satisfy the sharp decay estimate $\displaystyle
u(x)\approx|x|^{-\frac{n-p}{p-1}}$ and $\displaystyle|\nabla u|\leq
C|x|^{-\frac{n-1}{p-1}}$ for $\displaystyle|x|$ large. With this and the
results of [13] the symmetry results follows when $\displaystyle 1<p<2$. The
symmetry result for the case $\displaystyle p>2$ was completed by Sciunzi [30]
by establishing a precise pointwise gradient bound $\displaystyle|\nabla
u(x)|\approx|x|^{-\frac{n-1}{p-1}}$ for $\displaystyle|x|$ large. More
recently the classification result for (1.3) has also been established by a
technique independent of the moving plane method in conical domains as well as
the full Euclidean space $\displaystyle\mathbb{R}^{n}$ in [9]. We also refer
to [8], [27] and [38] for the classification result of (1.3) where the finite
energy assumption $\displaystyle u\in\mathcal{D}^{1,p}(\mathbb{R}^{n})$ has
been relaxed in certain range of $\displaystyle p$ and $\displaystyle n$.

Coming to the equation (1.1), when $\displaystyle p=2$ the symmetry of
solution has been studied in [1] for
$\displaystyle\lambda\leq\frac{1}{4}n(n-2)$ and subsequently extended to the
full range of $\displaystyle\lambda$ in [24] using the method of moving plane.
Consequently in [24] the solutions have been studied thoroughly by analysing
the corresponding ordinary differential equation. In this situation the moving
plane method can be applied as we are in the elliptic regime and we have
appropriate Sobolev inequalities. Thus the Euclidean method can be adopted to
this geometric situation by reflecting over foliations of the Hyperbolic
space.

Our main focus in this article is the case $\displaystyle p\not=2$. Like in
the Euclidean case here also the issues are lack of strong comparison
principles and also moving the plane from infinity. The issue of strong
comparison principle in bounded subsets of $\displaystyle\mathbb{H}^{n}$ can
be established by suitably adapting the ideas of [14], [15] to the Hyperbolic
space, the main challenge in this case is to establish the precise asymptotic
estimates at infinity.

The most important contribution of this article is to establish sharp
asymptotic estimates to the positive solutions of (1.1) and its gradient, so
that the moving plane method can be made to work in this case by suitably
adapting the Euclidean tools. In the Euclidean case the precise $\displaystyle
L^{\infty}$-estimates were obtained by crucially rescaling the equations.
However in our case we do not have any substitute for scaling. In Theorem-3.6
we will establish the $\displaystyle L^{\infty}$-estimates by finding
appropriate sub and super solutions combined with weak comparison tools using
the Picone identity. The estimates we obtained in Theorem-3.6 are new even in
the case of $\displaystyle p=2$, which in fact improves the estimates obtained
in [24] except in cases where explicit solutions are known. To prove the
estimates on the gradient, the upper bound follows from the $\displaystyle
L^{\infty}$-estimate and a use of Harnack inequality, however the lower bound
on the gradient at infinity is the difficult estimate to prove. In the
Euclidean case (1.3) this is achieved in [30] by a blow up analysis via
rescaling which leads to the limiting equation $\displaystyle-\Delta_{p}u=0$
in $\displaystyle\mathbb{R}^{n}\setminus\\{0\\}$ and $\displaystyle
u(x)\approx|x|^{-\frac{n-p}{p-1}}$ whose solutions have been classified in
[23] to be constant multiples of the fundamental solution $\displaystyle
u(x)=c|x|^{-\frac{n-p}{p-1}}$ (see also [30] for an alternative proof under
weaker assumptions). Also see [26] for a similar limiting equation with a
Hardy potential with singularity at origin in $\displaystyle\mathbb{R}^{n}$.
The main difference of our work when comparing with the Euclidean references
mentioned above is in the nature of the blow up and the limiting problem. In
our case the limiting problem is in fact an eigenvalue problem with a pole at
‘infinity’. We need to establish a classification result for positive
eigenfunctions of the following eigenvalue problem

$-\Delta_{p}^{\mathbb{H}^{n}}u=\lambda u^{p-1}$ (1.4)

which are comparable to certain eigenfunctions with desired behavior at
infinity. Here by an eigenfunction we means a $\displaystyle C^{1}$ function
which satisfy the relation

$\displaystyle\int_{\mathbb{H}^{n}}|\nabla_{g}u|_{g}^{p-2}g(\nabla_{g}u,\nabla_{g}\phi)\,dV_{g}=\int_{\mathbb{H}^{n}}\lambda
u^{p-1}\phi\,dV_{g}$

for all $\displaystyle\phi\in C_{c}^{1}(\mathbb{H}^{n})$.

Fix a point $\displaystyle O\in\mathbb{H}^{n}$ and
$\displaystyle(r,\theta)\in(0,\infty)\times\mathbb{S}^{n-1}$ be the geodesic
normal coordinates centred at $\displaystyle O$. For
$\displaystyle\xi\in\mathbb{S}^{n-1}$ define a function $\displaystyle
E_{\xi}$ by

$\displaystyle\displaystyle E_{\xi}(x)=\left(\cosh r-(\xi\cdot\theta)\sinh
r\right)^{-\alpha_{\lambda}}$ (1.5)

where, $\displaystyle(r,\theta)$ is the geodesic normal coordinate of
$\displaystyle x$ and $\displaystyle\alpha_{\lambda}$ is as in Theorem-1.1.
Note that $\displaystyle E_{\xi}$ extends as a $\displaystyle C^{1}$ function
to $\displaystyle\mathbb{H}^{n}$. With this definition we can state the
classification result as follows.

######  Theorem 1.2.

For any $\displaystyle\xi\in\mathbb{S}^{n-1}$, $\displaystyle E_{\xi}$ be as
in (1.5). If $\displaystyle E$ satisfies (1.4) and has the bounds
$\displaystyle E_{\xi}(x)\leq E(x)\leq CE_{\xi}(x)$ for some $\displaystyle
C>1$ and all $\displaystyle x\in\mathbb{H}^{n}$ then $\displaystyle E\equiv
cE_{\xi}$ for some $\displaystyle c\in[1,C]$.

We prove Theorem-1.2 in Section-3, Theorem-3.8 formulated in the equivalent
Poincaré ball model $\displaystyle\mathbb{B}^{n}$ of the Hyperbolic space.
The problem of classification of positive eigenfunctions in general may have
independent interest from a potential theoretic perspective, particularly in
the study of Martin boundary. We recall some relevant definitions. Let,
$\displaystyle\mathcal{L}$ be a homogeneous elliptic operator, meaning if
$\displaystyle u$ solves $\displaystyle\mathcal{L}u=0$ then so does
$\displaystyle cu$ for any constant $\displaystyle c$. Suppose, $\displaystyle
u\in C^{1}(\mathbb{H}^{n})$ be a positive solution to the equation
$\displaystyle\mathcal{L}u=0$ such that $\displaystyle u$ has vanishing
‘trace’ on $\displaystyle\partial_{\infty}\mathbb{H}^{n}\setminus\\{\xi\\}$
i.e., it vanishes in some generalized trace sense on the the ideal boundary
(or sphere at infinity) except at $\displaystyle\xi$ (see the definition of
ideal boundary or the sphere at infinity in Section-2). We call such a
solution $\displaystyle u$ to be a minimal function relative to the point
$\displaystyle\xi\in\partial_{\infty}\mathbb{H}^{n}$ (identified with
$\displaystyle\mathbb{S}^{n-1}$) if and only if any other solution
$\displaystyle\tilde{u}$ of $\displaystyle\mathcal{L}\tilde{u}=0$ satisfying
$\displaystyle 0\leq\tilde{u}\leq Cu$ for some positive constant
$\displaystyle C$, must be a positive constant multiple of $\displaystyle u$,
i.e., $\displaystyle\tilde{u}\equiv cu$ for some $\displaystyle c>0$. We say
that the $\displaystyle\mathcal{L}$-Martin boundary can be identified with
$\displaystyle\partial_{\infty}\mathbb{H}^{n}$ if each point
$\displaystyle\xi\in\partial_{\infty}\mathbb{H}^{n}$ has a unique (up to
constant multiple) positive minimal function relative to it.

In the linear case when $\displaystyle p=2$, the conclusion of Theorem-1.2 can
be obtained by only assuming the upper bound (see Remark-3.10). The minimality
of the eigenfunctions $\displaystyle E_{\xi}$ relative to
$\displaystyle\xi\in\mathbb{S}^{n-1}$ can be seen as a direct consequence of a
representation theorem based on its boundary behavior due to Helgason [21] and
Minemura [25]. The representation of eigenfunctions on non-compact symmetric
space through a hyperfunction on its minimal boundary, known as the Helgason
conjecture, was established in full generality in [22]. It would be
interesting to know if the eigenfunctions $\displaystyle E_{\xi}$ are minimal
relative to $\displaystyle\xi\in\mathbb{S}^{n-1}$ for a general $\displaystyle
1<p<n$, i.e., if the assumption of the lower bound in Theorem-1.2 can be
relaxed.

The structure of the article is as follows. In Section-2 we go over some of
the basic notations and definitions relating to the Hyperbolic space. In
Section-3 we prove the boundedness, sharp decay estimates for the solutions of
(1.1) and its gradient, as well as the classification of eigenfunctions
Theorem-1.2 (Theorem-3.8). Section-4 is dedicated to the proof of Hyperbolic
symmetry of (1.1) and in Section-5 we discuss the Sobolev regularity of
$\displaystyle u$ and the strong comparison theorem in Hyperbolic space.
Finally in Section-6 we discuss some existence and non-existence results
related to (1.1).

##  2\. preliminaries

In this section we will recall some of the basic notations and results which
we will be using in this article.
Notations : We will denote the $\displaystyle n$ dimensional real Hyperbolic
space by $\displaystyle(\mathbb{H}^{n},g)$. Fix a point $\displaystyle
O\in\mathbb{H}^{n}$ then in the normal coordinates based at $\displaystyle O$
we will identify the unit sphere in the tangent space of
$\displaystyle\mathbb{H}^{n}$ at $\displaystyle O$ by
$\displaystyle\mathbb{S}^{n-1}$. Also note that for
$\displaystyle\mathbb{H}^{n}$ we have the notion of the geometric boundary
called the ‘sphere at infinity’ or the ‘ideal boundary’ denoted
$\displaystyle\partial_{\infty}\mathbb{H}^{n}$ which is abstractly defined as
the equivalence classes of geodesic rays under the equivalence relation where
two geodesic rays $\displaystyle\gamma_{1},\gamma_{2}$ are equivalent if
$\displaystyle\operatorname{dist}_{g}(\gamma_{1}(t),\gamma_{2}(t))$ remains
bounded for $\displaystyle t\geq 0.$ Once we fix the point $\displaystyle
O\in\mathbb{H}^{n}$, we can identify this sphere at infinity (or the ideal
boundary) with $\displaystyle\mathbb{S}^{n-1}$. We refer the book [29],
Chapter-2 for details.
In geodesic normal coordinates the Hyperbolic metric takes the form
$\displaystyle g=dt^{2}+(\sinh t)^{2}g_{\mathbb{S}^{n-1}}$ where,
$\displaystyle t(x)=\operatorname{dist}_{g}(O,x)$ and $\displaystyle
g_{\mathbb{S}^{n-1}}$ is the standard metric on
$\displaystyle\mathbb{S}^{n-1}$. Consequently the Hyperbolic measure will be
given by $\displaystyle dV_{g}=(\sinh t)^{n-1}dV_{g_{\mathbb{S}^{n-1}}}dt$.

When we need to make precise calculations we will be working with either the
Poincaré ball model or the upper half-space model. In both the models we will
be using $\displaystyle g$ for the metric which will vary depending on the
model we are using. We will denote the gradient, divergence and modulus with
respect to the metric by $\displaystyle\nabla_{g},\operatorname{div}_{g}$ and
$\displaystyle|\cdot|_{g}$ respectively. While working in the Poincaré ball or
upper half-space models the corresponding quantities with respect to the
Euclidean metric will be denoted by $\displaystyle\nabla,\operatorname{div}$
and $\displaystyle|\cdot|$ respectively. Since the gradient and divergence are
intrinsic quantities (see [7], Chapter-4), we see that our equation (1.1) is
invariant under isometries of the Hyperbolic space, i.e., if $\displaystyle u$
solves (1.1) and $\displaystyle T$ is an isometry of
$\displaystyle\mathbb{H}^{n}$ then $\displaystyle u\circ T$ also solves (1.1).

We will denote the Poincaré ball model by $\displaystyle(\mathbb{B}^{n},g)$
where, $\displaystyle\mathbb{B}^{n}$ is the Euclidean unit ball in
$\displaystyle\mathbb{R}^{n}$ endowed with the metric
$\displaystyle\left(\frac{2}{1-|x|^{2}}\right)^{2}g_{E}$ and $\displaystyle
g_{E}$ denotes the Euclidean metric. By an abuse of notation, we will denote
the upper half-space model by $\displaystyle(\mathbb{H}^{n},g)$ where,
$\displaystyle\mathbb{H}^{n}=\\{x\in\mathbb{R}^{n}:x_{n}>0\\}$ and the metric
$\displaystyle g$ is given by $\displaystyle g=x_{n}^{-2}g_{E}$. Note that the
isometry group in these two models are respectively the Mobius groups of the
ball and upper half-space which we will denote by
$\displaystyle\mathcal{M}(\mathbb{B}^{n})$ and
$\displaystyle\mathcal{M}(\mathbb{H}^{n})$. For more details about the two
models and their isometry groups we refer to [34] and [28]. Throughout the
article unless otherwise stated $\displaystyle B_{R}(x_{0})$ will denote the
metric ball centered at $\displaystyle x_{0}$ with radius $\displaystyle R$,
meaning $\displaystyle
B_{R}(x_{0}):=\left\\{x\in\mathbb{H}^{n}:\operatorname{dist}_{g}(x,x_{0})<R\right\\}$.
We will end this section by recalling the expressions for the gradient,
divergence and the p-Laplacian in these coordinates. In the Poincaré ball
model $\displaystyle(\mathbb{B}^{n},g)$ they are given by

$\displaystyle\displaystyle\nabla_{g}u(x)$
$\displaystyle\displaystyle=\left(\frac{1-|x|^{2}}{2}\right)^{2}\nabla u(x)$
(2.1) $\displaystyle\displaystyle|\nabla_{g}u|_{g}(x)$
$\displaystyle\displaystyle=\left(\frac{1-|x|^{2}}{2}\right)|\nabla u(x)|$
(2.2) $\displaystyle\displaystyle\operatorname{div}_{g}X$
$\displaystyle\displaystyle=\left(\frac{1-|x|^{2}}{2}\right)^{n}\operatorname{div}\left(\left(\frac{2}{1-|x|^{2}}\right)^{n}X\right).$
(2.3)

The p-Laplacian is given by

$\displaystyle\displaystyle\Delta_{p}^{\mathbb{B}^{n}}u(x)=\operatorname{div}_{g}(|\nabla_{g}u|_{g}^{p-2}\nabla_{g}u)=\left(\frac{1-|x|^{2}}{2}\right)^{n}\operatorname{div}\left(\left(\frac{2}{1-|x|^{2}}\right)^{n-p}|\nabla
u|^{p-2}\nabla u\right).$ (2.4)

In the upper half-space model $\displaystyle(\mathbb{H}^{n},g)$ they are given
by

$\displaystyle\displaystyle\nabla_{g}u(x)$
$\displaystyle\displaystyle=x_{n}^{2}\nabla u(x)$ (2.5)
$\displaystyle\displaystyle|\nabla_{g}u|_{g}(x)$
$\displaystyle\displaystyle=x_{n}|\nabla u(x)|$ (2.6)
$\displaystyle\displaystyle\operatorname{div}_{g}X$
$\displaystyle\displaystyle=x_{n}^{n}\operatorname{div}\left(x_{n}^{-n}X\right).$
(2.7)

The p-Laplacian is given by

$\displaystyle\displaystyle\Delta_{p}^{\mathbb{H}^{n}}u(x)=\operatorname{div}_{g}(|\nabla_{g}u|_{g}^{p-2}\nabla_{g}u)=x_{n}^{n}\operatorname{div}\left(x_{n}^{p-n}|\nabla
u|^{p-2}\nabla u\right).$ (2.8)

##  3\. Decay estimate for solutions and its gradient

We begin by showing that the solutions to equation (1.1) have higher
integrability and are bounded by proceeding along the arguments in [37], [33].
Let us consider the auxiliary function

$\displaystyle\displaystyle f(\alpha):=|\alpha|^{p-2}\alpha(n-1-(p-1)\alpha)$
(3.1)

then, $\displaystyle\displaystyle f(0)=f\left(\frac{n-1}{p-1}\right)=0$ and
$\displaystyle f^{\prime}(\alpha)=(p-1)|\alpha|^{p-2}(n-1-p\alpha)$. Thus
$\displaystyle f$ has a unique maximum at
$\displaystyle\displaystyle\alpha_{c}=\frac{n-1}{p}$ and is concave and
increasing in $\displaystyle[0,\alpha_{c})$ and decreasing in
$\displaystyle\left(\alpha_{c},\dfrac{n-1}{p-1}\right]$. Thus the equation
$\displaystyle f(\alpha)=\lambda$ has two roots
$\displaystyle\alpha_{\lambda},\beta_{\lambda}$ satisfying

$\displaystyle\displaystyle
0\leq\beta_{\lambda}<\frac{n-1}{p}<\alpha_{\lambda}\leq\frac{n-1}{p-1}$ (3.2)

for any
$\displaystyle\displaystyle\lambda\in\left[0,\left(\frac{n-1}{p}\right)^{p}\right)$.

###  3.1. Boundedness of solution

In what follows we will use the notation
$\displaystyle\Omega_{m}=\left\\{x\in\mathbb{B}^{n}:|u|>m\right\\}$, its
characteristic function $\displaystyle\chi_{m}:=\chi_{\Omega_{m}}$ and the
characteristic function of complement will be denoted by
$\displaystyle\chi_{m}^{c}:=\chi_{\mathbb{B}^{n}\setminus\Omega_{m}}$.

######  Lemma 3.1.

Let $\displaystyle u$ be a solution to (1.1) when $\displaystyle q=p^{\ast}$.
Then $\displaystyle u\in L^{k}(\mathbb{B}^{n})$ for $\displaystyle p\leq
k<p^{\ast}\left(\frac{n-1}{p\beta_{\lambda}}\right)$ where,
$\displaystyle\beta_{\lambda}$ is as in (3.2). Hence, $\displaystyle u$ has
higher integrability than $\displaystyle p^{\ast}$.

###### Proof.

Note that since $\displaystyle u\in\mathcal{D}^{1,p}(\mathbb{B}^{n})$ by the
Poincaré-Sobolev inequality in $\displaystyle\mathbb{B}^{n}$ we know that
$\displaystyle u\in L^{k}(\mathbb{B}^{n})$ for $\displaystyle p\leq k\leq
p^{\ast}$. Let us denote by $\displaystyle u_{m}:=\min(|u|,m)$ for
$\displaystyle m>0$. Testing equation (1.1) with
$\displaystyle\varphi=u_{m}^{p\beta-p}u\in\mathcal{D}^{1,p}(\mathbb{B}^{n})$
where $\displaystyle\beta>1$, we have

$\displaystyle\displaystyle\int_{\mathbb{B}^{n}}|\nabla_{g}u|_{g}^{p-2}g(\nabla_{g}u,\nabla_{g}\varphi)\,dV_{g}-\lambda\int_{\mathbb{B}^{n}}|u|^{p-2}u\varphi\,dV_{g}=\int_{\mathbb{B}^{n}}|u|^{q-2}u\varphi\,dV_{g}.$
(3.3)

Then on the LHS of (3.3) we have

$\displaystyle\displaystyle\int_{\mathbb{B}^{n}}|\nabla_{g}u|_{g}^{p-2}g(\nabla_{g}u,\nabla_{g}\varphi)\,dV_{g}$
$\displaystyle\displaystyle=\int_{\mathbb{B}^{n}}|\nabla_{g}u|_{g}^{p}\left(\chi_{m}^{c}(p\beta-p+1)|u|^{p\beta-p}+\chi_{m}m^{p\beta-p}\right)\,dV_{g}$
$\displaystyle\displaystyle\geq\frac{(p\beta-p+1)}{\beta^{p}}\int_{\mathbb{B}^{n}}|\nabla_{g}u|_{g}^{p}\left(\chi_{m}^{c}\beta^{p}|u|^{p\beta-p}+\chi_{m}m^{p\beta-p}\right)\,dV_{g}$
$\displaystyle\displaystyle=\frac{(p\beta-p+1)}{\beta^{p}}\int_{\mathbb{B}^{n}}\left|\nabla_{g}\left(u_{m}^{\beta-1}u\right)\right|_{g}^{p}\,dV_{g}$
(3.4)

where, we used the Bernoulli’s inequality
$\displaystyle\beta^{p}=(1+(\beta-1))^{p}\geq 1+p(\beta-1)$. Again using
Poincaré inequality in $\displaystyle\mathbb{B}^{n}$ we have

$\displaystyle\displaystyle\left(\frac{p\beta-p+1}{\beta^{p}}-\frac{\lambda}{\lambda_{\text{max}}}\right)\int_{\mathbb{B}^{n}}\left|\nabla_{g}\left(u_{m}^{\beta-1}u\right)\right|_{g}^{p}\,dV_{g}\leq\int_{\mathbb{B}^{n}}|u|^{q}u_{m}^{p\beta-p}\,dV_{g}.$
(3.5)

Therefore, on the RHS of (3.5) for any $\displaystyle m>m_{0}$ we have

$\displaystyle\displaystyle\int_{\mathbb{B}^{n}}|u|^{q}u_{m}^{p\beta-p}\,dV_{g}\leq
m_{0}^{p\beta-p}\int_{\Omega_{m_{0}}^{c}}|u|^{q}\,dV_{g}+\int_{\Omega_{m_{0}}}|u|^{q}u_{m}^{p\beta-p}\,dV_{g}.$
(3.6)

Using Holder’s inequality followed by Sobolev Inequality on the second term on
RHS of (3.6) we get

$\displaystyle\displaystyle\int_{\Omega_{m_{0}}}|u|^{q}u_{m}^{p\beta-p}\,dV_{g}$
$\displaystyle\displaystyle\leq\left(\int_{\Omega_{m_{0}}}|u|^{q}\,dV_{g}\right)^{1-\frac{p}{q}}\left(\int_{\Omega_{m_{0}}}\left(u_{m}^{\beta-1}|u|\right)^{q}\,dV_{g}\right)^{\frac{p}{q}}$
$\displaystyle\displaystyle\leq
S\left(\int_{\Omega_{m_{0}}}|u|^{q}\,dV_{g}\right)^{1-\frac{p}{q}}\int_{\mathbb{B}^{n}}\left|\nabla_{g}\left(u_{m}^{\beta-1}u\right)\right|_{g}^{p}\,dV_{g}$
$\displaystyle\displaystyle\leq\frac{1}{2}\left(\frac{p\beta-p+1}{\beta^{p}}-\frac{\lambda}{\lambda_{\text{max}}}\right)\int_{\mathbb{B}^{n}}\left|\nabla_{g}\left(u_{m}^{\beta-1}u\right)\right|_{g}^{p}\,dV_{g}$
(3.7)

by choosing $\displaystyle m_{0}$ sufficiently large depending on
$\displaystyle\beta$. Therefore, combining (3.5), (3.6) and (3.1) we get

$\displaystyle\displaystyle\left(\frac{p\beta-p+1}{\beta^{p}}-\frac{\lambda}{\lambda_{\text{max}}}\right)\int_{\mathbb{B}^{n}}\left|\nabla_{g}\left(u_{m}^{\beta-1}u\right)\right|_{g}^{p}\,dV_{g}\leq
2m_{0}^{p\beta-p}\int_{\Omega_{m_{0}}^{c}}|u|^{q}\,dV_{g}$ (3.8)

followed by an application of Sobolev inequality and letting $\displaystyle
m\to\infty$ we get

$\displaystyle\displaystyle\left(\frac{p\beta-p+1}{\beta^{p}}-\frac{\lambda}{\lambda_{\text{max}}}\right)\left(\int_{\mathbb{B}^{n}}|u|^{\beta
q}\,dV_{g}\right)^{\frac{p}{q}}\leq
Cm_{0}^{p\beta-p}\int_{\Omega_{m_{0}}^{c}}|u|^{q}\,dV_{g}.$ (3.9)

Now, note that making the change of variable
$\displaystyle\displaystyle\beta=\frac{n-1}{p\alpha}$ we have

$\displaystyle\frac{p\beta-p+1}{\beta^{p}}=\frac{\alpha^{p-1}(n-1-(p-1)\alpha)}{\lambda_{\text{max}}}>\frac{\lambda}{\lambda_{\text{max}}}$

provided $\displaystyle\alpha\in(\beta_{\lambda},\alpha_{\lambda})$ as in
(3.2) i.e., if
$\displaystyle\beta\in\left(\frac{n-1}{p\alpha_{\lambda}},\frac{n-1}{p\beta_{\lambda}}\right)$.
This concludes the proof of higher integrability. ∎

######  Theorem 3.2.

Let $\displaystyle u$ be a solution of (1.1) then $\displaystyle u\in
L^{\infty}(\mathbb{B}^{n})$. Furthermore, $\displaystyle u(x)\to 0$ and
$\displaystyle|\nabla_{g}u|_{g}(x)\to 0$ as
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x)\to\infty$.

###### Proof.

The proof in the sub-critical case $\displaystyle q<p^{\ast}$ follows in a
similar way to the critical case. Note that in the critical case
$\displaystyle q=p^{\ast}$ we may rewrite equation (1.1) as

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}u=(\lambda+|u|^{p^{\ast}-p})|u|^{p-1}u$
(3.10)

where, $\displaystyle\lambda+|u|^{p^{\ast}-p}\in
L^{k}_{\text{loc}}(\mathbb{B}^{n})$ for some $\displaystyle\displaystyle
k>\frac{n}{p}$ by the higher integrability proved in Lemma-3.1.
Therefore, by usual Moser’s iteration (see [31], [36]) we have the estimate

$\displaystyle\displaystyle\sup_{B_{1}(0)}|u|\leq C\lVert
u\rVert_{L^{p^{\ast}}(B_{2}(0))}$ (3.11)

where, the constant $\displaystyle C$ only depends on
$\displaystyle\lVert\lambda+|u|^{p^{\ast}-p}\rVert_{L^{k}(B_{2})}\leq
C_{1}\left(\lambda+\lVert
u\rVert_{L^{p^{\ast}\beta_{0}}(\mathbb{B}^{n})}^{p^{\ast}-p}\right)$ for
$\displaystyle\displaystyle 1<\beta_{0}<\frac{n-1}{p\beta_{\lambda}}$. Since,
the equation (3.10) is invariant under isometries of the Hyperbolic space
$\displaystyle\mathcal{M}(\mathbb{B}^{n})$ then for any $\displaystyle
x\in\mathbb{B}^{n}$ consider
$\displaystyle\tau_{x}\in\mathcal{M}(\mathbb{B}^{n})$ such that
$\displaystyle\tau_{x}(0)=x$. Then using (3.11) on $\displaystyle
v:=(u\circ\tau_{x})$ we have $\displaystyle\displaystyle\sup_{B_{1}(0)}|v|\leq
C\lVert v\rVert_{L^{p^{\ast}}(B_{2}(0))}$ or equivalently
$\displaystyle\displaystyle\sup_{B_{1}(x)}|u|\leq C\lVert
u\rVert_{L^{p^{\ast}}(B_{2}(x))}$ for any $\displaystyle x\in\mathbb{B}^{n}$.
Combining this with standard $\displaystyle C^{1,\sigma}$ estimates (see [18],
[35]) we have
$\displaystyle\displaystyle\sup_{B_{1/2}(x)}|\nabla_{g}u|_{g}\leq
C^{\prime}\sup_{B_{1}(x)}|u|\leq C^{\prime\prime}\lVert
u\rVert_{L^{p^{\ast}}(B_{2}(x))}$ for all $\displaystyle x\in\mathbb{B}^{n}$
and the later goes to $\displaystyle 0$ as
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x)\to\infty$ since
$\displaystyle u\in L^{p^{\ast}}(\mathbb{B}^{n})$. ∎

###  3.2. Super-solution and sub-solutions to the equation

In the following lemma we identify the candidates for sub(super)-solutions to
our equation which will be used to establish the decay estimates.

######  Lemma 3.3.

The function $\displaystyle\displaystyle
v(x):=\left[\cosh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{m}\right)\right]^{-m\alpha_{\lambda}}$
for $\displaystyle m\geq 2$ is a super-solution to the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v-\lambda v^{p-1}\geq
c_{\lambda}e^{-\frac{2}{m}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}v^{p-1}$
(3.12)

in
$\displaystyle\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x)>R_{\lambda}$
for some $\displaystyle R_{\lambda}>0$ and some positive constant
$\displaystyle c_{\lambda}>0$.
Also, $\displaystyle\displaystyle
v(x):=\left[\sinh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{2}\right)\right]^{-2\alpha_{\lambda}}$
is a sub-solution to the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v-\lambda v^{p-1}\leq
0$ (3.13)

for
$\displaystyle\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x)>R_{\lambda}$
for some $\displaystyle R_{\lambda}>0$.

###### Proof.

Recall that in $\displaystyle\mathbb{B}^{n}$ we have
$\displaystyle\displaystyle
t_{x}:=\operatorname{dist}_{\mathbb{B}^{n}}(0,x)=\log\left(\frac{1+|x|}{1-|x|}\right)$
or equivalently $\displaystyle r_{x}:=|x|=\tanh(t_{x}/2)$. Therefore we may
express the Riemannian metric $\displaystyle g$ in geodesic polar coordinates
at the origin as

$\displaystyle\displaystyle
g=\frac{4dx^{2}}{(1-|x|^{2})^{2}}=\left(\frac{2}{1-r^{2}}\right)^{2}(dr^{2}+r^{2}g_{\mathbb{S}^{n-1}})=dt^{2}+(\sinh
t)^{2}g_{\mathbb{S}^{n-1}}.$ (3.14)

Then the $\displaystyle\Delta_{p}^{\mathbb{B}^{n}}$ of a radial function
$\displaystyle v(x)$, which we continue to denote as $\displaystyle v(t)$ with
$\displaystyle t=\operatorname{dist}_{\mathbb{B}^{n}}(0,x)$, is given by

$\displaystyle\displaystyle\Delta_{p}^{\mathbb{B}^{n}}v(x)$
$\displaystyle\displaystyle=\frac{1}{(\sinh t)^{n-1}}\left((\sinh
t)^{n-1}|v^{\prime}(t)|^{p-2}v^{\prime}(t)\right)^{\prime}$
$\displaystyle\displaystyle=|v^{\prime}(t)|^{p-2}\left[(p-1)v^{\prime\prime}(t)+(n-1)\coth(t)v^{\prime}(t)\right].$
(3.15)

We consider the function $\displaystyle\displaystyle
v(t):=\left[\cosh\left(\frac{t}{m}\right)\right]^{-m\alpha_{\lambda}}$ where,
$\displaystyle m\geq 2$. Then by a direct computation of the derivatives we
have

$\displaystyle
v^{\prime}(t)=-\alpha_{\lambda}\left[\cosh\left(\frac{t}{m}\right)\right]^{-m\alpha_{\lambda}}\tanh\left(\frac{t}{m}\right)=-\alpha_{\lambda}\tanh\left(\frac{t}{m}\right)v(t)$

and

$\displaystyle
v^{\prime\prime}(t)=-\frac{\alpha_{\lambda}}{m}\left[\cosh\left(\frac{t}{m}\right)\right]^{-2}v(t)+\alpha_{\lambda}^{2}\tanh^{2}\left(\frac{t}{m}\right)v(t).$

Substituting in the expression (3.2) we get

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v$
$\displaystyle\displaystyle=\alpha_{\lambda}^{p-2}\tanh^{p}\left(\frac{t}{m}\right)\left[(p-1)\frac{\alpha_{\lambda}}{m}\left[\sinh\left(\frac{t}{m}\right)\right]^{-2}-(p-1)\alpha_{\lambda}^{2}\right.$
$\displaystyle\displaystyle\quad\left.+(n-1)\alpha_{\lambda}\coth
t\coth\left(\frac{t}{m}\right)\right]v^{p-1}(t).$ (3.16)

Now we have the asymptotic expansions of the following functions and writing
$\displaystyle e^{-t/m}=s$ for brevity

$\displaystyle\displaystyle\left[\sinh\left(\frac{t}{m}\right)\right]^{-2}$
$\displaystyle\displaystyle=4e^{-2t/m}\left(1+2e^{-2t/m}+O(e^{-4t/m})\right)=4s^{2}+O(s^{4})$
$\displaystyle\displaystyle\coth\left(\frac{t}{m}\right)$
$\displaystyle\displaystyle=1+2e^{-2t/m}+O(e^{-4t/m})=1+2s^{2}+O(s^{4})$
$\displaystyle\displaystyle\coth t$
$\displaystyle\displaystyle=1+2e^{-2t}+O(e^{-4t})=1+2s^{2m}+O(s^{4m})$
$\displaystyle\displaystyle\tanh^{p}\left(\frac{t}{m}\right)$
$\displaystyle\displaystyle=\left(1-2e^{-2t/m}+O(e^{-4t/m})\right)^{p}=1-2pe^{-2t/m}+O(e^{-4t/m})=1-2ps^{2}+O(s^{4}).$
(3.17)

Substituting the asymptotic expressions in (3.2) we get

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v$
$\displaystyle\displaystyle=\alpha_{\lambda}^{p-2}(1-2ps^{2}+O(s^{4}))\left[(n-1)\alpha_{\lambda}(1+2s^{2}+O(s^{4}))(1+2s^{2m}+O(s^{4m}))\right.$
$\displaystyle\displaystyle\quad\left.-(p-1)\alpha_{\lambda}^{2}+4(p-1)\frac{\alpha_{\lambda}}{m}(s^{2}+O(s^{4}))\right]v^{p-1}(t)$
(3.18)
$\displaystyle\displaystyle=(1-2ps^{2}+O(s^{4}))\left[\underbrace{\alpha_{\lambda}^{p-1}(n-1-(p-1)\alpha_{\lambda})}_{=:f(\alpha_{\lambda})=\lambda}+2\alpha_{\lambda}^{p-1}(n-1)s^{2}\right.$
$\displaystyle\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\frac{4\alpha_{\lambda}^{p-1}}{m}(p-1)s^{2}+O(s^{4})\right]v^{p-1}(t)$
(3.19)
$\displaystyle\displaystyle=\left[\lambda+2\left(\alpha_{\lambda}^{p-1}(n-1)-p\lambda\right)s^{2}+\frac{4\alpha_{\lambda}^{p-1}}{m}(p-1)s^{2}+O(s^{4})\right]v^{p-1}(t)$
(3.20) $\displaystyle\displaystyle=\lambda
v^{p-1}(t)+2c_{\lambda}e^{-2t/m}\left(1+O(e^{-2t/m})\right)v^{p-1}(t)$ (3.21)

where, $\displaystyle f(\alpha)$ is as in (3.1), the constant
$\displaystyle\displaystyle
c_{\lambda}:=\left(\alpha_{\lambda}^{p-1}(n-1)-p\lambda\right)+\frac{2\alpha_{\lambda}^{p-1}}{m}(p-1)$
is positive since, $\displaystyle\displaystyle\alpha_{\lambda}>\frac{n-1}{p}$
and $\displaystyle\displaystyle\lambda<\left(\frac{n-1}{p}\right)^{p}$.
Therefore the function $\displaystyle\displaystyle
v(x):=\left[\cosh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{m}\right)\right]^{-m\alpha_{\lambda}}$
for $\displaystyle m\geq 2$ is a super-solution to the equation (3.12)
whenever $\displaystyle
t=\operatorname{dist}_{\mathbb{B}^{n}}(0,x)>R_{\lambda}$ is chosen large
enough.
By a similar calculation we also have $\displaystyle\displaystyle
v(x):=\left[\sinh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{2}\right)\right]^{-2\alpha_{\lambda}}$
is a sub-solution to (3.13) in
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x)>R_{\lambda}$. ∎

###  3.3. Proof of sharp decay

We first prove a weaker decay estimate for solutions to (1.1) which we will
subsequently reuse to prove the sharp decay estimate. Before proceeding with
the proof of decay estimates we recall the following Picone-inequality:

######  Lemma 3.4.

(Picone Inequality) Let $\displaystyle p\in(1,n)$ and $\displaystyle u,v\in
W^{1,p}_{\text{loc}}(\mathbb{B}^{n})$ be two positive functions then

$\displaystyle\displaystyle\mathscr{B}(u,v):=$
$\displaystyle\displaystyle|\nabla_{g}u|_{g}^{p-2}g\left(\nabla_{g}u,\nabla_{g}\left(u-\frac{v^{p}}{u^{p}}u\right)\right)+|\nabla_{g}v|_{g}^{p-2}g\left(\nabla_{g}v,\nabla_{g}\left(v-\frac{u^{p}}{v^{p}}v\right)\right)$
$\displaystyle\displaystyle\geq
C_{p}\min\left\\{u^{p},v^{p}\right\\}\left(|\nabla_{g}\log
u|_{g}+|\nabla_{g}\log v|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
v)|_{g}^{2}.$ (3.22)

###### Proof.

Directly follows from the $\displaystyle p$-homogeneity and multiplying by the
conformal factor in the Euclidean inequality from Lemma-3.1 in [39] when
$\displaystyle 1<p<2$ and Lemma-3.1 of [26] when $\displaystyle p>2$. ∎

######  Lemma 3.5.

Let, $\displaystyle u$ be a solution to (1.1). Then for each
$\displaystyle\varepsilon>0$ such that
$\displaystyle\lambda+\varepsilon<\lambda_{\text{max}}$ there is a constant
$\displaystyle C_{\lambda+\varepsilon}>0$ such that

$\displaystyle\displaystyle u(x)\leq
C_{\lambda+\varepsilon}\left[\cosh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{2}\right)\right]^{-2\alpha_{\lambda+\varepsilon}},\,\forall\,x\in\mathbb{B}^{n}.$
(3.23)

###### Proof.

Recall from Lemma-3.3, the function $\displaystyle\displaystyle
v_{\lambda+\varepsilon}(x):=C_{\lambda+\varepsilon}\left[\cosh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{2}\right)\right]^{-2\alpha_{\lambda+\varepsilon}}\in\mathcal{D}^{1,p}(\mathbb{B}^{n})$
is a super-solution to the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v_{\lambda+\varepsilon}-\lambda
v_{\lambda+\varepsilon}^{p-1}\geq\varepsilon v_{\lambda+\varepsilon}^{p-1}.$
(3.24)

Using Lemma-3.2 we may choose $\displaystyle R_{\lambda+\varepsilon}>0$ large
enough such that $\displaystyle u^{q-p}(x)\leq\varepsilon$ in
$\displaystyle\mathbb{B}^{n}\setminus B_{R_{\lambda+\varepsilon}}$. Then
$\displaystyle u$ is a subsolution to the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}u-\lambda
u^{p-1}\leq\varepsilon u^{p-1}.$ (3.25)

Finally we may fix $\displaystyle C_{\lambda+\varepsilon}>0$ large enough such
that $\displaystyle u\leq v_{\lambda+\varepsilon}$ in $\displaystyle
B_{R_{\lambda+\varepsilon}}$. Let $\displaystyle\eta\in
C_{c}^{\infty}(\mathbb{B}^{n})$ be a non-negative test function such that
$\displaystyle\eta\equiv 1$ in $\displaystyle B_{R}$ and
$\displaystyle\eta\equiv 0$ in $\displaystyle\mathbb{B}^{n}\setminus B_{R+1}$
where, $\displaystyle R>R_{\lambda+\varepsilon}$. Then testing (3.25) with
$\displaystyle\varphi_{1}:=\eta
u^{1-p}(u^{p}-v_{\lambda+\varepsilon}^{p})_{+}$ and (3.24) with
$\displaystyle\varphi_{2}:=\eta
v_{\lambda+\varepsilon}^{1-p}(u^{p}-v_{\lambda+\varepsilon}^{p})_{+}$ which
are admissible test functions supported in $\displaystyle B_{R+1}\setminus
B_{R_{\lambda+\varepsilon}}$ and subtracting we get

$\displaystyle\displaystyle\int_{\mathbb{B}^{n}\cap\left\\{u\geq
v_{\lambda+\varepsilon}\right\\}}\eta\mathscr{B}\left(u,v_{\lambda+\varepsilon}\right)\,dV_{g}$
$\displaystyle\displaystyle\leq\int_{\mathbb{B}^{n}\cap\left\\{u\geq
v_{\lambda+\varepsilon}\right\\}}\eta(u^{q-p}-\varepsilon)(u^{p}-v_{\lambda+\varepsilon}^{p})_{+}\,dV_{g}+\int_{B_{R+1}\setminus
B_{R}}|\nabla_{g}\eta|_{g}\left(|\nabla_{g}u|_{g}^{p-1}u+|\nabla_{g}\log
v_{\lambda+\varepsilon}|_{g}^{p-1}u^{p}\right)\,dV_{g}$
$\displaystyle\displaystyle\leq\int_{B_{R+1}\setminus
B_{R}}|\nabla_{g}\eta|_{g}\left(|\nabla_{g}u|_{g}^{p-1}u+|\nabla_{g}\log
v_{\lambda+\varepsilon}|_{g}^{p-1}u^{p}\right)\,dV_{g}$ (3.26)

where, $\displaystyle\mathscr{B}\left(u,v_{\lambda+\varepsilon}\right)$ is as
in (3.4). On the RHS of (3.3) we use the estimates
$\displaystyle|\nabla_{g}\eta|_{g}\leq c$ (for all $\displaystyle R>0$),

$\displaystyle\int_{B_{R+1}\setminus
B_{R}}|\nabla_{g}u|_{g}^{p-1}u\,dV_{g}\leq\lVert\nabla_{g}u\rVert_{L^{p}(B_{R+1}\setminus
B_{R})}^{p-1}\lVert u\rVert_{L^{p}(B_{R+1}\setminus B_{R})}\to 0$

as $\displaystyle R\to\infty$ and $\displaystyle|\nabla_{g}\log
v_{\lambda+\varepsilon}|_{g}\leq A_{\lambda+\epsilon}$ in
$\displaystyle\mathbb{B}^{n}$. Using the Picone inequality (3.4) on the LHS of
(3.3) and letting $\displaystyle R\to\infty$ on the RHS of (3.3) we get

$\displaystyle\displaystyle C_{p}\int_{\mathbb{B}^{n}\setminus
B_{R_{\lambda+\varepsilon}}\cap\left\\{u\geq
v_{\lambda+\varepsilon}\right\\}}v_{\lambda+\varepsilon}^{p}\left(|\nabla_{g}\log
u|_{g}+|\nabla_{g}\log
v_{\lambda+\varepsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
v_{\lambda+\varepsilon})|_{g}^{2}\,dV_{g}\leq 0.$ (3.27)

Since, $\displaystyle\nabla_{g}v_{\lambda+\varepsilon}\neq 0$ in
$\displaystyle\mathbb{B}^{n}\setminus\\{0\\}$, from (3.27) we must have
$\displaystyle\log u-\log v_{\lambda+\varepsilon}=C_{0}$ in
$\displaystyle\mathbb{B}^{n}\setminus
B_{R_{\lambda+\varepsilon}}\cap\left\\{u\geq
v_{\lambda+\varepsilon}\right\\}$. Since, $\displaystyle u\leq
v_{\lambda+\varepsilon}$ on $\displaystyle\partial
B_{R_{\lambda+\varepsilon}}$ we conclude $\displaystyle C_{0}=0$ i.e.,
$\displaystyle u=v_{\lambda+\varepsilon}$ in
$\displaystyle\mathbb{B}^{n}\setminus
B_{R_{\lambda+\varepsilon}}\cap\left\\{u\geq
v_{\lambda+\varepsilon}\right\\}$. Thus $\displaystyle u\leq
v_{\lambda+\varepsilon}$ in $\displaystyle\mathbb{B}^{n}\setminus
B_{R_{\lambda+\varepsilon}}$ as well. ∎

Now we proceed with the proof of sharp decay estimate and gradient estimate
for a solution of (1.1).

######  Theorem 3.6.

Let, $\displaystyle u$ be a solution of (1.1). Then there are constants
$\displaystyle C_{1},C_{2}>0$ and $\displaystyle m\geq 2$ (chosen large enough
depending on the exponent $\displaystyle q$) such that

$\displaystyle\displaystyle
C_{1}\left[\sinh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{2}\right)\right]^{-2\alpha_{\lambda}}\leq
u(x)\leq
C_{2}\left[\cosh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{m}\right)\right]^{-m\alpha_{\lambda}},\,\forall\,x\in\mathbb{B}^{n}\setminus
B_{\overline{R}}$ (3.28)

for some $\displaystyle\overline{R}>0$.

###### Proof.

We prove the upper bound. The proof of lower bound is much simpler and follows
similarly.
Recall from Lemma-3.3 that $\displaystyle\displaystyle
v(x):=C_{2}\left[\cosh\left(\frac{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}{m}\right)\right]^{-m\alpha_{\lambda}}$
is a super-solution to the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v-\lambda v^{p-1}\geq
c_{\lambda}e^{-\frac{2}{m}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}v^{p-1}\text{
in }\mathbb{B}^{n}\setminus B_{R_{\lambda}}$ (3.29)

for $\displaystyle C_{2}>0$ and $\displaystyle m\geq 2$.
Let $\displaystyle\eta\in C_{c}^{\infty}(\mathbb{B}^{n})$ be a non-negative
test function such that $\displaystyle\eta\equiv 1$ in $\displaystyle B_{R}$
and $\displaystyle\eta\equiv 0$ in $\displaystyle\mathbb{B}^{n}\setminus
B_{R+1}$ where, $\displaystyle R>\overline{R}>R_{\lambda}$
($\displaystyle\overline{R}$ to be determined later).
Then testing equation (1.1) with $\displaystyle\varphi_{1}:=\eta
u^{1-p}(u^{p}-v^{p})_{+}$ and equation (3.29) with
$\displaystyle\varphi_{2}:=\eta v^{1-p}(u^{p}-v^{p})_{+}$ and subtracting we
have

$\displaystyle\displaystyle\int_{\mathbb{B}^{n}\cap\left\\{u\geq
v\right\\}}\eta\mathscr{B}(u,v)\,dV_{g}$
$\displaystyle\displaystyle\leq\int_{\mathbb{B}^{n}\cap\left\\{u\geq
v\right\\}}\eta\left(u^{q-p}-c_{\lambda}e^{-\frac{2}{m}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}\right)(u^{p}-v^{p})_{+}\,dV_{g}$
$\displaystyle\displaystyle\quad+\int_{B_{R+1}\setminus
B_{R}}|\nabla_{g}\eta|_{g}\left(|\nabla_{g}u|_{g}^{p-1}u+|\nabla_{g}\log
v|_{g}^{p-1}u^{p}\right)\,dV_{g}:=I_{1}+I_{2},$ (3.30)

where, $\displaystyle\mathscr{B}(u,v)$ is as in (3.4).
Now, using the estimate from Lemma-3.5 we may choose
$\displaystyle\overline{R}>0$ large enough such that

$\displaystyle u^{q-p}(x)\leq
C_{\lambda+\varepsilon}^{q-p}e^{-(q-p)\alpha_{\lambda+\varepsilon}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}\leq
c_{\lambda}e^{-\frac{2}{m}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}$

for $\displaystyle x\in\mathbb{B}^{n}\setminus B_{\overline{R}}$ (the later is
ensured by choosing $\displaystyle
m>\frac{2}{(q-p)\alpha_{\lambda+\varepsilon}}$) and choose $\displaystyle
C_{2}>0$ large enough such that $\displaystyle u\leq v$ in $\displaystyle
B_{\overline{R}}$. Thus, $\displaystyle I_{1}\leq 0$ and estimating as in the
proof of Lemma-3.5 we have $\displaystyle I_{2}\to 0$ as $\displaystyle
R\to\infty$. Thus from the Picone inequality (3.4) we get

$\displaystyle\displaystyle C_{p}\int_{\mathbb{B}^{n}\setminus
B_{\overline{R}}\,\cap\,\left\\{u\geq v\right\\}}v^{p}\left(|\nabla_{g}\log
u|_{g}+|\nabla_{g}\log v|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
v)|_{g}^{2}\,dV_{g}\leq 0.$ (3.31)

Since, $\displaystyle\nabla_{g}v\neq 0$ in
$\displaystyle\mathbb{B}^{n}\setminus\left\\{0\right\\}$ we get the desired
upper bound (3.28) as before. ∎

######  Theorem 3.7.

Let, $\displaystyle u$ be a positive solution of (1.1). Then
$\displaystyle\exists\,A>0$ such that the gradient estimate

$\displaystyle\displaystyle A^{-1}u(x)\leq|\nabla_{g}u|_{g}(x)\leq Au(x)$
(3.32)

holds in $\displaystyle x\in\mathbb{B}^{n}\setminus B_{R_{0}}$ for some
$\displaystyle R_{0}$ large. In fact we have
$\displaystyle\lim\limits_{\operatorname{dist}_{\mathbb{B}^{n}}(0,x)\to\infty}\frac{|\nabla_{g}u|_{g}}{u}(x)=\alpha_{\lambda}$,
where, $\displaystyle\alpha_{\lambda}$ is as in (3.2).

###### Proof.

Using the sharp decay estimates Theorem-3.6 on a positive solution of (1.1) we
have

$\displaystyle
ce^{-\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}\leq u(x)\leq
Ce^{-\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}$

in $\displaystyle\mathbb{B}^{n}$ for some positive constants $\displaystyle
c,C>0$.
Let, $\displaystyle x_{0}\in\mathbb{B}^{n}\setminus B_{R_{0}+1}$ and define
$\displaystyle v_{0}(x)=(u\circ\tau_{0})$ where,
$\displaystyle\tau_{0}\in\mathcal{M}(\mathbb{B}^{n})$ is a Hyperbolic
reflection such that $\displaystyle\tau_{0}(0)=x_{0}$ and
$\displaystyle\tau_{0}(x_{0})=0$. Then from equation (1.1) we get

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v_{0}=(\lambda+v_{0}^{q-p})v_{0}^{p-1}=h_{0}v_{0}^{p-1}$
(3.33)

where, $\displaystyle h_{0}=\lambda+O(e^{-(q-p)\alpha_{\lambda}R_{0}})$ in
$\displaystyle B_{1}$. Then using $\displaystyle C^{1,\sigma}$ estimate (see
[18], [35]) and Harnack inequality (see [36]) for this equation we have

$\displaystyle\displaystyle\sup_{B_{1/2}}|\nabla_{g}v_{0}|_{g}\leq
C_{1}\sup_{B_{1}}\,v_{0}\leq C_{2}v_{0}(0).$ (3.34)

Reverting back to $\displaystyle u$ we get the estimate

$\displaystyle\displaystyle|\nabla_{g}u|_{g}(x_{0})\leq Au(x_{0})$ (3.35)

where the constant $\displaystyle A$ is independent of $\displaystyle
x_{0}\in\mathbb{B}^{n}\setminus B_{R_{0}+1}$.

Now to see the gradient estimate from below, suppose to the contrary there is
a sequence of points
$\displaystyle\\{x_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{B}^{n}$ such that
$\displaystyle|\nabla_{g}u|_{g}(x_{k})\leq\epsilon_{k}u(x_{k})$ and
$\displaystyle R_{k}:=\operatorname{dist}_{\mathbb{B}^{n}}(0,x_{k})\to\infty$
where, $\displaystyle\epsilon_{k}\to 0^{+}$ as $\displaystyle k\to\infty$.
Define $\displaystyle v_{k}:=\dfrac{(u\circ\tau_{k})}{u(x_{k})}$ where,
$\displaystyle\tau_{k}\in\mathcal{M}(\mathbb{B}^{n})$ is a Hyperbolic
reflection such that $\displaystyle\tau_{k}(0)=x_{k}$ and
$\displaystyle\tau_{k}(x_{k})=0$. Then $\displaystyle v_{k}(0)=1$,
$\displaystyle|\nabla_{g}v_{k}|_{g}(0)\leq\epsilon_{k}$ and $\displaystyle
v_{k}$ satisfies the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v_{k}=h_{k}v_{k}^{p-1}$
(3.36)

where, $\displaystyle h_{k}:=\lambda+(u\circ\tau_{k})^{q-p}\to\lambda$
uniformly in $\displaystyle B_{R}$ as $\displaystyle k\to\infty$ for each
$\displaystyle R>0$. Furthermore, using the sharp decay estimate of
$\displaystyle u$ we have $\displaystyle v_{k}$ satisfies the pointwise bounds

$\displaystyle\displaystyle C_{1}E_{k}(x)\leq v_{k}(x)\leq C_{2}E_{k}(x)$
(3.37)

for $\displaystyle x\in B_{R}$ and the positive constants $\displaystyle
C_{1},C_{2}>0$ (independent of $\displaystyle R$) where,

$\displaystyle\displaystyle
E_{k}(x):=\left(\sinh\left(\frac{1}{2}\operatorname{dist}_{\mathbb{B}^{n}}(0,x_{k})\right)\right)^{2\alpha_{\lambda}}\left(\sinh\left(\frac{1}{2}\operatorname{dist}_{\mathbb{B}^{n}}(x,x_{k})\right)\right)^{-2\alpha_{\lambda}}=\left(\frac{|x_{k}|^{2}(1-|x|^{2})}{|x-x_{k}|^{2}}\right)^{\alpha_{\lambda}}$
(3.38)

uniformly for all $\displaystyle k\geq K_{R}$ sufficiently large. Here we used
the fact that

$\displaystyle\displaystyle(u\circ\tau_{k})(x)\approx
e^{-\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,\tau_{k}(x))}=e^{-\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(\tau_{k}(0),x)}\approx\left(\sinh\left(\frac{1}{2}\operatorname{dist}_{\mathbb{B}^{n}}(x_{k},x)\right)\right)^{-2\alpha_{\lambda}}$
(3.39)

for $\displaystyle x\in B_{R}$ with uniform constants (independent of
$\displaystyle R$) for all $\displaystyle k\geq K_{R}$ sufficiently large and
the identity (see [34], equation (2.2.4))

$\displaystyle\displaystyle\sinh^{2}\left(\frac{1}{2}\operatorname{dist}_{\mathbb{B}^{n}}(x,y)\right)=\frac{|x-y|^{2}}{(1-|x|^{2})(1-|y|^{2})},\,\forall\,x,y\in\mathbb{B}^{n}.$
(3.40)

Then using $\displaystyle C^{1,\sigma}$ estimate on (3.36) in $\displaystyle
B_{R}$ we get $\displaystyle[v_{k}]_{C^{1,\sigma}(B_{R})}\leq
C_{R}\sup_{B_{R}}v_{k}$ where, the constant $\displaystyle C_{R}$ only depends
on $\displaystyle B_{R}$ for all $\displaystyle k\geq K_{R}$ sufficiently
large. Passing through a subsequence if necessary (which we still denote as
$\displaystyle\\{v_{k}\\}_{k\in\mathbb{N}}$) we get $\displaystyle v_{k}\to
v_{R,\infty}$ in $\displaystyle C^{1}(B_{R})$ and $\displaystyle
x_{k}\to\xi\in\partial_{\infty}\mathbb{B}^{n}(=\mathbb{S}^{n-1})$ as
$\displaystyle k\to\infty$. Then $\displaystyle v_{R,\infty}$ satisfies the
equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v_{R,\infty}=\lambda
v_{R,\infty}^{p-1}.$ (3.41)

Letting $\displaystyle R\to\infty$ and using a diagonalization argument we
have a subsequence $\displaystyle v_{k}\to v_{\infty}$ locally uniformly in
$\displaystyle C^{1}(\mathbb{B}^{n})$ such that $\displaystyle v_{\infty}$
satisfies the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v_{\infty}=\lambda
v_{\infty}^{p-1}$ (3.42)

in $\displaystyle\mathbb{B}^{n}$ and from (3.37) the pointwise bounds

$\displaystyle\displaystyle C_{1}E_{\xi}(x)\leq v_{\infty}(x)\leq
C_{2}E_{\xi}(x)$ (3.43)

where, $\displaystyle
E_{\xi}(x):=\left(\frac{1-|x|^{2}}{|x-\xi|^{2}}\right)^{\alpha_{\lambda}}$
with $\displaystyle v_{\infty}(0)=\lim_{k\to\infty}v_{k}(0)=1$ and
$\displaystyle|\nabla_{g}v_{\infty}|_{g}(0)=\lim\limits_{k\to\infty}|\nabla_{g}v_{k}|_{g}(0)=0$.

But from the classification of eigenfunctions from Theorem-3.8 we know
$\displaystyle v_{\infty}\equiv E_{\xi}$, which contradicts
$\displaystyle|\nabla_{g}E_{\xi}|_{g}(0)=\alpha_{\lambda}>0$. This completes
the proof of the gradient estimate from below. By the same argument let
$\displaystyle\\{x_{k}\\}_{k\in\mathbb{N}}$ be any sequence of points such
that $\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x_{k})\to\infty$,
then we can extract a subsequence $\displaystyle\\{x_{k_{j}}\\}$ such that
$\displaystyle\frac{|\nabla_{g}u|_{g}}{u}(x_{k_{j}})\to|\nabla_{g}E_{\xi}|_{g}(0)=\alpha_{\lambda}$
for some $\displaystyle\xi\in\mathbb{S}^{n-1}$. Therefore, the limit of
$\displaystyle\frac{|\nabla_{g}u|_{g}}{u}(x)$ as
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x)\to\infty$ exists and
equals $\displaystyle\alpha_{\lambda}$. ∎

Keeping in continuation with the previous notation we state Theorem-1.2 in the
Poincaré ball model $\displaystyle\mathbb{B}^{n}$. The proof is mostly carried
out in the equivalent upper half-space model of the Hyperbolic space, but we
expand upon the parallel picture in the ball model in Remark-3.9.

######  Theorem 3.8.

Let, $\displaystyle\lambda\in[0,\lambda_{\text{max}})$ and $\displaystyle v$
be a $\displaystyle C^{1}$ solution of the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}v=\lambda v^{p-1}$
(3.44)

in $\displaystyle\mathbb{B}^{n}$ such that there exists positive constant
$\displaystyle C>0$ and a point $\displaystyle\xi\in\mathbb{S}^{n-1}$ such
that

$\displaystyle\displaystyle E_{\xi}\leq v\leq CE_{\xi}$ (3.45)

where, $\displaystyle
E_{\xi}(x):=\left(\frac{1-|x|^{2}}{|x-\xi|^{2}}\right)^{\alpha_{\lambda}}$.
Then $\displaystyle\exists\,c_{0}\in[1,C]$ such that $\displaystyle v\equiv
c_{0}E_{\xi}$ in $\displaystyle\mathbb{B}^{n}$. This in particular implies
$\displaystyle\frac{|\nabla_{g}v|_{g}}{v}=\frac{|\nabla_{g}E_{\xi}|_{g}}{E_{\xi}}\equiv\alpha_{\lambda}$
in $\displaystyle\mathbb{B}^{n}$.

###### Proof.

It is more convenient to prove the result in the upper half-space model of the
Hyperbolic space. Using an orthogonal transformation in
$\displaystyle\mathbb{B}^{n}$ about origin without loss of generality we may
assume $\displaystyle\xi=-e_{n}$. Recall that the extended Mobius map
$\displaystyle\Phi:\overline{\mathbb{R}^{n}_{+}}\cup\\{\infty\\}\rightarrow\overline{\mathbb{B}^{n}}$
defined by

$\displaystyle\displaystyle\Phi(x):=\left(\frac{2x_{1}}{|x+e_{n}|^{2}},\cdots,\frac{2x_{n-1}}{|x+e_{n}|^{2}},\frac{1-|x|^{2}}{|x+e_{n}|^{2}}\right)\text{
and }\Phi(\infty):=-e_{n}$ (3.46)

defines an isometry between the Poincaré ball model
$\displaystyle\mathbb{B}^{n}$ and the upper half-space model
$\displaystyle\mathbb{H}^{n}$ of the Hyperbolic space. In fact
$\displaystyle\Phi$ is the inversion map w.r.t. the sphere centered at
$\displaystyle-e_{n}$ and radius $\displaystyle\sqrt{2}$ given by
$\displaystyle\Phi(x)=-e_{n}+\frac{2(x+e_{n})}{|x+e_{n}|^{2}}$ and it
satisfies $\displaystyle\Phi\circ\Phi=\operatorname{Id}$.

Let, $\displaystyle u:=v\circ\Phi$. Then the problem can be equivalently posed
in upper half-space model as

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{H}^{n}}u=\lambda u^{p-1}$
(3.47)

and satisfies the bounds

$\displaystyle\displaystyle x_{n}^{\alpha_{\lambda}}\leq u(x)\leq
Cx_{n}^{\alpha_{\lambda}}.$ (3.48)

We claim that $\displaystyle u(x)=c_{0}x_{n}^{\alpha_{\lambda}}$ for some
$\displaystyle c_{0}\in[1,C]$, from which our claim about the eigenfunction in
$\displaystyle\mathbb{B}^{n}$ model will follow. We start by showing that
$\displaystyle u$ has one dimensional symmetry in the $\displaystyle
e_{n}$-direction. This then reduces the problem to showing uniqueness (up to
constant multiples) of solutions to an ordinary differential equation.
Step-I: Note that due to the Hyperbolic isometry invariance and homogeneity of
the equation (3.44) by applying the standard $\displaystyle C^{1,\sigma}$
estimate and the Harnack inequality we have the global gradient estimate

$\displaystyle|\nabla_{g}u|_{g}(x)\leq Au(x)$

for all $\displaystyle x\in\mathbb{H}^{n}$.
For $\displaystyle x_{0}=(x_{0}^{\prime},0)\in\partial\mathbb{R}^{n}_{+}$ and
$\displaystyle R>0$, let us denote the Hyperbolic half-spaces

$\displaystyle
H_{R}(x_{0}):=\left\\{x\in\mathbb{R}^{n}_{+}:|x-x_{0}|<R,\,x_{n}>0\right\\}$

and its boundary the orthogonal sphere (orthogonal to the boundary
$\displaystyle\partial\mathbb{R}^{n}_{+}$)

$\displaystyle
S_{R}(x_{0}):=\left\\{x\in\mathbb{R}^{n}_{+}:|x-x_{0}|=R,\,x_{n}>0\right\\}.$

We start by noting that
$\displaystyle\lVert\nabla_{g}u\rVert_{L^{p}(H_{R}(x_{0}))}^{p}=\int_{H_{R}(x_{0})}|\nabla_{g}u|_{g}^{p}x_{n}^{-n}\,dx\leq(AC)^{p}\int_{H_{R}(x_{0})}x_{n}^{p\alpha_{\lambda}-n}\,dx<+\infty$
since, $\displaystyle\alpha_{\lambda}>\frac{n-1}{p}$.
Let $\displaystyle\tau\in\mathcal{M}(\mathbb{H}^{n})$ denote the reflection
across the orthogonal sphere $\displaystyle S_{R}(x_{0})$ i.e.,

$\displaystyle\tau(x):=x_{0}+\frac{R^{2}(x-x_{0})}{|x-x_{0}|^{2}}.$

Then $\displaystyle u_{\tau}:=(u\circ\tau)$ satisfies the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{H}^{n}}u_{\tau}=\lambda
u_{\tau}^{p-1}$ (3.49)

along with the bound

$\displaystyle\displaystyle\left(\frac{R^{2}x_{n}}{|x-x_{0}|^{2}}\right)^{\alpha_{\lambda}}\leq
u_{\tau}(x)\leq
C\left(\frac{R^{2}x_{n}}{|x-x_{0}|^{2}}\right)^{\alpha_{\lambda}}.$ (3.50)

Comparing (3.48) and (3.50) we have $\displaystyle u(x)<u_{\tau}(x)$ in the
set $\displaystyle H_{R^{\prime}}(x_{0})$ where, $\displaystyle
R^{\prime}:=RC^{-\frac{1}{2\alpha_{\lambda}}}<R$. Using the fact that
$\displaystyle\tau\in\mathcal{M}(\mathbb{H}^{n})$ is an isometry we also have

$\displaystyle\|\nabla_{g}u_{\tau}\rVert_{L^{p}(H_{R}(x_{0})\setminus
H_{R^{\prime}}(x_{0}))}=\|\nabla_{g}u\rVert_{L^{p}(H_{R^{\prime\prime}}(x_{0})\setminus
H_{R}(x_{0}))}\leq\|\nabla_{g}u\rVert_{L^{p}(H_{R^{\prime\prime}}(x_{0}))}<+\infty$

where, $\displaystyle
R^{\prime\prime}:=R^{2}/R^{\prime}=RC^{\frac{1}{2\alpha_{\lambda}}}>R$.
We note that the functions
$\displaystyle\phi_{1}:=u^{1-p}(u^{p}-u_{\tau}^{p})_{+}\chi_{H_{R}(x_{0})}$
and
$\displaystyle\phi_{2}:=u_{\tau}^{1-p}(u^{p}-u_{\tau}^{p})_{+}\chi_{H_{R}(x_{0})}$
are supported in $\displaystyle H_{R}(x_{0})\setminus H_{R^{\prime}}(x_{0})$,
since $\displaystyle u=u_{\tau}$ on $\displaystyle\partial H_{R}(x_{0})$ and
$\displaystyle u<u_{\tau}$ in $\displaystyle H_{R^{\prime}}(x_{0})$.
Furthermore, using the bounds on $\displaystyle u$ we have

$\displaystyle u(x)\approx
u_{\tau}(x)\approx\left(\frac{R^{2}x_{n}}{|x-x_{0}|^{2}}\right)^{\alpha_{\lambda}}\approx
x_{n}^{\alpha_{\lambda}}$

in $\displaystyle H_{R}(x_{0})\setminus
H_{R^{\prime}}(x_{0})=\left\\{x\in\mathbb{R}^{n}_{+}:R^{\prime}\leq|x-x_{0}|<R\right\\}$.
By Harnack’s inequality we also have $\displaystyle|\nabla_{g}u|_{g}\lesssim
u\lesssim x_{n}^{\alpha_{\lambda}}$ and
$\displaystyle|\nabla_{g}u_{\tau}|_{g}\lesssim u_{\tau}\lesssim
x_{n}^{\alpha_{\lambda}}$ in this set. Then we note that

$\displaystyle\displaystyle|\nabla_{g}\phi_{2}|_{g}$
$\displaystyle\displaystyle=\left|(u^{p}-u_{\tau}^{p})_{+}\nabla_{g}(u_{\tau}^{1-p})+u_{\tau}^{1-p}\nabla_{g}(u^{p}-u_{\tau}^{p})_{+}\right|_{g}$
$\displaystyle\displaystyle\leq(u^{p}-u_{\tau}^{p})_{+}\left|\nabla_{g}(u_{\tau}^{1-p})|_{g}+u_{\tau}^{1-p}|\nabla_{g}(u^{p}-u_{\tau}^{p})_{+}\right|_{g}$
$\displaystyle\displaystyle\leq(p-1)u_{\tau}^{-p}(u^{p}-u_{\tau}^{p})_{+}|\nabla_{g}u_{\tau}|_{g}+pu_{\tau}^{1-p}u^{p-1}|\nabla_{g}u|_{g}+pu_{\tau}^{1-p}u_{\tau}^{p-1}|\nabla_{g}u_{\tau}|_{g}$
$\displaystyle\displaystyle\leq\left((p-1)\left(\frac{u}{u_{\tau}}\right)^{p}+p\right)|\nabla_{g}u_{\tau}|_{g}+p\left(\frac{u}{u_{\tau}}\right)^{p-1}|\nabla_{g}u|_{g}\lesssim
u_{\tau}+u\lesssim x_{n}^{\alpha_{\lambda}}$

in $\displaystyle H_{R}(x_{0})\setminus H_{R^{\prime}}(x_{0})$. Similarly, we
have $\displaystyle|\nabla_{g}\phi_{1}|_{g}\lesssim x_{n}^{\alpha_{\lambda}}$
in $\displaystyle H_{R}(x_{0})\setminus H_{R^{\prime}}(x_{0})$. Therefore,
$\displaystyle\phi_{j}\in\mathcal{D}^{1,p}(\mathbb{H}^{n})$ for $\displaystyle
j=1,2$ are both admissible test functions.
We test equations (3.47) with
$\displaystyle\phi_{1}=u^{1-p}(u^{p}-u_{\tau}^{p})_{+}\chi_{H_{R}(x_{0})}$ and
(3.49) with
$\displaystyle\phi_{2}=u_{\tau}^{1-p}(u^{p}-u_{\tau}^{p})_{+}\chi_{H_{R}(x_{0})}$,
subtracting and combining with Lemma-3.4 we get

$\displaystyle\displaystyle C_{p}\int_{H_{R}(x_{0})\cap\\{u\geq
u_{\tau}\\}}u_{\tau}^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\tau}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log u_{\tau})|_{g}\,dV_{g}\leq
0.$ (3.51)

Then, $\displaystyle(\log u-\log u_{\tau})$ is a constant in $\displaystyle
H_{R}(x_{0})\cap\\{u\geq u_{\tau}\\}$. Since, $\displaystyle u=u_{\tau}$ on
$\displaystyle\partial H_{R}(x_{0})\cap\\{x_{n}>0\\}$ we conclude
$\displaystyle u=u_{\tau}$ in $\displaystyle H_{R}(x_{0})\cap\\{u\geq
u_{\tau}\\}$. Therefore, we have $\displaystyle u_{\tau}\geq u$ in
$\displaystyle H_{R}(x_{0})$.
This in particular implies the outward normal derivative of $\displaystyle u$
on the orthogonal sphere $\displaystyle S_{R}(x_{0})$ is non-negative i.e.,
$\displaystyle\frac{(x-x_{0})}{|x-x_{0}|}\cdot\nabla u(x)\geq 0$ for each
$\displaystyle x\in S_{R}(x_{0})$. Fixing the point $\displaystyle
x\in\mathbb{H}^{n}$ we may vary the orthogonal spheres $\displaystyle
S_{R}(x_{0})$ passing through $\displaystyle x$ instead where, $\displaystyle
x_{0}\in\partial\mathbb{R}^{n}_{+}$ and $\displaystyle R>0$. Then let us
denote $\displaystyle
x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}_{+}$ and for each
$\displaystyle\xi\in\mathbb{S}^{n-2}$ consider the inequality on the
orthogonal sphere $\displaystyle
S_{(x_{n}^{2}+t^{2})^{1/2}}((x^{\prime}-t\xi,0))$ with centre at
$\displaystyle(x^{\prime}-t\xi,0)\in\partial\mathbb{R}^{n}_{+}$ and passing
through $\displaystyle x$, we get
$\displaystyle\frac{(t\xi,x_{n})}{(x_{n}^{2}+t^{2})^{1/2}}\cdot\nabla u(x)\geq
0$. Letting $\displaystyle t\to+\infty$ we get
$\displaystyle(\xi,0)\cdot\nabla u(x)\geq 0$ for all
$\displaystyle\xi\in\mathbb{S}^{n-2}$ i.e., $\displaystyle\nabla
u=(0^{\prime},u_{x_{n}})$ or the horizontal component of gradient of
$\displaystyle u$ is zero. Hence, $\displaystyle u(x):=w(x_{n})$ is a function
of a single variable and looking at $\displaystyle t=0$ we get $\displaystyle
u_{x_{n}}\geq 0$.

Step-II: As a result the equation (3.47) reduces to the following ordinary
differential equation

$\displaystyle\displaystyle-t^{n}(t^{p-n}(w^{\prime})^{p-1})^{\prime}=\lambda
w^{p-1}$ (3.52)

on $\displaystyle\mathbb{R}_{+}$ where, $\displaystyle w^{\prime}\geq 0$ and
satisfies the global bounds

$\displaystyle\displaystyle t^{\alpha_{\lambda}}\leq w(t)\leq
Ct^{\alpha_{\lambda}}.$ (3.53)

Note that when $\displaystyle\lambda=0$ we can directly integrate both sides
of the ODE and conclude $\displaystyle w(t)=c_{0}t^{\frac{n-1}{p-1}}$ as
$\displaystyle\alpha_{0}=\frac{n-1}{p-1}$. Therefore, it remains to prove the
result for $\displaystyle\lambda>0$ i.e., when
$\displaystyle\frac{n-1}{p}<\alpha_{\lambda}<\frac{n-1}{p-1}$.
From the estimate $\displaystyle|\nabla_{g}u|_{g}\leq Au$ we get
$\displaystyle tw^{\prime}(t)\leq Aw(t)\leq C^{\prime}t^{\alpha_{\lambda}}$.
We have $\displaystyle t^{p-n}(w^{\prime})^{p-1}\leq
C^{\prime\prime}t^{p-n+(p-1)(\alpha_{\lambda}-1)}=C^{\prime\prime}t^{(p-1)\alpha_{\lambda}-(n-1)}\to
0$ as $\displaystyle t\to\infty$ for $\displaystyle\lambda>0$. Integrating
equation (3.52) from $\displaystyle t$ to $\displaystyle\infty$ we get

$\displaystyle\displaystyle
t^{p-n}(w^{\prime}(t))^{p-1}=-\int_{t}^{\infty}(s^{p-n}(w^{\prime})^{p-1})^{\prime}\,ds$
$\displaystyle\displaystyle=\lambda\int_{t}^{\infty}w^{p-1}s^{-n}\,ds$
$\displaystyle\displaystyle\geq\lambda\int_{t}^{\infty}s^{(p-1)\alpha_{\lambda}-n}\,ds$
$\displaystyle\displaystyle=\alpha_{\lambda}^{p-1}t^{(p-1)\alpha_{\lambda}-(n-1)}$
(3.54)

i.e., $\displaystyle w^{\prime}(t)\geq\alpha_{\lambda}t^{\alpha_{\lambda}-1}$.
Therefore using the bounds of $\displaystyle w$ we conclude that

$\displaystyle\displaystyle
0<\alpha:=\inf_{t>0}\frac{tw^{\prime}(t)}{w(t)}\leq\frac{tw^{\prime}(t)}{w(t)}\leq\sup_{t>0}\frac{tw^{\prime}(t)}{w(t)}:=\beta<\infty.$
(3.55)

We note from (5) that the linearized equation corresponding to (3.47) is
satisfied by $\displaystyle(x.\nabla u)$ (since, $\displaystyle u(x)=w(x_{n})$
has no critical points in $\displaystyle\mathbb{H}^{n}$) and is given by

$\displaystyle\displaystyle\int_{\mathbb{H}^{n}}x_{n}^{p-n}\left(|\nabla
u|^{p-2}(\nabla(x.\nabla u),\nabla\varphi)+(p-2)|\nabla
u|^{p-4}(\nabla(x.\nabla u),\nabla u)(\nabla u,\nabla\varphi)\right)\,dx$
$\displaystyle\displaystyle=(p-1)\lambda\int_{\mathbb{H}^{n}}u^{p-2}(x.\nabla
u)\varphi x_{n}^{-n}\,dx,\,\forall\,\varphi\in
C_{c}^{\infty}(\mathbb{H}^{n}).$ (3.56)

Since, $\displaystyle u(x)=w(x_{n})$ the above PDE reduces to the ordinary
differential equation

$\displaystyle\displaystyle-t^{n}(t^{p-n}(w^{\prime})^{p-2}(tw^{\prime})^{\prime})^{\prime}=\lambda
w^{p-2}(tw^{\prime}).$ (3.57)

Let, $\displaystyle\tilde{w}_{1}:=tw^{\prime}-\alpha w$ and
$\displaystyle\tilde{w}_{2}:=\beta w-tw^{\prime}$, then from (3.55) these are
non-negative functions. Combining (3.52) and (3.57) we see that
$\displaystyle\tilde{w}_{j}$ for $\displaystyle j=1,2$ satisfies

$\displaystyle\displaystyle-t^{n}(t^{p-n}(w^{\prime})^{p-2}(\tilde{w}_{j})^{\prime})^{\prime}=\lambda
w^{p-2}(\tilde{w}_{j}).$ (3.58)

From the strong minimum-principle (see Lemma-5.7) we conclude either
$\displaystyle\tilde{w}_{j}\equiv 0$ in $\displaystyle\mathbb{R}_{+}$ or has
strict inequality $\displaystyle\tilde{w}_{j}>0$ in
$\displaystyle\mathbb{R}_{+}$. In the first case clearly we must have
$\displaystyle w(t)=c_{0}t^{\alpha_{\lambda}}$ keeping in mind (3.53). In the
second case neither the infimum or supremum in (3.55) are attained in interior
of $\displaystyle\mathbb{R}_{+}$.
Therefore, without loss of generality if we assume
$\displaystyle\alpha:=\inf_{t>0}\frac{tw^{\prime}(t)}{w(t)}=\lim\limits_{t\to
0^{+}}\frac{tw^{\prime}(t)}{w(t)}$ then we have for $\displaystyle\epsilon>0$
the inequality

$\displaystyle\alpha-\epsilon<\frac{tw^{\prime}}{w}<\alpha+\epsilon$

for all $\displaystyle 0<t<\delta$ (say). In particular then from
$\displaystyle\left(\frac{w}{t^{\alpha-\epsilon}}\right)^{\prime}=\frac{tw^{\prime}-(\alpha-\epsilon)w}{t^{\alpha-\epsilon+1}}\geq
0$ we have

$\displaystyle\limsup_{t\to
0^{+}}\frac{w(t)}{t^{\alpha-\epsilon}}\leq\frac{w(t)}{t^{\alpha-\epsilon}}<+\infty$

for all $\displaystyle t<\delta$. Since, $\displaystyle w(t)\geq
t^{\alpha_{\lambda}}$ we must have
$\displaystyle\alpha-\epsilon\leq\alpha_{\lambda}$ for each
$\displaystyle\epsilon>0$ and hence $\displaystyle\alpha\leq\alpha_{\lambda}$.
Similarly from
$\displaystyle\left(\frac{w}{t^{\alpha+\epsilon}}\right)^{\prime}=\frac{tw^{\prime}-(\alpha+\epsilon)w}{t^{\alpha+\epsilon+1}}\leq
0$ we have

$\displaystyle\liminf_{t\to
0^{+}}\frac{w(t)}{t^{\alpha+\epsilon}}\geq\frac{w(t)}{t^{\alpha+\epsilon}}>0$

for all $\displaystyle t<\delta$. Using the fact that $\displaystyle w(t)\leq
Ct^{\alpha_{\lambda}}$ we must have
$\displaystyle\alpha+\epsilon\geq\alpha_{\lambda}$ for each
$\displaystyle\epsilon>0$, i.e., $\displaystyle\alpha\geq\alpha_{\lambda}$.
Hence, $\displaystyle\alpha=\alpha_{\lambda}$. By a similar consideration as
$\displaystyle t\to+\infty$ we also have
$\displaystyle\beta=\alpha_{\lambda}$. Therefore,
$\displaystyle\alpha=\beta=\alpha_{\lambda}$, which now implies $\displaystyle
w(t)=c_{0}t^{\alpha_{\lambda}}$. This completes the proof of classification of
solutions to the ordinary differential equation. ∎

######  Remark 3.9.

It is also interesting to see the proof of symmetry in Theorem-3.8 directly in
the ball model $\displaystyle\mathbb{B}^{n}$. Let $\displaystyle H$ denote an
open half-space in $\displaystyle\mathbb{B}^{n}$, i.e. the boundary of
$\displaystyle H$ is an orthogonal sphere $\displaystyle\Sigma:=\partial H$
that splits $\displaystyle\mathbb{B}^{n}$ into two isometric components
$\displaystyle H$ and $\displaystyle\mathbb{B}^{n}\setminus\overline{H}$. Let
us denote the Hyperbolic reflection w.r.t. $\displaystyle\Sigma$ as
$\displaystyle\sigma\in\mathcal{M}(\mathbb{B}^{n})$, then
$\displaystyle\mathbb{B}^{n}\setminus\overline{H}=\sigma(H)$. We define the
ideal-boundary of $\displaystyle\Sigma$ as
$\displaystyle\partial_{\infty}\Sigma:=\partial_{\infty}\overline{H}\setminus\partial_{\infty}H$,
that is the part of the ideal boundary
$\displaystyle\partial_{\infty}\mathbb{B}^{n}$ formed by geodesic rays
restricted to $\displaystyle\Sigma$.
Now, given $\displaystyle\xi\in\partial_{\infty}\mathbb{B}^{n}$ as in
statement of Theorem-3.8, if we denote $\displaystyle
u_{\sigma}=(u\circ\sigma)$, then the weak-comparison argument in the first
half of the proof shows that $\displaystyle u_{\sigma}\geq u$ in
$\displaystyle\overline{H}$, for any half-space $\displaystyle H$ such that
$\displaystyle\xi\not\in\partial_{\infty}\overline{H}$. Let,
$\displaystyle\Sigma_{1}$ be an orthogonal sphere with
$\displaystyle\xi\in\partial_{\infty}\Sigma_{1}$, that splits
$\displaystyle\mathbb{B}^{n}$ into the two half-spaces $\displaystyle H_{1}$
and $\displaystyle H_{2}=\mathbb{B}^{n}\setminus\overline{H_{1}}$. We denote
the reflection w.r.t. $\displaystyle\Sigma_{1}$ as
$\displaystyle\sigma_{1}\in\mathcal{M}(\mathbb{B}^{n})$. Note that
$\displaystyle u_{\sigma}\geq u$ in all half-spaces
$\displaystyle\overline{H}\subset H_{1}$ such that
$\displaystyle\xi\not\in\partial_{\infty}\overline{H}$. So by continuity of
$\displaystyle u$ we have $\displaystyle u_{\sigma_{1}}\geq u$ in
$\displaystyle H_{1}$. Similarly, we also have $\displaystyle
u_{\sigma_{1}}\geq u$ in $\displaystyle H_{2}$. Combining this information we
get $\displaystyle u=u_{\sigma_{1}}$ in $\displaystyle\mathbb{B}^{n}$ for all
$\displaystyle\Sigma_{1}$ such that
$\displaystyle\xi\in\partial_{\infty}\Sigma_{1}$. Now the set of reflections
in orthogonal spheres $\displaystyle\Sigma_{1}$ with
$\displaystyle\xi\in\partial_{\infty}\Sigma_{1}$ generate the parabolic
subgroup of isometries in $\displaystyle\mathcal{M}(\mathbb{B}^{n})$ that fix
$\displaystyle\xi\in\partial_{\infty}\mathbb{B}^{n}$ (for more details we
refer to Section-4.7: Classification of Transformations in [28]). Hence the
level sets of $\displaystyle u$ are horospheres with geometric center
$\displaystyle\xi\in\partial_{\infty}\mathbb{B}^{n}$, i.e. $\displaystyle u$
has horospherical symmetry.
Under an isometric map that sends $\displaystyle\mathbb{B}^{n}$ to the upper
half-space model $\displaystyle\mathbb{H}^{n}$ of the Hyperbolic space such
that $\displaystyle\xi\in\partial_{\infty}\mathbb{B}^{n}$ is mapped to
$\displaystyle+\infty\in\partial_{\infty}\mathbb{H}^{n}$, we note that
horospheres with geometric center $\displaystyle\xi$ are mapped to Euclidean
hyperplanes $\displaystyle\\{x_{n}=c\\}$ in
$\displaystyle\mathbb{R}^{n}_{+}(=\mathbb{H}^{n})$. This is indeed why we saw
the 1-dimensional symmetry of $\displaystyle u$ in the proof of Theorem-3.8 in
the upper half-space model of the Hyperbolic space.

######  Remark 3.10.

For $\displaystyle\lambda\in[0,\lambda_{\text{max}})$, each
$\displaystyle\xi\in\partial_{\infty}\mathbb{B}^{n}(=\mathbb{S}^{n-1})$ has at
least two distinct classes (upto constant multiple) of corresponding positive
eigenfunctions solving (3.44), given by

$\displaystyle\displaystyle
E(x,\xi)=\left(\frac{1-|x|^{2}}{|x-\xi|^{2}}\right)^{\alpha_{\lambda}}\text{
and }F(x,\xi)=\left(\frac{1-|x|^{2}}{|x-\xi|^{2}}\right)^{\beta_{\lambda}}$
(3.59)

where, $\displaystyle\alpha_{\lambda}$ and $\displaystyle\beta_{\lambda}$ are
as in (3.2). The important distinction between them is that
$\displaystyle\lVert\nabla_{g}E(.,\xi)\rVert_{L^{p}(\overline{H})}<+\infty$
for every Hyperbolic closed half-space $\displaystyle\overline{H}$ of
$\displaystyle\mathbb{B}^{n}$ such that
$\displaystyle\xi\not\in\partial_{\infty}\overline{H}$ i.e., $\displaystyle
E(.,\xi)$ has finite energy in each closed half-space
$\displaystyle\overline{H}$ which does not contain the ‘singularity’
$\displaystyle\xi$ on its ideal boundary. On the other hand
$\displaystyle\lVert\nabla_{g}F(.,\xi)\rVert_{L^{p}(\overline{H})}=+\infty$
for all $\displaystyle\overline{H}$. The finiteness of energy of
$\displaystyle E(.,\xi)$ in each half-space $\displaystyle\overline{H}$ away
from the ‘singularity’ at $\displaystyle\xi$ plays a crucial role in the
moving plane argument in proof of Theorem-3.8.

We only state an extract of the representation theorem in the linear case
$\displaystyle p=2$ due to Helgason [21] and Minemura [25] relevant to our
particular case. We denote the normalized measure on
$\displaystyle\mathbb{S}^{n-1}$ with $\displaystyle d\sigma$, the space of all
analytic functions on $\displaystyle\mathbb{S}^{n-1}$ by $\displaystyle
C^{\omega}(\mathbb{S}^{n-1})$ and its dual the space of analytic functionals
(hyperfunctions) by $\displaystyle C^{-\omega}(\mathbb{S}^{n-1})$. Let us
denote by
$\displaystyle\mathcal{L}_{\lambda}:=-\Delta^{\mathbb{B}^{n}}-\lambda$.

######  Theorem 3.11.

The Poisson-Helgason transform $\displaystyle
P_{\lambda}:C^{-\omega}(\mathbb{S}^{n-1})\to C^{\omega}(\mathbb{B}^{n})$
defined by

$\displaystyle\displaystyle
P_{\lambda}(T)(x):=\int_{\mathbb{S}^{n-1}}E(x,\xi)T(\xi)\,d\sigma(\xi),\text{
for }T\in C^{-\omega}(\mathbb{S}^{n-1})$ (3.60)

is an isomorphism between $\displaystyle C^{-\omega}(\mathbb{S}^{n-1})$ and
$\displaystyle\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$, with
inverse $\displaystyle B_{\lambda}$ given by a generalized boundary value. For
$\displaystyle v\in\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$ the
hyperfunction $\displaystyle B_{\lambda}(v)$ is given by

$\displaystyle\displaystyle\left<B_{\lambda}(v),\varphi\right>:=c_{n,\lambda}\lim\limits_{r\to
1^{-}}(1-r^{2})^{\alpha_{\lambda}-(n-1)}\int_{\mathbb{S}^{n-1}}v(r\xi)\varphi(\xi)\,d\sigma(\xi),\text{
for }\varphi\in C^{\omega}(\mathbb{S}^{n-1})$ (3.61)

where, $\displaystyle
c_{n,\lambda}=\frac{\pi^{\frac{1}{2}}\Gamma\left(\alpha_{\lambda}\right)}{\Gamma\left(\alpha_{\lambda}-\frac{n-1}{2}\right)}$.

For the sake of completeness we include a short proof of the minimality of
$\displaystyle E(.,\xi)$ relative to $\displaystyle\xi\in\mathbb{S}^{n-1}$ in
the linear case $\displaystyle p=2$. Let us consider the following weighted
analogue of a Hardy Space

$\displaystyle\displaystyle\mathrm{H}^{1}_{\lambda}(\mathbb{B}^{n}):=\left\\{f\in
L^{1}_{\text{loc}}(\mathbb{B}^{n}):\sup_{r\in[0,1)}(1-r^{2})^{\alpha_{\lambda}-(n-1)}\int_{\mathbb{S}^{n-1}}|f(r\xi)|\,d\sigma(\xi)<+\infty\right\\}.$
(3.62)

We note that if $\displaystyle
v\in\mathrm{H}^{1}_{\lambda}(\mathbb{B}^{n})\cap\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$,
then the set of Borel measures

$\displaystyle
d\mu_{r}(\xi):=c_{n,\lambda}(1-r^{2})^{\alpha_{\lambda}-(n-1)}v(r\xi)\,d\sigma(\xi)\in
M(\mathbb{S}^{n-1})$

for $\displaystyle r\in[0,1)$ has uniformly bounded total variation norm.
Extracting a $\displaystyle w^{\ast}$-convergent subsequence
$\displaystyle\mu_{r_{j}}\overset{w^{\ast}}{\rightharpoonup}\mu$ as
$\displaystyle r_{j}\to 1^{-}$ we note that in (3.61) we have

$\displaystyle\displaystyle\left<B_{\lambda}(v),\varphi\right>=\int_{\mathbb{S}^{n-1}}\varphi\,d\mu,\,\forall\,\varphi\in
C^{\omega}(\mathbb{S}^{n-1}).$ (3.63)

Thus (3.63) extends by density to all $\displaystyle\varphi\in
C(\mathbb{S}^{n-1})$ and hence $\displaystyle B_{\lambda}(v)=\mu\in
M(\mathbb{S}^{n-1})$ is given by a Borel measure. This establishes an
isomorphism of
$\displaystyle\mathrm{H}^{1}_{\lambda}(\mathbb{B}^{n})\cap\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$
and $\displaystyle M(\mathbb{S}^{n-1})$. In particular we note that positive
functions in
$\displaystyle\mathrm{H}^{1}_{\lambda}(\mathbb{B}^{n})\cap\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$
corresponds to non-negative measures in $\displaystyle M(\mathbb{S}^{n-1})$.
However we note that if $\displaystyle
v\in\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$ is a positive
eigenfunction then by the mean-value equality in Hyperbolic space (see [34],
Theorem-5.5.5) we have

$\displaystyle\displaystyle\int_{\mathbb{S}^{n-1}}v(r\xi)\,d\sigma(\xi)=g_{\alpha_{\lambda}}(r\eta)v(0)$
(3.64)

where, $\displaystyle r\in[0,1),\eta\in\mathbb{S}^{n-1}$ and $\displaystyle
g_{\alpha_{\lambda}}\in\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$
is the unique bounded radial eigenfunction with $\displaystyle
g_{\alpha_{\lambda}}(0)=1$ given by

$\displaystyle\displaystyle
g_{\alpha_{\lambda}}(x):=\int_{\mathbb{S}^{n-1}}E(x,\eta^{\prime})\,d\sigma(\eta^{\prime})$
(3.65)

with asymptotic boundary behavior (see [34], Theorem-5.5.7 and
Corollary-5.5.8) given by

$\displaystyle\displaystyle
g_{\alpha_{\lambda}}(r\eta)\approx(1-r^{2})^{n-1-\alpha_{\lambda}}.$ (3.66)

This in particular implies that $\displaystyle\lVert
v\rVert_{\mathrm{H}^{1}_{\lambda}(\mathbb{B}^{n})}=\sup_{r\in[0,1)}(1-r^{2})^{\alpha_{\lambda}-(n-1)}\int_{\mathbb{S}^{n-1}}v(r\xi)\,d\sigma(\xi)\leq
C^{\prime}v(0)$, i.e., positive eigenfunctions are in
$\displaystyle\mathrm{H}^{1}_{\lambda}(\mathbb{B}^{n})$ and hence have non-
negative measure for a generalized boundary value. As a consequence if
$\displaystyle v\in\operatorname{Ker}\left(\mathcal{L}_{\lambda}\right)$ is a
positive eigenfunction satisfying the estimate from above

$\displaystyle\displaystyle 0\leq v(x)\leq CE(x,\xi)$ (3.67)

for some $\displaystyle\xi\in\mathbb{S}^{n-1}$, then

$\displaystyle\left<B_{\lambda}(v),\varphi\right>=c_{n,\lambda}\lim\limits_{r\to
1^{-}}(1-r^{2})^{\alpha_{\lambda}-(n-1)}\int_{\mathbb{S}^{n-1}}v(r\eta)\varphi(\eta)\,d\sigma(\eta)=0$

for all $\displaystyle\varphi\in C(\mathbb{S}^{n-1})$ with
$\displaystyle\operatorname{supp}(\varphi)\cap\\{\xi\\}=\emptyset$. Thus
$\displaystyle B_{\lambda}(v)$ is a non-negative measure supported at
$\displaystyle\xi\in\mathbb{S}^{n-1}$ and hence must be a dirac mass,
$\displaystyle B_{\lambda}(v)=c_{0}\delta_{\xi}$ for some $\displaystyle
c_{0}\geq 0$. Therefore, $\displaystyle
v=P_{\lambda}(c_{0}\delta_{\xi})=c_{0}E(.,\xi).$

##  4\. Proof of symmetry

We will use the Alexandrov Moving Plane method to prove symmetry of positive
energy solutions to (1.1) in the Poincaré ball model
$\displaystyle(\mathbb{B}^{n},g)$ of the Hyperbolic space. Recall that the
Hyperbolic reflections are isometries of the Hyperbolic space
$\displaystyle\mathbb{B}^{n}$ given by spherical inversions (or reflections)
with respect to generalized Euclidean spheres that are orthogonal to
$\displaystyle\mathbb{S}^{n-1}$. In particular these are either orthogonal
Euclidean spheres $\displaystyle S(a,r):=\\{x:|x-a|=r\\}$ where
$\displaystyle|a|>1$ and $\displaystyle r^{2}=|a|^{2}-1$ or are Euclidean
hyperplanes passing through the origin of $\displaystyle\mathbb{B}^{n}$.
Given a direction $\displaystyle\xi\in\mathbb{S}^{n-1}$ we perform the moving
plane method in direction $\displaystyle\xi$ by reflecting along the family of
orthogonal spheres parametrized by $\displaystyle
S\left(\mu^{-1}\xi,\sqrt{|\mu|^{-2}-1}\right)$ for
$\displaystyle\mu\in(-1,1)\setminus\\{0\\}$. The limiting case
$\displaystyle\mu=0$ corresponds to reflecting along the Euclidean hyperplane
passing through $\displaystyle 0\in\mathbb{B}^{n}$ and orthogonal to
$\displaystyle\xi$. When $\displaystyle\mu\neq 0$ we denote the reflection
across $\displaystyle S\left(\mu^{-1}\xi,\sqrt{|\mu|^{-2}-1}\right)$ by
$\displaystyle\sigma_{\mu,\xi}\in\mathcal{M}(\mathbb{B}^{n})$,

$\displaystyle\displaystyle\sigma_{\mu,\xi}(x):=\mu^{-1}\xi+\frac{\left(|\mu|^{-2}-1\right)\left(x-\mu^{-1}\xi\right)}{|x-\mu^{-1}\xi|^{2}}$
(4.1)

and when $\displaystyle\mu=0$ we denote

$\displaystyle\displaystyle\sigma_{0,\xi}(x):=x-2(x,\xi)\xi.$ (4.2)

Thus $\displaystyle S\left(\mu^{-1}\xi,\sqrt{|\mu|^{-2}-1}\right)$ separates
$\displaystyle\mathbb{B}^{n}$ into two isometric connected components (half-
spaces). We denote by $\displaystyle\Sigma_{\mu}(\xi)$ the half-space in
$\displaystyle\mathbb{B}^{n}$ containing $\displaystyle\xi$ in its boundary
shared with $\displaystyle\partial_{\infty}\mathbb{B}^{n}$
($\displaystyle=\mathbb{S}^{n-1}$). We also denote the part of the sphere
$\displaystyle S\left(\mu^{-1}\xi,\sqrt{|\mu|^{-2}-1}\right)$ in
$\displaystyle\mathbb{B}^{n}$ as $\displaystyle
T_{\mu}(\xi):=S\left(\mu^{-1}\xi,\sqrt{|\mu|^{-2}-1}\right)\cap\mathbb{B}^{n}$.
In this notation we have
$\displaystyle\sigma_{\mu,\xi}\left(\Sigma_{\mu}(\xi)\right)=\mathbb{B}^{n}\setminus\overline{\Sigma_{\mu}(\xi)}=\Sigma_{-\mu}(-\xi)$.

Before proceeding with the proof of symmetry we need the following simple
lemma comparing the image and object distances of a reflection from an
observer in Hyperbolic space. The proof is similar to that of the same fact in
the Euclidean space.

######  Lemma 4.1.

Let $\displaystyle T$ be a generalized orthogonal sphere in
$\displaystyle\mathbb{B}^{n}$ which splits the space into two isometric
components (half-spaces) $\displaystyle\Sigma$ and
$\displaystyle\mathbb{B}^{n}\setminus\overline{\Sigma}$. Let $\displaystyle E$
(observer) be a fixed point in $\displaystyle\overline{\Sigma}$. Let
$\displaystyle P$ (object) be a point in $\displaystyle\Sigma$ and we denote
its reflection in $\displaystyle T$ as $\displaystyle\sigma_{T}(P)=P^{\prime}$
(image). Then
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P,E)\leq\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},E)$
with equality if and only if $\displaystyle E$ lies on $\displaystyle T$.

###### Proof.

The proof can be seen as a corollary of the Pythagoras’ theorem (or more
generally the law of cosines in Hyperbolic space, see Theorem-3.5.3 in [28]).
Let $\displaystyle\ell$ denote the Hyperbolic line passing through
$\displaystyle P$ and $\displaystyle P^{\prime}$ which intersects
$\displaystyle T$ at $\displaystyle O$. Then clearly
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P,O)=\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},O)$
i.e., $\displaystyle O$ is the mid-point of $\displaystyle P$ and
$\displaystyle P^{\prime}$ on the line $\displaystyle\ell$. It is easier to
visualize it after applying a Hyperbolic isometry to bring $\displaystyle O$
to the origin of $\displaystyle\mathbb{B}^{n}$. Then in this new frame,
$\displaystyle T$ is a Euclidean hyperplane passing through $\displaystyle O$
splitting $\displaystyle\mathbb{B}^{n}$ into two half-spaces
$\displaystyle\Sigma$ and
$\displaystyle\mathbb{B}^{n}\setminus\overline{\Sigma}$. $\displaystyle\ell$
is an Euclidean line passing through $\displaystyle O$ and orthogonal to
$\displaystyle T$. Let $\displaystyle F$ be the orthogonal projection of
$\displaystyle E$ on the line $\displaystyle\ell$ (i.e., $\displaystyle F$ is
intersection of $\displaystyle\ell$ with the unique Hyperbolic line
$\displaystyle\ell_{1}$ through $\displaystyle E$ which is orthogonal to
$\displaystyle\ell$). Then applying Hyperbolic Pythagoras’ theorem to the
triangles $\displaystyle\Delta PFE$ and $\displaystyle\Delta P^{\prime}FE$
respectively we get

$\displaystyle\displaystyle\cosh(\operatorname{dist}_{\mathbb{B}^{n}}(P,E))$
$\displaystyle\displaystyle=\cosh(\operatorname{dist}_{\mathbb{B}^{n}}(P,F))\cosh(\operatorname{dist}_{\mathbb{B}^{n}}(E,F))$
$\displaystyle\displaystyle\cosh(\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},E))$
$\displaystyle\displaystyle=\cosh(\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},F))\cosh(\operatorname{dist}_{\mathbb{B}^{n}}(E,F))$

respectively. Therefore the inequality
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P,E)\leq\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},E)$
will follow if we see that
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P,F)\leq\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},F)$.
Note that the orthogonal projection $\displaystyle F$ of the point
$\displaystyle E$ on the line $\displaystyle\ell$ must lie on the same half-
space $\displaystyle\overline{\Sigma}$ as $\displaystyle E$ (the unique
Hyperbolic line $\displaystyle\ell_{1}$ passing through $\displaystyle
E\in\overline{\Sigma}$ which is orthogonal to $\displaystyle\ell$ must be
entirely contained in $\displaystyle\overline{\Sigma}$). Since,
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},F)=\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},O)+\operatorname{dist}_{\mathbb{B}^{n}}(O,F)$
(as $\displaystyle O=T\cap\ell$ lies in between the points $\displaystyle
P^{\prime}\in\mathbb{B}^{n}\setminus\overline{\Sigma}$ and $\displaystyle
F\in\overline{\Sigma}$ on the line $\displaystyle\ell$) and
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},O)=\operatorname{dist}_{\mathbb{B}^{n}}(P,O)$,
using triangle inequality we get

$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},F)=\operatorname{dist}_{\mathbb{B}^{n}}(P,O)+\operatorname{dist}_{\mathbb{B}^{n}}(O,F)\geq\operatorname{dist}_{\mathbb{B}^{n}}(P,F)$

and consequently
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(P,E)\leq\operatorname{dist}_{\mathbb{B}^{n}}(P^{\prime},E)$.
Equality holds if and only if $\displaystyle F=O$ i.e., $\displaystyle E$ lies
on $\displaystyle T$. This completes the proof of the claim. ∎

We now proceed with the proof of Theorem-1.1.

###### Proof of Theorem-1.1:.

Without loss of generality we fix the direction $\displaystyle\xi=e_{1}$ and
denote $\displaystyle\Sigma_{\mu}(e_{1})$ simply as
$\displaystyle\Sigma_{\mu}$ and $\displaystyle T_{\mu}(e_{1})$ as
$\displaystyle T_{\mu}$. For $\displaystyle x\in\mathbb{B}^{n}$ denote
$\displaystyle x_{\mu}:=\sigma_{\mu,e_{1}}(x)$ and $\displaystyle
u_{\mu}(x):=u\circ\sigma_{\mu,e_{1}}(x)=u(x_{\mu})$, where $\displaystyle
u\in\mathcal{D}^{1,p}(\mathbb{B}^{n})$ is a positive solution to (1.1) i.e.,

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}u-\lambda
u^{p-1}=u^{q-1}.$ (4.3)

Since $\displaystyle\sigma_{\mu,e_{1}}\in\mathcal{M}(\mathbb{B}^{n})$ we have
$\displaystyle u_{\mu}$ is also a positive solution of (1.1) i.e.,

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}u_{\mu}-\lambda
u_{\mu}^{p-1}=u_{\mu}^{q-1}$ (4.4)

and $\displaystyle u_{\mu}\in\mathcal{D}^{1,p}(\mathbb{B}^{n})$.

Step-I: To start the Moving plane method we will show that $\displaystyle
u\leq u_{\mu}$ in $\displaystyle\Sigma_{\mu}$ for all $\displaystyle\mu$
sufficiently close to $\displaystyle 1$.
Let $\displaystyle\varphi\in C_{c}^{\infty}(\mathbb{B}^{n})$ be a non-negative
test function such that $\displaystyle 0\leq\varphi\leq 1$ with
$\displaystyle\varphi\equiv 1$ in $\displaystyle B_{R}$ and
$\displaystyle\text{supp}(\varphi)\subset B_{R+1}$,
$\displaystyle|\nabla_{g}\varphi|_{g}\leq c$ for all $\displaystyle R$
sufficiently large.
Consider the test functions
$\displaystyle\psi_{1}:=\varphi^{2}u^{1-p}(u^{p}-u_{\mu}^{p})_{+}\,\chi_{\Sigma_{\mu}}$
and
$\displaystyle\psi_{2}:=\varphi^{2}u_{\mu}^{1-p}(u^{p}-u_{\mu}^{p})_{+}\,\chi_{\Sigma_{\mu}}$.
Note that the functions are admissible as test functions since $\displaystyle
u=u_{\mu}$ on $\displaystyle T_{\mu}$, therefore testing (4.3) with
$\displaystyle\psi_{1}$ and (4.4) with $\displaystyle\psi_{2}$ respectively
and subtracting we have

$\displaystyle\displaystyle\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}|\nabla_{g}u|_{g}^{p-2}g\left(\nabla_{g}u,\nabla_{g}\psi_{1}\right)-|\nabla_{g}u_{\mu}|_{g}^{p-2}g\left(\nabla_{g}u_{\mu},\nabla_{g}\psi_{2}\right)\,dV_{g}$
$\displaystyle\displaystyle=\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}(u^{q-p}-u_{\mu}^{q-p})(u^{p}-u_{\mu}^{p})\,dV_{g}.$
(4.5)

In the set $\displaystyle\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}$ using the sharp
decay estimate on $\displaystyle u$ we also have the inequality

$\displaystyle\displaystyle 1\leq\frac{u}{u_{\mu}}\leq
C^{\prime}e^{-\alpha_{\lambda}(\operatorname{dist}_{\mathbb{B}^{n}}(0,x)-\operatorname{dist}_{\mathbb{B}^{n}}(0,x_{\mu}))}\leq
C^{\prime}$ (4.6)

where the final inequality is a consequence of Lemma-4.1, i.e.,
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,x_{\mu})\leq\operatorname{dist}_{\mathbb{B}^{n}}(0,x)$
for all $\displaystyle x\in\Sigma_{\mu}$ and $\displaystyle\mu\geq 0$ and
hence is true in particular for all $\displaystyle\mu$ close to $\displaystyle
1$ (here we have applied Lemma-4.1 with $\displaystyle E=0$, $\displaystyle
P=x_{\mu}$ and $\displaystyle P^{\prime}=x$).
Note that in $\displaystyle\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}$ using
Lagrange’s Mean Value theorem we have

$\displaystyle\displaystyle(u^{q-p}-u_{\mu}^{q-p})(u^{p}-u_{\mu}^{p})$
$\displaystyle\displaystyle\leq
u^{-p}\left(\frac{u^{q}-u_{\mu}^{q}}{u-u_{\mu}}\right)\left(\frac{u^{p}-u_{\mu}^{p}}{u-u_{\mu}}\right)(u-u_{\mu})^{2}$
$\displaystyle\displaystyle\leq qp\,u^{q-2}(u-u_{\mu})^{2}.$

Furthermore using the inequality $\displaystyle\frac{b-a}{b}\leq\log b-\log
a\leq\frac{b-a}{a}$ for $\displaystyle b\geq a>0$ we may write

$\displaystyle\displaystyle(u-u_{\mu})^{2}\leq u^{2}(\log u-\log
u_{\mu})^{2}.$

Combining these two inequalities we have

$\displaystyle\displaystyle(u^{q-p}-u_{\mu}^{q-p})(u^{p}-u_{\mu}^{p})\leq
qp\,u^{q}(\log u-\log u_{\mu})^{2}$

which we use on RHS of (4) to write

$\displaystyle\displaystyle\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}|\nabla_{g}u|_{g}^{p-2}g\left(\nabla_{g}u,\nabla_{g}\psi_{1}\right)-|\nabla_{g}u_{\mu}|_{g}^{p-2}g\left(\nabla_{g}u_{\mu},\nabla_{g}\psi_{2}\right)\,dV_{g}$
$\displaystyle\displaystyle\leq qp\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}u^{q}(\log u-\log u_{\mu})^{2}\,dV_{g}.$ (4.7)

On the other hand from the LHS of (4) we have

$\displaystyle\displaystyle\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}\left(|\nabla_{g}u|_{g}^{p-2}g\left(\nabla_{g}u,\nabla_{g}\left(u-\frac{u_{\mu}^{p}}{u^{p}}u\right)\right)+|\nabla_{g}u_{\mu}|_{g}^{p-2}g\left(\nabla_{g}u_{\mu},\nabla_{g}\left(u_{\mu}-\frac{u^{p}}{u_{\mu}^{p}}u_{\mu}\right)\right)\right)\,dV_{g}$
$\displaystyle\displaystyle\leq 2\underbrace{\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi|\nabla_{g}u|_{g}^{p-1}|\nabla_{g}\varphi|_{g}u^{1-p}(u^{p}-u_{\mu}^{p})\,dV_{g}}_{=I_{1}}$
$\displaystyle\displaystyle\quad+2\underbrace{\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi|\nabla_{g}u_{\mu}|_{g}^{p-1}|\nabla_{g}\varphi|_{g}u_{\mu}^{1-p}(u^{p}-u_{\mu}^{p})\,dV_{g}}_{=I_{2}}$
$\displaystyle\displaystyle\quad+qp\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}u^{q}(\log u-\log u_{\mu})^{2}\,dV_{g}$ (4.8)

In order to estimate $\displaystyle I_{1}$ and $\displaystyle I_{2}$ we may
use the global gradient estimate from above in Theorem-3.7 for $\displaystyle
u$ to write $\displaystyle|\nabla_{g}u|\leq Au$,
$\displaystyle|\nabla_{g}u_{\mu}|\leq Au_{\mu}$ and the fact that
$\displaystyle|\nabla_{g}\varphi|_{g}\leq c$ with
$\displaystyle\text{supp}(|\nabla_{g}\varphi|_{g})\subset B_{R+1}\setminus
B_{R}$ to write

$\displaystyle\displaystyle I_{1}+I_{2}\leq 4cA^{p-1}\int_{B_{R+1}\setminus
B_{R}}u^{p}\,dV_{g}\leq Ce^{-(p\alpha_{\lambda}-(n-1))R}\to 0^{+}$ (4.9)

as $\displaystyle R\to\infty$ since, $\displaystyle p\alpha_{\lambda}>n-1$.
Then using the inequality from Lemma-3.4 to LHS of (4) we have

$\displaystyle\displaystyle C_{p}\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}\min\left\\{u^{p},u_{\mu}^{p}\right\\}\left(|\nabla_{g}\log
u|_{g}+|\nabla_{g}\log u_{\mu}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
u_{\mu})|_{g}^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
I_{1}+I_{2}+qp\int_{\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}}\varphi^{2}u^{q}(\log
u-\log u_{\mu})^{2}\,dV_{g}.$ (4.10)

Now in case $\displaystyle p>2$ we may use the gradient estimate from below on
$\displaystyle u$ along with the global gradient estimate from above for both
$\displaystyle u$ and $\displaystyle u_{\mu}$ from Theorem-3.7 to write

$\displaystyle\displaystyle(2A)^{p-2}\geq\left(|\nabla_{g}\log
u|_{g}+|\nabla_{g}\log u_{\mu}|_{g}\right)^{p-2}\geq|\nabla_{g}\log
u|_{g}^{p-2}\geq A^{2-p}$ (4.11)

in case
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,\Sigma_{\mu})>R_{0}$
which is true for all $\displaystyle\mu$ sufficiently close to $\displaystyle
1$. Else if $\displaystyle p<2$ then we have

$\displaystyle\displaystyle A^{2-p}\geq|\nabla_{g}\log
u|_{g}^{p-2}\geq\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu}|_{g}\right)^{p-2}\geq(2A)^{p-2}.$ (4.12)

Combining these along with inequality (4.6), i.e.,
$\displaystyle\frac{u}{C^{\prime}}\leq u_{\mu}=\min\\{u,u_{\mu}\\}\leq u$ in
$\displaystyle\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}$, on the LHS of (4) we have

$\displaystyle\displaystyle C_{p,A}^{\prime}\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}u^{p}|\nabla_{g}(\log u-\log
u_{\mu})|_{g}^{2}\,dV_{g}\leq I_{1}+I_{2}+qp\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}\varphi^{2}u^{q}(\log u-\log u_{\mu})^{2}\,dV_{g}.$ (4.13)

Therefore using Cauchy-Schwartz inequality we have

$\displaystyle\displaystyle C_{p,A}^{\prime}\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}u^{p}|\nabla_{g}(\varphi(\log u-\log u_{\mu}))|_{g}^{2}\,dV_{g}$
$\displaystyle\displaystyle\leq
2I_{1}+2I_{2}+2qp\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}u^{q}\varphi^{2}(\log u-\log u_{\mu})^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+2C_{p,A}^{\prime}\underbrace{\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}|\nabla_{g}\varphi|_{g}^{2}u^{p}(\log u-\log
u_{\mu})^{2}\,dV_{g}}_{=I_{3}}.$ (4.14)

Recall also from (4.6) we have the inequality $\displaystyle 0\leq\log u-\log
u_{\mu}\leq\frac{u-u_{\mu}}{u_{\mu}}\leq(C^{\prime}-1)$ in
$\displaystyle\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}$. So we may estimate
$\displaystyle I_{3}$ similar to $\displaystyle I_{1},I_{2}$ as before

$\displaystyle\displaystyle I_{3}\leq
2C_{p,A}^{\prime}(C^{\prime}-1)^{2}c^{2}\int_{B_{R+1}\setminus
B_{R}}u^{p}\,dV_{g}\to 0^{+}$ (4.15)

as $\displaystyle R\to\infty$. Therefore using the sharp decay estimate of
$\displaystyle u$ in (4) we have

$\displaystyle\displaystyle\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}e^{-p\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}|\nabla_{g}(\varphi(\log
u-\log u_{\mu}))|_{g}^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
C^{\prime\prime}e^{-(q-p)\operatorname{dist}_{\mathbb{B}^{n}}(0,\Sigma_{\mu})}\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}e^{-p\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}\varphi^{2}(\log
u-\log u_{\mu})^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+2I_{1}+2I_{2}+I_{3}.$ (4.16)

Recall from Lemma-3.3 we have $\displaystyle
w(x):=\left[\sinh\left(\frac{1}{2}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)\right)\right]^{-2(n-1)}$
is a subsolution to $\displaystyle-\Delta_{g}w\leq 0$ in
$\displaystyle\mathbb{B}^{n}\setminus B_{r_{0}}$ for some $\displaystyle
r_{0}=r_{0}(n)>0$. Furthermore $\displaystyle\frac{|\nabla_{g}w|_{g}}{w}\geq
c_{n}$ in $\displaystyle\mathbb{B}^{n}\setminus B_{r_{0}}$ for some
$\displaystyle c_{n}>0$. Since
$\displaystyle\alpha:=\frac{p\alpha_{\lambda}}{n-1}>1$ we may apply the Hardy
type inequality from [16], Theorem-3.1 to write

$\displaystyle\displaystyle\left(\frac{|1-\alpha|}{2}\right)^{2}\int_{\mathbb{B}^{n}\setminus
B_{r_{0}}}w^{\alpha}\frac{|\nabla_{g}w|_{g}^{2}}{w^{2}}v^{2}\,dV_{g}\leq\int_{\mathbb{B}^{n}\setminus
B_{r_{0}}}w^{\alpha}|\nabla_{g}v|_{g}^{2}\,dV_{g},\,\forall\,v\in
C_{c}^{\infty}(\mathbb{B}^{n}\setminus B_{r_{0}}).$ (4.17)

This combined with the observation

$\displaystyle
w(x)\approx_{n,r_{0}}e^{-(n-1)\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}\text{
in }\mathbb{B}^{n}\setminus B_{r_{0}}$

gives the inequality

$\displaystyle\displaystyle
C_{n,\alpha_{\lambda}}\int_{\mathbb{B}^{n}}e^{-p\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}v^{2}\,dV_{g}\leq\int_{\mathbb{B}^{n}}e^{-p\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}|\nabla_{g}v|_{g}^{2}\,dV_{g},\,\forall\,v\in
C_{c}^{\infty}(\mathbb{B}^{n}\setminus B_{r_{0}}).$ (4.18)

We remark that the above inequality (4.18) remains valid for all functions
supported away from $\displaystyle B_{r_{0}}$ and with right decay property at
infinity.
Then using inequality (4.18) on RHS of (4) with $\displaystyle v:=\varphi(\log
u-\log u_{\mu})_{+}$ we get

$\displaystyle\displaystyle\left(1-C^{\prime\prime\prime}e^{-(q-p)\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,\Sigma_{\mu})}\right)\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}e^{-p\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}|\nabla_{g}(\varphi(\log
u-\log u_{\mu}))|_{g}^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq\quad
2I_{1}+2I_{2}+I_{3}.$ (4.19)

Finally letting $\displaystyle R\to\infty$ from (4.9) and (4.15) we get
$\displaystyle 2I_{1}+2I_{2}+I_{3}\to 0^{+}$, i.e.,

$\displaystyle\displaystyle\left(1-C^{\prime\prime\prime}e^{-(q-p)\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,\Sigma_{\mu})}\right)\int_{\Sigma_{\mu}\cap\\{u\geq
u_{\mu}\\}}e^{-p\alpha_{\lambda}\operatorname{dist}_{\mathbb{B}^{n}}(0,x)}|\nabla_{g}(\log
u-\log u_{\mu})|_{g}^{2}\,dV_{g}\leq 0.$ (4.20)

Therefore $\displaystyle\nabla_{g}(\log u-\log u_{\mu})\equiv 0$ in
$\displaystyle\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}$ i.e., $\displaystyle
u=u_{\mu}$ in $\displaystyle\Sigma_{\mu}\cap\\{u\geq u_{\mu}\\}$ for all
$\displaystyle\mu$ sufficiently close to $\displaystyle 1$. This completes the
proof of the claim $\displaystyle u\leq u_{\mu}$ in
$\displaystyle\Sigma_{\mu}$ for all $\displaystyle\mu$ close to $\displaystyle
1$.

Step-II: Let us denote

$\displaystyle M_{u}:=\left\\{\mu\in(-1,1):u\leq u_{\tilde{\mu}}\text{ in
}\Sigma_{\tilde{\mu}},\,\forall\,\mu\leq\tilde{\mu}<1\right\\}.$

By the previous step $\displaystyle M_{u}\neq\emptyset$ and let
$\displaystyle\mu_{0}:=\inf M_{u}$. To keep consistent with the previous step,
wlog we may assume $\displaystyle\mu_{0}>0$, this may be achieved by making a
change in coordinate frame by a Hyperbolic translation
$\displaystyle\tau_{-te_{1}}\in\mathcal{M}(\mathbb{B}^{n})$ (which fixes the
points $\displaystyle-e_{1}$ and $\displaystyle e_{1}$) for $\displaystyle
t>0$ chosen sufficiently large if necessary and working with $\displaystyle
v:=(u\circ\tau_{-te_{1}})$ instead so that $\displaystyle\mu_{0}^{\prime}=\inf
M_{v}>0$.

We claim that $\displaystyle u\equiv u_{\mu_{0}}$ in
$\displaystyle\Sigma_{\mu_{0}}$. Suppose to the contrary $\displaystyle
u\not\equiv u_{\mu_{0}}$ in $\displaystyle\Sigma_{\mu_{0}}$, we will reach a
contradiction on infimality of $\displaystyle\mu_{0}$ by showing that there is
a $\displaystyle\overline{\epsilon}>0$ such that $\displaystyle u\leq
u_{\mu_{0}-\epsilon}$ in $\displaystyle\Sigma_{\mu_{0}-\epsilon}$ for all
$\displaystyle 0<\epsilon\leq\overline{\epsilon}$.

The following argument is made exactly as in [26]. Note that we have
$\displaystyle u\leq u_{\mu_{0}}$ in $\displaystyle\Sigma_{\mu_{0}}$ by
continuity. Then by strong comparison principle Lemma-5.8, we have either
$\displaystyle u\equiv u_{\mu_{0}}$ or $\displaystyle u<u_{\mu_{0}}$ in each
connected component of
$\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$, where,
$\displaystyle\mathcal{C}_{u}$ is the critical point set of $\displaystyle u$.
Note that by the gradient estimate of $\displaystyle u$ from below we know
$\displaystyle\mathcal{C}_{u}$ is a compact set and furthermore from Lemma-5.4
we have $\displaystyle|\mathcal{C}_{u}|=0$ ($\displaystyle\mathcal{C}_{u}$ is
a set of Lebesgue measure zero).
If $\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$ is connected then
by our assumption $\displaystyle u\not\equiv u_{\mu_{0}}$ in
$\displaystyle\Sigma_{\mu_{0}}$ by using strong comparison in
$\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$ we have $\displaystyle
u<u_{\mu_{0}}$ in $\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$.
Else if $\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$ has at least
two connected components then compactness of $\displaystyle\mathcal{C}_{u}$
implies there is exactly one unbounded component which we denote by
$\displaystyle\Omega_{1}$. Suppose, $\displaystyle u\equiv u_{\mu_{0}}$ in
$\displaystyle\Omega_{1}$ then it is easy to see that
$\displaystyle(\Sigma_{\mu_{0}}\setminus\Omega_{1})\cup\sigma_{\mu_{0}}(\Sigma_{\mu_{0}}\setminus\Omega_{1})$
contains a bounded connected component of
$\displaystyle\mathbb{B}^{n}\setminus\mathcal{C}_{u}$ which we denote by
$\displaystyle\omega$. Note that
$\displaystyle\partial\omega\subset\mathcal{C}_{u}$. Recall from the
regularity of $\displaystyle u$ from Remark-5.2 we have $\displaystyle|\nabla
u|^{p-2}\nabla u\in W^{1,2}_{\text{loc}}(\mathbb{B}^{n})$ so that this vector
field is continuous and equals $\displaystyle 0$ on
$\displaystyle\partial\omega$. Since the equality
$\displaystyle-\Delta_{p}^{\mathbb{B}^{n}}u=f(u)$ with $\displaystyle
f(u)=\lambda u^{p-1}+u^{q-1}$ holds pointwise a.e. in $\displaystyle\omega$ we
may integrate the expression in $\displaystyle\omega$ to write

$\displaystyle\int_{\omega}-\operatorname{div}\left(\left(\frac{2}{1-|x|^{2}}\right)^{n-p}|\nabla
u|^{p-2}\nabla
u\right)\,dx=\int_{\omega}f(u)\left(\frac{2}{1-|x|^{2}}\right)^{n}\,dx.$

Applying divergence theorem we have

$\displaystyle
0=-\int_{\partial\omega}\left(\frac{2}{1-|x|^{2}}\right)^{n-p}|\nabla
u|^{p-2}(\nabla
u,\hat{n})\,d\mathcal{H}^{n-1}=\int_{\omega}f(u)\left(\frac{2}{1-|x|^{2}}\right)^{n}\,dx>0$

which leads to a contradiction as $\displaystyle f(u)>0$ in
$\displaystyle\mathbb{B}^{n}$. Else if $\displaystyle u\equiv u_{\mu_{0}}$ in
a bounded connected component $\displaystyle\Omega_{2}$ of
$\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$ then again
$\displaystyle\Omega_{2}\cup\sigma_{\mu_{0}}(\Omega_{2})$ must contain a
bounded connected component of
$\displaystyle\mathbb{B}^{n}\setminus\mathcal{C}_{u}$ which is absurd as seen
earlier. Thus by appealing to strong comparison we must have $\displaystyle
u<u_{\mu_{0}}$ in each component of
$\displaystyle\Sigma_{\mu_{0}}\setminus\mathcal{C}_{u}$ as well.

Let us consider $\displaystyle\overline{R}>0$ to be chosen sufficiently large
later such that $\displaystyle\mathcal{C}_{u}\subset\subset B_{\overline{R}}$.
Furthermore using the fact that $\displaystyle|\mathcal{C}_{u}|=0$ (see
Lemma-5.4), we let $\displaystyle N:=N_{\kappa}(\mathcal{C}_{u})$ be a open
neighborhood of $\displaystyle\mathcal{C}_{u}$ such that
$\displaystyle|N|<\kappa$ has small measure. Then for
$\displaystyle\overline{\epsilon},\epsilon,\delta,\kappa>0$ let us define the
sets

$\displaystyle\displaystyle\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}:=\Sigma_{\mu_{0}-\epsilon}\setminus
B_{\overline{R}},$ $\displaystyle\displaystyle
S_{\delta}^{\epsilon}(\overline{R}):=\left((\Sigma_{\mu_{0}-\epsilon}\setminus\Sigma_{\mu_{0}+\delta})\cap
B_{\overline{R}}\right)\cup(N\cap\Sigma_{\mu_{0}+\delta}),$
$\displaystyle\displaystyle
K_{\delta}(\overline{R}):=\overline{\Sigma_{\mu_{0}+\delta}\cap
B_{\overline{R}}}\cap(\mathbb{B}^{n}\setminus N)$ (4.21)

so that we have

$\displaystyle\displaystyle\Sigma_{\mu_{0}-\epsilon}=\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}\cup
S_{\delta}^{\epsilon}(\overline{R})\cup K_{\delta}(\overline{R}).$ (4.22)

We choose $\displaystyle\epsilon$ small enough such that $\displaystyle
0\not\in\Sigma_{\mu_{0}-\epsilon}$ (i.e.,
$\displaystyle\Sigma_{\mu_{0}-\epsilon}$ still remains to the right of the
origin in $\displaystyle\mathbb{B}^{n}$) and $\displaystyle\delta$ small
enough such that $\displaystyle K_{\delta}$ is non-empty. Since,
$\displaystyle
K_{\delta}(\overline{R}+1):=\overline{\Sigma_{\mu_{0}+\delta}\cap
B_{\overline{R}+1}}\cap(\mathbb{B}^{n}\setminus N)$ is compact, by uniform
continuity of $\displaystyle u$ and $\displaystyle u_{\mu}$ we have
$\displaystyle u<u_{\mu_{0}-\epsilon}$ in $\displaystyle
K_{\delta}(\overline{R}+1)$ for each
$\displaystyle\epsilon\leq\overline{\epsilon}$ for some
$\displaystyle\overline{\epsilon}>0$.
Then as in the last step for $\displaystyle R>\overline{R}$, let
$\displaystyle\varphi\in C_{c}^{\infty}(\mathbb{B}^{n})$ be a non-negative
test function such that $\displaystyle 0\leq\varphi\leq 1$ with
$\displaystyle\varphi\equiv 1$ in $\displaystyle B_{R}$ and
$\displaystyle\text{supp}(\varphi)\subset B_{R+1}$,
$\displaystyle|\nabla_{g}\varphi|_{g}\leq c$. We consider the test functions
$\displaystyle\psi_{1}:=\varphi^{2}u^{1-p}\left(u^{p}-u_{\mu_{0}-\epsilon}^{p}\right)_{+}\,\chi_{\Sigma_{\mu_{0}-\epsilon}}$
and
$\displaystyle\psi_{2}:=\varphi^{2}u_{\mu_{0}-\epsilon}^{1-p}\left(u^{p}-u_{\mu_{0}-\epsilon}^{p}\right)_{+}\,\chi_{\Sigma_{\mu_{0}-\epsilon}}$
as before. Testing the PDE for $\displaystyle u$ with $\displaystyle\psi_{1}$
and the PDE for $\displaystyle u_{\mu_{0}-\epsilon}$ with
$\displaystyle\psi_{2}$, subtracting them and following the steps from Step-I
we get

$\displaystyle\displaystyle C_{p}\int_{\Sigma_{\mu_{0}-\epsilon}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}\varphi^{2}u^{p}\left(|\nabla_{g}\log
u|_{g}+|\nabla_{g}\log u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log
u-\log u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
Ce^{-(p\alpha_{\lambda}-(n-1))R}$
$\displaystyle\displaystyle\quad+qp\int_{\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}\varphi^{2}u^{q}(\log u-\log
u_{\mu_{0}-\epsilon})^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+qp\int_{S_{\delta}^{\epsilon}(\overline{R})\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}\varphi^{2}u^{q}(\log u-\log
u_{\mu_{0}-\epsilon})^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
Ce^{-(p\alpha_{\lambda}-(n-1))R}$
$\displaystyle\displaystyle\quad+2qp\int_{\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}\varphi^{2}u^{q}\eta^{2}(\log u-\log
u_{\mu_{0}-\epsilon})^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+2qp\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}\varphi^{2}u^{q}(1-\eta)^{2}(\log u-\log
u_{\mu_{0}-\epsilon})^{2}\,dV_{g}$
$\displaystyle\displaystyle=Ce^{-(p\alpha_{\lambda}-(n-1))R}+J_{1}+J_{2}$
(4.23)

where, $\displaystyle\eta\in C^{\infty}(\mathbb{B}^{n})$ is such that
$\displaystyle 0\leq\eta\leq 1$ with $\displaystyle\eta\equiv 1$ in
$\displaystyle\mathbb{B}^{n}\setminus B_{\overline{R}+1}$ and
$\displaystyle\eta\equiv 0$ in $\displaystyle B_{\overline{R}}$.
Let,

$\displaystyle\displaystyle
J_{0}:=C_{p}\int_{\Sigma_{\mu_{0}-\epsilon}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g},$ (4.24)

we claim that $\displaystyle J_{0}\leq 0$. Suppose to the contrary
$\displaystyle J_{0}>0$.
Note that we have the gradient estimates (4.11) and (4.12) in
$\displaystyle\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}$. Using the decay
estimate of $\displaystyle u$ and the inequality (4.18) followed by the
gradient estimates of $\displaystyle u$ we may estimate $\displaystyle J_{1}$
as

$\displaystyle\displaystyle J_{1}$ $\displaystyle\displaystyle\leq
Ce^{-(q-p)\alpha_{\lambda}\overline{R}}\int_{\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\eta(\log u-\log
u_{\mu_{0}-\epsilon}))|_{g}^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
2Ce^{-(q-p)\alpha_{\lambda}\overline{R}}\int_{\Sigma_{\mu_{0}-\epsilon}^{\overline{R}}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+2C_{A}e^{-(q-p)\alpha_{\lambda}\overline{R}}\int_{B_{\overline{R}+1}\setminus
B_{\overline{R}}}u^{p}\,dV_{g}.$ (4.25)

In the second line we have used the Cauchy-Schwartz inequality followed by the
fact that $\displaystyle|\nabla_{g}\eta|_{g}\leq c$ and is supported in
$\displaystyle B_{\overline{R}+1}\setminus B_{\overline{R}}$ and
$\displaystyle(\log u-\log u_{\mu_{0}-\epsilon})_{+}$ is bounded from above in
$\displaystyle\Sigma_{\mu_{0}-\epsilon}$.
Again in order to estimate $\displaystyle J_{2}$ we split it into two cases.
In case $\displaystyle p>2$ we may use the admissibility of
$\displaystyle|\nabla_{g}u|_{g}^{p-2}$ as a weight for a 2-Poincaré inequality
as stated in Remark-5.6 to write

$\displaystyle\displaystyle J_{2}$ $\displaystyle\displaystyle\leq
C_{u}\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}(1-\eta)^{2}(\log u-\log
u_{\mu_{0}-\epsilon})^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}|\nabla_{g}u|_{g}^{p-2}|\nabla_{g}((1-\eta)(\log
u-\log u_{\mu_{0}-\epsilon}))|_{g}^{2}\,dV_{g}$
$\displaystyle\displaystyle\leq
2C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}|\nabla_{g}u|_{g}^{p-2}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+8C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}|\nabla_{g}u|_{g}^{p-2}|\nabla_{g}\eta|_{g}^{2}(\log
u-\log u_{\mu_{0}-\epsilon})^{2}\,dV_{g}$
$\displaystyle\displaystyle:=J_{2}^{\prime}+J_{2}^{\prime\prime}.$ (4.26)

where, $\displaystyle C_{u}=2qp\left(\sup_{S_{\delta}^{\epsilon}}u^{q}\right)$
and $\displaystyle\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))$ is the
constant for the weighted 2-Poincaré inequality. Since $\displaystyle p>2$ we
have

$\displaystyle\displaystyle J_{2}^{\prime}\leq
2C_{u}^{\prime}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}|\nabla_{g}\log u|_{g}^{p-2}|\nabla_{g}(\log
u-\log u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$ $\displaystyle\displaystyle\leq
2C_{u}^{\prime}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}(\overline{R}+1)\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$ (4.27)

where, $\displaystyle
C_{u}^{\prime}=2qp\left(\frac{\sup_{S_{\delta}^{\epsilon}}u^{q}}{\inf_{S_{\delta}^{\epsilon}}u^{2}}\right)$.
Also, using the sharp decay, global gradient estimate from above of
$\displaystyle u$ along with the fact that
$\displaystyle|\nabla_{g}\eta|_{g}\leq c$ and is supported in $\displaystyle
B_{\overline{R}+1}\setminus B_{\overline{R}}$, followed by the fact that
$\displaystyle(\log u-\log u_{\mu_{0}-\epsilon})_{+}$ is bounded from above in
the set, we have

$\displaystyle\displaystyle J_{2}^{\prime\prime}\leq
C^{\prime\prime}C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))e^{-(p-2)\alpha_{\lambda}\overline{R}}\operatorname{Vol}_{g}(S_{\delta}^{\epsilon}(\overline{R}+1)\cap
B_{\overline{R}+1}\setminus B_{\overline{R}}).$ (4.28)

Note that the constant $\displaystyle C_{u}^{\prime}$ does not increase if we
let $\displaystyle\epsilon,\delta\to 0^{+}$, thus remains uniformly bounded
for all $\displaystyle\epsilon\leq\overline{\epsilon}$ and
$\displaystyle\delta<\overline{\delta}$. Furthermore from (5.33) we have the
Poincaré constant
$\displaystyle\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\to 0^{+}$ as
$\displaystyle|S_{\delta}^{\epsilon}(\overline{R}+1)|\to 0^{+}$.
In case $\displaystyle p<2$ similar to the last case we may start by using the
usual unweighted Poincaré inequality instead and the global gradient bound

$\displaystyle\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{2-p}\leq(2A)^{2-p}$

to write

$\displaystyle\displaystyle J_{2}$ $\displaystyle\displaystyle\leq
2C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+8C^{\prime\prime\prime}C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\operatorname{Vol}_{g}(S_{\delta}^{\epsilon}(\overline{R}+1)\cap
B_{\overline{R}+1}\setminus B_{\overline{R}})$ $\displaystyle\displaystyle\leq
2C_{u}^{\prime}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\int_{S_{\delta}^{\epsilon}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}$
$\displaystyle\displaystyle\quad+8C^{\prime\prime\prime}C_{u}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))\operatorname{Vol}_{g}(S_{\delta}^{\epsilon}(\overline{R}+1)\cap
B_{\overline{R}+1}\setminus B_{\overline{R}}).$ (4.29)

where, the constant $\displaystyle
C_{u}^{\prime}=(2A)^{2-p}qp\left(\frac{\sup_{S_{\delta}^{\epsilon}}u^{q}}{\inf_{S_{\delta}^{\epsilon}}u^{p}}\right)$
does not increase as we let $\displaystyle\epsilon,\delta\to 0+$ and hence
remains uniformly bounded as in the previous case.
Thus letting $\displaystyle R\to\infty$ in (4) and combining with the fact
that with $\displaystyle\overline{R}$ chosen large enough we have
$\displaystyle J_{1}\leq J_{0}/4$ and in the estimates for $\displaystyle
J_{2}$ we can choose $\displaystyle\epsilon\leq\overline{\epsilon}$,
$\displaystyle\delta$ and $\displaystyle\kappa$ small enough s.t.,
$\displaystyle
2C_{u}^{\prime}\Lambda(S_{\delta}^{\epsilon}(\overline{R}+1))<1/4$,
$\displaystyle J_{2}^{\prime\prime}<J_{0}/4$ we get

$\displaystyle\displaystyle
J_{0}/4=\frac{C_{p}}{4}\int_{\Sigma_{\mu_{0}-\epsilon}\cap\\{u\geq
u_{\mu_{0}-\epsilon}\\}}u^{p}\left(|\nabla_{g}\log u|_{g}+|\nabla_{g}\log
u_{\mu_{0}-\epsilon}|_{g}\right)^{p-2}|\nabla_{g}(\log u-\log
u_{\mu_{0}-\epsilon})|_{g}^{2}\,dV_{g}\leq 0$ (4.30)

which leads to a contradiction. Thus we must have $\displaystyle J_{0}\leq 0$
to begin with.
This in turn implies as in Step-I, $\displaystyle u\leq u_{\mu_{0}-\epsilon}$
in $\displaystyle\Sigma_{\mu_{0}-\epsilon}$ for each
$\displaystyle\epsilon\leq\overline{\epsilon}$. But this contradicts the
minimality of $\displaystyle\mu_{0}$, completing the proof.

Step-III: Finally applying another Hyperbolic translation
$\displaystyle\tau_{te_{1}}\in\mathcal{M}(\mathbb{B}^{n})$ for some
$\displaystyle t\in\mathbb{R}$ we may choose a coordinate frame s.t.,
$\displaystyle\mu_{0}=0$. Thus we have $\displaystyle
u\equiv(u\circ\sigma_{0,e_{1}}$) in $\displaystyle\Sigma_{0}(e_{1})$.
Similarly repeating this in $\displaystyle e_{j}$-directions for
$\displaystyle j=2,\cdots,n$ we also have $\displaystyle
u\equiv(u\circ\sigma_{0,e_{j}}$) in $\displaystyle\Sigma_{0}(e_{j})$. Owing to
the fact that $\displaystyle u$ is non-decreasing in direction of the moving
plane we must have $\displaystyle u(0)=\sup_{\mathbb{B}^{n}}u=M$ (say). Since,
$\displaystyle u$ is symmetric about the $\displaystyle n$ orthogonal planes
passing through $\displaystyle 0\in\mathbb{B}^{n}$, it must also have point
symmetry about $\displaystyle 0$, meaning $\displaystyle u(x)=u(-x)$ by
successively reflecting along these $\displaystyle n$ planes. We claim that
for any direction $\displaystyle\overline{e}\in\mathbb{S}^{n-1}$, the moving
plane in the direction of $\displaystyle\overline{e}$ must stop only after it
reaches the origin $\displaystyle 0\in\mathbb{B}^{n}$. Otherwise, if we have
$\displaystyle u\equiv(u\circ\sigma_{\mu,\overline{e}})$ in
$\displaystyle\Sigma_{\mu}(\overline{e})$ for some $\displaystyle\mu\neq 0$,
i.e., $\displaystyle T_{\mu}(\overline{e})$ stops at an optimal position
before reaching $\displaystyle 0$, by reflecting $\displaystyle 0$ about
$\displaystyle T_{\mu}(\overline{e})$ we must have $\displaystyle u\equiv M$
along the line segment joining $\displaystyle 0$ and its reflection
$\displaystyle O_{1}=\sigma_{\mu,\overline{e}}(0)$. Again due to the point
symmetry about $\displaystyle 0$ we must have $\displaystyle u(-O_{1})=M$. By
reflecting $\displaystyle-O_{1}$ about $\displaystyle T_{\mu}(\overline{e})$
we have $\displaystyle u\equiv M$ on the line segment joining $\displaystyle-
O_{1}$ and its reflection $\displaystyle
O_{2}=\sigma_{\mu,\overline{e}}(-O_{1})$. If we denote
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,O_{1})=2\ell$, since
reflections are isometries, successively iterating this process with
$\displaystyle O_{k}:=\sigma_{\mu,\overline{e}}(-O_{k-1})$ we get
$\displaystyle\operatorname{dist}_{\mathbb{B}^{n}}(0,O_{k})=2k\ell$.
Therefore, $\displaystyle u\equiv M$ along the entire line passing through
$\displaystyle 0$ in direction $\displaystyle\overline{e}$, which contradicts
the decay of $\displaystyle u$. Therefore, $\displaystyle
u\equiv(u\circ\sigma_{0,\overline{e}})$ for any
$\displaystyle\overline{e}\in\mathbb{S}^{n-1}$, which implies $\displaystyle
u$ is a radially symmetric function in $\displaystyle\mathbb{B}^{n}$ with
respect to the origin $\displaystyle 0$. To see that $\displaystyle u$ is
strictly decreasing in radial direction (i.e.,
$\displaystyle\mathcal{C}_{u}=\\{0\\}$), we may either look at the ordinary
differential equation satisfied by $\displaystyle u$ or argue as in beginning
of Step-II to conclude that
$\displaystyle\mathbb{B}^{n}\setminus\mathcal{C}_{u}$ cannot have a bounded
connected component since, level sets of $\displaystyle u$ are spheres. This
completes the proof of Hyperbolic symmetry of $\displaystyle u$. Finally we
note that if we denote $\displaystyle
u(x)=\Phi(\operatorname{dist}_{\mathbb{B}^{n}}(0,x))$ then
$\displaystyle\Phi^{\prime}(t)<0$ for all $\displaystyle t>0$ and by standard
regularity argument $\displaystyle\Phi(t)$ is smooth for $\displaystyle t>0$.
∎

##  5\. Strong comparison for Hyperbolic $\displaystyle p$-Laplace equation

In this section we establish the Sobolev regularity of
$\displaystyle|\nabla_{g}u|_{g}^{p-2}\nabla_{g}u$ and local integrability of
$\displaystyle|\nabla_{g}u|_{g}^{-(p-1)r}$ for $\displaystyle r<1$, of
positive solutions $\displaystyle u\in\mathcal{D}^{1,p}(\mathbb{H}^{n})\cap
C_{\text{loc}}^{1,\alpha}(\mathbb{H}^{n})$ of the equation

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{H}^{n}}u$
$\displaystyle\displaystyle=f(u)$ (5.1)

where, $\displaystyle f$ is a positive locally Lipschitz function such that
$\displaystyle f(t)>0$ for $\displaystyle t>0$ and the set of critical points
of $\displaystyle u$ is compact. These estimates play a crucial role in
proving a strong comparison theorem when $\displaystyle p>2$. We will proceed
in similar lines of [14], [15]. We remark that these results have also been
proved in a more general homogeneous compact Riemannian manifolds with
boundary in case $\displaystyle p>2$ in [2]. In the prior sections we saw that
when $\displaystyle f(t)=\lambda t^{p-1}+t^{q-1}$ for $\displaystyle p<q\leq
p^{\ast}$ and $\displaystyle\displaystyle
0\leq\lambda<\lambda_{\text{max}}:=\left(\frac{n-1}{p}\right)^{p}$, the
positive finite energy solutions have non-vanishing gradient in complement of
a large ball. For the sake of simplicity of calculations we will be using the
upper half-space model $\displaystyle(\mathbb{H}^{n},g)$ of the Hyperbolic
space.

We begin by recalling that the equation (5.1) is invariant under the action of
the Mobius transformations $\displaystyle\mathcal{M}(\mathbb{H}^{n})$
(isometries of $\displaystyle\mathbb{H}^{n}$). Let,
$\displaystyle\tau\in\mathcal{M}(\mathbb{H}^{n})$ then $\displaystyle
u_{\tau}:=(u\circ\tau)\in\mathcal{D}^{1,p}(\mathbb{H}^{n})$ also solves

$\displaystyle\displaystyle-\Delta_{p}^{\mathbb{H}^{n}}u_{\tau}=f(u_{\tau}).$
(5.2)

We will linearize the equation (5.1) along the flows of the Hyperbolic
translations. Note that for $\displaystyle t>0$,
$\displaystyle\tau_{t}\in\mathcal{M}(\mathbb{H}^{n})$ given by
$\displaystyle\tau_{t}(x):=tx$ is a translation in the upper half-space model
of the Hyperbolic space that fixes $\displaystyle 0$ and
$\displaystyle\infty$. Note that by classical regularity theory $\displaystyle
u\in C^{2,\sigma}$ in $\displaystyle\mathbb{H}^{n}\setminus\mathcal{C}_{u}$
where,
$\displaystyle\mathcal{C}_{u}:=\\{x\in\mathbb{H}^{n}:\nabla_{g}u(x)=0\\}$ is
the critical point set. Let $\displaystyle\varphi\in
C_{c}^{\infty}(\mathbb{H}^{n})$ be such that
$\displaystyle\text{supp}(\varphi)\subset\mathbb{H}^{n}\setminus\mathcal{C}_{u}$.
Testing equation (5.2) with $\displaystyle\varphi$ and differentiating at
$\displaystyle t=1$ we have

$\displaystyle\displaystyle\left.\frac{d}{dt}\right|_{t=1}\int_{\mathbb{H}^{n}}|\nabla_{g}(u\circ\tau_{t})|_{g}^{p-2}g(\nabla_{g}(u\circ\tau_{t}),\nabla_{g}\varphi)\,dV_{g}=\left.\frac{d}{dt}\right|_{t=1}\int_{\mathbb{H}^{n}}f(u\circ\tau_{t})\varphi\,dV_{g}.$
(5.3)

Note that we have successively for each $\displaystyle j=1,\cdots,n$

$\displaystyle\displaystyle\left.\frac{d}{dt}\right|_{t=1}(u\circ\tau_{t})=(x,\nabla
u):=V_{n}(u).$ (5.4)
$\displaystyle\displaystyle\frac{\partial(u\circ\tau_{t})}{\partial
x_{j}}=tu_{x_{j}}(tx)$ (5.5)
$\displaystyle\displaystyle\left.\frac{d}{dt}\right|_{t=1}\frac{\partial(u\circ\tau_{t})}{\partial
x_{j}}=u_{x_{j}}+\left(x,\nabla u_{x_{j}}\right)=\frac{\partial
V_{n}(u)}{\partial x_{j}}$ (5.6)
$\displaystyle\displaystyle\left.\frac{d}{dt}\right|_{t=1}|\nabla(u\circ\tau_{t})|^{2}=2(\nabla
u,\nabla V_{n}(u)).$ (5.7)

Therefore, (5.3) becomes

$\displaystyle\displaystyle\int_{\mathbb{H}^{n}}x_{n}^{p-n}\left(|\nabla
u|^{p-2}(\nabla V_{n}(u),\nabla\varphi)+(p-2)|\nabla u|^{p-4}(\nabla
V_{n}(u),\nabla u)(\nabla u,\nabla\varphi)\right)\,dx$
$\displaystyle\displaystyle=\int_{\mathbb{H}^{n}}f^{\prime}(u)V_{n}(u)\varphi
x_{n}^{-n}\,dx,\,\forall\,\varphi\in
C_{c}^{\infty}(\mathbb{H}^{n}\setminus\mathcal{C}_{u}).$ (5.8)

Keeping consistent with the notations in [14], we define the linearized PDE
corresponding to (5.1) to be, $\displaystyle L_{u}(v,\varphi)=0$ for
$\displaystyle\varphi\in C_{c}^{\infty}(\mathbb{H}^{n})$ where

$\displaystyle\displaystyle L_{u}(v,\varphi)$
$\displaystyle\displaystyle:=\int_{\mathbb{H}^{n}}|\nabla_{g}u|_{g}^{p-2}g(\nabla_{g}v,\nabla_{g}\varphi)+(p-2)|\nabla_{g}u|_{g}^{p-4}g(\nabla_{g}v,\nabla_{g}u)g(\nabla_{g}u,\nabla_{g}\varphi)\,dV_{g}$
$\displaystyle\displaystyle\quad-\int_{\mathbb{H}^{n}}f^{\prime}(u)v\varphi\,dV_{g}.$
(5.9)

Similarly considering horizontal translations as well in upper half-space
(Euclidean translations $\displaystyle x\mapsto x+te_{j}$ in $\displaystyle
e_{j}$-direction for $\displaystyle j=1,\cdots,n-1$, which are also isometries
in the upper half-space model of the Hyperbolic space) we have

$\displaystyle\displaystyle L_{u}(V_{j}(u),\varphi)=0$ (5.10)

$\displaystyle\forall\,\varphi\in
C_{c}^{\infty}(\mathbb{H}^{n}\setminus\mathcal{C}_{u})$ where,

$\displaystyle\displaystyle V_{j}(u)$
$\displaystyle\displaystyle:=u_{x_{j}}\text{ for }j=1,\cdots,n-1$
$\displaystyle\displaystyle V_{n}(u)$ $\displaystyle\displaystyle=(x,\nabla
u).$ (5.11)

By density argument the above equation remains valid $\displaystyle\varphi\in
W^{1,2}_{\text{loc}}(\mathbb{H}^{n})$ with compact support in
$\displaystyle\mathbb{H}^{n}\setminus\mathcal{C}_{u}$. We note that the vector
field $\displaystyle V(u):=(V_{1}(u),\cdots,V_{n}(u))^{T}$ is given by

$\displaystyle\displaystyle V(u)=A(x)\nabla u$ (5.12)

where, $\displaystyle\nabla u=(u_{x_{1}},\cdots,u_{x_{n}})^{T}$,
$\displaystyle A(x):=[e_{1},\cdots,e_{n-1},x]^{T}$ (where, $\displaystyle
e_{1},e_{2},\cdots,e_{n},x$ are the column vectors of
$\displaystyle[e_{1},\cdots,e_{n-1},x]$) is an invertible matrix with its
inverse $\displaystyle A(x)^{-1}=[e_{1},\cdots,e_{n-1},\overline{x}]^{T}$
where
$\displaystyle\overline{x}=\left(-\frac{x_{1}}{x_{n}},\cdots,-\frac{x_{n-1}}{x_{n}},\frac{1}{x_{n}}\right)^{T}$.
We also have $\displaystyle\lVert A(x)^{-1}\rVert_{2}^{-1}|\nabla
u|\leq|V(u)|\leq\lVert A(x)\rVert_{2}|\nabla u|$ where, $\displaystyle\lVert
A(x)\rVert_{2}^{2}=n-1+|x|^{2}$ and $\displaystyle\lVert
A(x)^{-1}\rVert_{2}^{2}=n-1+\frac{|x|^{2}-x_{n}^{2}+1}{x_{n}^{2}}$.
Keeping the linearized equation (5.10) in mind we proceed to state the
regularity results for $\displaystyle V(u)$. The proofs are verbatim
adaptations from [14] and [15], some of which we present here and summarize
the rest for the sake of completeness. See Theorem-2.2 in [14] for the
corresponding Euclidean analogue of the following lemma.

######  Lemma 5.1.

Let, $\displaystyle 1<p<\infty$ and $\displaystyle
u\in\mathcal{D}^{1,p}(\mathbb{H}^{n})\cap
C^{1,\sigma}_{\text{loc}}(\mathbb{H}^{n})$ be a solution of (5.1) such that
the critical point set
$\displaystyle\mathcal{C}_{u}:=\left\\{x\in\mathbb{H}^{n}:\nabla_{g}u=0\right\\}$
is compact and contained in a bounded domain
$\displaystyle\Omega\subset\mathbb{H}^{n}$. Let us denote by $\displaystyle
Z_{k}:=\left\\{x\in\mathbb{H}^{n}:V_{k}(u)(x)=0\right\\}$ for each
$\displaystyle k=1,2,\cdots,n$. Then for each compact set $\displaystyle E$
such that $\displaystyle\mathcal{C}_{u}\subset\subset E\subset\subset\Omega$
we have

$\displaystyle\displaystyle\sup_{x\in\Omega}\int_{E\setminus
Z_{k}}\frac{|\nabla_{g}u|_{g}^{p-2}\,|\nabla_{g}V_{k}(u)|_{g}^{2}}{|V_{k}(u)|^{\beta}\,\operatorname{dist}_{\mathbb{H}^{n}}(x,y)^{\gamma}}\,dV_{g}(y)<C,\text{
for }k=1,2,\cdots,n$ (5.13)

for some positive constant $\displaystyle C=C(u,\beta,\gamma,E)$ (depending on
$\displaystyle u$) where, $\displaystyle\beta,\gamma$ are non-negative real
numbers such that $\displaystyle 0\leq\beta<1$ and $\displaystyle
0\leq\gamma<n-2$ when $\displaystyle n\geq 3$ ($\displaystyle\gamma=0$ when
$\displaystyle n=2$). This in turn implies

$\displaystyle\displaystyle\sup_{x\in\Omega}\int_{E\setminus\mathcal{C}_{u}}\frac{|\nabla_{g}u|_{g}^{p-2-\beta}\,|\nabla_{g}^{2}u|_{g}^{2}}{\operatorname{dist}_{\mathbb{H}^{n}}(x,y)^{\gamma}}\,dV_{g}(y)<C.$
(5.14)

###### Proof.

Wlog we may assume $\displaystyle x\in E$. Indeed, suppose we prove

$\displaystyle\sup_{x\in E}\int_{E\setminus
Z_{k}}\frac{|\nabla_{g}u|_{g}^{p-2}\,|\nabla_{g}V_{k}(u)|_{g}^{2}}{|V_{k}(u)|^{\beta}\,d_{\mathbb{H}^{n}}(x,y)^{\gamma}}\,dV_{g}(y):=K(\beta,\gamma,E)<C.$

Then splitting between the two cases $\displaystyle x\in
E_{\delta}:=\left\\{x\in\Omega:\operatorname{dist}_{\mathbb{B}^{n}}(x,E)<\delta\right\\}$
with
$\displaystyle\displaystyle\delta<\frac{1}{2}\operatorname{dist}_{\mathbb{B}^{n}}(E,\partial\Omega)$
and $\displaystyle x\in\Omega\setminus E_{\delta}$ we have

$\displaystyle\sup_{x\in\Omega}\int_{E\setminus
Z_{k}}\frac{|\nabla_{g}u|_{g}^{p-2}\,|\nabla_{g}V_{k}(u)|_{g}^{2}}{|V_{k}(u)|^{\beta}\,d_{\mathbb{H}^{n}}(x,y)^{\gamma}}\,dV_{g}(y)\leq
K(\beta,\gamma,E_{\delta})+\frac{1}{\delta^{\gamma}}K(\beta,0,E).$

Let $\displaystyle\displaystyle G_{\varepsilon}(s)=\begin{cases}0&\text{ when
}|s|\leq\varepsilon\\\ s&\text{ when }|s|\geq 2\varepsilon\end{cases}$ and
linear continuous when $\displaystyle\varepsilon<|s|<2\varepsilon$. Then
$\displaystyle G_{\varepsilon}$ is a Lipschitz function with
$\displaystyle|G_{\varepsilon}^{\prime}|\leq 2$. Furthermore we have
$\displaystyle\displaystyle\left(G_{\varepsilon}^{\prime}(s)-\beta\frac{G_{\varepsilon}(s)}{s}\right)\geq
0$ for all $\displaystyle s\in\mathbb{R}$ and $\displaystyle\beta<1$.

Case-1: Let $\displaystyle x\in E\cap\mathcal{C}_{u}$ and let
$\displaystyle\phi\in C_{c}^{\infty}(\Omega)$ be a non-negative test function
such that $\displaystyle\phi\equiv 1$ in a neighborhood containing
$\displaystyle
E_{\delta}:=\left\\{y\in\Omega:\operatorname{dist}_{\mathbb{B}^{n}}(y,E)<\delta\right\\}$
of $\displaystyle E$ and $\displaystyle\text{supp}(\phi)\subset\Omega$ with
$\displaystyle\delta<\frac{1}{2}\operatorname{dist}_{\mathbb{H}^{n}}(E,\partial\Omega)$.
Consider the test function
$\displaystyle\displaystyle\psi_{\varepsilon,x}(y):=\frac{G_{\varepsilon}(V_{k}(u)(y))}{|V_{k}(u)(y)|^{\beta}}\frac{\phi(y)}{\operatorname{dist}_{\mathbb{H}^{n}}(x,y)^{\gamma}}$,
for the linearized equation (5.10). Then we have

$\displaystyle\displaystyle\int_{\Omega\setminus