# Existence of high energy positive solutions for a class of elliptic
equations in the hyperbolic space

Debdip Ganguly  Department of Mathematics, Indian Institute of Technology
Delhi, Hauz Khas New Delhi 110016, India<EMAIL_ADDRESS>,  Diksha
Gupta  Department of Mathematics, Indian Institute of Technology Delhi, Hauz
Khas New Delhi 110016, India<EMAIL_ADDRESS>and  K. Sreenadh
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas New
Delhi 110016, India<EMAIL_ADDRESS>
###### Abstract.

We study the existence of positive solutions for the following class of scalar
field problem on the hyperbolic space

$-\Delta_{\mathbb{H}^{N}}u-\lambda
u=a(x)|u|^{p-1}\,u\;\;\text{in}\;\mathbb{B}^{N},\quad u\in
H^{1}{(\mathbb{B}^{N})},$

where $\mathbb{B}^{N}$ denotes the hyperbolic space,
$1<p<2^{*}-1:=\frac{N+2}{N-2}$, if $N\geqslant 3;1<p<+\infty$, if
$N=2,\;\lambda<\frac{(N-1)^{2}}{4}$, and $0<a\in L^{\infty}(\mathbb{B}^{N}).$
We prove the existence of a positive solution by introducing the min-max
procedure in the spirit of Bahri-Li in the hyperbolic space and using a series
of new estimates involving interacting hyperbolic bubbles.

###### Key words and phrases:

Hyperbolic space, hyperbolic bubbles, Palais-Smale decomposition, semilinear
elliptic problem

###### 2010 Mathematics Subject Classification:

Primary: 35J20, 35J60, 58E30

##  1\. Introduction

In this paper, we investigate the existence of positive solutions for the
following semilinear elliptic problem on the hyperbolic space $\mathbb{B}^{N}$

$(P_{\lambda})\;\;\left\\{-\Delta_{\mathbb{B}^{N}}u\,-\,\lambda
u\,=\,a(x)\,|u|^{p-1}u,\quad u\in H^{1}\left(\mathbb{B}^{N}\right),\right.$

where $1<p<2^{*}-1:=\frac{N+2}{N-2}$, if $N\geqslant 3;1<p<+\infty$, if
$N=2,\;\lambda<\frac{(N-1)^{2}}{4},$ $H^{1}\left(\mathbb{B}^{N}\right)$
denotes the Sobolev space on the disc model of the hyperbolic space
$\mathbb{B}^{N},$ $\Delta_{\mathbb{B}^{N}}$ denotes the Laplace Beltrami
operator on $\mathbb{B}^{N},$ $\frac{(N-1)^{2}}{4}$ being the bottom of the
$L^{2}-$ spectrum of $-\Delta_{\mathbb{B}^{N}},$ and $a(x)\in
L^{\infty}(\mathbb{B}^{N}).$ Moreover, we investigate the existence of
solutions under the following hypotheses

  1. (a1)

$a(x)>0,\forall\ x\in\mathbb{\mathbb{B}^{N}}$ and
$\displaystyle\lim_{d(x,0)\rightarrow\infty}a(x)=1,$

  2. (a2)

there exist some positive constants $C$ and $\delta>0$ such that

$a(x)\geqslant 1-\operatorname{C\,exp}(-\delta\,d(x,0))\quad\forall\
d(x,0)\rightarrow\infty,$

where $d$ is the hyperbolic distance.

The problem, $(P_{\lambda})$ when posed in all of $\mathbb{R}^{N},$ with
$\lambda=-1,$ has been the subject of intense research in the past few
decades, starting from the seminal papers by Berestycki-Lions [7, 8], Bahri-
Berestycki [4], Bahri-Li [5], Bahri-Lions [6]. Apart from purely mathematical
interest, such semilinear elliptic equations also arise in several physical
phenomena, e.g., nonlinear Schrödinger and Klein-Gordan equations (see [18,
19, 20]). The main difficulty in studying such equations in all of
$\mathbb{R}^{N}$ comes from the lack of compactness due to the unboundedness
of the domain $\mathbb{R}^{N},$ i.e., through the vanishing of mass in the
sense of concentration compactness of Lions [21]. Non-compact subcritical
problems have been studied thoroughly and brought to a high level of
sophistication by many authors. We refer to the papers [1, 2, 3, 12, 13, 14,
15, 23, 24, 25] and references quoted therein without any claim of
completeness. The asymptotic of $a(x)$ in $(a1)$ suggests that the
corresponding “limiting problem,” i.e.,

$-\Delta_{\mathbb{R}^{N}}u\,+\,u\,=\,u^{p}\quad\mbox{in}\ \mathbb{R}^{N},\quad
u\in H^{1}(\mathbb{R}^{N}),\quad u>0\quad\mbox{in}\ \mathbb{R}^{N},$ (1.1)

will play an essential role. In fact, the proof of the existence of positive
solutions introduced by Bahri-Li in [5] essentially depends on the uniqueness
(up to translation) and decay estimate of the unique solution, and it is worth
mentioning that Bahri-Li’s solution to the equation $(P_{\lambda})$ does not
correspond to the mountain pass critical point. More precisely, consider the
energy functional $E_{a}:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ defined
by

$E_{a}(u):=\frac{\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{2})\,+\,u^{2}\right)\,{\rm
d}x}{\left(\int_{\mathbb{R}^{N}}a(x)\,|u|^{p+1}\,{\rm
d}x\right)^{\frac{2}{p+1}}}.$

and it was shown in [5] that the Palais-Smale condition holds for $E_{a}$ in
the range $(S_{1},2^{\frac{p-1}{p+1}}S_{1}),$ where

$S_{1}:=\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\\{0\\}}\,E_{a}(u).$

We note that if $a(x)\geq 1$ in $\mathbb{R}^{N},$ then the mountain pass level
for $E_{a}$ is strictly less than $S_{1}$ and hence achieved (see [1]). But
assumptions (a1) and (a2), in general, do not ensure $a(x)\geq 1.$ So Bahri-Li
constructed solutions in higher energy level by using energy estimates
involving interacting (translated) solutions to the limiting problem.

As concerns the analogous equation in the hyperbolic space, Sandeep-Mancini,
in their seminal paper [22], have shown the existence and uniqueness of finite
energy positive solutions of the following homogeneous elliptic equation:

$-\Delta_{\mathbb{B}^{N}}u\,-\,\lambda u\,=\,u^{p},\quad u\in
H^{1}\left(\mathbb{B}^{N}\right),$ (1.2)

where $\lambda\leq\frac{(N-1)^{2}}{4},$ $1<p\leq\frac{N+2}{N-2}$ if $N\geq 3;$
$1<p<\infty$ if $N=2.$ Among many other results, they established in the
subcritical case, and for $p>1,$ if $N=2$ and $1<p<2^{\star}-1$ if $N\geq 3,$
the problem (1.2) has a positive solution if and only if
$\lambda<\frac{(N-1)^{2}}{4}.$ These positive solutions are also shown to be
unique up to hyperbolic isometries, except possibly for $N=2$ and
$\lambda>\frac{2(p+1)}{(p+3)^{2}}.$ The question of finite energy solutions
was fully resolved in [22] and subsequently, the existence of sign-changing
solutions, compactness, and non-degeneracy were studied in [11, 16, 17] and on
the other hand, authors in [9, 10] showed the existence of infinite energy
solutions of (1.2) and determined the exact asymptotic behavior of wide
classes of finite and infinite energy solutions.

One should notice that for the existence of finite energy solutions, one looks
for the energy functional associated with (1.2) which is defined as

$E_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{B}^{N}}\left(|\nabla_{\mathbb{B}^{N}}u|^{2}-\lambda
u^{2}\right)\,\mathrm{~{}d}V_{\mathbb{B}^{N}}\,-\,\frac{1}{p+1}\int_{\mathbb{B}^{N}}|u|^{p+1}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

In the subcritical case, the variational problem lacks compactness because of
the hyperbolic translation (see section 3 for more details), and so it cannot
be solved by the standard minimization method. Moreover, in [11], a detailed
analysis of the Palais-Smale decomposition is performed. One can easily see
that if $U$ is a solution of (1.2), then

$u:=U\circ\tau,\quad\mbox{for}\ \tau\in I(\mathbb{B}^{N}),$

where $I(\mathbb{B}^{N})$ is the group of isometries on the hyperbolic space,
is also a solution, and hence if we define a sequence by varying $\tau_{n}\in
I(\mathbb{B}^{N}),$ then $u_{n}$ is also a Palais-Smale (PS) sequence for
$E_{\lambda}.$ In fact, it was shown in [11, Theorem 3.3] that in the
subcritical case, i.e., when $1<p<\frac{N+2}{N-2},$ non-compact PS sequences
are made of finitely many sequences of the form $u_{n}.$ We call this $u$ a
hyperbolic bubble (See Section 3 below for more details). In our analysis,
these hyperbolic bubbles will play a major role.

Inspired by the above mentioned papers, our main goal in this article is to
study whether positive solutions can still exist for a perturbed problem as in
$(P_{\lambda}).$ Moreover, we seek solutions in the higher energy range
following Bahri-Li in [5]. In particular, we prove the following theorem.

######  Theorem 1.1.

Assume $a(x)$ satisfies (a1)- (a2). Then $(P_{\lambda})$ admits a positive
solution for all $\lambda$ in the range

$\lambda\in\begin{cases}\left(-\infty,\frac{2(p+1)}{(p+3)^{2}}\right],&N=2,\\\
\left(-\infty,\frac{(N-1)^{2}}{4}\right),&N\geq 3.\end{cases}$

###  1.1. Main Novelty and the strategy of the Proof

As discussed earlier, we follow the approach of Bahri-Li. We first show that
the energy functional associated with $(P_{\lambda})$ satisfies the Palais-
Smale condition in the range $(S_{1,\lambda},S_{2,\lambda})$ where
$S_{i,\lambda},\;i=1,2$ are defined in (4.3), and a detailed PS decomposition
is provided in Proposition 3.1. The proof of proposition 3.1 is a
starightforward adaptation of [11, Theorem 3.3] to the problem
$(P_{\lambda})$. In the next step, we prove the key energy estimates. The main
novelty lies in this step, where we compute the energy estimates involving the
convex combination of translated hyperbolic bubbles  with two different
centers. To this end, we need precise estimates on the interacting hyperbolic
bubbles,  and since the hyperbolic volume grows exponentially, i.e.,

$\mathrm{~{}d}V_{\mathbb{B}^{N}}\asymp e^{(N-1)r},\quad r\rightarrow\infty,$

where $r$ denotes the geodesic distance $d(x,0)$, and
$\mathrm{~{}d}V_{\mathbb{B}^{N}}$ denotes the hyperbolic volume form, and to
compensate for the exponential volume growth of the hyperbolic space, one
requires a new way to tackle the integrals involving interacting bubbles.
Unlike in the Euclidean space, where solutions to (1.1) decays exponentially
and the euclidean volume grows polynomially. We compute the effect of
interacting bubbles in the integral in a novel way by breaking the integral in
different sub-regions, and using the precise asymptotic estimate on the
solution of (1.2) obtained in [22]. We establish a series of new estimates for
interacting terms, which are crucial in proving the corresponding energy of
the convex combination is strictly less than $S_{2,\lambda}.$ Finally, the
existence of a positive solution is proved using the min-max  procedure of
Bahri-Li by suitably defining the center of mass for functions in
$H^{1}(\mathbb{B}^{N})$ on the hyperbolic space. We refer section 5 below for
more details.

The paper is organized as follows: In Section 2, we introduce some of the
notations and some geometric definitions and preliminaries concerning
hyperbolic space. In Section 3, we state and prove the Palais-Smale
decomposition theorem associated with $(P_{\lambda})$. Section 4 contains the
proof of key estimates on the energy, while Section 5 contains the proof of
the main Theorem 1.1.

##  2\. Notations and Functional Analytic Preliminaries

In this section, we will introduce some of the notations and definitions used
in this paper and also recall some of the embeddings related to the Sobolev
space on the hyperbolic space.

We will denote by $\mathbb{B}^{N}$ the disc model of the hyperbolic space,
i.e., the unit disc equipped with the Riemannian metric
$g_{\mathbb{B}^{N}}:=\sum\limits_{i=1}^{N}\left(\frac{2}{1-|x|^{2}}\right)^{2}\,{\rm
d}x_{i}^{2}$. To simplify our notations, we will denote $g_{\mathbb{B}^{N}}$
by $g$. The corresponding volume element is given by
$\mathrm{~{}d}V_{\mathbb{B}^{N}}=\big{(}\frac{2}{1-|x|^{2}}\big{)}^{N}{\rm
d}x,$ where ${\rm d}x$ denotes the Lebesgue measure on $\mathbb{R}^{N}$.

Hyperbolic distance on $\mathbb{B}^{N}$. The hyperbolic distance between two
points $x$ and $y$ in $\mathbb{B}^{N}$ will be denoted by $d(x,y).$ For the
hyperbolic distance between $x$ and the origin we write

$\rho:=\,d(x,0)=\int_{0}^{r}\frac{2}{1-s^{2}}\,{\rm
d}s\,=\,\log\frac{1+r}{1-r},$

where $r=|x|$, which in turn implies that $r=\tanh\frac{\rho}{2}.$ Moreover,
the hyperbolic distance between $x,y\in\mathbb{B}^{N}$ is given by

$d(x,y)=\cosh^{-1}\left(1+\dfrac{2|x-y|^{2}}{(1-|x|^{2})(1-|y|^{2})}\right).$

It easily follows that a subset $S$ of $\mathbb{B}^{N}$ is a hyperbolic sphere
in $\mathbb{B}^{N}$ iff $S$ is a Euclidean sphere in $\mathbb{R}^{N}$ and
contained in $\mathbb{B}^{N}$, probably with a different center and different
radius, which can be computed. Geodesic balls in $\mathbb{B}^{N}$ of radius
$a$ centered at the origin will be denoted by

$B(0,a):=\\{x\in\mathbb{B}^{N}:d(x,0)<a\\}.$

We also need some information on the isometries of $\mathbb{B}^{N}$. Below we
recall the definition of a particular type of isometry, namely the hyperbolic
translation. For more details on the isometry group of $\mathbb{B}^{N}$, we
refer [26].

Hyperbolic Translation. For $b\in\mathbb{B}^{N},$ define

$\tau_{b}(x)=\dfrac{(1-|b|^{2})x+(|x|^{2}+2x.b+1)b}{|b|^{2}|x|^{2}+2x.b+1},$
(2.1)

then $\tau_{b}$ is an isometry of $\mathbb{B}^{N}$ with $\tau_{b}(0)=b.$ The
map $\tau_{b}$ is called the hyperbolic translation of $\mathbb{B}^{N}$ by
$b.$ It can also be seen that $\tau_{-b}=\tau_{b}^{-1}.$

The hyperbolic gradient $\nabla_{\mathbb{B}^{N}}$ and the hyperbolic Laplacian
$\Delta_{\mathbb{B}^{N}}$ are given by

$\displaystyle\nabla_{\mathbb{B}^{N}}=\left(\frac{1-|x|^{2}}{2}\right)^{2}\nabla,\
\ \
\Delta_{\mathbb{B}^{N}}=\left(\frac{1-|x|^{2}}{2}\right)^{2}\Delta+(N-2)\frac{1-|x|^{2}}{2}\,\langle
x,\nabla\,\rangle.$

A sharp Poincaré-Sobolev inequality. (see [22])

Sobolev Space : We will denote by ${H^{1}}(\mathbb{B}^{N})$ the Sobolev space
on the disc model of the hyperbolic space $\mathbb{B}^{N}$, equipped with norm
$\|u\|=\left(\int_{\mathbb{B}^{N}}|\nabla_{\mathbb{B}^{N}}u|^{2}\right)^{\frac{1}{2}},$
where $|\nabla_{\mathbb{B}^{N}}u|$ is given by
$|\nabla_{\mathbb{B}^{N}}u|:=\langle\nabla_{\mathbb{B}^{N}}u,\nabla_{\mathbb{B}^{N}}u\rangle^{\frac{1}{2}}_{\mathbb{B}^{N}}.$

For $N\geq 3$ and every $p\in\left(1,\frac{N+2}{N-2}\right]$ there exists an
optimal constant $S_{N,p}>0$ such that

$S_{N,p}\left(\int_{\mathbb{B}^{N}}|u|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}\leq\int_{\mathbb{B}^{N}}\left[|\nabla_{\mathbb{B}^{N}}u|^{2}-\frac{(N-1)^{2}}{4}u^{2}\right]\,\mathrm{~{}d}V_{\mathbb{B}^{N}},$

for every $u\in C^{\infty}_{0}(\mathbb{B}^{N}).$ If $N=2$, then any $p>1$ is
allowed.

A basic information is that the bottom of the spectrum of
$-\Delta_{\mathbb{B}^{N}}$ on $\mathbb{B}^{N}$ is

$\frac{(N-1)^{2}}{4}=\inf_{u\in
H^{1}(\mathbb{B}^{N})\setminus\\{0\\}}\dfrac{\int_{\mathbb{B}^{N}}|\nabla_{\mathbb{B}^{N}}u|^{2}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}}{\int_{\mathbb{B}^{N}}|u|^{2}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}}.$
(2.2)

######  Remark 2.1.

A consequence of (2.2) is that if $\lambda<\frac{(N-1)^{2}}{4},$ then

$||u||_{H_{\lambda}}:=||u||_{\lambda}:=\left[\int_{\mathbb{B}^{N}}\left(|\nabla_{\mathbb{B}^{N}}u|^{2}-\lambda\,u^{2}\right)\,\mathrm{~{}d}V_{\mathbb{B}^{N}}\right]^{\frac{1}{2}},\quad
u\in C_{c}^{\infty}(\mathbb{B}^{N})$

is a norm, equivalent to the $H^{1}(\mathbb{B}^{N})$ norm and the
corresponding inner product is given by $\langle u,v\rangle_{H_{\lambda}}.$

##  3\. Palais-Smale characterization

In this section, we study the Palais-Smale sequences (PS sequences)
corresponding to the problem $(P_{\lambda})$. To be precise, define the
associated energy functional $I_{\lambda,a}$ as

$I_{\lambda,a}(u)=\frac{1}{2}\int_{\mathbb{B}^{N}}\left[\left|\nabla_{\mathbb{B}^{N}}u\right|_{\mathbb{B}^{N}}^{2}-\lambda
u^{2}\right]\mathrm{~{}d}V_{\mathbb{B}^{N}}-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)|u|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

We say a sequence $u_{n}\in H^{1}\left(\mathbb{B}^{N}\right)$ is a Palais-
Smale sequence for $I_{\lambda,a}$ at a level $d$ if
$I_{\lambda,a}\left(u_{n}\right)\rightarrow d$ and
$I_{\lambda,a}^{\prime}\left(u_{n}\right)\rightarrow 0$ in
$H^{-1}\left(\mathbb{B}^{N}\right).$ One can easily see that PS sequences are
bounded. Throughout this section, we assume $a(x)\rightarrow 1$ as
$d(x,0)\rightarrow\infty$.

###  3.1. Compactness and Palais-Smale sequences

Before introducing the Palais-Smale characterization, we shall remark on the
compactness properties of $(P_{\lambda})$ posed in the hyperbolic space. To
see this, let $u\in H^{1}\left(\mathbb{B}^{N}\right)$ and
$b_{n}\in\mathbb{B}^{N}$ such that $b_{n}\rightarrow\infty$ and let $\tau_{n}$
be the hyperbolic translation given by (2.1) such that $\tau_{n}(0)=b_{n}.$
Define $u_{n}=u\circ\tau_{n},$ then
$||u_{n}||_{H^{1}\left(\mathbb{B}^{N}\right)}=||u||_{H^{1}\left(\mathbb{B}^{N}\right)}$
but $u_{n}\rightharpoonup 0$ weakly in $H^{1}(\mathbb{B}^{N}).$ This shows
that the embedding $H^{1}(\mathbb{B}^{N})\hookrightarrow
L^{p}(\mathbb{B}^{N})$ is not compact for any $2<p<\frac{2N}{N-2}$ even in the
subcritical case.

It is worth noticing that if $v$ is a solution of $(P_{\lambda})$ with
$a(x)\equiv 1,$ then $v\circ\tau$ is also a solution for any $\tau\in
I(\mathbb{B}^{N}),$ where $I(\mathbb{B}^{N})$ is an isometry group. In
particular, if we define

$v_{n}=v\circ\tau_{n},$

where $\tau_{n}$ as defined in (2.1) with $\tau_{n}(0)\rightarrow\infty,$ then
$v_{n}$ is a PS sequence converging weakly to zero. Thus in the limiting case,
i.e., $a(x)\rightarrow 1,$ as $d(x,0)\rightarrow\infty,$ the functional
$I_{\lambda,a}$ for the Palais-Smale sequences exhibits sequences of the form
$v_{n}.$ To be precise, we state the following proposition.

######  Proposition 3.1.

Let $\left\\{u_{n}\right\\}\subset H^{1}\left(\mathbb{B}^{N}\right)$ be a PS
sequence for $I_{\lambda,a}$ at a level $d\geq 0.$ Then there exists a
subsequence (still denoted by $\left\\{u_{n}\right\\}$) for which the
following holds: there exists an integer $m\geq 0$, sequences $\tau_{n}^{i}$
$\in I\left(\mathbb{B}^{N}\right)$, functions $\bar{u},w_{i}\in
H^{1}(\mathbb{B}^{N})$ for $1\leq i\leq m$ such that

$-\Delta_{\mathbb{B}^{N}}\bar{u}-\lambda\bar{u}=a(x)|\bar{u}|^{p-1}\bar{u}\;\text{in}\;H^{-1}(\mathbb{B}^{N}),$

$\begin{gathered}-\Delta_{\mathbb{B}^{N}}w_{i}-\lambda
w_{i}=|w_{i}|^{p-1}w_{i}\;\text{in}\;H^{-1}(\mathbb{B}^{N}),w_{i}\not\equiv
0,\end{gathered}$
$\begin{gathered}u_{n}-\left(\bar{u}+\sum_{i=1}^{m}w_{i}\left(\tau_{n}^{i}(\bullet)\right)\right)\rightarrow
0\text{ as }n\rightarrow\infty,\\\ I_{\lambda,a}\left(u_{n}\right)\rightarrow
I_{\lambda,a}(\bar{u})+\sum_{i=1}^{m}I_{\lambda,1}\left(w_{i}\right)\text{ as
}n\rightarrow\infty,\\\
\tau_{n}^{i}(0)\rightarrow\infty,\;d(\tau_{n}^{i}(0),\tau_{n}^{j}(0))\rightarrow\infty\text{
as }n\rightarrow\infty,\text{ for }1\leq i\neq j\leq m.\end{gathered}$

To prove the above proposition, we need the following auxiliary lemmas.

######  Lemma 3.2.

Let $\left\\{u_{n}\right\\}$ be a bounded sequence in
$H^{1}\left(\mathbb{B}^{N}\right)$ such that

$\sup_{y\in\mathbb{B}^{N}}\int_{B(y,r)}|u_{n}|^{q}\mathrm{~{}d}V_{\mathbb{B}^{N}}\rightarrow
0\quad\text{as}\quad n\rightarrow\infty,$

for some $r>0$ and $2\leq q<2^{*}$. Then $u_{n}\rightarrow 0$ strongly in
$L^{p}\left(\mathbb{B}^{N}\right)$ for all $p\in\left(2,2^{*}\right)$. In
addition, if $u_{n}$ satisfies

$-\Delta_{\mathbb{B}^{N}}u_{n}-\lambda u_{n}-a(x)|u_{n}|^{p-1}u_{n}\rightarrow
0\quad\text{ in }\quad H^{-1}\left(\mathbb{B}^{N}\right),$ (3.1)

then $u_{n}\rightarrow 0$ strongly in $H^{1}\left(\mathbb{B}^{N}\right)$.

###### Proof.

For any $s\in\left(q,2^{*}\right),$ we have from the interpolation inequality

$\|u_{n}\|_{L^{s}(B(y,r))}\leq\|u_{n}\|_{L^{q}(B(y,r))}^{1-\lambda}\|u_{n}\|_{L^{2^{*}}(B(y,r))}^{\lambda},$
(3.2)

where $\lambda=\frac{s-q}{2^{*}-q}\frac{2^{*}}{s}.$ The Sobolev inequalities
on geodesic balls in the hyperbolic space imply that there exists a positive
constant $C>0$ independent of $y\in\mathbb{B}^{N}$ such that

$\int_{B(y,r)}|u_{n}|^{2^{*}}dV_{\mathbb{B}^{N}}\leq
C\,\left(\int_{B(y,r)}\left(|u_{n}|^{2}+\left|\nabla_{\mathbb{B}^{N}}u_{n}\right|^{2}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2^{*}}{2}}.$
(3.3)

Using (3.2) and (3.3), we get

$\int_{B(y,r)}|u_{n}|^{s}\mathrm{~{}d}V_{\mathbb{B}^{N}}\leq
C||u_{n}||_{L^{q}(B(y,r))}^{s(1-\lambda)}\left(\int_{B(y,r)}\left(|u_{n}|^{2}+\left|\nabla_{\mathbb{B}^{N}}u\right|^{2}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{\lambda
s}{2}}.$

Now covering $\mathbb{B}^{N}$ by balls of radius $r$, in such a way that each
point of $\mathbb{B}^{N}$ is contained inside at most $N_{0}$ balls, we find

$\int_{\mathbb{B}^{N}}|u_{n}|^{s}\mathrm{~{}d}V_{\mathbb{B}^{N}}\leq
C\sup_{y\in\mathbb{B}^{N}}\|u_{n}\|_{L^{q}(B(y,r))}^{s(1-\lambda)}\int_{\mathbb{B}^{N}}\left(|u_{n}|^{2}+\left|\nabla_{\mathbb{B}^{N}}u_{n}\right|^{2}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Therefore, utilizing the lemma hypothesis, we get for the sequence
$\left\\{u_{n}\right\\}$ that
$\left\|u_{n}\right\|_{L^{s}\left(\mathbb{B}^{N}\right)}\rightarrow 0$ for all
$s\in\left(q,2^{*}\right)$. This proves the first part of the lemma if $q=2$;
otherwise, if $q>2$, then again, one can argue similarly by choosing
$s\in(2,q)$. In addition, if (3.1) is satisfied, then we obtain

$\left|\left\langle-\Delta_{\mathbb{B}^{N}}u_{n}-\lambda
u_{n}-a(x)|u_{n}|^{p-1}u_{n},u_{n}\right\rangle_{H_{\lambda}}\right|=o(1)\left\|u_{n}\right\|_{\lambda}.$
(3.4)

Since $\left\\{u_{n}\right\\}$ is bounded in
$H^{1}\left(\mathbb{B}^{N}\right)$, the RHS is $o(1).$ On the other hand, for
the LHS we notice that because $u_{n}$ is bounded in
$H^{1}\left(\mathbb{B}^{N}\right)$ and $u_{n}\rightarrow 0$ strongly in
$L^{r}\left(\mathbb{B}^{N}\right)$, for $r\in\left(2,2^{*}\right)$, we must
have $u_{n}\rightharpoonup 0$ weakly in $H^{1}\left(\mathbb{B}^{N}\right)$.
Moreover, by the previous part, $u_{n}\rightarrow 0$ strongly in
$L^{p+1}\left(\mathbb{B}^{N}\right).$ Thus by (3.4), we get $u_{n}\rightarrow
0$ strongly in $H^{1}\left(\mathbb{B}^{N}\right).$ ∎

######  Lemma 3.3.

Let $\phi_{k}\rightharpoonup\phi$ weakly in
$H^{1}\left(\mathbb{B}^{N}\right)$, then we have

$a\left|\phi_{k}\right|^{p-1}\phi_{k}-a|\phi|^{p-1}\phi\rightarrow
0\quad\text{in }\quad H^{-1}\left(\mathbb{B}^{N}\right).$

###### Proof.

Defining $\psi_{k}=\phi_{k}-\phi$, gives $\psi_{k}\rightharpoonup 0$ weakly in
$H^{1}\left(\mathbb{B}^{N}\right).$ Particularly, $\left\\{\psi_{k}\right\\}$
is bounded in $H^{1}\left(\mathbb{B}^{N}\right).$ Thus, up to a subsequence,
$\psi_{k}\rightarrow 0$ strongly in $L_{\text{loc }}^{q}(\mathbb{B}^{N})$ for
all $2<q<2^{*}$ and $\psi_{k}\rightarrow 0$ a.e.. As a result,
$a\left|\phi+\psi_{k}\right|^{p-1}\left(\phi+\psi_{k}\right)-a|\phi|^{p-1}\phi\rightarrow
0$ a.e.. Therefore, it follows from Vitaly’s convergence theorem that
$a\left|\phi+\psi_{k}\right|^{p-1}\left(\phi+\psi_{k}\right)-a|\phi|^{p-1}\phi\rightarrow
0$ strongly in $L_{\text{loc }}^{\frac{p+1}{p}}\left(\mathbb{B}^{N}\right).$
Also, we can see that for every $\varepsilon>0$, there exists
$K_{\varepsilon}>0$ such that

$\left|a|\phi+\psi_{k}|^{p-1}\left(\phi+\psi_{k}\right)-a|\phi|^{p-1}\phi\right|^{\frac{p+1}{p}}\leq\varepsilon\left|\psi_{k}\right|^{p+1}+K_{\varepsilon}|\phi|^{p+1}.$
(3.5)

Furthermore, as $\psi_{k}\rightarrow 0$ weakly in
$H^{1}\left(\mathbb{B}^{N}\right)$ implies $\psi_{k}$ is uniformly bounded in
$L^{p+1}\left(\mathbb{B}^{N}\right)$ and the fact that $|\phi|^{p+1}\in
L^{1}\left(\mathbb{B}^{N}\right)$, it can be immediately observed from (3.5)
that given $\varepsilon>0$, there exists $R>0$ for which

$\int_{\mathbb{R}^{N}\backslash
B(0,R)}|a|\phi+\left.\psi_{k}\right|^{p-1}\left(\phi+\psi_{k}\right)-\left.a|\phi|^{p-1}\phi\right|^{\frac{p+1}{p}}\mathrm{~{}d}V_{\mathbb{B}^{N}}<\varepsilon.$

Consequently,
$a\left|\phi+\psi_{k}\right|^{p-1}\left(\phi+\psi_{k}\right)-a|\phi|^{p-1}\phi\rightarrow
0$ strongly in $L^{\frac{p+1}{p}}\left(\mathbb{B}^{N}\right).$ Hence as
$H^{1}\left(\mathbb{B}^{N}\right)$ is continuously embedded in
$L^{p+1}\left(\mathbb{B}^{N}\right)$, which is the dual space of
$L^{\frac{p+1}{p}}\left(\mathbb{B}^{N}\right)$, it follows that
$a\left|\phi+\psi_{k}\right|^{p-1}\left(\phi+\psi_{k}\right)-a|\phi|^{p-1}\phi\rightarrow
0$ strongly in $H^{-1}\left(\mathbb{B}^{N}\right).$ ∎

######  Lemma 3.4.

For each $c_{0}\geq 0$, there exists $\delta>0$ such that if $v\in
H^{1}\left(\mathbb{B}^{N}\right)$ solves

$-\Delta_{\mathbb{B}^{N}}v-\lambda v=|v|^{p-1}v\text{ in
}H^{-1}(\mathbb{B}^{N}),$

and $\|v\|_{H^{1}\left(\mathbb{B}^{N}\right)}\leq
c_{0},\|v\|_{L^{2}\left(\mathbb{B}^{N}\right)}\leq\delta$, then $v\equiv 0$.

###### Proof.

Taking $v$ as a test function yields

$C^{\prime}\,\|v\|_{H^{1}(\mathbb{B}^{N})}^{2}\leq\|v\|_{\lambda}^{2}\,=\,\int_{\mathbb{B}^{N}}|v|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\leq\|v\|_{L^{2}\left(\mathbb{B}^{N}\right)}^{\alpha}\left(\int_{\mathbb{B}^{N}}|v|^{\frac{2N}{N-2}}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\beta}\leq
C\delta^{\alpha}\|v\|_{H^{1}(\mathbb{B}^{N})}^{\gamma},$

with $\gamma=\frac{2N}{N-2}\beta,$ by Hölder and Sobolev inequalities, where
$C^{\prime},C$ denote non-negative constants independent of $c_{0},\delta$ and
$\alpha=\left(\frac{2N}{N-2}-(p+1)\right)\left(\frac{2N}{N-2}-2\right)^{-1},\;\beta=(p-1)\left(\frac{2N}{N-2}-2\right)^{-1}$.
Now, if $p\geq 1+\frac{4}{N}$ then $\gamma\geq 2$ and we conclude easily if
$\delta$ is small enough. On the other hand if $p<1+\frac{4}{N}$, we deduce

$\|v\|_{H^{1}\left(\mathbb{B}^{N}\right)}\leq C\delta^{k}\text{ for
}k=\alpha(2-\gamma)^{-1}>0.$

Therefore, choosing $\delta>0$ small enough, we can conclude the lemma. ∎

Proof of Proposition 3.1:

The boundedness of a PS sequence in $H^{1}\left(\mathbb{B}^{N}\right)$ follows
from the standard arguments. To be precise,
$I_{\lambda,a}\left(u_{n}\right)=d+o(1)$ and
$I_{\lambda,a}^{\prime}\left(u_{n}\right)(u_{n})=o(1)\left\|u_{n}\right\|_{H^{1}\left(\mathbb{B}^{N}\right)}$,
evaluating
$I_{\lambda,a}\left(u_{n}\right)-\frac{1}{p+1}I_{\lambda,a}^{\prime}\left(u_{n}\right)(u_{n})$,
we obtain

$\left\|u_{n}\right\|_{H^{1}\left(\mathbb{B}^{N}\right)}^{2}\leq
C+o(1)\left\|u_{n}\right\|_{H^{1}\left(\mathbb{B}^{N}\right)},$

and hence boundedness follows. As a result, we can assume
$u_{n}\rightharpoonup u$ weakly in $H^{1}\left(\mathbb{B}^{N}\right)$ up to a
subsequence. Moreover, $|I_{\lambda,a}^{\prime}(u_{n})(v)|\rightarrow 0$ as
$n\rightarrow$ $\infty\;\;\forall v\in H^{1}\left(\mathbb{B}^{N}\right)$
implies

$\int_{\mathbb{B}^{N}}\nabla_{\mathbb{B}^{N}}u_{n}\nabla_{\mathbb{B}^{N}}v\mathrm{~{}d}V_{\mathbb{B}^{N}}\,-\,\lambda\int_{\mathbb{B}^{N}}u_{n}v\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}a(x)\left|u_{n}\right|^{p-1}u_{n}v\mathrm{~{}d}V_{\mathbb{B}^{N}}\rightarrow
0,$ (3.6)

as $n\rightarrow\infty$ for all $v\in H^{1}\left(\mathbb{B}^{N}\right)$.

Furthermore, using $\ref{lem3},$ we deduce that weak limit $u$ satisfies

$-\Delta_{\mathbb{B}^{N}}u-\lambda u=a(x)|u|^{p-1}u\text{ in
}H^{-1}(\mathbb{B}^{N}).$

Next, we show that $u_{n}\,-\,u$ is a PS sequence for $I_{\lambda,a}$ at the
level $d-I_{\lambda,a}(u).$ First of all, applying Brezis-Lieb lemma, we
obtain the following equations

$\displaystyle\int_{\mathbb{B}^{N}}\left|\nabla_{\mathbb{B}^{N}}u_{n}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}\left|\nabla_{\mathbb{B}^{N}}u\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}=\int_{\mathbb{B}^{N}}\left|\nabla_{\mathbb{B}^{N}}\left(u_{n}-u\right)\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}+o(1),$
$\displaystyle\int_{\mathbb{B}^{N}}\left|u_{n}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}|u|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}=\int_{\mathbb{B}^{N}}\left|u_{n}-u\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}+o(1),$
$\displaystyle\int_{\mathbb{B}^{N}}a(x)\left|u_{n}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle-\int_{\mathbb{B}^{N}}a(x)|u|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=\int_{\mathbb{B}^{N}}a(x)\left|u_{n}-u\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)+o(1).$

Thus applying the above equations, we obtain

$\displaystyle I_{\lambda,a}\left(u_{n}-u\right)=$
$\displaystyle\frac{1}{2}\left(\left\|u_{n}\right\|_{\lambda}^{2}-\|u\|_{\lambda}^{2}\right)$
$\displaystyle-\frac{1}{p+1}\left(\int_{\mathbb{B}^{N}}a(x)\left|u_{n}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)-\int_{\mathbb{B}^{N}}a(x)|u|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)+o(1)$
$\displaystyle\longrightarrow{d-I_{\lambda,a}(u)}\quad\text{ as
}n\rightarrow\infty.$

Now for $\phi\in H^{1}\left(\mathbb{B}^{N}\right)$,

$\displaystyle I_{\lambda,a}^{\prime}(u_{n}-u)(\phi)=$
$\displaystyle\int_{\mathbb{B}^{N}}\nabla_{\mathbb{B}^{N}}\left(u_{n}-u\right)\nabla_{\mathbb{B}^{N}}\phi\,\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle-\lambda\int_{\mathbb{B}^{N}}\left(u_{n}-u\right)\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}a(x)\left|u_{n}-u\right|^{p-1}\left(u_{n}-u\right)\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Using (3.6), we get

${I_{\lambda,a}^{\prime}(u_{n}-u),(\phi)}=\int_{\mathbb{B}^{N}}a(x)\left[\left|u_{n}\right|^{p-1}u_{n}-\left|u\right|^{p-1}u-\left|u_{n}\,-\,u\right|^{p-1}\left(u_{n}\,-\,u\right)\right]\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}+o(1).\\\
$

Applying the Hölder inequality, the above quantity can be estimated as

$\left|\phi\right|_{L^{2^{*}}\left(\mathbb{B}^{N}\right)}\left[\int_{\mathbb{B}^{N}}\left|a(x)\left(\left|u_{n}\right|^{p-1}u_{n}-|u|^{p-1}u-\left|u_{n}-u\right|^{p-1}\left(u_{n}-u\right)\right)\right|^{\frac{2N}{N+2}}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right]^{\frac{N+2}{2N}}.\\\
$

Now, we can observe that the term inside the bracket is of $o(1)$, which
follows from the standard arguments using Vitali’s convergence theorem. Hence
we get $I_{\lambda,a}^{\prime}\left(u_{n}-u\right)=o(1)$.

Therefore, in view of the Lemma 3.2 we have, either $u_{n}-u\rightarrow 0$
strongly in $H^{1}\left(\mathbb{B}^{N}\right)$, in that case we are done or
there exists $\alpha>0$, such that up to a subsequence

$Q_{n}(1):=\sup_{y\in\mathbb{B}^{N}}\int_{B(y,1)}\left|u_{n}-u\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}>\alpha>0.$

Consequently, we can find a sequence
$\left\\{y_{n}\right\\}\subset\mathbb{B}^{N}$ such that

$\int_{B\left(y_{n},1\right)}\left|u_{n}-u\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}\geq\alpha.$
(3.7)

Now define $v_{n}(x):=\left(u_{n}-u\right)\left(T_{n}(x)\right)$, where
$T_{n}(x)=\tau_{y_{n}}(x)$ and $\tau_{y_{n}}$ is the hyperbolic translation of
$\mathbb{B}^{N}$ by $y_{n}$. Now, as $\tau_{y_{n}}$ is an isometry, so $v_{n}$
will form a PS sequence at the same level as $u_{n}-u$ which implies ${v}_{n}$
is also bounded in $H^{1}\left(\mathbb{B}^{N}\right)$, and hence converges
weakly in $H^{1}\left(\mathbb{B}^{N}\right)$ upto a subsequence to say $v$.
The compact embedding of $H^{1}\left(B\left(0,1\right)\right)$ into
$L^{2}\left(B\left(0,1\right)\right)$ and (3.7) implies $v\not\equiv 0$. Also,
as $u_{n}-u\rightharpoonup 0$ weakly in $H^{1}\left(\mathbb{B}^{N}\right)$ and
using (3.7) it follows that

$\tau_{y_{n}}(0)\rightarrow\infty\quad\text{ as }n\rightarrow\infty.$

Let us now define,

$w_{n}:=v_{n}-v.$

Clearly, the fact that $v_{n}\rightharpoonup v$ implies $w_{n}\rightharpoonup
0$ weakly in $H^{1}\left(\mathbb{B}^{N}\right).$ Applying this and Lemma 3.3,
in the definition of $I_{\lambda,1}^{\prime}\left(w_{n}\right)$ results in

$I_{\lambda,1}^{\prime}\left(w_{n}\right)=o(1)\text{ in
}H^{-1}\left(\mathbb{B}^{N}\right),$

i.e.,

$-\Delta_{\mathbb{B}^{N}}w_{n}-\lambda w_{n}-|w_{n}|^{p-1}w_{n}\rightarrow
0\quad\text{ in }H^{-1}\left(\mathbb{B}^{N}\right).$

Next, we show that $v$ satisfies

$-\Delta_{\mathbb{B}^{N}}v-\lambda v=|v|^{p-1}v\quad\text{ in
}H^{-1}\left(\mathbb{B}^{N}\right),\quad v\in
H^{1}\left(\mathbb{B}^{N}\right).$ (3.8)

To prove this, let $\tilde{v}\in C_{0}^{\infty}\left(\mathbb{B}^{N}\right)$.
Since, $v_{n}\rightharpoonup v$ weakly in $H^{1}(\mathbb{B}^{N})$, we estimate
as follows:

$\displaystyle\langle v,\tilde{v}\rangle_{H_{\lambda}}$
$\displaystyle=\lim_{n\rightarrow\infty}\left\langle
v_{n},\tilde{v}\right\rangle_{H_{\lambda}}$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{B}^{N}}\nabla_{\mathbb{B}^{N}}v_{n}\nabla_{\mathbb{B}^{N}}\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\lambda\int_{\mathbb{B}^{N}}v_{n}\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{B}^{N}}\nabla_{\mathbb{B}^{N}}\left(u_{n}-u\right)(T_{n}(x))\nabla_{\mathbb{B}^{N}}\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\lambda\int_{\mathbb{B}^{N}}\left(u_{n}-u\right)\left(T_{n}(x)\right)\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{B}^{N}}a(y)\left|\left(u_{n}-u\right)(y)\right|^{p-1}\left(u_{n}-u\right)(y)\tilde{v}\left({T_{n}}^{-1}(y)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{B}^{N}}a\left(T_{n}(x)\right)\left|v_{n}(x)\right|^{p-1}v_{n}(x)\tilde{v}(x)\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Also,

$\displaystyle\left|\int_{\mathbb{B}^{N}}a\left(T_{n}(x)\right)\left|v_{n}(x)\right|^{p-1}v_{n}(x)\tilde{v}(x)\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}|v|^{p-1}v\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right|$
$\displaystyle\leq\left|\int_{\mathbb{B}^{N}}a\left(T_{n}(x)\right)\left(\left|v_{n}\right|^{p-1}v_{n}-\left|v\right|^{p-1}v\right)\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right|$
$\displaystyle\;\;+\left|\int_{\mathbb{B}^{N}}\left(a\left(T_{n}(x)\right)-1\right)\left|v\right|^{p-1}v\tilde{v}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right|$
$\displaystyle:=I_{n}^{1}+J_{n}^{1}.$

Since $T_{n}(0)\rightarrow\infty,\left|v\right|^{p-1}v\tilde{v}\in
L^{1}\left(\mathbb{B}^{N}\right),a\in L^{\infty}\left(\mathbb{B}^{N}\right)$
and $a(x)\rightarrow 1$ as $d(x,0)\rightarrow\infty$, the dominated
convergence theorem yields

$\lim_{n\rightarrow\infty}J_{n}^{1}=0.$

On the other hand, since $\tilde{v}$ has a compact support; applying Vitaly’s
convergence theorem gives

$\lim_{n\rightarrow\infty}I_{n}^{1}\leq\lim_{n\rightarrow\infty}\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\int_{\text{supp
}v}\left|\left|v_{n}\right|^{p-1}v_{n}-\left|v\right|^{p-1}v\right|\left|\tilde{v}\right|\mathrm{~{}d}V_{\mathbb{B}^{N}}=0.$

The above two estimates help us to conclude that $v$ satisfies (3.8).

Further, applying Brezis-Lieb Lemma, we get

$\begin{gathered}\int_{\mathbb{B}^{N}}\left|\nabla_{\mathbb{B}^{N}}v_{n}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}\left|\nabla_{\mathbb{B}^{N}}v\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}\left|\nabla_{\mathbb{B}^{N}}w_{n}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}\rightarrow
0,\\\
\int_{\mathbb{B}^{N}}\left|v_{n}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}|v|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}\left|w_{n}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}\rightarrow
0,\end{gathered}$

as $n\rightarrow\infty$. In view of the above steps, we rerun the procedure
for the Palais-Smale (PS) sequence $v_{n}-v$ to end up in either of the two
cases. If it converges to zero, we stop, or else we repeat the steps. But it
is worth noticing that this process has to terminate in finitely many steps,
and we obtain $v_{1},v_{2},\ldots,v_{n}$ which denotes the limit solutions of
(3.8) obtained through the procedure, we have

$\sum_{i=1}^{n}\int_{\mathbb{B}^{N}}\left|v_{i}\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}\leq\liminf_{n\rightarrow\infty}\int_{\mathbb{B}^{N}}\left|u_{n}-u\right|^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Thus in view of Lemma 3.4, $n$ can not approach infinity.

##  4\. Key Lemma : Energy estimates involving hyperbolic bubbles

In this section, we derive key energy estimates of the functional associated
with $(P_{\lambda})$ involving hyperbolic bubbles. To this end, let us recall
the asymptotic estimates of positive solutions to the corresponding
homogeneous problem

$\begin{gathered}-\Delta_{\mathbb{B}^{N}}w-\lambda w=|w|^{p-1}w,\;w>0\;\text{
in }\mathbb{B}^{N},w\in H^{1}\left(\mathbb{B}^{N}\right).\end{gathered}$ (4.1)

Then by elliptic regularity, any solution, $w\in H^{1}(\mathbb{B}^{N}),$ is
also in $C^{\infty}$ and satisfies the decay property (See [22, Lemma 3.4]):
for every $\varepsilon>0,$ there exist positive constants
$C_{1}^{\varepsilon}$ and $C_{2}^{\varepsilon}$ such that there holds

$C_{1}^{\varepsilon}{\text{e}}^{-(c(N,\lambda)+\varepsilon)\,d(x,0)}\leq
w(x)\leq
C_{2}^{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)\,d(x,0)},\quad\hbox{for
all}\ x\in\mathbb{B}^{N},$

where $c(N,\lambda)\,=\,\frac{1}{2}(N-1+\sqrt{(N-1)^{2}-4\lambda}).$ This
decay property will play a pivotal role in the subsequent analysis.

We define the functionals
$J,J_{\infty}:H^{1}(\mathbb{B}^{N})\rightarrow\mathbb{R}$ as

$J(u):=\frac{\|u\|_{\lambda}^{2}}{\left(\int_{\mathbb{B}^{N}}a(x)|u(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{2}{p+1}}},\quad
J_{\infty}(u):=\frac{\|u\|_{\lambda}^{2}}{\left(\int_{\mathbb{B}^{N}}|u(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{2}{p+1}}}\\\
$ (4.2)

and the energy levels

$S_{1,\lambda}:=\inf_{u\in
H^{1}\left(\mathbb{B}^{N}\right)\backslash\\{0\\}}J_{\infty}(u),\quad
S_{m,\lambda}:=m^{\frac{p-1}{p+1}}S_{1,\lambda},\quad m=2,3,4,\cdots$ (4.3)

Let us recall a convex inequality.

######  Lemma 4.1.

Let $p>1$ be any real number. Then for any non-negative real numbers $a,b$
there holds

$(a+b)^{p+1}\geq a^{p+1}+b^{p+1}\,+\,p\left(a^{p}b+ab^{p}\right).$

For the proof of the above lemma, we refer [12].

######  Lemma 4.2.

Let $a(x)$ satisfies (a1)-(a2) and let $w$ be a unique radial solution of
(4.1). Then, there exists a large number $R_{0}$, such that for any $R\geq
R_{0},\;d(x_{1},0)\geq R^{\alpha},\;d(x_{2},0)\geq
R^{\alpha},\;R^{\alpha^{\prime}}\leq d\left(x_{1},x_{2}\right)\leq
R^{\alpha^{\prime}-\alpha}\min\left\\{d(x_{1},0),\;d(x_{2},0)\right\\}$, where
$\alpha>\alpha^{\prime}>1$, it holds

$J\left(tu_{1}+(1-t)u_{2}\right)<S_{2,\lambda},$ (4.4)

where $0\leq t\leq 1$ and $u_{i}=w\left(\tau_{-x_{i}}(\bullet)\right),i=1,2$.

###### Proof.

  1. Step 1:

In this step, we will prove $\eqref{3h}$ for $t=\frac{1}{2}$. Set,

$A:=\|w\|_{H_{\lambda}}^{2}.$

Then by taking $w$ as a test function in $\eqref{3d}$, we obtain

$A=\int_{\mathbb{B}^{N}}w(x)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Therefore,

$\displaystyle J\left(\frac{1}{2}(u_{1}+u_{2})\right)$
$\displaystyle=J\left(u_{1}+u_{2}\right)$ (4.5)
$\displaystyle=\frac{\left\|u_{1}\right\|_{H_{\lambda}}^{2}+\left\|u_{2}\right\|_{H_{\lambda}}^{2}+2\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(\int_{\mathbb{B}^{N}}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}(1-a(x))\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}}$
$\displaystyle\leq\frac{2A+2\left\langle
u_{1},u_{2}\right\rangle_{{H_{\lambda}}}}{\left(\int_{\mathbb{B}^{N}}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}(1-a(x))_{+}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}}.$

Firstly, we estimate the integral,
$\int_{\mathbb{B}^{N}}u_{1}^{p}\,u_{2}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}.$
Since $w$ is positive, radially symmetric, symmetric decreasing and smooth, we
get

$\displaystyle\int_{\mathbb{B}^{N}}u_{1}^{p}u_{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\int_{\mathbb{B}^{N}}w\left(d(\tau_{-x_{1}}(x),0)\right)^{p}w\left(d(\tau_{-x_{2}}(x),0)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\geq\int_{d(x,x_{1})\leq
1}w\left(d(\tau_{-x_{1}}(x),0)\right)^{p}w\left(d(\tau_{-x_{2}}(x),0)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\geq C\int_{d(x,x_{1})\leq
1}w\left(d(\tau_{-x_{2}}(x),0)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=\int_{d(x,x_{1})\leq
1}w\left(d(x,x_{2})\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

In the above calculations, we have used the following observations

$d(x,x_{1})\leq 1\Longrightarrow d(x,x_{2})\leq d(x,x_{1})+d(x_{1},x_{2})\leq
1+d(x_{1},x_{2}).$

Consequently, $w\left(d(x,x_{2}\right)\geq w\left(1+d(x_{1},x_{2})\right)$.
Therefore, for $d(x_{1},x_{2})>>1$,

$\int_{\mathbb{B}^{N}}u_{1}^{p}\,u_{2}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}\geq
C_{\varepsilon}w\left(1+d(x_{1},x_{2})\right)\geq
C_{\varepsilon}{\text{e}}^{-(c(N,\lambda)+\varepsilon)^{+}d(x_{1},x_{2})},$

where $C_{\varepsilon}$ is a positive constant independent of $R$, and for
$\gamma\in\mathbb{R},$ $\gamma^{+}\left(\right.$respectively
$\left.\gamma^{-}\right)$ stands for any $\gamma+\delta$ (respectively
$\gamma-\delta$ ) with $\delta>0.$

Next, we estimate

$\displaystyle\int_{\mathbb{B}^{N}}(1-a)_{+}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\leq
C\sum_{i=1}^{2}\int_{\mathbb{B}^{N}}(1-a)_{+}u_{i}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle\leq
C\sum_{i=1}^{2}\int_{\mathbb{B}^{N}}\frac{w\left(d(\tau_{-x_{i}}(x),0)\right)^{p+1}}{{\text{e}}^{\delta
d(x,0)}}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=C\sum_{i=1}^{2}\left(\underbrace{\int_{d(x,x_{i})>d(x_{i},0)}\frac{w\left(d(\tau_{-x_{i}}(x),0)\right)^{p+1}}{{\text{e}}^{\delta
d(x,0)}}\mathrm{~{}d}V_{\mathbb{B}^{N}}}_{I_{1}^{i}}+\underbrace{\int_{d(x,x_{i})\leq
d(x_{i},0)}\frac{w\left(d(\tau_{-x_{i}}(x),0)\right)^{p+1}}{{\text{e}}^{\delta
d(x,0)}}\mathrm{~{}d}V_{\mathbb{B}^{N}}}_{I_{2}^{i}}\right).$

Now we shall estimate $I_{1}^{i}+I_{2}^{i}$ with
$\delta\in\left(Kc(N,\lambda)+(N-1),(p+1)c(N,\lambda)\right),$ and choose
$\varepsilon>0$ such that $\delta<(p+1)(c(N,\lambda)-\varepsilon),$ where $K$
is fixed, and is such that $0<K<(p+1)-\frac{N-1}{c(N,\lambda)}.$ Before moving
further, we note that $\frac{c(N,\lambda)}{N-1}>\frac{1}{2}$ and $p+1>2$
implies $(p+1)-\frac{N-1}{c(N,\lambda)}>0$. Let us first consider $I_{1}^{i}:$

$\displaystyle I_{1}^{i}$
$\displaystyle=\int_{d(x,x_{i})>d(x_{i},0)}\frac{w\left(d(\tau_{-x_{i}}(x),0)\right)^{p+1}}{{\text{e}}^{\delta
d(x,0)}}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq\int_{d(x,x_{i})>d(x_{i},0)}\frac{w\left(d(x_{i},0)\right)^{p+1}}{{\text{e}}^{\delta
d(x,0)}}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$ $\displaystyle\leq
C_{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)d(x_{i},0)(p+1)}\int_{d(x,x_{i})>d(x_{i},0)}{\text{e}}^{-\delta
d(x,0)}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$ $\displaystyle\leq
C_{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)d(x_{i},0)(p+1)}\int_{d(x,0)>0}{\text{e}}^{-\delta
d(x,0)}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=C_{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)d(x_{i},0)(p+1)}\int_{r=0}^{\infty}{\text{e}}^{-\delta
r}{\text{e}}^{(N-1)r}\mathrm{~{}d}r$ $\displaystyle\leq
C_{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)d(x_{i},0)(p+1)},$

where passing from the second inequality to third we have used
$d(x,x_{i})>d(x_{i},0)$ which in turn implies, $d(x,0)\geq
d(x,x_{i})-d(x_{i},0)>0$, and the last step follows from the choice of
$\delta$ as $\delta>Kc(N,\lambda)+(N-1)>N-1.$

Now, we estimate $I_{2}^{i}$ for $\delta<(p+1)(c(N,\lambda)-\varepsilon),$ and
using triangle inequality $d(x_{i},0)\leq d(x_{i},x)+d(x,0),$

$\displaystyle I_{2}^{i}$ $\displaystyle=\int_{d(x,x_{i})\leq
d(x_{i},0)}\frac{w\left(d(\tau_{-x_{i}}(x),0)\right)^{p+1}}{{\text{e}}^{\delta
d(x,0)}}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$ $\displaystyle\leq
C_{\varepsilon}\int_{d(x,x_{i})\leq
d(x_{i},0)}{\text{e}}^{-(c(N,\lambda)-\varepsilon)d(x,x_{i})(p+1)}{\text{e}}^{-\delta
d(x,0))}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$ $\displaystyle\leq
C_{\varepsilon}\int_{d(x,x_{i})\leq d(x_{i},0)}{\text{e}}^{-\delta
d(x,x_{i})}{\text{e}}^{-\delta d(x,0))}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq C_{\varepsilon}\int_{d(x,x_{i})\leq
d(x_{i},0)}{\text{e}}^{-\delta d(x_{i},0)}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq C_{\varepsilon}{\text{e}}^{-\delta
d(x_{i},0)}\int_{d(x,0)\leq d(x_{i},0)}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq C_{\varepsilon}{\text{e}}^{-\delta
d(x_{i},0)}{\text{e}}^{(N-1)d(x_{i},0)}.$

Combining the above estimates results in

$\displaystyle\int_{\mathbb{B}^{N}}(1-a)_{+}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq
C_{\varepsilon}\sum_{i=1}^{2}\left({\text{e}}^{-(c(N,\lambda)-\varepsilon)(p+1)d(x_{i},0)}+{\text{e}}^{-\delta
d(x_{i},0)}{\text{e}}^{(N-1)d(x_{i},0)}\right)$ $\displaystyle\leq
C_{\varepsilon}\sum_{i=1}^{2}\left({\text{e}}^{-(\delta-(N-1))d(x_{i},0)}\right)\text{
since }\delta<(p+1)(c(N,\lambda)-\varepsilon)$ $\displaystyle\leq
C_{\varepsilon}{\text{e}}^{\left(-\frac{Kc(N,\lambda)d(x_{1},x_{2})}{K+R^{\alpha^{\prime}-\alpha}}\right)}$
$\displaystyle\leq
C_{\varepsilon}{\text{e}}^{\left(-\frac{\left(c(N,\lambda)+c_{K}\right)d(x_{1},x_{2})}{1+\frac{1}{K}R^{\alpha^{\prime}-\alpha}}\right)}$
$\displaystyle\leq
C_{\varepsilon}{\text{e}}^{-\left(c(N,\lambda)+c_{K}\right)d(x_{1},x_{2})}$
$\displaystyle\leq
o(1)\left(\int_{\mathbb{B}^{N}}u_{1}^{p}u_{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)$
(4.6) for large $R>0$.

In the above calculations we have used our hypothesis,
$d\left(x_{1},x_{2}\right)\leq(K+R^{\alpha^{\prime}-\alpha})d(x_{i},0)$ for
$i=1,2.$ Further, as $p>1$, we can use Lemma 4.1 to get

$\int_{\mathbb{B}^{N}}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\geq\int_{\mathbb{B}^{N}}\left(u_{1}^{p+1}+u_{2}^{p+1}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}+p\int_{\mathbb{B}^{N}}\left(u_{1}^{p}u_{2}+u_{1}u_{2}^{p}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}.$
(4.7)

Since, $u_{1},u_{2}$ solves (4.1), that implies

$\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}=\int_{\mathbb{B}^{N}}u_{1}^{p}u_{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}=\int_{\mathbb{B}^{N}}u_{2}^{p}u_{1}\mathrm{~{}d}V_{\mathbb{B}^{N}},\;\left\|u_{i}\right\|_{H_{\lambda}}=\int_{\mathbb{B}^{N}}u_{i}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}},\;i=1,2.$
(4.8)

Therefore, using (4.8) in (4.7), we obtain

$\int_{\mathbb{B}^{N}}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\geq
2A+2p\left\langle u_{1},u_{2}\right\rangle_{H_{\lambda}}.$ (4.9)

Combining (4.9) and (4.6) with (4.5) yields

$\displaystyle J\left(u_{1}+u_{2}\right)$
$\displaystyle\leq\frac{2A+2\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(\int_{\mathbb{B}^{N}}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}(1-a)_{+}\left(u_{1}+u_{2}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}}$
$\displaystyle\leq\frac{2A+2\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(2A+\left(2p-o(1)\right)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)^{\frac{2}{p+1}}}$
$\displaystyle=\frac{(2A)^{\frac{p-1}{p+1}}\left(1+\frac{1}{A}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)}{\left(1+\frac{p-o(1)}{A}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)^{\frac{2}{p+1}}}$

From [22], it is known that $J_{\infty}$ is achieved by $w$, which is a
solution of (4.1). This in turn implies $S_{1,\lambda}=A^{\frac{p-1}{p+1}}$
and $S_{2,\lambda}=(2A)^{\frac{p-1}{p+1}}$. Hence,

$\displaystyle J\left(u_{1}+u_{2}\right)$ $\displaystyle\leq
S_{2,\lambda}\frac{1+\frac{1}{A}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(1+\frac{p-o(1)}{A}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)^{\frac{2}{p+1}}}$
$\displaystyle\leq S_{2,\lambda}\frac{1+\frac{1}{A}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(1+\frac{2p-o(1)}{(p+1)A}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)}.$

Now, $\frac{1}{A}<\frac{2p-o(1)}{(p+1)A}$ for $R$ large and hence
$J\left(u_{1}+u_{2}\right)<S_{2,\lambda}.$

  2. Step 2:

We will complete the proof of the lemma in this step. Suppose,

$v_{1}=tu_{1},\quad v_{2}=(1-t)u_{2},\quad\text{ where }\quad t\in[0,1].$

Doing a straight forward computation as in (4.5), it is easy to check that

$J\left(v_{1}+v_{2}\right)\leq\frac{\left(t^{2}+(1-t)^{2}\right)A+2t(1-t)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(\int_{\mathbb{B}^{N}}\left|v_{1}+v_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}(1-a)_{+}\left|v_{1}+v_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}}$

Observing that for $t$ or $1-t$ tending to zero, $v_{1}+v_{2}$ tends to
$u_{2}$ or $u_{1}$, and consequently, $J\left(v_{1}+v_{2}\right)$ converges to
$S_{1,\lambda}$. Therefore, there exists some $\delta^{\prime}>0$ such that
any $\min\\{t,1-t\\}\leq\delta^{\prime}$, then
$J\left(v_{1}+v_{2}\right)<S_{2,\lambda}$, and this $\delta^{\prime}$ is
independent of (large) $R$. Moreover,
$J(\frac{u_{1}+u_{2}}{2})<S_{2,\lambda},$ i.e.,
$J(tu_{1}+(1-t)u_{2})<S_{2,\lambda}$ for $t=\frac{1}{2}.$ Therefore there
exists a neighbourhood, say $N(\frac{1}{2})$ of $t=\frac{1}{2}$ such that
$J(tu_{1}+(1-t)u_{2})<S_{2,\lambda}$ for all $t\in N(\frac{1}{2}).$

From here onward, we will assume $\min\\{t,1-t\\}\geq\delta^{\prime}$ and
$t\in([0,1]\setminus N(\frac{1}{2})).$

Using (4.6), we can get the subsequent inequality

$\displaystyle\int_{\mathbb{B}^{N}}(1-a)_{+}\left|v_{1}+v_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\int_{\mathbb{B}^{N}}(1-a)_{+}\left|tu_{1}+(1-t)u_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
(4.10)
$\displaystyle\leq\int_{\mathbb{B}^{N}}(1-a)_{+}\left|u_{1}+u_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle\leq o(1)\left\langle u_{1},u_{2}\right\rangle_{H_{\lambda}}.$

Using Lemma 4.1 and equations in (4.8), we obtain

$\displaystyle\int_{\mathbb{B}^{N}}\left|v_{1}+v_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}=$
$\displaystyle\int_{\mathbb{B}^{N}}\left|tu_{1}+(1-t)u_{2}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
(4.11) $\displaystyle\geq$ $\displaystyle
t^{p+1}\int_{\mathbb{B}^{N}}u_{1}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}+(1-t)^{p+1}\int_{\mathbb{B}^{N}}u_{2}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle+p\int_{\mathbb{B}^{N}}\left[t^{p}(1-t)u_{1}^{p}u_{2}+t(1-t)^{p}u_{1}u_{2}^{p}\right]\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=$
$\displaystyle\left(t^{p+1}+(1-t)^{p}\right)A+p\left\\{t^{p}(1-t)+t(1-t)^{p}\right\\}\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}.$

We claim that

$\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}=\int_{\mathbb{B}^{N}}u_{1}^{p}u_{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}\rightarrow
0\quad\text{as}\quad R\rightarrow\infty.$
$\displaystyle\int_{\mathbb{B}^{N}}u_{1}^{p}u_{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\int_{\mathbb{B}^{N}}w\left(d(\tau_{-x_{1}}(x),0)\right)^{p}w\left(d(\tau_{-x_{2}}(x),0)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=\int_{\mathbb{B}^{N}}w\left(d(x,0)\right)^{p}w\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=\int_{d(x,0)\leq\frac{d(\bar{x},0)}{2}}w\left(d(x,0)\right)^{p}w\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle+\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}w\left(d(x,0)\right)^{p}w\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\text{where}\quad\bar{x}=\tau_{-x_{1}}(x_{2})$
$\displaystyle:=I_{1}+I_{2}.$

Now,

$\displaystyle I_{1}:$
$\displaystyle=\int_{d(x,0)\leq\frac{d(\bar{x},0)}{2}}w\left(d(x,0)\right)^{p}w\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$
$\text{As}\quad d(\bar{x},x)\geq d(\bar{x},0)-d(x,0)\geq
d(\bar{x},0)-\frac{d(\bar{x},0)}{2}=\frac{d(\bar{x},0)}{2}$

Consequently, $w\left(d(\bar{x},x)\right)\leq
w\left(\frac{d(\bar{x},0)}{2}\right)$. Therefore,

$\displaystyle I_{1}$
$\displaystyle\leq\int_{d(x,0)\leq\frac{d(\bar{x},0)}{2}}w\left(d(x,0)\right)^{p}w\left(\frac{d(\bar{x},0)}{2}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq
C_{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)\frac{d(\bar{x},0)}{2}}\int_{0}^{\frac{d(\bar{x},0)}{2}}{\text{e}}^{-(c(N,\lambda)-\varepsilon)pr}{\text{e}}^{(N-1)r}\mathrm{~{}d}r$
$\displaystyle\leq
C_{\varepsilon}\left[{\text{e}}^{\left(-(c(N,\lambda)-\varepsilon)(p+1)+(N-1)\right)\frac{d(\bar{x},0)}{2}}-{\text{e}}^{-(c(N,\lambda)-\varepsilon)\frac{d(\bar{x},0)}{2}}\right]\,=\,o(1),$

since $c(N,\lambda)(p+1)>N-1$, we can choose $\varepsilon$ such that
$(c(N,\lambda)-\varepsilon)(p+1)>N-1$ and where $o(1)$ tends to $0$ as
$R\rightarrow\infty$.

Next we shall estimate $I_{2}:$ to estimate $I_{2}$ we distinguish into the
following two cases.

Case 1 : $1<p<2.$

$\displaystyle I_{2}:$
$\displaystyle=\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}w\left(d(x,0)\right)^{p}w\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq\left(\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}w^{2}\left(d(x,0)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{p}{2}}\left(\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}w^{\frac{2}{2-p}}\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{2-p}{2}}$
$\displaystyle\leq\left(\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}w^{2}\left(d(x,0)\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{p}{2}}\left(\int_{\mathbb{B}^{N}}w^{\frac{2}{2-p}}\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{2-p}{2}}$
$\displaystyle\leq
C_{\varepsilon}\left(\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}{\text{e}}^{-(c(N,\lambda)-\varepsilon)2r}{\text{e}}^{(N-1)r}\,{\rm
d}r\right)^{\frac{p}{2}}\underbrace{\left(\int_{0}^{\infty}{\text{e}}^{-(c(N,\lambda)-\varepsilon)\frac{2}{2-p}r}{\text{e}}^{(N-1)r}\,{\rm
d}r\right)^{\frac{2-p}{2}}}_{\leq\,C}\,=\,o(1)$

where $o(1)$ tends to $0$ as $R\rightarrow\infty.$ i.e.,
$d(x_{1},x_{2})\rightarrow\infty,$ in the second last step we have used the
fact that $\frac{2}{2-p}>2$ whenever $1<p<2,$ and choose $\varepsilon$ such
that $(c(N,\lambda)-\varepsilon)\frac{2}{2-p}>N-1.$

Case 2 : $p\geq 2$

$\displaystyle I_{2}:$
$\displaystyle=\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}w\left(d(x,0)\right)^{p}w\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq\left(\int_{d(x,0)>\frac{d(\bar{x},0)}{2}}(w\left(d(x,0))\right)^{\frac{2Np}{N+2}}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{N+2}{2N}}$
$\displaystyle\times\underbrace{\left(\int_{\mathbb{B}^{N}}\,w^{2^{\star}}\left(d(x,\tau_{-x_{1}}(x_{2}))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{1}{2^{\star}}}}_{\leq\,C}$
$\displaystyle\leq
C_{\varepsilon}\,\left(\int_{\frac{d(\bar{x},0)}{2}}^{\infty}\,e^{-(c(N,\lambda)-\varepsilon)\,\frac{2Npr}{N+2}+(N-1)r}\,{\rm
d}r\right)^{\frac{N+2}{2N}}=o(1),$

where in the last step we used the fact that
$(c(N,\lambda)-\varepsilon)\,\frac{2Np}{N+2}-(N-1)\,>\,0$ for $p\geq 2.$ Now
by choosing minimum among all finitely many $\varepsilon,$ (in step 1 and step
2) we can make all the above constants uniform.

Therefore, using (4.10) and (LABEL:4j.j) we can evaluate the following:

$\displaystyle J\left(v_{1}+v_{2}\right)$ $\displaystyle\leq$
$\displaystyle\frac{\left(t^{2}+(1-t)^{2}\right)A+2t(1-t)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left[\left(t^{p+1}+(1-t)^{p+1}\right)A+p\left(t^{p}(1-t)+t(1-t)^{p}-o(1)\right)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right]^{\frac{2}{p+1}}}$
$\displaystyle\leq\frac{\left(t^{2}+(1-t)^{2}\right)A}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}A^{\frac{2}{p+1}}}$
$\displaystyle+\frac{2t(1-t)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}A^{\frac{2}{p+1}}}\times\left(1+p\left(\frac{\left(t^{p}(1-t)+t(1-t)^{p}\right)}{\left(t^{p+1}+(1-t)^{p+1}\right)A}-o(1)\right)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)^{\frac{-2}{p+1}}.$

Since
$\frac{A}{A^{\frac{2}{p+1}}}=S_{1,\lambda}=2^{\frac{1-p}{1+p}}S_{2,\lambda}$
we have,

$\displaystyle J\left(v_{1}+v_{2}\right)$ $\displaystyle\leq
S_{1,\lambda}\times\frac{t^{2}+(1-t)^{2}}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}}$
$\displaystyle+\frac{2t(1-t)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}A^{\frac{2}{p+1}}}$
$\displaystyle\times\left(1+p\left(\frac{\left(t^{p}(1-t)+t(1-t)^{p}\right)}{\left(t^{p+1}+(1-t)^{p+1}\right)A}-o(1)\right)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\right)^{\frac{-2}{p+1}}$
$\displaystyle=S_{1,\lambda}\,\frac{t^{2}+(1-t)^{2}}{(t^{p+1}\,+\,(1-t)^{p+1})^{\frac{2}{p+1}}}\,+\,\frac{2t(1-t)\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}}{(t^{p+1}\,+\,(1-t)^{p+1})^{\frac{2}{p+1}}A^{\frac{2}{p+1}}}\times$
$\displaystyle\left(1-\frac{2p}{p+1}\frac{(t^{p}(1-t)+t(1-t)^{p})}{(t^{p+1}+(1-t)^{p+1})A}\,\left\langle
u_{1},u_{2}\right\rangle_{H_{\lambda}}\,+\,o(1)\right)$
$\displaystyle=S_{1,\lambda}\times\frac{t^{2}+(1-t)^{2}}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}}+\,C\,g(t)\,\langle
u_{1},u_{2}\rangle_{H_{\lambda}}\,+\,o(\langle
u_{1}u_{2}\rangle_{H_{\lambda}}),$

where $g(t)$ is a bounded function and $C>0$ is a constant. Thus we obtain

$J\left(v_{1}+v_{2}\right)\leq
S_{1,\lambda}\times\frac{t^{2}+(1-t)^{2}}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}}+\,g(t)\,\langle
u_{1},u_{2}\rangle_{H_{\lambda}}\,+\,o(\langle
u_{1}u_{2}\rangle_{H_{\lambda}}).$

Moreover, we have

$\mbox{max}\,\frac{t^{2}+(1-t)^{2}}{\left(t^{p+1}+(1-t)^{p+1}\right)^{\frac{2}{p+1}}}<2^{\frac{p-1}{p+1}},\quad\forall\
t\in([0,1]\setminus N(\frac{1}{2})).$

Therefore using the above fact and letting $R\rightarrow\infty,$ we conclude

$J(v_{1}+v_{2})\,<\,2^{\frac{p-1}{p+1}}S_{1,\lambda}=S_{2,\lambda}.$

This completes the proof of Lemma 4.2. ∎

##  5\. Proof of Theorem 1.1

In this section, we shall show the existence of a solution by employing the
min-max procedure in the spirit of Bahri-Li [5]. Before going further, let us
define some notations and functional.

$\Sigma:=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right):\|u\|_{\lambda}=1\right\\},\quad\Sigma^{+}:=\left\\{u\in\Sigma:u\geq
0\quad\text{ a.e. in }\quad\mathbb{B}^{N}\right\\}.$

We shall now define the center of mass corresponding to $\Sigma$ on the
hyperbolic space. To this end, it is worth noticing that

$\dfrac{|x|}{d(x,0)}:=\dfrac{|x|}{\log\left(\frac{1+|x|}{1-|x|}\right)}\rightarrow\begin{cases}\frac{1}{2},\quad\quad
x\rightarrow 0\\\ 0,\quad\quad x\rightarrow 1.\end{cases}$

Moreover, the function is bounded in $\mathbb{B}^{N},$ i.e., the unit disc in
$\mathbb{R}^{N}$. Therefore let
$k:=\sup\left\\{\dfrac{|x|}{d(x,0)}:x\in\mathbb{R}^{N}\text{ with
}|x|<1\right\\}<\infty.$ Now we are in a situation to define the following
quantity:
Let $m:\Sigma\rightarrow B(0,1)$ defined as

$m(u):=\frac{\tanh(1/2)}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}}\int_{\mathbb{B}^{N}}\frac{x}{k\,d(x,0)}|u|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Clearly $|m(u)|<1.$ As discussed in Section 2, and using elementary
computation we can also conclude that $d(m(u),0)\leq 1.$ Moreover, $m$ is
continuous from $\Sigma$ to $\mathbb{B}^{N}.$

We can now prove Theorem 1.1 as follows:

###### Proof.

Let $J$ as defined in (4.2), we define

$I_{z}:=\inf_{m(u)=z,\;u\in\Sigma}J(u)\quad\text{for}\quad
z\in\mathbb{B}^{N}\quad\text{such that}\quad d(z,0)<1.$

It is straightforward to note that if inf ${}_{u\in\Sigma}J(u)<S_{1,\lambda},$
then exploiting the standard variational arguments (see [27]), the existence
of a positive solution of $(P_{\lambda})$ can be proven, which in fact, is a
constant multiple of the global minimum of the functional $J$.

Henceforth we are left with the case when $\inf_{u\in\Sigma}J(u)\geq
S_{1,\lambda}$.

If $I_{z}=S_{1,\lambda}$ for some $z\in\mathbb{B}^{N}$ with $d(z,0)<1$ then
there exists some $u\in\Sigma^{+},m(u)=z$, such that $J(u)=S_{1,\lambda}$. The
theorem, under this condition, follows from the concentration compactness
principle established in [27] in the spirit of [5], with minor modifications
needed for the hyperbolic space and again, in this case, the solution obtained
is a constant multiple of the global minimum of $J.$

Thus we are only left with the following possibility

$I_{z}>S_{1,\lambda}\text{ for every}\quad z\in\mathbb{B}^{N}\text{ with
}d(z,0)<1.$

We now claim the existence of some positive constant $R_{1}$ for which the
following holds

$J(w(\tau_{-y}(\bullet)))\leq S_{1,\lambda}\quad\text{for every}\quad
y\in\mathbb{B}^{N}\quad\text{with}\quad d(y,0)\geq R_{1}.$

To prove the above claim, we can observe the following by utilizing (a2):

$\liminf\limits_{d(y,0)\rightarrow\infty}a(\tau_{y}(x))\geq 1\quad\mbox{for
every}\ x\in\mathbb{B}^{N}.$

Further, applying Fatou’s lemma and the above equation yields

$\displaystyle\int_{\mathbb{B}^{N}}|w(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle\leq\int_{\mathbb{B}^{N}}\liminf\limits_{d(y,0)\rightarrow\infty}a(\tau_{y}(x))|w(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
(5.1)
$\displaystyle\leq\liminf\limits_{d(y,0)\rightarrow\infty}\int_{\mathbb{B}^{N}}a(\tau_{y}(x))|w(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Thus we have,

$\displaystyle\lim_{d(y,0)\rightarrow\infty}J(w(\tau_{-y}(\bullet)))$
$\displaystyle=\frac{\|w\|_{\lambda}^{2}}{\liminf\limits_{d(y,0)\rightarrow\infty}\left(\int_{\mathbb{B}^{N}}a(\tau_{y}(x))|w(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}}$
$\displaystyle\leq\frac{\|w\|_{\lambda}^{2}}{\left(\int_{\mathbb{B}^{N}}|w(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{\frac{2}{p+1}}}$
$\displaystyle=J_{\infty}(w)=S_{1,\lambda},$

where we have used (5.1) to get the last inequality. Hence the claim follows.

Fixing any $z\in\mathbb{B}^{N}$ such that $d(z,0)<1$ and
$I_{z}>S_{1,\lambda}$. Consequently,
$S_{1,\lambda}<\frac{1}{2}\left(I_{z}+S_{1,\lambda}\right)<I_{z}$. Thus, using
above the claim, there exists some positive constant $\bar{R}_{1}$, such that

$I_{z}>\frac{1}{2}\left(I_{z}+S_{1,\lambda}\right)>S_{1,\lambda}\geq
J(w(\tau_{-y}(\bullet)))\quad\forall y\quad\text{with}\quad
d(y,0)\geq\bar{R}_{1}.$

Assume $\alpha,\alpha^{\prime}$ and $R>R_{0}$ be the same constants as were in
lemma 4.2. Choose $R_{2}>\max\left\\{R^{\alpha},\bar{R}_{1}\right\\}$be very
large and

$x_{2}=\left(0,0,\ldots,\tanh{\left(\frac{R_{2}-R_{2}^{\frac{\alpha^{\prime}}{\alpha}}}{2}\right)}\right).$

Define a map $h_{0}:\partial B(0,R_{2})\rightarrow\Sigma^{+}$ as

$h_{0}\left(x_{1}\right)=\frac{w\left(\tau_{-x_{1}}(\bullet)\right)}{\left\|w\left(\tau_{-x_{1}}(\bullet)\right)\right\|_{\lambda}},\text{
where }x_{1}\in\partial B(0,R_{2}).$

By the choice of $R_{2}$, we have $R_{2}>\bar{R}_{1}$, so we get

$J\left(h_{0}\left(x_{1}\right)\right)<\frac{1}{2}\left(I_{z}+S_{1,\lambda}\right)<I_{z}\quad\text{for}\quad
x_{1}\in\partial B(0,R_{2}).$

Now we define another map $h^{*}:B(0,R_{2})\rightarrow\Sigma^{+}$ by

$h^{*}\left(tx_{1}+(1-t)x_{2}\right)=\frac{tw\left(\tau_{-x_{1}}(\bullet)\right)+(1-t)w\left(\tau_{-x_{2}}(\bullet)\right)}{\left\|tw\left(\tau_{-x_{1}}(\bullet)\right)+(1-t)w\left(\tau_{-x_{2}}(\bullet)\right)\right\|_{\lambda}},$

for $0\leq t\leq 1,\;x_{1}\in\partial B(0,R_{2})$. It can be observed that
$\left.h^{*}\right|_{\partial B(0,R_{2})}=h_{0}$.
Also, from Lemma 4.2, we obtain

$J\left(h^{*}(y)\right)<S_{2,\lambda}\text{ for all }y\in B(0,R_{2}).$ (5.2)

Next, we define some min-max value. Let

$\Gamma:=\left\\{h:B(0,R_{2})\rightarrow\Sigma^{+}:h\text{ is continuous,
}\left.h\right|_{\partial B(0,R_{2})}=h_{0}\right\\}$

and

$\mu_{0}:=\inf_{h\in\Gamma}\max_{y\in B(0,R_{2})}J(h(y)).$

We now claim that the following holds

$S_{1,\lambda}<I_{z}\leq\mu_{0}<S_{2,\lambda}.$ (5.3)

Using (5.2), it is evident that $\mu_{0}<S_{2,\lambda}$. Furthermore, consider
the map

$m\circ h:B(0,R_{2})\rightarrow\mathbb{B}^{N},\text{ where }h\in\Gamma.$

Then one can see that

$\lim_{R_{2}\rightarrow\infty}m\circ
h_{0}\left(x_{1}\right)=\frac{x_{1}}{kd(x_{1},0)}\tanh{\left(1/2\right)}\text{
uniformly for }x_{1}\in\partial B(0,R_{2}).$

Thus, the above convergence immediately implies that for any
$z\in\mathbb{B}^{N}$ with $d(z,0)<d(m\circ h_{0}\left(x_{1}\right),0)$ for
$x_{1}\in\partial B(0,R_{2}),$ $z\notin m\circ h\left(\partial
B(0,R_{2})\right)$ and hence $\operatorname{deg}\left(m\circ
h,\;B(0,R_{2}),\;z\right)$ is well defined.

Let us define $I:\bar{B}(0,R_{2})\rightarrow B(0,1)$ by

$I(x):=\frac{x}{kR_{2}}\tanh(1/2).$

Moreover, for $R_{2}$ large enough, $m\circ h\left(\partial
B(0,R_{2})\right)=I\left(\partial B(0,R_{2})\right)$. Therefore applying
degree theory, for $R_{2}$ large enough, we have

$\operatorname{deg}\left(m\circ
h,\;B(0,R_{2}),\;z\right)=\operatorname{deg}\left(I,\;B(0,R_{2}),\;z\right)=1.$

In particular, using the solution property of degree, we can guarantee the
existence of some $y\in B(0,R_{2})$, such that, $m\circ h(y)=z$. Hence
$\mu_{0}\geq I_{z}>S_{1,\lambda}$.

From Proposition 3.1, we can conclude that $I_{\lambda,a}$ satisfies PS
condition for $\frac{p-1}{2(p+1)}A<c<\frac{p-1}{p+1}A.$ Equivalently,
$\left.J\right|_{\Sigma+}$ satisfies PS condition for
$S_{1,\lambda}<c<S_{2,\lambda}$, i.e.,
$A^{\frac{p-1}{p+1}}<c<(2A)^{\frac{p-1}{p+1}}$. Therefore, by (5.3),
$\left.J\right|_{\Sigma^{+}}$satisfies PS condition at $\mu_{0}$. As a result,
using deformation lemma, $\mu_{0}$ is a critical value of
$\left.J\right|_{\Sigma+}$ with some corresponding critical point $0\not\equiv
u\geq 0$ i.e., $\left.J^{\prime}\right|_{\Sigma+}(u)=0.$ Hence we obtain a
non-negative solution of $(P_{\lambda})$ by scaling $u$. Furthermore, $u$ is a
positive solution to $(P_{\lambda})$ follows from the maximum principle. ∎

Acknowledgments. D. Ganguly is partially supported by the INSPIRE faculty
fellowship (IFA17-MA98). D. Gupta is supported by the PMRF. D. Ganguly is
grateful to D. Karmakar for useful discussions.

## References

  * [1] S. Adachi, Shinji, A positive solution of a nonhomogeneous elliptic equation in $\mathbb{R}^{N}$ with G-invariant nonlinearity. Comm. Partial Differential Equations, 27 (2002), no. 1-2, 1–22.
  * [2] Adachi. S, Tanaka, K., Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^{p}+f(x)$ in $\mathbb{R}^{N}$. Calc. Var. Partial Differential Equations 11 (2000), no. 1, 63–95.
  * [3] Adachi. S, Tanaka. K., Existence of positive solutions for a class of nonhomogeneous elliptic equations in $\mathbb{R}^{N},$ Nonlinear Anal. 48 (2002), no. 5, Ser. A: Theory Methods, 685–705.
  * [4] A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1–32.
  * [5] A. Bahri, Li. Y. Y, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}^{N}$. Rev. Mat. Iberoamericana. 6 (1990), no. 1-2, 1–15.
  * [6] A. Bahri. A, Lions. P-L., On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire. 14 (1997), no. 3, 365–413.
  * [7] H. Berestycki, P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345.
  * [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 347–375.
  * [9] E. Berchio, A. Ferrero, G. Grillo, Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models. J. Math. Pures Appl, (9) 102 (2014), no. 1, 1–35.
  * [10] M. Bonforte, Matteo, F. Gazzola, G. Grillo, J. L. Vázquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space. Calc. Var. Partial Differential Equations 46 (2013), no. 1-2, 375–401.
  * [11] M. Bhakta, Mousomi, K. Sandeep, Poincaré-Sobolev equations in the hyperbolic space. Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 247–269.
  * [12] Cerami. G, Passaseo. D, Existence and multiplicity results for semilinear elliptic Dirichlet problems in exterior domains. Nonlinear Analysis TMA 24 (1995), 1533–1547.
  * [13] M. Clapp, T. Weth, Multiple solutions of nonlinear scalar field equations. Comm. Partial Differential Equations, 29 (2004), no. 9-10, 1533–1554.
  * [14] W. Y. Ding, W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal, 91 (1986), no. 4, 283–308.
  * [15] W. Y. Ding, W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), no. 4, 283–308.
  * [16] D. Ganguly, Debdip, K. Sandeep, Sign changing solutions of the Brezis-Nirenberg problem in the hyperbolic space. Calc. Var. Partial Differential Equations 50 (2014), no. 1-2, 69–91.
  * [17] D. Ganguly, Debdip, K. Sandeep, Kunnath Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space. Commun. Contemp. Math. 17 (2015), no. 1, 1450019, 13 pp.
  * [18] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J. Funct. Analysis 69 (1986), 397–408
  * [19] Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_{\alpha}.$ Commun. Partial Diff. Eqns . Eqns 14 (1988), 1499–1519.
  * [20] Y. G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131 (1990), 223–253.
  * [21] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II.  Ann. Inst. H. Poincaré Anal. Non Linéaire 1. (1984), no. 4, 223–283.
  * [22] Mancini. G, Sandeep. K, On a semilinear elliptic equation in $\mathbb{B}^{N}.$ Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008), no. 4, 635–671.
  * [23] A. Malchiodi, , Multiple positive solutions of some elliptic equations in $\mathbb{R}^{N},$ Nonlinear Anal. 43 (2001), no. 2, Ser. A: Theory Methods, 159–172.
  * [24] R. Molle, M. Musso, D. Passaseo, Positive solutions for a class of nonlinear elliptic problems in $\mathbb{R}^{N}$. Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), no. 1, 141–166
  * [25] M. Musso, D. Passaseo, Some nonlinear elliptic equations in $\mathbb{R}^{N}$. Nonlinear Anal. 39 (2000), no. 7, Ser. A, Theory Methods, 837–860.
  * [26] Ratcliffe. J .G, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149.(1994) Springer, New York.
  * [27] Struwe. M, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, (1990) Springer-Verlag, Berlin.


# Search for the primordial gravitational waves with Very Long Baseline
Interferometry

Oleg Titov
Geoscience Australia, PO Box 378, Canberra, ACT, Australia
Sébastien Lambert
SYRTE, Observatoire de Paris, PSL Research University, CNRS,
Sorbonne Universités, UPMC Univ. Paris 06, LNE

(February 15, 2016)

###### Abstract

Some models of the expanding Universe predict that the astrometric proper
motion of distant radio sources embedded in space-time are non-zero as the
radial distance from observer to the source grows. Systematic proper motion
effects would produce a predictable quadrupole pattern on the sky that could
be detected using Very Long Baseline Interferometry (VLBI) technique. This
quadrupole pattern can be interpreted either as an anisotropic Hubble
expansion, or as a signature of the primordial gravitational waves in the
early Universe. We present our analysis of a large set of geodetic VLBI data
spanning 1979–2015 to estimate the dipole and quadrupole harmonics in the
expansion of the vector field of the proper motions of quasars in the sky. The
analysis is repeated for different redshift zones.

##  1 Introduction

Very long baseline interferometry (VLBI) measures the differential arrival
times of signals from extragalactic radio sources. It is currently the most
powerful technique for measuring absolute positions of thousands of radio
sources, the orientation and rotation speed of the Earth and ground-based
station coordinates, with an accuracy of about 1 cm (or 0.1 mas). VLBI has
been extensively used for astrometry and geodesy for about 30 years and, since
1998, is operated by the International VLBI Service (IVS, Schuh & Behrend
2012). VLBI allows an astrometric precision of $\sim 40$ $\mu$as (Fey et al.
2015).

A dipole systematic caused by the galactocentric acceleration of the Solar
System Barycentre was detected for the first time in VLBI data by Titov et al.
(2011, 2013). The measured amplitude of the aberration drift is in good
agreement with the value predicted by Galactic models (e.g., Reid et al.
2009). The quadrupole component presents some systematics that have to be
clarified. A major interest of this component is that it can constrain Hubble
Constant anisotropy or the amplitude of primordial gravitational waves
(Kristian & Sachs 1966; Gwinn et al 1997).

##  2 The Galactic Aberration

The Galactic aberration, or secular aberration drift, is a small proper motion
of a few microarc seconds affecting distant bodies induced by the rotation of
the Solar system about the Galactic center, which takes about 250 Myr
(Kovalevsky 2003). This systematic effect appears as a dipolar deformation of
the proper motion field towards the Galactic center ($\alpha=266^{\circ}$,
$\delta=-29^{\circ}$) with a magnitude about 6 $\mu$as/yr, corresponding to a
Solar system acceleration of $3\times 10^{-13}$ km/$s^{2}$ in accordance with
the equation as follows

$\displaystyle\Delta\mu_{\alpha}\cos\delta$ $\displaystyle=$ $\displaystyle-
d_{1}\sin\alpha+d_{2}\cos\alpha,$ (1) $\displaystyle\Delta\mu_{\delta}$
$\displaystyle=$ $\displaystyle-d_{1}\cos\alpha\sin\delta-
d_{2}\sin\alpha\sin\delta+d_{3}\cos\delta,$ (2)

where the $d_{i}$ are the components of the acceleration vector in units of
the proper motion, and which corresponds to the degree 1 spheroidal (or
electric) development of (e.g., Mignard & Morando 1990)

$\vec{\mu}=\sum_{l,m}\left(a_{l,m}^{E}\vec{Y}_{l,m}^{E}+a_{l,m}^{M}\vec{Y}_{l,m}^{M}\right),$
(3)

where $d_{1}=a_{1,1}^{E}$, $d_{2}=a_{1,-1}^{E}$, and $d_{3}=a_{1,0}^{E}$, and
$Y_{l,m}^{E}$ and $Y_{l,m}^{M}$ the vector spherical harmonics of electric and
magnetic types of degree $l$ and order $m$.

##  3 Rotation, Primordial Gravitational Waves, and Anisotropic Expansion of
the Universe

In addition to the aberration distortion, there may also be a small global
rotation which can be described by the toroidal (or magnetic) harmonics of
degree 1:

$\displaystyle\Delta\mu_{\alpha}\cos\delta$ $\displaystyle=$ $\displaystyle
r_{1}\cos\alpha\sin\delta+r_{2}\sin\alpha\sin\delta-r_{3}\cos\delta,$ (4)
$\displaystyle\Delta\mu_{\delta}$ $\displaystyle=$ $\displaystyle-
r_{1}\sin\alpha+r_{2}\cos\alpha,$ (5)

where the $r_{i}$ can be expressed in terms of vector spherical harmonics
coefficients as $r_{1}=a_{1,1}^{M}$, $r_{2}=a_{1,-1}^{M}$, and
$r_{3}=a_{1,0}^{M}$.

Along with the aberration and rotation, more advanced cosmological effects may
be detected using the proper motion of distant quasars. In particular, the
anisotropic expansion of the Universe would result in the degree 2 vector
spherical harmonics of electric type, and the primordial gravitation waves
would be an origin of the degree 2 harmonics of electric and magnetic types.
To investigate a possible quadrupolar anisotropy of the velocity field, let us
give the development of the degree 2 vector spherical harmonics (i.e., $l=2$
in Eq. (3)):

$\displaystyle\Delta\mu_{\alpha}\cos\delta$ $\displaystyle=$
$\displaystyle-(a_{2,2}^{E,\rm Re}\sin 2\alpha-a_{2,2}^{E,\rm Im}\cos
2\alpha)\cos\delta+(a_{2,1}^{E,\rm Re}\sin\alpha-a_{2,1}^{E,\rm
Im}\cos\alpha)\sin\delta$ (6) $\displaystyle+(a_{2,2}^{M,\rm Re}\sin 2\alpha-
a_{2,2}^{M,\rm Im}\cos 2\alpha)\sin\delta\cos\delta+(a_{2,1}^{M,\rm
Re}\sin\alpha-a_{2,1}^{M,\rm Im}\cos\alpha)\cos 2\delta$ $\displaystyle-
a_{2,0}^{M}\sin\delta\cos\delta,$ $\displaystyle\Delta\mu_{\delta}$
$\displaystyle=$ $\displaystyle-(a_{2,2}^{E,\rm Re}\cos 2\alpha+a_{2,2}^{E,\rm
Im}\sin 2\alpha)\sin\delta\cos\delta-(a_{2,1}^{E,\rm
Re}\cos\alpha+a_{2,1}^{E,\rm Im}\sin\alpha)\cos 2\delta$ (7)
$\displaystyle+(a_{2,2}^{M,\rm Re}\cos 2\alpha+a_{2,2}^{M,\rm Im}\sin
2\alpha)\cos\delta-(a_{2,1}^{M,\rm Re}\cos\alpha+a_{2,1}^{M,\rm
Im}\sin\alpha)\sin\delta$ $\displaystyle+a_{2,0}^{E}\sin\delta\cos\delta.$

Figure 1: The proper motion in right ascension (left) and dipole systematics (right) for the radio sources observed in more than 1000 sessions. The green dot indicates the position of the Galactic center. Table 1: Parameters estimated from different subsets of radio sources. | Dipole only | Dipole + rotation | 16 parameters
---|---|---|---
55 radio sources observed in more than 1000 sessions.
Amplitude ($\mu$as/yr) | $5.8\pm 1.5$ | $5.9\pm 1.5$ | $6.3\pm 2.1$
$\alpha$ (∘) | $257\pm 19$ | $277\pm 20$ | $270\pm 21$
$\delta$ (∘) | $-54\pm 14$ | $-49\pm 15$ | $-31\pm 17$
Rotation ($\mu$as/yr) |  | $4.4\pm 1.6$ | $4.6\pm 2.0$
Second harmonics ($\mu$as/yr) |  |  | $5.1\pm 1.8$
All 617 radio sources.
Amplitude ($\mu$as/yr) | $5.9\pm 1.0$ | $6.0\pm 1.0$ | $6.4\pm 2.1$
$\alpha$ (∘) | $273\pm 13$ | $278\pm 14$ | $289\pm 13$
$\delta$ (∘) | $-56\pm 9$ | $-54\pm 9$ | $-41\pm 11$
Rotation ($\mu$as/yr) |  | $2.2\pm 0.8$ | $2.9\pm 1.0$
Second harmonics ($\mu$as/yr) |  |  | $4.4\pm 1.1$
378 ‘distant’ radio sources with $z>0.9$.
Amplitude ($\mu$as/yr) | $7.9\pm 1.4$ | $8.1\pm 1.4$ | $8.3\pm 1.7$
$\alpha$ (∘) | $267\pm 15$ | $277\pm 14$ | $297\pm 19$
$\delta$ (∘) | $-59\pm 9$ | $-54\pm 10$ | $-49\pm 12$
Rotation ($\mu$as/yr) |  | $3.1\pm 1.5$ | $3.7\pm 1.9$
Second harmonics ($\mu$as/yr) |  |  | $4.8\pm 1.6$
256 ‘close’ radio sources with $z<0.9$.
Amplitude ($\mu$as/yr) | $3.7\pm 1.3$ | $3.5\pm 1.4$ | $4.1\pm 1.5$
$\alpha$ (∘) | $285\pm 23$ | $284\pm 31$ | $283\pm 22$
$\delta$ (∘) | $-44\pm 20$ | $-48\pm 22$ | $-17\pm 23$
Rotation ($\mu$as/yr) |  | $3.1\pm 1.1$ | $3.6\pm 1.2$
Second harmonics ($\mu$as/yr) |  |  | $7.8\pm 1.5$

##  4 Observations and results

About 10 million VLBI observations since 1979 were analyzed with the geodetic
VLBI analysis software Calc/Solve to generate astrometric coordinate time
series of about 3800 radio sources. Amplitudes and direction of dipole,
rotation and second order harmonics are displayed in Table 1 (below) for
various subsets of radio sources. The figures show the proper motion in right
ascension and dipole systematics for the most observed sources, as well as the
electric part of the quadrupole systematics for closest sources.

Figure 2: The electric part of the quadrupole systematics for closest sources
($z<0.9$).

Table 1 displays the magnitude of the dipole, rotation and second order
harmonics for different sets of reference radio sources. As the magnitude
estimates of the second order harmonics exceed the 3-$\sigma$ formal errors in
some cases, they vary from one solution to another. More VLBI data needs to be
collected to obtain more reliable results.

##  5 Acknowledgment

The paper is published with the permission of the CEO, Geoscience Australia.

## References

  * [1] Fey, A, et al (2015) AJ 150, 58
  * [2] Gwinn, C et al (1997) ApJ 485, 87
  * [3] Kovalevsky, J (2003) A&A 404, 7439
  * [4] Kristian, J & Sachs R. (1966) ApJ 143, 37
  * [5] Mignard, F., & Morando, B. 1990, in Proc. Journées 1990 Systèmes de Référence Spatio-Temporels, Observatoire de Paris, ed. N. Capitaine & S. Débarbat, 151
  * [6] Reid, M, et al (2009) ApJ 700, 137
  * [7] Schuh, H & Behrend, D (2012) J Geodyn 61, 68
  * [8] Titov, O et al (2011) A&A 529, 91
  * [9] Titov, O & Lambert, S (2013) A&A 559, 95