# On a class of elliptic equations with Critical Perturbations 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 and non-existence of positive solutions for the
following class of nonlinear elliptic problems in the hyperbolic space

$-\Delta_{\mathbb{B}^{N}}u-\lambda u=a(x)u^{p-1}\,+\,\varepsilon
u^{2^{*}-1}\,\;\;\text{in}\;\mathbb{B}^{N},\quad u\in
H^{1}{(\mathbb{B}^{N})},$

where $\mathbb{B}^{N}$ denotes the hyperbolic space,
$2<p<2^{*}:=\frac{2N}{N-2}$, if $N\geqslant 3;2<p<+\infty$, if
$N=2,\;\lambda<\frac{(N-1)^{2}}{4}$, and $0<a\in L^{\infty}(\mathbb{B}^{N}).$
We first prove the existence of a positive radially symmetric ground-state
solution for $a(x)\equiv 1.$ Next, we prove that for $a(x)\geq 1$, there
exists a ground-state solution for $\varepsilon$ small. For proof, we employ
“conformal change of metric” which allows us to transform the original
equation into a singular equation in a ball in $\mathbb{R}^{N}$. Then by
carefully analysing the energy level using blow-up arguments, we prove the
existence of a ground-state solution. Finally, the case $a(x)\leq 1$ is
considered where we first show that there is no ground-state solution, and
prove the existence of a bound-state solution (high energy solution) for
$\varepsilon$ small. We employ variational arguments in the spirit of Bahri-Li
to prove the existence of high energy-bound-state solutions in the hyperbolic
space.

###### 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 solutions for the following
class of critical elliptic problem in the hyperbolic space $\mathbb{B}^{N}$

$-\Delta_{\mathbb{B}^{N}}u-\lambda u=a(x)u^{p-1}+\varepsilon u^{2^{*}-1}\text{
in }\mathbb{B}^{N},\;u>0\text{ in }\mathbb{B}^{N},\;u\in
H^{1}(\mathbb{B}^{N}),$ ($P_{\varepsilon}$)

where $2<p<2^{*}:=\frac{2N}{N-2}$, if $N\geqslant 3;2<p<+\infty$, if
$N=2,\;\lambda<\frac{(N-1)^{2}}{4}$, $\varepsilon$ is a real parameter,
$H^{1}\left(\mathbb{B}^{N}\right)$ 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 $\mathbb{B}^{N}.$ Further, let $\mu$ denotes
the hyperbolic volume measure, and $d(x,0)=\log(\frac{1+|x|}{1-|x|})$ is the
hyperbolic distance of $x$ from $0$. Moreover, we investigate the existence of
solutions for appropriately chosen $\varepsilon$ under the following
hypotheses separately:

$\displaystyle\left(\mathbf{A}_{1}\right):$ $\displaystyle\;\;a(x)\geq
1\;\;\forall x\in\mathbb{B}^{N},\;\;\mu(\\{x:a(x)\not\equiv 1\\})>0,\;\;a\in
L^{\infty}(\mathbb{B}^{N})\text{ and }a(x)\rightarrow 1$ $\displaystyle\text{
as }d(x,0)\rightarrow\infty.$ $\displaystyle\left(\mathbf{A}_{2}\right):$
$\displaystyle\;\;a(x)\in(0,1]\;\;\forall
x\in\mathbb{B}^{N},\;\;\mu(\\{x:a(x)\not\equiv
1\\})>0,\;\;\inf_{x\in\mathbb{B}^{N}}a(x)>0,\text{ and }$
$\displaystyle\;\;a(x)\rightarrow 1\text{ as }d(x,0)\rightarrow\infty.$
$\displaystyle\left(\mathbf{A}_{3}\right):$ $\displaystyle\;\;a(x)\equiv
1\;\;\forall x\in\mathbb{B}^{N}.$

Further, let us prescribe the following assumption on the parameter $\lambda:$

$\lambda\in\begin{cases}\left(-\infty,\frac{2(p+1)}{(p+3)^{2}}\right],&N=2,\\\
\left(-\infty,\frac{(N-1)^{2}}{4}\right),&N\geq 3.\end{cases}$ (1.1)

Here, $\frac{(N-1)^{2}}{4}$ is the bottom of the $L^{2}-$ spectrum of
$-\Delta_{\mathbb{B}^{N}}.$

We recall that the solutions of ($P_{\varepsilon}$) are the critical points of
the corresponding energy functional
$E_{\varepsilon}:H^{1}(\mathbb{B}^{N})\rightarrow\mathbb{R}$ defined as

$E_{\varepsilon}(u)=\frac{1}{2}\int_{\mathbb{B}^{N}}\left(|\nabla_{\mathbb{B}^{N}}u|^{2}-\lambda
u^{2}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}-\frac{1}{p}\int_{\mathbb{B}^{N}}a(x)|u|^{p}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\frac{\varepsilon}{2^{*}}\int_{\mathbb{B}^{N}}|u|^{2^{*}}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$
(1.2)

Then $E_{\varepsilon}$ is a well-defined $C^{1}$ functional on
$H^{1}(\mathbb{B}^{N}).$ We use variational, refined energy estimates and
blow-up arguments to prove the existence of solutions. The intriguing nature
of the problem is related to the fact that the equation ($P_{\varepsilon}$) is
non-compact, so standard variational methods fail. The problem studied in this
article is in continuation to our study on scalar-field type equations on the
hyperbolic space (see [28, 29]) where we only dealt with the purely
subcritical problem, i.e., when $\varepsilon=0,$ the unperturbed problem. In
the subcritical case, the variational problem lacks compactness because of the
hyperbolic translation (see section 2 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 ($P_{\varepsilon}$), with $\varepsilon=0$ and
$a(x)\equiv 1,$ 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. Hence if we define a sequence by varying $\tau_{n}\in
I(\mathbb{B}^{N}),$ then for a Palais-Smale (PS) sequence $u_{n}$,
$u_{n}\circ\tau_{n}$ is also a Palais-Smale (PS) sequence for
($P_{\varepsilon}$) with $\varepsilon=0$ and $a(x)\equiv 1$. In fact, it was
shown in [11, Theorem 3.3] that in the subcritical case, i.e., when
$2<p<\frac{2N}{N-2},$ non-compact PS sequences are made of finitely many
sequences of the form $U\circ\tau_{n}$.

The problem we considered in this article also has a critical nonlinearity for
$\varepsilon>0$. The Palais-Smale decomposition established in [11, Theorem
3.3] reveals that for critical exponent problems in the hyperbolic space, loss
of compactness can happen along two different profiles, one along the
hyperbolic translations and the other along concentration of Aubin-Talenti
bubble (locally). This makes the problem ($P_{\varepsilon}$) very fascinating.
There will be an interplay between subcritical and critical nonlinearity.
Indeed, some concentration phenomena can happen owing to critical
nonlinearity. Although we will only consider $\varepsilon$ small enough so
that the critical nonlinearity can be seen as a perturbation of the
subcritical problem. Before analysing the difficulties and methodology we
adopt to restore compactness, we first discuss the “state of the art” of such
problems when posed in the Euclidean space.

There has been intensive research over the past few decades on
($P_{\varepsilon}$) when $\varepsilon=0$ in the Euclidean space after the
seminal papers by Berestycki-Lions [9, 10], Bahri-Berestycki [5], Bahri-Li
[4], Bahri-Lions [6]. Many authors have contributed to a much deeper
understanding of the problem in the framework of existence and multiplicity,
we name a few, e.g.,[1, 2, 3, 15, 22, 26, 30, 32, 39, 42], and this list is
far from being complete. The fundamental challenge in dealing with such
problems in unbounded domains in $\mathbb{R}^{N}$ is the lack of compactness,
even in the subcritical case, thus preventing the typical variational
approaches from succeeding. As a result, various authors have studied these
equations by deploying various conditions on $a(x)$ and presented new tools
and methodologies to overcome this difficulty. For example, if $a(x)=a(|x|)$,
the compactness of the embedding $H_{r}\left(\mathbb{R}^{N}\right)$, the
subspace of $H^{1}\left(\mathbb{R}^{N}\right)$ consisting of radially
symmetric functions into
$L^{p}\left(\mathbb{R}^{N}\right),\;p\in\left(2,2^{*}\right)$ helps to restore
the usage of standard variational arguments([9, 10]). However, if the symmetry
restriction on $a(x)$ is dropped, the problem becomes more exciting and
complex. In particular, the authors in [4, 6] have established the existence
of positive solutions by considering the asymptotic condition on $a(x)$, i.e.,
$a(x)\rightarrow a_{\infty}$ as $|x|\rightarrow\infty$ and appropriate decay
estimates. They carefully examined the levels of failure of PS condition, then
searched for high energy solutions when ground state solution did not exist
and used delicate variational and topological arguments. The subject of the
multiplicity of solutions has also been investigated in [17, 19, 34, 35, 45]
under different circumstances like some suitable assumption on
$|a(x)-a_{\infty}|$ or some order relation between $a(x)$ and $a_{\infty},$
i.e., $a(x)$ goes to $a_{\infty}$ from above or below or a certain periodicity
assumption on $a$. Furthermore, in [16] a more general equation $-\Delta
u+\alpha(x)u=\beta(x)|u|^{p-1}u$ has been studied in $\mathbb{R}^{N}$ where
$\alpha$ and $\beta$ are positive functions such that
$\lim_{|x|\rightarrow\infty}\alpha(x)=a_{\infty}>0$ and
$\lim_{|x|\rightarrow\infty}\beta(x)=b_{\infty}>0$. They have proven the
existence and non-existence of ground state solutions under a variety of
hypotheses like $\alpha(x)\rightarrow a_{\infty}$ from below and
$\beta(x)\rightarrow b_{\infty}$ from above and vice-versa, $\alpha(x)$ decays
faster or slower than $\beta(x).$

Further, the scalar field equations in $\mathbb{R}^{N}$ involving the critical
exponent provide an even greater mathematical challenge because of the loss of
compactness in two profiles, translation in $\mathbb{R}^{N}$ and the presence
of the critical exponent. The following class of Dirichlet problems in
different domains in $\mathbb{R}^{N}$ with sufficiently smooth boundary
conditions has been the focus of much research over the past several years

$-\Delta u-\eta u=|u|^{2^{*}-2}u\quad\mbox{in}\quad\Omega$

where $\eta$ is a real parameter, and $\Omega$ is a domain in
$\mathbb{R}^{N}.$ Brezis-Nirenberg, in their commendable work [14] have shown
the existence and non-existence of positive solutions in bounded domains for
$\eta>0$. They identified the first critical level below which compactness can
be restored with the help of Aubin-Talenti functions, popularly known as
_bubbles_. On the other hand, for $\eta\leq 0,$ the shape of the domain comes
into play, and it is well known using the Pohozaev identity that solutions
cease to exist for star-shaped domains. Following that, attempts were made to
discover solutions either by altering the domain’s shape ([25, 38, 36, 41]) or
experimenting with the lower order terms ([8, 40]). Also, see [13, 23, 24] and
references therein.

Thereafter the authors in [31] studied a problem involving critical and
subcritical non-linearities (also, see [37]). This motivated us to study the
related problem ($P_{\varepsilon}$) in the hyperbolic space. It makes sense
that subcritical analysis ([28, 29, 33]) for $\varepsilon=0$ cannot be applied
given the loss of compactness in two profiles, one of which is caused by the
presence of the critical exponent. Moreover, because of the hyperbolic
translations, the other profile can be owed to the following limiting problems

$-\Delta_{\mathbb{B}^{N}}u-\lambda u=|u|^{p-2}u\text{ in
}\mathbb{B}^{N},\;\;\\\ u\in H^{1}\left(\mathbb{B}^{N}\right)$ ($P_{\infty}$)

and

$-\Delta_{\mathbb{B}^{N}}u-\lambda
u=|u|^{p-2}u+\varepsilon|u|^{2^{*}-2}u\text{ in }\mathbb{B}^{N},\;\;u\in
H^{1}\left(\mathbb{B}^{N}\right).$ ($P_{\varepsilon,\infty}$)

Since the PS decomposition for the problems of the type
($P_{\varepsilon,\infty}$) is yet to be discovered, we can no longer employ
the standard tools and techniques.

###  1.1. Methodologies and strategy

The Nehari set for a functional $J$ defined on a function space $X$ is defined
as

$\mathbbmss{N}=\\{u\in X,\;J^{\prime}(u)[u]=0\\}.$

It is easy to show that this set is a manifold for a large class of
functionals associated with elliptic problems such as ($P_{\varepsilon}$),
($P_{\varepsilon,\infty}$), ($P_{\infty}$). Additionally, it can also be
proven that the functionals are bounded below on this Nehari manifold. Suppose
$E_{\infty}$ and $\mathcal{N}_{\infty}$ are the functional and Nehari
manifold, respectively, associated with the problem ($P_{\infty}$). Then the
minimization problem (see [33])

$m=\inf_{\mathcal{N}_{\infty}}E_{\infty}$

has a solution, and $m$ is achieved by some $w\in\mathcal{N}_{\infty}$, thus
solving ($P_{\infty}$). To be precise, authors in [33] established that in the
subcritical case, and for $p>2,$ if $N=2$ and $2<p<2^{\star}$ if $N\geq 3,$
the problem ($P_{\infty}$) 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 above discussion and the hypotheses $\left(\mathbf{A}_{1}\right)$ and
$\left(\mathbf{A}_{2}\right)$ indicate that the corresponding limiting
problems will play a vital role in studying ($P_{\varepsilon}$). We shall
recall from [11] that the solutions to the following problem can be attributed
to the loss of compactness due to the critical exponent problem

$-\Delta V=|V|^{2^{*}-2}V,\quad V\in D^{1,2}\left(\mathbb{R}^{N}\right).$
($CP_{\infty}$)

We know that ($CP_{\infty}$) and ($P_{\infty}$) have been thoroughly and
extensively studied in [14, 33], respectively. However, to our knowledge,
$(P_{\varepsilon,\infty})$ still needs to be explored. So we first establish
the existence of its solutions, particularly the ground state solution (refer
Section 3) using the standard variational methods. For this, we define the
functional
$E_{\varepsilon,\infty}:H^{1}\left(\mathbb{B}^{N}\right)\rightarrow\mathbb{R}^{N}$
corresponding to $(P_{\varepsilon,\infty})$ as

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

and $\mathcal{N}_{\varepsilon,\infty}$ denotes the associated Nehari manifold.
We address the following minimization problem

$m_{\varepsilon}:=\inf_{\mathcal{N}_{\varepsilon,\infty}}E_{\varepsilon,\infty},$

and exploit the radial symmetry of ($P_{\varepsilon,\infty}$) , then make use
of the compactness of embedding and finally establish the existence of
solution using the Ekeland Variational principle. The solution thus obtained
by solving such a minimization problem is referred to as a _ground-state
solution_. Then we move on to search for the solutions to our main problem
($P_{\varepsilon}$). Also, as the uniqueness and the decay estimates on the
solutions of the problem at infinity ($P_{\varepsilon,\infty}$) are still
unknown, we fail to use the techniques used in [4, 28, 29] to establish the
solutions of ($P_{\varepsilon}$). However, we are able to recover the
compactness under the hypothesis $\left(\mathbf{A}_{1}\right)$ below a level.
In restoring the compactness, we test the sequence of scaled solutions of
($CP_{\infty}$) on $E_{\varepsilon,\infty}$, but this is not feasible since
there is no scaling in the hyperbolic space. Thus to perform this blow-up
analysis, we conformally transformed the problem ($P_{\varepsilon,\infty}$) to
$\mathbb{R}^{N}$ (refer Section 2). Then we obtained a series of estimates for
the required integrals in $\mathbb{R}^{N}$, which resulted in the restoration
below the level $m_{\varepsilon}$ (Proposition 3.4). Additionally, we want to
point out to the readers that these estimations are valid for $N\geq 4$ and
$\frac{N(N-2)}{4}<\lambda<\frac{(N-1)^{2}}{4}$. Further, note that we obtained
the following two estimates on $m_{\varepsilon}$

$m_{\varepsilon}<\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}\text{for
$\varepsilon$ small}$

where $S$ is the best Sobolev constant that occurs in the Sobolev inequality
in $\mathbb{R}^{N}$, and

$\lim_{\varepsilon\rightarrow 0}m_{\varepsilon}=m.$

Also, in (3.1), we deduce that $m_{\epsilon}\leq m$. Therefore, in a way, we
can say that we retrieved the compactness (for small $\varepsilon$) below the
level where we have managed to avoid all the anticipated bubbles. Finally,
with the help of the solution of ($P_{\varepsilon,\infty}$) as determined in
Theorem 1.1 and the restored compactness, we find the ground state solution of
($P_{\varepsilon}$) (Theorem 1.2) under the hypothesis
$\left(\mathbf{A}_{1}\right)$. After that, we look for positive solutions of
($P_{\varepsilon}$) assuming $\left(\mathbf{A}_{2}\right)$. However, we prove
the non-existence of ground state solution in this case (Proposition 4.1).
Thus we look for high-energy solutions by assuming a decay estimate on $a(x)$
(Theorem 1.3). We call such a solution as a _high energy-bound-state solution_
because it has an energy level above the level of the ground state and is
located in an interval. The extra assumption on $a(x)$ helps regain the
compactness locally (Proposition 4.3) using the problem studied in our
previous work [28], i.e., $\eqref{mainprob}$ when $\varepsilon=0.$ For this,
the crucial step is to define an appropriate barycentric map ([7, 12]) to
prove some auxiliary lemmas for the existence of bound state high-energy
solutions. The usual barycentric map in $\mathbb{R}^{N}$ enjoys a nice
property under the action of translation. However, the highly non-linear
nature of hyperbolic translation makes it difficult to achieve such a
characteristic in our context. We conclude this article by proving the
existence of bound state high-energy solutions by delicately applying energy
estimates, barycentric maps and topological degree arguments.

###  1.2. Main results

Now we shall discuss and state our main results in this article. We shall
prove the existence of solutions under the assumptions
$\left(\mathbf{A}_{1}\right),\left(\mathbf{A}_{2}\right)$ and
$\left(\mathbf{A}_{3}\right).$ First, we start with the simplest case, i.e.,
when $a(x)$ satisfies $\left(\mathbf{A}_{3}\right).$

######  Theorem 1.1.

Let $a(x)$ satisfies $\left(\mathbf{A}_{3}\right),$ i.e., $a(x)\equiv 1$ for
all $x\in\mathbb{B}^{N}.$ Then there exists $\varepsilon_{0}>0$ such that for
any $\varepsilon\in\left(0,\varepsilon_{0}\right)$ the problem
($P_{\varepsilon,\infty}$) has a positive radially symmetric ground-state
solution $w_{\varepsilon}$.

Then, assuming $a(x)\geq 1,$ we prove the following result concerning the
existence of a ground-state solution:

######  Theorem 1.2.

Let $a(x)\in C\left(\mathbb{B}^{N}\right)\cap L^{\infty}(\mathbb{B}^{N})$
satisfies $\left(\mathbf{A}_{1}\right).$ Assume $N\geq 4$ and
$\frac{N(N-2)}{4}<\lambda<\frac{(N-1)^{2}}{4}.$ Then there exists
$\varepsilon^{\prime}>0$ such that the problem ($P_{\varepsilon}$) has a
ground-state solution for every
$\varepsilon\in\left(0,\varepsilon^{\prime}\right)$.

Now an obvious question arises whether a ground-state solution does exist
under the assumption $\left(\mathbf{A}_{2}\right)$ for the problem
($P_{\varepsilon}$). In fact, in Proposition 4.1, we prove the non-existence
of ground-state solutions for ($P_{\varepsilon}$). Furthermore, when $a(x)\leq
1,$ a bound-state solution exists in this case, particularly a high-energy
solution in the spirit of Bahri-Li. We shall borrow the ideas of Bahri-Li in
their seminal paper [4] to establish the existence of a bound-state solution.
However, we shall describe later the many nontrivial difficulties that arise
in the hyperbolic space to achieve solutions compared to the Euclidean case.
In particular, we prove the following theorem:

######  Theorem 1.3.

Let $a(x)\in C\left(\mathbb{B}^{N}\right)$ satisfies
$\left(\mathbf{A}_{2}\right).$ In addition, assume that $a(x)$ also satisfies

$a(x)\geqslant 1-\operatorname{C\,exp}(-\delta\,d(x,0))\quad\forall
x\in\mathbb{B}^{N},$ (1.3)

for some positive constants $C$ and $\delta.$ Then there exists
$\widehat{\varepsilon}>0$ such that for any
$0<\varepsilon<\widehat{\varepsilon}$ the problem ($P_{\varepsilon}$) has at
least one positive solution, that is a high energy-bound-state solution.

The paper is organized as follows: In Section 2, we introduce some of the
notations, geometric definitions, and preliminaries concerning the hyperbolic
space and derive a conformal equivalent problem on the Euclidean ball. Section
3 begins with the proof of Theorem 1.1 and further establishes auxiliary
propositions, which include an upper estimate on $m_{\varepsilon},$ Palais-
Smale decomposition and finally completes the proof of Theorem 1.2. Finally,
Section 4 is devoted to the proof of Theorem 1.3.

##  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
$g_{\mathbb{B}^{N}}:=\sum\limits_{i=1}^{N}\left(\frac{2}{1-|x|^{2}}\right)^{2}\,{\rm
d}x_{i}^{2}$. The Euclidean unit ball $B(0,1):=\\{x\in\mathbb{R}^{N}:|x|<1\\}$
equipped with the Riemannian metric

$\displaystyle{\rm d}s^{2}=\left(\frac{2}{1-|x|^{2}}\right)^{2}\,{\rm d}x^{2}$

constitute the ball model for the hyperbolic $N$-space, where ${\rm d}x$ is
the standard Euclidean metric and
$|x|=\left(\sum_{i=1}^{N}x_{i}^{2}\right)^{1/2}$ is the standard Euclidean
length. 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 $\mathbb{B}^{N}$ if and only if $S$ is a Euclidean sphere in
$\mathbb{R}^{N}$ and contained in $\mathbb{B}^{N}$, probably with a different
centre and different radius, which can be computed. Geodesic balls in
$\mathbb{B}^{N}$ of radius $r$ centred at the origin will be denoted by

$B_{r}(y):=\\{x\in\mathbb{B}^{N}:d(x,y)<r\\}.$

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 to [43].

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 [33])

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

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

is a norm, equivalent to the $H^{1}(\mathbb{B}^{N})$ norm and the
corresponding inner product is given by $\langle u,v\rangle_{H_{\lambda}}.$
Also, throughtout the article, we use $|\cdot|_{p}$ to denote the
$L^{p}(\mathbb{B}^{N})$ norm.

######  Remark 2.2.

It is interesting to note that there exists $c>0$ independent of small
$\varepsilon$ such that

$\|u\|_{\lambda}\geq c\quad\forall u\in\mathcal{N}_{\varepsilon,\infty}.$
(2.3)

Indeed using the Poincaré-Sobolev inequality on the hyperbolic space, we have

$0=\|u\|^{2}_{\lambda}-|u|_{p}^{p}-\varepsilon|u|_{2^{*}}^{2^{*}}\geq\|u\|^{2}_{\lambda}-c_{1}\|u\|^{p}_{\lambda}-c_{1}\varepsilon\|u\|^{2^{*}}_{\lambda},\quad\forall
u\in\mathcal{N}_{\varepsilon,\infty}.$

Conformal change of metric. We want to conformally transform the problem
($P_{\varepsilon,\infty}$) to the Euclidean Space. For that, define
$P_{1,\mathbb{B}^{N}}:=-\Delta_{\mathbb{B}^{N}}+$
$\frac{(N-2)}{4(N-1)}S_{\mathbb{B}^{N}}=-\Delta_{\mathbb{B}^{N}}-\frac{N(N-2)}{4}$
is the first order conformally invariant Laplacian operator where
$S_{\mathbb{B}^{N}}:=-N(N-1)$ is the scalar curvature of ${\mathbb{B}^{N}}$.
Therefore, for a conformal change in metric
$\tilde{g}=e^{2\psi}g_{\mathbb{B}^{N}}$, we have
$P_{1,\tilde{g}}(u)=e^{-\left(\frac{N}{2}+1\right)\psi}P_{1,\mathbb{B}^{N}}\left(e^{\left(\frac{N}{2}-1\right)\psi}u\right)$
for every smooth function $u$. As a consequence of the Poincaré metric being
conformal to the Euclidean metric with
$\psi(x)=\ln\left(\frac{1-|x|^{2}}{2}\right)$ we can transform
($P_{\varepsilon,\infty}$) to the Euclidean space as follows: Let
$H_{0}^{1}\left(\mathrm{B}^{N}\right)$ is the Sobolev space on
$\mathrm{B}^{N}$ characterized by zero traces on the boundary
$\partial\mathrm{B}^{N}$ where $\mathrm{B}^{N}$ is the open Euclidean ball
with centre at the origin and unit radius. Suppose $u$ be a solution to
($P_{\varepsilon,\infty}$). Set $\varphi:=$
$\left(\frac{2}{1-|x|^{2}}\right)^{\frac{N-2}{2}}$, then $v:=\varphi u$ solves

$-\Delta v-b_{\lambda}(x)v=c_{p}(x)v^{p-1}+\varepsilon v^{2^{\star}-1},\quad
v\in H_{0}^{1}\left(\mathrm{B}^{N}\right),$ (2.4)

where $b_{\lambda}(x)=\frac{4\lambda-N(N-2)}{\left(1-|x|^{2}\right)^{2}}$,
$c_{p}(x)=\left(\frac{2}{1-|x|^{2}}\right)^{N-{\left(\frac{N-2}{2}\right)p}}$.
Observe that $b_{\lambda}(x)>0$ in $\mathrm{B}^{N}$ whenever
$\lambda>\frac{N(N-2)}{4}$, and $N-\left(\frac{N-2}{2}\right)p>0$ for
$p<2^{*}$.

##  3\. Existence of a ground-state solution

This section is devoted to the existence of solutions of ($P_{\varepsilon}$)
when the potential $a(x)$ satisfies $\left(\mathbf{A}_{1}\right)$ and
$\left(\mathbf{A}_{3}\right).$ We first begin with the simplest case, when
$a(x)\equiv 1.$ The proof is a straightforward adaption of standard
variational arguments ([31]) in the hyperbolic setting and restoring
compactness for (hyperbolic) radial functions in the subcritical case. Let us
recall a Strauss-type lemma in the hyperbolic space (see [11, Theorem 3.1]):

Let $H^{1}_{r}(\mathbb{B}^{N})$ denotes the subspace,

$H^{1}_{r}(\mathbb{B}^{N}):=\\{u\in H^{1}(\mathbb{B}^{N}):\mbox{ $u$ is
radial}\\}.$

######  Lemma 3.1 ([11]).

The embedding $H^{1}_{r}(\mathbb{B}^{N})\hookrightarrow L^{p}(\mathbb{B}^{N})$
for $2<p<2^{\star}$ is compact.

###### Proof of Theorem 1.1.

First, we obtain that $m_{\varepsilon}\leq m\;\forall\varepsilon>0$ as
follows: let $\tau_{\varepsilon}>0$ be such that
$\tau_{\varepsilon}w\in\mathcal{N}_{\varepsilon,\infty}$, then

$m_{\varepsilon}\leq
E_{\varepsilon,\infty}\left(\tau_{\varepsilon}w\right)\leq
E_{\infty}\left(\tau_{\varepsilon}w\right)\leq E_{\infty}(w)=m,$ (3.1)

where the first and second inequality follows from the definition of
$m_{\varepsilon},E_{\varepsilon,\infty},E_{\infty}$, and the last inequality
follows from $w$ being a unique (upto hyperbolic translations), positive
radial solution of ($P_{\infty}$).
We confine our analysis to the following sets in order to solve the
minimization problem for $m_{\varepsilon}$

$H_{r}^{1}\left(\mathbb{B}^{N}\right),\;\;\mathcal{N}_{r}^{\varepsilon}=\mathcal{N}_{\varepsilon,\infty}\cap
H_{r}^{1}\left(\mathbb{B}^{N}\right).$

Let $\left\\{u_{n}^{\varepsilon}\right\\}_{n}$ in
$\mathcal{N}_{r}^{\varepsilon}$ be a minimizing sequence, i.e.,

$\left\|u_{n}^{\varepsilon}\right\|_{\lambda}^{2}=\left|u_{n}^{\varepsilon}\right|_{p}^{p}+\varepsilon\left|u_{n}^{\varepsilon}\right|_{2^{*}}^{2^{*}},$
(3.2)
$E_{\varepsilon,\infty}\left(u_{n}^{\varepsilon}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\left\|u_{n}^{\varepsilon}\right\|_{\lambda}^{2}+\left(\frac{1}{p}-\frac{1}{2^{*}}\right)\varepsilon\left|u_{n}^{\varepsilon}\right|_{2^{*}}^{2^{*}}=m_{\varepsilon}+o(1).$
(3.3)

Using the inequalities (3.1) and (3.3), we get

$\left\|u_{n}^{\varepsilon}\right\|_{\lambda}^{2}\leq\left(\frac{1}{2}-\frac{1}{p}\right)^{-1}m_{\varepsilon}+o(1)\leq\left(\frac{1}{2}-\frac{1}{p}\right)^{-1}m+o(1).$
(3.4)

Notice that from $\eqref{a.1},\eqref{ub},\eqref{a.3}$ and the Sobolev
embedding theorem, we get the existence of $\varepsilon_{0}>0$ such that, for
all $n\in\mathbb{N}$,

$\left|u_{n}^{\varepsilon}\right|_{p}^{p}\geq\left\|u_{n}^{\varepsilon}\right\|_{\lambda}^{2}-c\;\varepsilon\left\|u_{n}^{\varepsilon}\right\|_{\lambda}^{2^{*}}\geq\text{
constant }>0\quad\forall\varepsilon\in\left(0,\varepsilon_{0}\right).$ (3.5)

Further, Lemma (3.1) implies $H_{r}^{1}\left(\mathbb{B}^{N}\right)$ embeds
compactly in $L^{p}(\mathbb{B}^{N})$. Thus there exists $w_{\varepsilon}\in
H_{r}^{1}(\mathbb{B}^{N})$ such that, up to a subsequence,

$u_{n}^{\varepsilon}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}w_{\varepsilon}\left\\{\begin{array}[]{l}\text{
strongly in }L^{p}\left(\mathbb{B}^{N}\right)\text{ for }2<p<2^{*}\\\ \text{
weakly in }H^{1}\left(\mathbb{B}^{N}\right)\text{ and in
}L^{2^{*}}\left(\mathbb{B}^{N}\right).\end{array}\right.$ (3.6)

Moreover, $w_{\varepsilon}\not\equiv 0$ follows from (3.5). Ekeland’s
variational principle (see section 3 of [27]) allows choosing the minimizing
sequence $\left\\{u_{n}^{\varepsilon}\right\\}_{n}$ in
$\mathcal{N}_{r}^{\varepsilon}$ such that

$E_{\varepsilon,\infty}^{\prime}\left(u_{n}^{\varepsilon}\right)[v]=\lambda_{n}G^{\prime}\left(u_{n}^{\varepsilon}\right)[v]+o(1)\|v\|\quad\forall
v\in H_{r}^{1}\left(\mathbb{B}^{N}\right).$ (3.7)

where, for all $n\in\mathbb{N},\lambda_{n}\in\mathbb{R}$ is the Lagrange
multiplier and $G(u)=E_{\varepsilon,\infty}^{\prime}(u)[u]$. Thus
$G\left(u_{n}^{\varepsilon}\right)=0$ for all $n\in\mathbb{N}$, and (3.7)
implies

$0=G\left(u_{n}^{\varepsilon}\right)=E_{\varepsilon,\infty}^{\prime}\left(u_{n}^{\varepsilon}\right)\left[u_{n}^{\varepsilon}\right]=\lambda_{n}G^{\prime}\left(u_{n}^{\varepsilon}\right)\left[u_{n}^{\varepsilon}\right]+o(1)\left\|u_{n}^{\varepsilon}\right\|_{\lambda}.$

Hence, we get $\lambda_{n}=o(1)$ using
$\left\|u_{n}^{\varepsilon}\right\|_{\lambda}$ is bounded and
$G^{\prime}\left(u_{n}^{\varepsilon}\right)\left[u_{n}^{\varepsilon}\right]\leq
c<0$ on $\mathcal{N}_{r}^{\varepsilon}.$ Choosing $v=w_{\varepsilon}$ in
(3.7), by (3.6), we can deduce
$w_{\varepsilon}\in\mathcal{N}_{\varepsilon,\infty}$.
By utilising once more, we obtain

$m_{\varepsilon}\leq
E_{\varepsilon,\infty}\left(w_{\varepsilon}\right)\leq\liminf_{n\rightarrow\infty}\left[\left(\frac{1}{2}-\frac{1}{2^{*}}\right)\left\|u_{n}^{\varepsilon}\right\|_{\lambda}^{2}-\left(\frac{1}{p}-\frac{1}{2^{*}}\right)\left|u_{n}^{\varepsilon}\right|_{p}^{p}\right]=m_{\varepsilon}.$

This means that the minimizing function we are seeking for is
$w_{\varepsilon}$. Consequently, $w_{\varepsilon}$ solves

$-\Delta_{\mathbb{B}^{N}}u-\lambda
u=|u|^{p-2}u+\varepsilon|u|^{2^{*}-2}u\quad\text{ in }\mathbb{B}^{N}\text{. }$

Furthermore, as a result of the maximum principle, $w_{\varepsilon}$ is
strictly positive. ∎

###  3.1. Blow-up argument

In this section, we shall find a ground-state solution to our aimed problem,
i.e., ($P_{\varepsilon}$) under the assumption $\left(\mathbf{A}_{1}\right).$
We prove the existence of a least energy positive solution $\tilde{u}$ of
($P_{\varepsilon}$), i.e., $\tilde{u}\in\mathcal{N}_{\varepsilon}$ satisfies
$E_{\varepsilon}(\tilde{u})=\inf_{\mathcal{N}_{\varepsilon}}E_{\varepsilon}$
where $\mathcal{N}_{\varepsilon}$ denotes the Nehari manifold corresponding to
($P_{\varepsilon}$). The following auxiliary results are required to prove
Theorem 1.2. We first begin with the blow-up argument, which controls
$m_{\varepsilon}.$

######  Proposition 3.2.

The estimate mentioned below holds:

$m_{\varepsilon}\leq\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}\quad\forall\varepsilon>0$
(3.8)

where $S$ is the best Sobolev constant that occurs in the Sobolev inequality
in $\mathbb{R}^{N}$.

###### Proof.

The energy functional corresponding to (2.4) is given by

$\displaystyle J_{\varepsilon}(v)$
$\displaystyle=\frac{1}{2}\int_{\mathrm{B}^{N}}\left[|\nabla
v|^{2}-\left(\lambda-\frac{N(N-2)}{4}\right)\left(\frac{2}{1-|x|^{2}}\right)^{2}v^{2}\right]\mathrm{d}x$
(3.9)
$\displaystyle-\frac{1}{p}\int_{\mathrm{B}^{N}}\left(\frac{2}{1-|x|^{2}}\right)^{N-\left(\frac{N-2}{2}\right)p}|v|^{p}\mathrm{d}x-\frac{\varepsilon}{2^{*}}\int_{\mathrm{B}^{N}}|v|^{2^{*}}\mathrm{d}x.$
(3.10)

Then for any $u\in H^{1}\left(\mathbb{B}^{N}\right)$ if $\tilde{u}$ is defined
as $\tilde{u}(x)=\left(\frac{2}{1-|x|^{2}}\right)^{\frac{N-2}{2}}u(x)$ then
$E_{\varepsilon}(u)=$ $J_{\varepsilon}(v)$. Moreover, $\left\langle
E_{\varepsilon}^{\prime}(u),v\right\rangle=\left\langle
J_{\varepsilon}^{\prime}(\tilde{u}),\tilde{v}\right\rangle$ where $\tilde{v}$
is defined in the same way as $\tilde{u}$.
Consequently, we obtain

$m_{\varepsilon}:=\inf_{\mathcal{N}_{\varepsilon,\infty}}E_{\varepsilon,\infty}=\inf_{\mathcal{N}_{J}^{\varepsilon}}J_{\varepsilon}$

where $\mathcal{N}_{J}^{\varepsilon}$ denotes the Nehari manifold associated
with the functional $J_{\varepsilon}$.
Now to prove (3.8), we exhibit a sequence $v_{n}$ in
$\mathcal{N}_{J}^{\varepsilon}$ such that $J_{\varepsilon}\left(v_{n}\right)$
converges to
$\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}.$
Before moving further, observe that the value
$\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}$ can be
characterised as

$\displaystyle\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}=\min\left\\{\right.$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\mathrm{d}x-\frac{\varepsilon}{2^{*}}\int_{\mathbb{R}^{N}}|u|^{2^{*}}\mathrm{d}x:u\in\mathcal{D}^{1,2}\left(\mathbb{R}^{N}\right),$
(3.11) $\displaystyle\left.\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\mathrm{d}x=\varepsilon\int_{\mathbb{R}^{N}}|u|^{2^{*}}\mathrm{d}x\right\\}.$

We define the following sequence by scaling the Aubin-Talenti bubble:

$U_{n}(x):=n^{\frac{N-2}{2}}U(nx)=(N(N-2))^{\frac{N-2}{4}}\frac{n^{\frac{N-2}{2}}}{\left(1+|nx|^{2}\right)^{\frac{N-2}{2}}},\quad
x\in\mathbb{R}^{N}$

where $U\in\mathcal{D}^{1,2}\left(\mathbb{R}^{N}\right)$ is a fixed radial
function that realizes the minimum in (3.11).
Let $\eta\in C_{c}^{\infty}\left(\mathrm{B}^{N}\right)$ be a cut-off function
such that $\eta(x)\equiv 1$ in a neighborhood of 0. Then consider $u_{n}\in
D^{1,2}(\mathbb{R}^{N})$ to be a sequence of test functions defined by
$u_{n}(x):=\eta(x)U_{n}(x)$ for $x\in\mathrm{B}^{N}$. Further, define a
sequence of functions $v_{n}:=t_{n}u_{n},n\in\mathbb{N}$, where $t_{n}$ is
such that $v_{n}\in\mathcal{N}_{J}^{\varepsilon}$, that is

$\displaystyle\int_{\mathrm{B}^{N}}\left[|\nabla
u_{n}|^{2}-\left(\lambda-\frac{N(N-2)}{4}\right)\left(\frac{2}{1-|x|^{2}}\right)^{2}u_{n}^{2}\right]\mathrm{d}x$
(3.12)
$\displaystyle=t_{n}^{p-2}\int_{\mathrm{B}^{N}}\left(\frac{2}{1-|x|^{2}}\right)^{N-\left(\frac{N-2}{2}\right)p}|u_{n}|^{p}\mathrm{d}x+\varepsilon
t_{n}^{2^{*}-2}\int_{\mathrm{B}^{N}}|u_{n}|^{2^{*}}\mathrm{d}x.$

Now we further compute each of the above terms separately and let
$n\rightarrow\infty.$

$\displaystyle\int_{\mathrm{B}^{N}}u_{n}^{2^{\star}}\mathrm{d}x$
$\displaystyle=\int_{B(0,\frac{1}{2})}u_{n}^{2^{\star}}\mathrm{d}x+\int_{\mathrm{B}^{N}\setminus
B(0,\frac{1}{2})}u_{n}^{2^{\star}}\mathrm{d}x=\int_{B(0,\frac{1}{2}n)}U^{2^{\star}}~{}\mathrm{d}x+\int_{B(0,\frac{1}{2}n)^{c}}\eta\left(\frac{x}{n}\right)^{2^{\star}}U^{2^{\star}}\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{N}}U^{2^{\star}}\mathrm{d}x+O\left((1/n)^{N}\right).$
(3.13)

$\displaystyle\int_{\mathrm{B}^{N}}\left(\frac{1-|x|^{2}}{2}\right)^{\left(\frac{N-2}{2}\right)p-N}u_{n}^{p}$
$\displaystyle\mathrm{~{}d}x=\left(\int_{B(0,\frac{1}{2})}+\int_{\mathrm{B}^{N}\setminus
B(0,\frac{1}{2})}\right)\left(\frac{2}{1-|x|^{2}}\right)^{N-\left(\frac{N-2}{2}\right)p}u_{n}^{p}\mathrm{~{}d}x$
$\displaystyle=\left(\frac{1}{n}\right)^{N-\left(\frac{N-2}{2}\right)p}\int_{B(0,\frac{1}{2}n)}\left(\frac{2}{1-|\frac{x}{n}|^{2}}\right)^{N-\left(\frac{N-2}{2}\right)p}U^{p}\mathrm{d}x$
$\displaystyle+\left(\frac{1}{n}\right)^{N-\left(\frac{N-2}{2}\right)p}\int_{B(0,\frac{1}{2}n)^{c}}\left(\frac{2}{1-|\frac{x}{n}|^{2}}\right)^{N-\left(\frac{N-2}{2}\right)p}\eta\left(\frac{x}{n}\right)^{p}U^{p}\mathrm{d}x$
$\displaystyle=\left(\frac{1}{n}\right)^{N-\left(\frac{N-2}{2}\right)p}\int_{\mathbb{R}^{N}}U^{p}\mathrm{d}x+O\left(\left(\frac{1}{n}\right)^{\left(\frac{N-2}{2}\right)p}\right).$
(3.14)

Similarly, for the gradient term, we have the following estimate

$\displaystyle\int_{\mathrm{B}^{N}}|\nabla u_{n}|^{2}\mathrm{d}x$
$\displaystyle=\left(\int_{B(0,\frac{1}{2})}+\int_{\mathrm{B}^{N}\setminus
B(0,\frac{1}{2})}\right)|\nabla u_{n}|^{2}\mathrm{d}x$
$\displaystyle=\int_{B(0,\frac{1}{2}n)}|\nabla
U|^{2}~{}\mathrm{d}x+O\left((1/n)^{N-2}\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}|\nabla
U|^{2}\mathrm{~{}d}x+O\left((1/n)^{N-2}\right).$ (3.15)

Further, for the lower-order terms, we can obtain the following as
$n\rightarrow\infty$

$\displaystyle\int_{\mathrm{B}^{N}}\left[\frac{4\lambda-N(N-2)}{\left(1-|x|^{2}\right)^{2}}\right]u_{n}^{2}\mathrm{~{}d}x=\left\\{\begin{array}[]{lc}\left(4\lambda-N(N-2)\right)\left(\frac{1}{n}\right)^{2}\left[~{}\displaystyle\int_{\mathbb{R}^{n}}U^{2}\mathrm{~{}d}x+o(1)\right]&\hbox{
for }~{}N\geq 5,\\\
\left(4\lambda-N(N-2)\right)\left(\frac{1}{n}\right)^{2}\log(n)\left[\omega_{N-1}+o(1)\right]&\hbox{
for }~{}N=4.\end{array}\right.$ (3.16)

Hence, taking $n\rightarrow\infty$ in (3.12), applying the estimates (3.1) -
(3.16) and using the fact that $U$ realizes the minimum in (3.11),
$t_{n}\rightarrow 1$ follows and so

$J_{\varepsilon}\left(v_{n}\right)-\left(\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
U|^{2}\mathrm{~{}d}x-\frac{\varepsilon}{2^{*}}\int_{\mathbb{R}^{N}}U^{2^{\star}}\mathrm{d}x\right)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0.$

Hence (3.8) follows since

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
U|^{2}\mathrm{~{}d}x-\frac{\varepsilon}{2^{*}}\int_{\mathbb{R}^{N}}U^{2^{\star}}\mathrm{d}x=\frac{1}{N}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}S^{N/2}.$

∎

In fact, it can be deduced that:

$m_{\varepsilon}<\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}\text{for
$\varepsilon$ small. }$ (3.17)

For this, observe that by conformally transforming the sequence found in
Proposition (3.2) to $\mathbb{B}^{N}$, we can exhibit a sequence
$\left\\{\hat{u}_{n}\right\\}$ radial functions in
$\mathcal{N}_{\varepsilon,\infty}$ that converges weakly to 0 in
$L^{2^{*}}(\mathbb{B}^{N})$, and such that
$E_{\varepsilon,\infty}\left(\hat{u}_{n}\right)\rightarrow\frac{1}{N}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}S^{N/2}$.
On the other hand, we have already shown in the proof of Theorem (1.1) that,
up to a subsequence, any minimising sequence of radial functions weakly
converges to a nonzero minimising function of $E_{\varepsilon,\infty}$ on
$\mathcal{N}_{\varepsilon,\infty}$ for small $\varepsilon$. Therefore, (3.17)
follows.
The following lemma provides another, more accurate estimate of
$m_{\varepsilon}$ for small $\varepsilon$ and examines its asymptotic
behaviour.

######  Lemma 3.3.

The relation $m_{\varepsilon}\leq m$ is true for any $\varepsilon>0$.
$m_{\varepsilon}\leq m$. Moreover, the following holds

$\lim_{\varepsilon\rightarrow 0}m_{\varepsilon}=m.$

###### Proof.

We have already established the inequality $m_{\varepsilon}\leq m$ in (3.1).
Now, for $\varepsilon\in\left(0,\varepsilon_{0}\right)$ assume
$w_{\varepsilon}$ be the minimizing function found in Theorem 1.1 and
$t_{\varepsilon}>0$ be such that
$t_{\varepsilon}w_{\varepsilon}\in\mathcal{N}_{\infty}$, i.e.,

$t_{\varepsilon}=\left(\frac{\left\|w_{\varepsilon}\right\|_{\lambda}^{2}}{\left|w_{\varepsilon}\right|_{p}^{p}}\right)^{\frac{1}{p-2}}\text{.
}$ (3.18)

Besides, we have the following expression for
$E_{\varepsilon,\infty}\left(w_{\varepsilon}\right)$

$E_{\varepsilon,\infty}\left(w_{\varepsilon}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\left\|w_{\varepsilon}\right\|_{\lambda}^{2}+\left(\frac{1}{p}-\frac{1}{2^{*}}\right)\varepsilon\left|w_{\varepsilon}\right|_{2^{*}}^{2^{*}}=m_{\varepsilon}\leq
m$

implying $\left\|w_{\varepsilon}\right\|_{\lambda}$ is bounded, uniformly with
respect to $\varepsilon\in\left(0,\varepsilon_{0}\right)$. Moreover,
$\left|w_{\varepsilon}\right|_{p}^{p}\geq c>0$ holds using (3.5).
Consequently, (3.18) suggests that $t_{\varepsilon}$ is bounded and

$\displaystyle m$ $\displaystyle\leq
E_{\infty}\left(t_{\varepsilon}w_{\varepsilon}\right)=E_{\varepsilon,\infty}\left(t_{\varepsilon}w_{\varepsilon}\right)+\frac{\varepsilon}{2^{*}}\int_{\mathbb{B}^{N}}\left(t_{\varepsilon}w_{\varepsilon}\right)^{2^{*}}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle\leq
E_{\varepsilon,\infty}\left(w_{\varepsilon}\right)+\frac{\varepsilon}{2^{*}}\int_{\mathbb{B}^{N}}\left(t_{\varepsilon}w_{\varepsilon}\right)^{2^{*}}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=m_{\varepsilon}+o(1).$

With this, we have completed our proof. ∎

In the subsequent proposition, we analyse the Palais-Smale sequences at a
level $c\left((\mathrm{PS})_{c}\right.$-sequences) that will be beneficial in
establishing the existence of a ground-state solution. We adapt the approach
as described in [31, Proposition 3.2] in our setting.

######  Proposition 3.4.

Let $a(x)\in C\left(\mathbb{B}^{N}\right)\cap L^{\infty}(\mathbb{B}^{N})$
satisfies $\left(\mathbf{A}_{1}\right).$ Let $\varepsilon>0$ and
$\left\\{u_{n}\right\\}_{n}$ be a $(PS)_{c}$-sequence for $E_{\varepsilon}$
constrained on $\mathcal{N}_{\varepsilon}$. If $c<m_{\varepsilon},$ then
$\left\\{u_{n}\right\\}_{n}$ is relatively compact.

###### Proof.

The proof is divided into several steps.

  1. Step 1:

Let us commence by noticing that the sequence
$\left\\{\left\|u_{n}\right\|_{\lambda}\right\\}_{n}$ is bounded above since
it is a PS sequence, and is bounded away from 0 as it belongs to
$\mathcal{N}_{\varepsilon}$. Then, using the same approach as in Theorem 1.1,
we can show that $\left\\{u_{n}\right\\}_{n}$ is a $(PS)_{c}$-sequence for the
functional $E_{\varepsilon}$ as well, that is, $\forall v\in
H^{1}(\mathbb{B}^{N})$

$\displaystyle\int_{\mathbb{B}^{N}}\nabla_{\mathbb{B}^{N}}u_{n}\cdot\nabla_{\mathbb{B}^{N}}v\mathrm{~{}d}V_{\mathbb{B}^{N}}-\lambda\int_{\mathbb{B}^{N}}u_{n}v\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}a(x)\left|u_{n}\right|^{p-2}u_{n}v\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle-\varepsilon\int_{\mathbb{B}^{N}}\left|u_{n}\right|^{2^{*}-2}u_{n}v\mathrm{~{}d}V_{\mathbb{B}^{N}}=o(1)\|v\|_{\lambda},$
(3.19)

and $E_{\varepsilon}(u_{n})\rightarrow c$ as $n\rightarrow\infty$.
In the subsequent steps, $\left\\{u_{n}\right\\}_{n}$ will denote the sequence
and its subsequences.
The boundedness of $\left\\{u_{n}\right\\}_{n}$ in $H^{1}(\mathbb{B}^{N})$
implies existence of a function $\bar{u}\in H^{1}(\mathbb{B}^{N})$ such that

$u_{n}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\bar{u}\quad\left\\{\begin{array}[]{l}\text{
weakly in }H^{1}(\mathbb{B}^{N})\text{ and in }L^{2^{*}}(\mathbb{B}^{N})\\\
\text{ strongly in }L_{\mathrm{loc}}^{p}(\mathbb{B}^{N})\text{ and in
}L_{\mathrm{loc}}^{2}(\mathbb{B}^{N})\\\ \text{ a.e. in
}\mathbb{B}^{N}.\end{array}\right.$ (3.20)

Then using (3.20) and (3.19), we get $\bar{u}$ is a weak solution of
($P_{\varepsilon}$), hence

$\|\bar{u}\|_{\lambda}=|a^{1/p}\bar{u}|_{p}^{p}+\varepsilon|\bar{u}|_{2^{*}}^{2^{*}}.$
(3.21)

  2. Step 2:

We intend to prove that $u_{n}\rightarrow\bar{u}$ in $H^{1}(\mathbb{B}^{N})$.
We will argue by contradiction that if $u_{n}\nrightarrow\bar{u}$ in
$H^{1}(\mathbb{B}^{N})$, then the sequence $v_{n}:=u_{n}-\bar{u}$ verifies
$\left\|v_{n}\right\|_{\lambda}\geq\hat{c}>0,\forall n\in\mathbb{N}$.
Utilising (3.20) and the Brezis-Lieb Lemma,

$E_{\varepsilon}\left(u_{n}\right)=E_{\varepsilon}(\bar{u})+E_{\varepsilon}\left(v_{n}\right)+o(1).$
(3.22)

Additionally, because $\bar{u}$ is a solution of
($P_{\varepsilon}$)$,\left\\{v_{n}\right\\}_{n}$ also becomes a (PS)-sequence
for $E_{\varepsilon}$. We now claim the following

$\left|v_{n}\right|_{2^{*}}^{2^{*}}\geq\tilde{c}>0.$ (3.23)

If not, $u_{n}\rightarrow\bar{u}$ in $L^{2^{*}}(\mathbb{B}^{N})$ and via
interpolation in $L^{p}(\mathbb{B}^{N})$, since $\left\\{u_{n}\right\\}_{n}$
is bounded in $L^{2}(\mathbb{B}^{N})$. Further, by
$\left\|u_{n}\right\|_{\lambda}=\left|a^{1/p}u_{n}\right|_{p}^{p}+\varepsilon\left|u_{n}\right|_{2^{*}}^{2^{*}}$
and (3.21), we have

$\lim_{n\rightarrow\infty}\left\|u_{n}\right\|_{\lambda}=|a^{1/p}\bar{u}|_{p}^{p}+\varepsilon|\bar{u}|_{2^{*}}^{2^{*}}=\|\bar{u}\|_{\lambda}$

implying $u_{n}\rightarrow\bar{u}$ in $H^{1}(\mathbb{B}^{N})$, leading to a
contradiction.

  3. Step 3:

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. The fact
that $v_{n}\in L^{2^{*}}\left(\mathbb{B}^{N}\right)$ allows us to define

$d_{n}=\sup_{y\in\mathbb{B}^{N}}|v_{n}|_{L^{2^{*}}(B_{r}(y))}\quad\forall
n\in\mathbb{N}.$

Using (3.23) and the boundedness of $\left\\{u_{n}\right\\}_{n}$ in
$H^{1}(\mathbb{B}^{N})$, we get

$\displaystyle 0<\tilde{c}\leq\left|v_{n}\right|_{2^{*}}^{2^{*}}$
$\displaystyle=\sum_{y\in\mathbb{B}^{N}}\left|v_{n}\right|_{L^{2^{*}}\left(B_{r}(y)\right)}^{2^{*}}$
(3.24) $\displaystyle\leq
d_{n}^{2^{*}-2}\sum_{y\in\mathbb{B}^{N}}\left|v_{n}\right|_{L^{2^{*}}\left(B_{r}(y)\right)}^{2}\leq
cd_{n}^{2^{*}-2}\sum_{y\in\mathbb{B}^{N}}\left\|v_{n}\right\|_{H^{1}\left(B_{r}(y)\right)}^{2}$
$\displaystyle\leq c^{\prime}d_{n}^{2^{*}-2}.$

Thus $d_{n}\geq\gamma\;\;\forall n\in\mathbb{N}$, where $\gamma>0$.
Consequently, we can find a sequence
$\left\\{z_{n}\right\\}\subset\mathbb{B}^{N}$ such that

$\int_{B_{r}(z_{n})}\left|v_{n}\right|^{2^{*}}\mathrm{~{}d}V_{\mathbb{B}^{N}}\geq\gamma>0.$
(3.25)

Further, define

$w_{n}(x):=v_{n}\left(\tau_{z_{n}}(x)\right)$

where $\tau_{z_{n}}$ is the hyperbolic translation of $\mathbb{B}^{N}$ by
$z_{n}$. Since the sequence $\left\\{w_{n}\right\\}_{n}$ is bounded in
$H^{1}(\mathbb{B}^{N})$, there exists some $\bar{w}\in
H^{1}\left(\mathbb{B}^{N}\right)$ such that

$w_{n}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\bar{w}\quad\left\\{\begin{array}[]{l}\text{
weakly in }H^{1}\left(\mathbb{B}^{N}\right)\text{ and in
}L^{2^{*}}\left(\mathbb{B}^{N}\right)\\\ \text{ strongly in
}L_{\text{loc}}^{p}\left(\mathbb{B}^{N}\right)\text{ and in
}L_{\mathrm{loc}}^{2}\left(\mathbb{B}^{N}\right)\\\ \text{ a.e. in
}\mathbb{B}^{N}.\end{array}\right.$ (3.26)

Setting $L_{0}=B_{1}(0)$, one of the following two scenarios can happen:

$(a)\int_{L_{0}}\left|w_{n}(x)\right|^{p}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\geq
c>0$
$(b)\int_{L_{0}}\left|w_{n}(x)\right|^{p}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0.$ (3.27)

  4. Step 4:

First of all, let’s assume that (3.27)$(a)$ is true. Then, as
$v_{n}\rightarrow 0$ in $L_{\text{loc}}^{p}\left(\mathbb{B}^{N}\right)$ will
imply $\tau_{z_{n}}(0)\rightarrow\infty\text{ as }n\rightarrow\infty$.
Further, combining the facts that $\left\\{u_{n}\right\\}_{n}$ is a
$(PS)$-sequence and $\bar{u}$ is a solution of ($P_{\varepsilon}$), we can
obtain the following

$\displaystyle\int_{\mathbb{B}^{N}}\nabla_{\mathbb{B}^{N}}w_{n}\cdot\nabla_{\mathbb{B}^{N}}\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}-\lambda\int_{\mathbb{B}^{N}}w_{n}\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}-\int_{\mathbb{B}^{N}}\left|w_{n}\right|^{p-2}w_{n}\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}$
(3.28) $\displaystyle\hskip
256.0748pt-\varepsilon\int_{\mathbb{B}^{N}}\left|w_{n}\right|^{2^{*}-2}w_{n}\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle=\int_{\mathbb{B}^{N}}\left[a\left(\tau_{z_{n}}(\cdot)\right)-1\right]\left|w_{n}\right|^{p-2}w_{n}\phi\mathrm{~{}d}V_{\mathbb{B}^{N}}+o(1)\|\phi\|=o(1)\|\phi\|,\quad\forall\phi\in\mathcal{C}_{0}^{\infty}\left(\mathbb{B}^{N}\right).$

As a result of (3.27)$(a)$, (3.28), (3.26), we are able to conclude that
$\bar{w}$ is a nonzero solution of $\eqref{limprob}$. Then
$\left\\{w_{n}-\bar{w}\right\\}_{n}$ is a $(PS)$-sequence for
$E_{\varepsilon,\infty}$ and
$E_{\varepsilon,\infty}\left(w_{n}-\bar{w}\right)\geq o(1)$. Consequently, by
using the Brezis-Lieb Lemma, we derive

$\displaystyle c$
$\displaystyle=E_{\varepsilon}\left(u_{n}\right)+o(1)=E_{\varepsilon}(\bar{u})+E_{\varepsilon,\infty}(\bar{w})$
$\displaystyle+E_{\varepsilon,\infty}\left(w_{n}-\bar{w}\right)+o(1)\geq
E_{\varepsilon,\infty}(\bar{w})+o(1)\geq m_{\varepsilon}+o(1)$

that contradicts the assumption that $c<m_{\varepsilon}$. Thus we can conclude
that (3.27)$(a)$ can not hold.

  5. Step 5:

Finally, we suppose that (3.27)$(b)$ holds and get a contradiction again.
Notice that in this case, we can also assume the following

$\tilde{d}_{n}=\sup_{y\in\mathbb{B}^{N}}|v_{n}|_{L^{p}(B_{r}(y))}=\sup_{y\in\mathbb{B}^{N}}|w_{n}|_{L^{p}(B_{r}(y))}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0.$ (3.29)

Indeed, if it is false, we can land in the case (3.27)$(a)$ by replacing
$B_{r}(y)$ with a ball $B_{r}(\tilde{y})$ that satisfies
$\left|v_{n}\right|_{L^{p}\left(B_{r}(\tilde{y})\right)}\geq c_{1}>0$. So, it
is possible to assume (3.29). Furthermore, repeating the inequalities in
(3.24) with the $L^{p}$-norm instead of the $L^{2^{*}}$-norm, and
$\tilde{d}_{n}$ instead of $d_{n}$ yield

$\left|v_{n}\right|_{p}=\left|w_{n}\right|_{p}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0.$ (3.30)

Thus implying $\bar{w}=0$. Also, observe that (3.30) gives

$w_{n}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0\;\;\text{ in
}L_{\text{loc}}^{q}(\mathbb{B}^{N})\text{ for $q<p$}.$ (3.31)

Now let’s consider that $\left\\{z_{n}\right\\}_{n}$ is bounded so that in the
subsequent argument we can consider $z_{n}=0,\forall n\in\mathbb{N}$. Further,
fix $R>0$ such that $|a(x)-1|<\eta\;\forall x\in\mathbb{B}^{N}\setminus
B_{R}(0)$, where $\eta$ is a suitable small constant that will be chosen
later.
Consider the functionals $I_{\varepsilon}$:
$\mathcal{D}^{1,2}\left(\mathbb{B}^{N}\right)\rightarrow\mathbb{R}$ and
$J_{\epsilon,\infty}$:
$\mathcal{D}^{1,2}\left(\mathbb{R}^{N}\right)\rightarrow\mathbb{R}$ defined by

$\begin{gathered}I_{\varepsilon}(u)=\frac{1}{2}\int_{\mathbb{B}^{N}}\left(|\nabla_{\mathbb{B}^{N}}u|^{2}-\lambda
u^{2}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}-\frac{\varepsilon}{2^{*}}\int_{\mathbb{B}^{N}}|u|^{2^{*}}\mathrm{~{}d}V_{\mathbb{B}^{N}},\\\
J_{\varepsilon,\infty}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\mathrm{d}x-\frac{\varepsilon}{2^{*}}\int_{\mathbb{R}^{N}}|u|^{2^{*}}\mathrm{d}x.\end{gathered}$

Firstly, observe that the Euler Lagrange equation (say $(AP_{\infty})$)
corresponding to the functional $I_{\varepsilon}$ is invariant under
hyperbolic isometries. Thus for a solution $U$ of $(AP_{\infty})$, if we
define

$u_{n}=U\circ\tau_{n}$ (3.32)

where $\tau_{n}\in$ $I\left(\mathbb{B}^{N}\right)$ with
$\tau_{n}(0)\rightarrow\infty$ and $I\left(\mathbb{B}^{N}\right)$ denotes the
isometry group of $\mathbb{B}^{N}$, then $\\{u_{n}\\}_{n}$ is a PS sequence
converging weakly to zero. Moreover, in the case of critical exponent, another
PS sequence can be exhibited emerging from the concentration phenomenon as
follows ([11]).
Let $V$ be a solution of the Euler-Lagrange equation corresponding to the
functional $J_{\epsilon,\infty}$. Fix $x_{0}\in\mathbb{B}^{N}$ and $\phi\in
C_{c}^{\infty}\left(\mathbb{B}^{N}\right)$ such that $0\leq\phi\leq 1$ and
$\phi\equiv 1$ in a neighborhood of $x_{0}$. Define

$v_{n}(x)=\left(\frac{1-|x|^{2}}{2}\right)^{\frac{N-2}{2}}\phi(x)\varepsilon_{n}^{\frac{2-N}{2}}V\left(\left(x-x_{0}\right)/\varepsilon_{n}\right)$
(3.33)

where $\varepsilon_{n}>0$ and $\varepsilon_{n}\rightarrow 0$, then $v_{n}$ is
also a PS sequence.

Then using (3.19), (3.30), and (3.31), we can get that
$\left\\{w_{n}\right\\}_{n}$ is a (PS)-sequence for $I_{\varepsilon}(u)$. So,
recalling Theorem $3.3$ of [11]: there exist $n_{1},n_{2}\in\mathbb{N}$ and
functions $u_{n}^{j}\in H^{1}(\mathbb{B}^{N}),0\leq j\leq n_{1},v_{n}^{k}\in
H^{1}(\mathbb{B}^{N}),0\leq k\leq n_{2}$ and $w\in
H^{1}\left(\mathbb{B}^{N}\right)$ such that upto a subsequence

$w_{n}=w+\sum_{j=1}^{n_{1}}u_{n}^{j}+\sum_{k=1}^{n_{2}}v_{n}^{k}+o(1)$

where $I^{\prime}_{\varepsilon}(w)=0,u_{n}^{j},v_{n}^{k}$ are $PS$ sequences
of the form (3.32) and (3.33), respectively and $o(1)\rightarrow 0$ in
$H^{1}(\mathbb{B}^{N})$. Moreover

$I_{\varepsilon}(w_{n})=I_{\varepsilon}(w)+\sum_{j=1}^{n_{1}}I_{\varepsilon}\left(U_{j}\right)+\sum_{k=1}^{n_{2}}J_{\varepsilon,\infty}\left(V_{k}\right)+o(1)$
(3.34)

where $U_{j},V_{k}$ are the solutions Euler Lagrange equation associated with
$I_{\varepsilon}$ and $J_{\varepsilon,\infty}$, respectively, corresponding to
$u_{n}^{j}$, and $v_{n}^{k}$.
Moreover, (3.11) implies

$J_{\varepsilon,\infty}\left(V_{k}\right)\geq\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}$
(3.35)

where $S$ is the best Sobolev constant that occurs in the Sobolev inequality
in $\mathbb{R}^{N}$.
Finally, by (3.22),(3.35),(3.30),(3.34) and Proposition 3.2, we have

$\displaystyle E_{\varepsilon}\left(u_{n}\right)$
$\displaystyle=E_{\varepsilon}(\bar{u})+E_{\varepsilon}\left(v_{n}\right)+o(1)$
$\displaystyle\geq
E_{\varepsilon}(\bar{u})+I_{\varepsilon}\left(w_{n}\right)-\frac{1}{p}\int_{\mathbb{B}^{N}\setminus
B_{R}(0)}(a(x)-1)|v_{n}|^{p}\mathrm{~{}d}V_{\mathbb{B}^{N}}+o(1)$
$\displaystyle\geq
E_{\varepsilon}(\bar{u})+\sum_{k=1}^{n_{2}}J_{\varepsilon,\infty}\left(V_{k}\right)-\eta\hat{c}+o(1)$
$\displaystyle\geq\frac{1}{N}S^{N/2}\left(\frac{1}{\varepsilon}\right)^{\frac{N-2}{2}}-\eta\hat{c}+o(1)$
$\displaystyle\geq m_{\varepsilon}-\eta\hat{c}+o(1)$ $\displaystyle>c$

for $n$ large and $\eta$ sufficiently small enough. Therefore, it contradicts
our assumption $E_{\varepsilon}\left(u_{n}\right)\rightarrow c$. To conclude,
let’s take into account the case $d(z_{n},0)\rightarrow\infty$. In this case,
it is simpler to repeat the argument implemented in the case
$\left\\{z_{n}\right\\}_{n}$ bounded.

∎

###### Proof of Theorem 1.2.

The following inequality will be put to use for establishing a solution

$\inf_{\mathcal{N}_{\varepsilon}}E_{\varepsilon}<m_{\varepsilon}.$ (3.36)

In order to prove the above inequality, take into account the minimising
function $w_{\varepsilon}$ provided by the Theorem 1.1, and $t>0$ be such that
$tw_{\varepsilon}\in\mathcal{N}_{\varepsilon}$. Then

$\inf_{\mathcal{N}_{\varepsilon}}E_{\varepsilon}\leq
E_{\varepsilon}\left(tw_{\varepsilon}\right)<E_{\varepsilon,\infty}\left(tw_{\varepsilon}\right)\leq
E_{\varepsilon,\infty}\left(w_{\varepsilon}\right)=m_{\varepsilon}.$

It is implied by (3.36) and the Proposition 3.4 that a minimising function
$\tilde{u}$ exists for the functional $E_{\varepsilon}$ constrained on
$\mathcal{N_{\varepsilon}}$. Further, one can verify that $\tilde{u}$ is a
constant sign function that can be chosen strictly positive by employing the
same argument used in the proof of Theorem 1.1. ∎

##  4\.  bound-state solution: Existence and Non-existence

This section is devoted to the proof of Theorem 1.3 concerning the solutions
of ($P_{\varepsilon}$) for $a(x)\leq 1$. Firstly, we show the non-existence of
a ground-state solution. Then we restore (local) compactness in a range of
functional values and prove Theorem 1.3. In particular, we prove the following
proposition, which led us to a non-existence result:

######  Proposition 4.1.

Let $\varepsilon\in\left[0,\varepsilon_{0}\right)$. Assume $a(x)\leq 1$,
$a(x)\not\equiv 1$, then

$\inf_{\mathcal{N}_{\varepsilon}}E_{\varepsilon}=m_{\varepsilon}.$ (4.1)

Furthermore, there is no solution to the minimization problem (4.1). Moreover,
here $E_{0}:=E,\;\mathcal{N}_{0}:=\mathcal{N},\;m_{0}:=m$.

First, note that when $\varepsilon=0,$ ($P_{\varepsilon}$) becomes

$-\Delta_{\mathbb{B}^{N}}u\,-\,\lambda u\,=\,a(x)u^{p-1}\text{ in
}\mathbb{B}^{N},\;\;u>0\text{ in }\mathbb{B}^{N},\;\;u\in
H^{1}(\mathbb{B}^{N}),$ ($P$)

and the corresponding energy functional is
$E:H^{1}(\mathbb{B}^{N})\rightarrow\mathbb{R}$ defined by

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

and $\mathcal{N}$ denotes the corresponding Nehari Manifold.

###### Proof of Proposition 4.1.

Let $u\in\mathcal{N}_{\varepsilon}$ and $t_{u}\in\mathbb{R}$ be such that
$t_{u}u\in\mathcal{N}_{\varepsilon,\infty}$. Using the assumption that
$a(x)\leq 1$ a.e. in $\mathbb{B}^{N}$, we have

$m_{\varepsilon}\leq E_{\varepsilon,\infty}\left(t_{u}u\right)\leq
E_{\varepsilon}\left(t_{u}u\right)\leq E_{\varepsilon}(u),$

which in turn implies, $\inf_{\mathcal{N}_{\varepsilon}}E_{\varepsilon}\geq
m_{\varepsilon}$. Now we shall exhibit a sequence
$\\{v_{n}\\}_{n}\in\mathcal{N}_{\varepsilon}$ such that
$E_{\varepsilon}\left(v_{n}\right)\rightarrow m_{\varepsilon}$ as
$n\rightarrow\infty.$ This will prove our desired result.

Define $v_{n}(x)=t_{n}w_{\varepsilon}(\tau_{n}(x))$ where $\tau_{n}$ is the
hyperbolic translation with $\tau_{n}(0)=b_{n}$, and $b_{n}\in\mathbb{B}^{N}$
such that $b_{n}\rightarrow\infty$, $w_{\varepsilon}$ is the minimizing
function (solution) established in Theorem 1.1 and $t_{n}>0$ is such that
$v_{n}(x)=t_{n}w_{\varepsilon}(\tau_{n}(x))\in\mathcal{N}_{\varepsilon}$.
Now $v_{n}\in\mathcal{N}_{\varepsilon}$ gives

$\left\|w_{\varepsilon}(\tau_{n}(\cdot))\right\|_{\lambda}^{2}-t_{n}^{p-2}|a^{\frac{1}{p}}w_{\varepsilon}(\tau_{n}(\cdot))|_{p}^{p}-\varepsilon
t_{n}^{2^{*}-2}\left|w_{\varepsilon}(\tau_{n}(\cdot))\right|_{2^{*}}^{2^{*}}=0,$
(4.2)

that is

$t_{n}^{p-2}|a^{\frac{1}{p}}w_{\varepsilon}(\tau_{n}(\cdot))|_{p}^{p}+\varepsilon
t_{n}^{2^{*}-2}\left|w_{\varepsilon}\right|_{2^{*}}^{2^{*}}=\left\|w_{\varepsilon}\right\|_{\lambda}^{2}.$

Consequently, $\left\\{t_{n}\right\\}_{n}$ is bounded and, up to a
subsequence, $t_{n}\rightarrow t$. Letting $n\rightarrow\infty$ in (4.2)
yields

$\left\|w_{\varepsilon}\right\|_{\lambda}^{2}-t^{p-2}\left|w_{\varepsilon}\right|_{p}^{p}-\varepsilon
t^{2^{*}-2}\left|w_{\varepsilon}\right|_{2^{*}}^{2^{*}}=0,$

giving us $tw_{\varepsilon}\in\mathcal{N}_{\varepsilon,\infty}$. But
$w_{\varepsilon}\in\mathcal{N}_{\varepsilon,\infty}$, therefore, $t=1$. Then
it is not difficult to see that $E_{\varepsilon}\left(v_{n}\right)\rightarrow$
$E_{\varepsilon,\infty}\left(w_{\varepsilon}\right)=m_{\varepsilon}$ and we
can conclude $\inf_{\mathcal{N}_{\varepsilon}}E_{\varepsilon}\leq
m_{\varepsilon}$.

Furthermore, we want to show that $m_{\varepsilon}$ is not attained in
$\mathcal{N}_{\varepsilon}$. We argue by contradiction, let
$u\in\mathcal{N}_{\varepsilon}$ be such that
$E_{\varepsilon}(u)=m_{\varepsilon}$. Let $t>0$ be such that
$tu\in\mathcal{N}_{\varepsilon,\infty}$, then

$m_{\varepsilon}=E_{\varepsilon}(u)\geq
E_{\varepsilon}(tu)>E_{\varepsilon,\infty}(tu)\geq m_{\varepsilon},$

which leads to a contradiction, and hence the proof is complete. ∎

###  4.1. Bound-state solutions

Proposition 4.1 tells us that the problem ($P_{\varepsilon}$) for $a(x)\leq 1$
does not admit a ground-state solution. This tempts us to look for high-energy
solutions in the spirit of Bahri-Li. This idea has already been exploited for
(purely) subcritical problems in the hyperbolic space, see e.g., [28, 29]. It
hinges on delicate interaction estimates of two hyperbolic bubbles. Let us
first recall some of the well-known facts from the previous papers.

We recall the following standard compactness result in the subcritical case as
a consequence of ([28, Proposition 3.1]) for $a(x)$ satisfying the conditions
mentioned in Theorem 1.3.

######  Proposition 4.2.

Assume $\left\\{v_{n}\right\\}_{n}$ to be a $(PS)_{c}$-sequence of $E$ for
$c\in(m,2m)$, then $\left\\{v_{n}\right\\}_{n}$ is relatively compact and, up
to a subsequence, converges to a nonzero function $\bar{v}\in
H^{1}(\mathbb{B}^{N})$ such that $E(\bar{v})\in(m,2m)$.

The above proposition led us to the following compactness result corresponding
to the critical perturbation problem.

######  Proposition 4.3.

For every $\delta\in(0,m/2)$ there corresponds $\varepsilon_{\delta}>0$
satisfying the following property:
$\forall\varepsilon\in\left(0,\varepsilon_{\delta}\right),\forall
c\in(m+\delta,2m-\delta)$, if $\left\\{u_{n}\right\\}_{n}$ is a $(PS)_{c}-$
sequence of $E_{\varepsilon}$ constrained on $\mathcal{N}_{\varepsilon}$, then
$u_{n}\rightharpoonup\bar{u}\not\equiv 0$ weakly in $H^{1}(\mathbb{B}^{N})$.
Furthermore, $\bar{u}$ is a critical point of $E_{\varepsilon}$ on
$\mathcal{N}_{\varepsilon}$ and $E_{\varepsilon}(\bar{u})\leq c$.

###### Proof.

First of all, note that every $(\mathrm{PS})_{c}$ \- sequence for the
constrained functional is also a $(\mathrm{PS})_{c}$ \- sequence for the free
functional, and its weak limit is a critical point which can be seen by
following similar arguments as in Proposition 3.4. Also, (PS) sequence is
bounded in $H^{1}(\mathbb{B}^{N})$, so it ensures the existence of a weak
limit in $H^{1}\left(\mathbb{B}^{N}\right)$.
Now to prove the Proposition in hand, we argue by contradiction. We can assume
that there exist $\bar{\delta}\in(0,m/2)$, a sequence
$\left\\{c_{n}\right\\}_{n}$ in $(m+\bar{\delta},2m-\bar{\delta})$, a sequence
$\left\\{\varepsilon_{n}\right\\}_{n}$ in $(0,+\infty)$, with
$\varepsilon_{n}\rightarrow 0$, and, for every $n\in\mathbb{N}$, a sequence
$\left\\{u_{k}^{n}\right\\}_{k}$ in $H^{1}(\mathbb{B}^{N})$ such that

$\begin{gathered}E_{\varepsilon_{n}}\left(u_{k}^{n}\right)\stackrel{{\scriptstyle
k\rightarrow\infty}}{{\longrightarrow}}c_{n},\quad
E_{\varepsilon_{n}}^{\prime}\left(u_{k}^{n}\right)\stackrel{{\scriptstyle
k\rightarrow\infty}}{{\longrightarrow}}0,\\\ u_{k}^{n}\stackrel{{\scriptstyle
k\rightarrow\infty}}{{\rightharpoonup}}0\quad\text{ weakly in
}H^{1}(\mathbb{B}^{N}).\end{gathered}$

Additionally, as $p<2^{\star},$ we can also assume

$u_{k}^{n}\stackrel{{\scriptstyle
k\rightarrow\infty}}{{\longrightarrow}}0\quad\text{ in
}L_{\text{loc}}^{p}\left(\mathbb{B}^{N}\right).$

Up to a subsequence,
$c_{n}\rightarrow\bar{c}\in[m+\bar{\delta},2m-\bar{\delta}]$, and by using a
diagonalization argument, we construct the sequence
$\left\\{v_{n}\right\\}_{n}:=\left\\{u_{k_{n}}^{n}\right\\}_{n}$ that verifies

$E_{\varepsilon_{n}}\left(v_{n}\right)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\bar{c},\quad
E_{\varepsilon_{n}}^{\prime}\left(v_{n}\right)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0,\quad v_{n}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0\quad\text{ in }L_{\text{loc
}}^{p}\left(\mathbb{B}^{N}\right).$ (4.3)

Moreover, the following equality guarantees
$\left\\{\left\|v_{n}\right\|_{\lambda}\right\\}_{n}$ is bounded

$E_{\varepsilon_{n}}\left(v_{n}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\left\|v_{n}\right\|_{\lambda}^{2}+\left(\frac{1}{p}-\frac{1}{2^{*}}\right)\varepsilon_{n}\left|v_{n}\right|_{2^{*}}^{2^{*}}=\bar{c}+o(1).$
(4.4)

Thus we can deduce that

$\begin{gathered}E\left(v_{n}\right)=E_{\varepsilon_{n}}\left(v_{n}\right)+\frac{\varepsilon_{n}}{2^{*}}\int_{\mathbb{B}^{N}}\left|v_{n}\right|^{2^{*}}\mathrm{d}x\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}\bar{c}\\\
\left\|E^{\prime}\left(v_{n}\right)\right\|_{H^{-1}(\mathbb{B}^{N})}\leq\left\|E_{\varepsilon_{n}}^{\prime}\left(v_{n}\right)\right\|_{H^{-1}(\mathbb{B}^{N})}+C\varepsilon_{n}\left\|v_{n}\right\|_{\lambda}^{2^{*}-1}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0,\end{gathered}$

and the above steps imply $\left\\{v_{n}\right\\}_{n}$ is a $(PS)_{\bar{c}}$
\- sequence of $E$, with $\bar{c}\in(m,2m)$. Then, according to Proposition
4.2$,\bar{v}\in H^{1}(\mathbb{B}^{N}),\bar{v}\not\equiv 0$, exists such that
$v_{n}\rightarrow\bar{v}$, in contrast to (4.3).

To conclude, if $\left\\{u_{n}\right\\}_{n}$ is a $(\mathrm{PS})_{c}$ \-
sequence for $E_{\varepsilon}$ constrained on $\mathcal{N}_{\varepsilon}$, and
$u_{n}\rightharpoonup\bar{u}$, then replacing $\varepsilon_{n}$ with
$\varepsilon$, and $\bar{c}$ with $c$ in (4.4), we can obtain
$E_{\varepsilon}(\bar{u})\leq c$. ∎

###  4.2. Energy Estimates

In this subsection, we recall and establish some energy estimates for
interacting hyperbolic bubbles. In addition, these functions are also used to
analyse the sublevels of the functional $E_{\varepsilon}$. Subsequently, we
construct barycenter-type maps to study a few properties of these sublevels.

We first recollect the following energy estimate corresponding to
$\eqref{prob1}$, purely subcritical problem ([28, Lemma 4.2]):

######  Proposition 4.4.

Let $a(x)$ satisfy the conditions of Theorem 1.3, and let $w$ be the unique
radial solution of ($P_{\infty}$). Then, there exists a large number $R_{0}$,
such that for any $R\geq R_{0}$, and for any $x_{1},x_{2}$ satisfying

$d(x_{1},0)\geq R^{\alpha},\;d(x_{2},0)\geq
R^{\alpha},\;R^{\alpha^{\prime}}\leq d\left(x_{1},x_{2}\right)\leq
R^{\alpha^{\prime}-\alpha}\min\left\\{d(x_{1},0),d(x_{2},0)\right\\},$

where $\alpha>\alpha^{\prime}>1$, it holds

$J\left(tu_{1}+(1-t)u_{2}\right)<S_{2,\lambda},$ (4.5)

where $0\leq t\leq 1,$ and $u_{i}=w\left(\tau_{-x_{i}}(\cdot)\right),i=1,2$,
$J,J_{\infty}:H^{1}(\mathbb{B}^{N})\rightarrow\mathbb{R}$ are defined as

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

and the energy levels as

$S_{1,\lambda}:=\inf_{u\in
H^{1}(\mathbb{B}^{N})\backslash\\{0\\}}J_{\infty}(u),\quad
S_{2,\lambda}:=2^{\frac{p-2}{p}}S_{1,\lambda}.$

With the above proposition in mind, let us introduce some notations. Fix
$x_{1}\in\mathbb{B}^{N}$ such that $x_{1}$ satisfies the condition in
Proposition 4.4. Precisely, choose $x_{1}$ such that
$d(x_{1},0)=2R_{0}^{\alpha}$. Also, for all the notations as in the above
proposition fix $R^{\prime}=R_{0}^{\alpha^{\prime}}$.

Now set $\Sigma=\partial B_{R^{{}^{\prime}}}\left(x_{1}\right)$. Further, for
any $R>0$ as in the above Proposition 4.4, define the map
$\psi_{R}:[0,1]\times\Sigma\longrightarrow H^{1}(\mathbb{B}^{N})$ by

$\psi_{R}[s,y](x)=(1-s)w\left(\tau_{-x_{1}}(x)\right)+sw\left(\tau_{-y}(x)\right),$

where $w$ is the ground-state solution of ($P_{\infty}$). Further, let
$t_{R,s,y},t_{R,s,y}^{\prime}>0$ such that
$t_{R,s,y}\psi_{R}[s,y]\in\mathcal{N}_{\varepsilon}$ and
$t_{R,s,y}^{\prime}\psi_{R}[s,y]\in\mathcal{N}$.

We can deduce an energy estimate using Proposition 4.4 in our case here. In
particular, we have the following proposition.

######  Lemma 4.5.

There exists $\bar{R}>0$ and $\mathcal{A}\in(m,2m)$ such that for any
$R>\bar{R}$ and for any $\varepsilon>0$

$\mathcal{A}_{\varepsilon,R}=\max\left\\{E_{\varepsilon}\left(t_{R,s,y}\psi_{R}[s,y]\right):s\in[0,1],y\in\Sigma\right\\}<\mathcal{A}<2m.$

###### Proof.

For the sake of simplicity, we omit $s,y$ and write
$t_{R}=t_{R,s,y},t_{R}^{\prime}=t_{R,s,y}^{\prime}$ and
$\psi_{R}=\psi_{R}[s,y]$. Using $t_{R}^{\prime}\psi_{R}\in$ $\mathcal{N},$ we
have

$\left\|t_{R}^{\prime}\psi_{R}\right\|_{\lambda}^{2}=\left|a^{1/p}t_{R}^{\prime}\psi_{R}\right|_{p}^{p},\quad
t_{R}^{\prime}=\left(\frac{\left\|\psi_{R}\right\|_{\lambda}^{2}}{\left|a^{1/p}\psi_{R}\right|_{p}^{p}}\right)^{1/p-2}$

Now for every $\varepsilon>0$, we have

$\displaystyle E_{\varepsilon}\left(t_{R}\psi_{R}\right)$ $\displaystyle\leq
E\left(t_{R}\psi_{R}\right)\leq E\left(t_{R}^{\prime}\psi_{R}\right)$ (4.6)
$\displaystyle=\frac{1}{2}\left\|t_{R}^{\prime}\psi_{R}\right\|_{\lambda}^{2}-\frac{1}{p}\left|a^{1/p}t_{R}^{\prime}\psi_{R}\right|_{p}^{p}$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p}\right)\left(t_{R}^{\prime}\right)^{2}\left\|\psi_{R}\right\|_{\lambda}^{2}$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p}\right)\left(\frac{\left\|\psi_{R}\right\|_{\lambda}^{2}}{\left|a^{1/p}\psi_{R}\right|_{p}^{2}}\right)^{\frac{p}{p-2}}.$

So to prove the lemma we are left to estimate the above ratio. Note that

$\left(\frac{\left\|\psi_{R}\right\|_{\lambda}^{2}}{\left|a^{1/p}\psi_{R}\right|_{p}^{2}}\right)^{\frac{p}{p-2}}=\left(J(\psi_{R})\right)^{\frac{p}{p-2}},$

Moreover, using the following strict inequality in the last steps of the proof
of Proposition 4.4,

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

where $N(\frac{1}{2})$ denotes a neighbourhood of $\frac{1}{2}$, it can be
easily shown that for $R$ sufficiently large

$\max\left\\{J(\psi_{R}):s\in[0,1],y\in\Sigma\right\\}<S_{2,\lambda}.$ (4.7)

Also, from [33], it is known that $S_{1,\lambda}$ is achieved by $w$, which is
a solution of ($P_{\infty}$). This in turn implies
$S_{1,\lambda}=\left(\|w\|_{\lambda}^{2}\right)^{\frac{p-2}{p}}$ and
$S_{2,\lambda}=(2\|w\|_{\lambda}^{2})^{\frac{p-2}{p}}$.
Further,

$m=E_{\infty}(w)=\left(\frac{1}{2}-\frac{1}{p}\right)\|w\|_{\lambda}^{2}=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)S_{2,\lambda}^{\frac{p}{p-2}}.$

Then by (4.6), (4.7) and using
$\left(\frac{1}{2}-\frac{1}{p}\right)S_{2,\lambda}^{\frac{p}{p-2}}=2m$, the
lemma holds. ∎

######  Corollary 4.6.

There exist $\bar{R},\bar{\varepsilon}>0$ such that for any $R>\bar{R}$ and
for any $\varepsilon\in$ $(0,\bar{\varepsilon})$

$\mathcal{A}_{\varepsilon,R}=\max\left\\{E_{\varepsilon}\left(t_{R,s,y}\psi_{R}[s,y]\right):s\in[0,1],y\in\Sigma\right\\}<2m_{\varepsilon}.$

###### Proof.

It results directly from the Lemmas 3.3 and 4.5. Indeed, using Lemma 4.5, we
have an existence of $\bar{R}>0$ such that $\mathcal{A}_{\varepsilon,R}<2m$
for any $R>\bar{R}$. Also, from (3.1) we have $m_{\varepsilon}\leq m$. Now if
$\mathcal{A}_{\varepsilon,R}>2m_{\varepsilon}\;\forall\varepsilon>0$, then
$|m-m_{\varepsilon}|>m-\frac{\mathcal{A}_{\varepsilon,R}}{2}\;\forall\varepsilon>0$
which contradicts Lemma 3.3. ∎

###  4.3. Barycentric map

We now introduce a barycentric type function as follows: for $u\in
H^{1}(\mathbb{B}^{N})\backslash\\{0\\}$, we set

$\mu(u)(x)=\frac{1}{\left|B_{1}(0)\right|}\int_{B_{1}(x)}|u(y)|\mathrm{~{}d}V_{\mathbb{B}^{N}}(y)\quad
x\in\mathbb{B}^{N},$

and we observe that $\mu(u)$ is bounded and continuous, allowing us to
introduce the function

$\hat{u}(x)=\left[\mu(u)(x)-\frac{1}{2}\max\mu(u)\right]^{+}\quad
x\in\mathbb{B}^{N}.$

The function defined above is continuous and has compact support.
Consequently, we can set
$\beta:H^{1}\left(\mathbb{B}^{N}\right)\backslash\\{0\\}$
$\rightarrow\mathbb{R}^{N}$ as

$\beta(u)=\frac{1}{|\hat{u}|_{1}}\int_{\mathbb{B}^{N}}\frac{x}{1+|x|}\hat{u}(x)\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Further, define

$C_{0}=\inf\\{E(u):u\in\mathcal{N},\beta(u)=0\\},\,C_{0,\varepsilon}=\inf\left\\{E_{\varepsilon}(u):u\in\mathcal{N}_{\varepsilon},\,\beta(u)=0\right\\}.$

######  Lemma 4.7.

The following facts hold:

  * $(a)$

$C_{0}>m$;

  * $(b)$

$\displaystyle\lim_{\varepsilon\rightarrow 0}C_{0,\varepsilon}=C_{0}$.

###### Proof.

We shall first prove the inequality $(a)$. Clearly, Proposition 4.1 gives
$C_{0}\geq m$. If possible, let us assume that $C_{0}=m$ and we will obtain a
contradiction. Let $\left\\{u_{n}\right\\}_{n}$ be a sequence in $\mathcal{N}$
with $\beta\left(u_{n}\right)=0$ such that $E\left(u_{n}\right)\rightarrow m$
and $t_{n}>0$ be such that $t_{n}u_{n}\in\mathcal{N}_{\infty},\forall
n\in\mathbb{N}$. Using the assumption that $a(x)\leq 1$ a.e. in
$\mathbb{B}^{N}$, we deduce that

$m\leq E_{\infty}\left(t_{n}u_{n}\right)\leq E\left(t_{n}u_{n}\right)\leq
E\left(u_{n}\right)=m+o(1),$ (4.8)

which in turn implies that $\left\\{t_{n}u_{n}\right\\}_{n}$ is a minimizing
sequence for $E_{\infty}$ on $\mathcal{N}_{\infty}$. Therefore there exists a
sequence $\left\\{y_{n}\right\\}_{n}$ in $\mathbb{B}^{N}$ such that

$t_{n}u_{n}(x)=w\left(\tau_{-y_{n}}(x)\right)+\phi_{n}(x),\quad\phi_{n}\rightarrow
0\quad\text{ strongly in }H^{1}\left(\mathbb{B}^{N}\right),$

where $w$ denotes the solution of $(P_{\infty})$ and is radially symmetric
with respect to origin. Moreover, this sequence $\left\\{y_{n}\right\\}_{n}$
must be bounded because, if (up to a subsequence)
$\lim_{n\rightarrow\infty}d(y_{n},0)=+\infty$, i.e.,
$\left|y_{n}\right|\rightarrow 1$ as $n\rightarrow\infty$ then

$\lim_{n\rightarrow\infty}\left|\beta\left(t_{n}u_{n}\right)-\frac{y_{n}}{1+|y_{n}|}\right|=0,$
(4.9)

which will give us to a contradiction, as
$\beta\left(t_{n}u_{n}\right)=\beta\left(u_{n}\right)=0$ for all
$n\in\mathbb{N}$. To prove (4.9), consider

$\displaystyle\left|\beta\left(t_{n}u_{n}\right)-\frac{y_{n}}{1+|y_{n}|}\right|$
$\displaystyle=\frac{1}{|\hat{w}|_{1}}\left|\int_{\mathbb{B}^{N}}\left(\frac{\tau_{y_{n}}(z)}{1+|\tau_{y_{n}}(z)|}-\frac{y_{n}}{1+|y_{n}|}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(z)\right|+o(1)$
$\displaystyle=\frac{1}{|\hat{w}|_{1}}\left|\int_{\mathbb{B}^{N}}\left(\frac{\tau_{y_{n}}(z)+|y_{n}|\tau_{y_{n}}(z)-y_{n}-y_{n}|\tau_{y_{n}}(z)|}{(1+|\tau_{y_{n}}(z)|)(1+|y_{n}|)}\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(z)\right|+o(1).$

Moreover, it follows from the expression of the hyperbolic translation (2.1)
that $\tau_{y_{n}}(x)=y_{n}+\circ(1),$ as $|y_{n}|\rightarrow 1,$ for each
fixed $x\in\mathbb{B}^{N}.$ Now using this along with dominated convergence
theorem in (LABEL:trancal), (4.9) follows.

Thus, up to a subsequence, $y_{n}\rightarrow\bar{y}$ for some
$\bar{y}\in\mathbb{B}^{N}$; indeed, $\bar{y}=0$ because $w$ has radial
symmetry with respect to origin. Hence $t_{n}u_{n}\rightarrow w$ strongly in
$H^{1}\left(\mathbb{B}^{N}\right)$. Moreover, $a(x)\not\equiv 1$ and
considering (4.8), we obtain

$m=E_{\infty}(w)<E(w)=\lim_{n\rightarrow\infty}E\left(t_{n}u_{n}\right)\leq\lim_{n\rightarrow\infty}E\left(u_{n}\right)=m,$

hence a contradiction. This proves $(a)$.

We shall now prove ($b$). Let $\varepsilon>0$ be fixed and for every $\eta>0,$
let $u_{\eta}\in\mathcal{N}$ be such that $\beta\left(u_{\eta}\right)=0$ and
$E\left(u_{\eta}\right)\leq C_{0}+\eta$. Also, let $s_{\eta}>0$ be such that
$s_{\eta}u_{\eta}\in\mathcal{N}_{\varepsilon}$. Then

$C_{0,\varepsilon}\leq E_{\varepsilon}\left(s_{\eta}u_{\eta}\right)\leq
E\left(s_{\eta}u_{\eta}\right)\leq E\left(u_{\eta}\right)\leq C_{0}+\eta.$

Since $\eta$ is arbitrary, we get

$C_{0,\varepsilon}\leq C_{0}\quad\forall\varepsilon>0.$ (4.11)

Let $v_{\varepsilon}\in\mathcal{N}_{\varepsilon}$ be such that
$\beta\left(v_{\varepsilon}\right)=0$ and
$E_{\varepsilon}\left(v_{\varepsilon}\right)\leq
C_{0,\varepsilon}+\varepsilon$, and let $t_{\varepsilon}>0$ such that
$t_{\varepsilon}v_{\varepsilon}\in\mathcal{N}$. Thus

$\displaystyle C_{0}$ $\displaystyle\leq
E\left(t_{\varepsilon}v_{\varepsilon}\right)=E_{\varepsilon}\left(t_{\varepsilon}v_{\varepsilon}\right)+\frac{\varepsilon}{2^{*}}\left|t_{\varepsilon}v_{\varepsilon}\right|_{2^{*}}^{2^{*}}$
(4.12) $\displaystyle\leq
E_{\varepsilon}\left(v_{\varepsilon}\right)+\frac{\varepsilon}{2^{*}}\left|t_{\varepsilon}v_{\varepsilon}\right|_{2^{*}}^{2^{*}}$
$\displaystyle\leq
C_{0,\varepsilon}+\varepsilon+\frac{\varepsilon}{2^{*}}t_{\varepsilon}^{2^{*}}\left|v_{\varepsilon}\right|_{2^{*}}^{2^{*}}.$

Furthermore, taking into account (4.11) gives

$E_{\varepsilon}\left(v_{\varepsilon}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\left\|v_{\varepsilon}\right\|_{\lambda}^{2}+\varepsilon\left(\frac{1}{p}-\frac{1}{2^{*}}\right)\left|v_{\varepsilon}\right|_{2^{*}}^{2^{*}}\leq
C_{0,\varepsilon}+\varepsilon\leq C_{0}+\varepsilon$

yielding $\left|v_{\varepsilon}\right|_{2^{*}}^{2^{*}}$ is bounded. Moreover,
taking into account $a(x)\leq 1$ a.e. in $\mathbb{B}^{N}$ and
$v_{\varepsilon}\in\mathcal{N}_{\varepsilon}$, and applying Sobolev
inequalities, we can deduce

$\left\|v_{\varepsilon}\right\|_{\lambda}^{2}[1-S_{N,p}^{-p}\left\|v_{\varepsilon}\right\|_{\lambda}^{p-2}-\varepsilon
S_{N,2^{*}}^{-2^{*}}\left\|v_{\varepsilon}\right\|_{\lambda}^{2^{*}-2}]\leq
0.$

Then we can get a $c>0$ independent of small $\varepsilon$ such that
$\|v_{\varepsilon}\|_{\lambda}\geq c>0$. Thus we obtain
$\left\|v_{\varepsilon}\right\|_{\lambda}\not\rightarrow 0$. Hence, performing
similar calculations as to obtain (3.5), we can get that
$\left|a^{1/p}v_{\varepsilon}\right|_{p}\not\rightarrow 0$. Further,
$t_{\varepsilon}v_{\varepsilon}\in\mathcal{N}$ $\implies
t_{\varepsilon}=\left(\frac{\left\|v_{\varepsilon}\right\|_{\lambda}}{\left|a^{1/p}v_{\varepsilon}\right|_{p}}\right)^{\frac{1}{p-2}}$
giving $\left\\{t_{\varepsilon}\right\\}$ is bounded. Thus, using (4.12), we
get $\liminf_{\varepsilon\rightarrow 0}C_{0,\varepsilon}\geq C_{0}$ that,
along with (4.11), gives ($b$). ∎

######  Corollary 4.8.

There exists $\widetilde{\varepsilon}>0$ such that the inequality
$C_{0,\varepsilon}>\frac{C_{0}+m}{2}$ is true for every
$\varepsilon\in(0,\widetilde{\varepsilon})$

###### Proof.

The statement can be easily proved by appropriately using $(a)$ and $(b)$ of
Lemma 4.7. ∎

######  Lemma 4.9.

Let $\mathcal{A}_{\varepsilon,R}$ be as in Lemma 4.5. Then $\widehat{R}>0$
exists such that $C_{0,\varepsilon}\leq$
$\mathcal{A}_{\varepsilon,R}\;\;\forall R>\widehat{R},\forall\varepsilon>0$.

###### Proof.

Using the similar calculations performed in (LABEL:trancal), we can deduce
that for $y\in\Sigma$,

$\beta(w\left(\tau_{-y}(\cdot)\right))=\frac{y}{1+|\tau_{-x_{1}}(y)|}+o(1)\text{,
}$

where $o(1)\rightarrow 0$ as $R^{\prime}\rightarrow\infty$. Then following the
arguments similar to in [28, Section 5] and defining
$J:\bar{B}_{R^{\prime}}(x_{1})\rightarrow\mathbb{R}^{N}$ by

$K(x):=\frac{x}{1+\tanh{\frac{R^{\prime}}{2}}},$

we have for $R^{\prime}$ large enough, and $\forall y\in\Sigma=\partial
B_{R^{{}^{\prime}}}\left(x_{1}\right)$,
$\beta\circ\psi_{R^{\prime}}[1,y]=K(y)$. Therefore, applying the invariance of
the topological degree by homotopy, and the solution property of degree, we
can ensure the existence of some $(s_{R},y_{R})\in[0,1]\times\Sigma$ such that
$\beta(\psi_{R}[s_{R},y_{R}])=0$. Thus
$\beta\left(t_{R,s_{R},y_{R}}\psi_{R}\left[s_{R},y_{R}\right]\right)=0$. Since
$t_{R,s_{R},y_{R}}\psi_{R}\left[s_{R},y_{R}\right]\in$
$\mathcal{N}_{\varepsilon}$, the assertion follows. ∎

######  Lemma 4.10.

Assume $\tilde{\varepsilon}$ as in Corollary 4.8 and
$\varepsilon\in(0,\tilde{\varepsilon})$. There exists $\tilde{R}>0$ such that
for any $R>\tilde{R}$

$\mathcal{B}_{\varepsilon,R}:=\max\left\\{E_{\varepsilon}\left(t_{R,1,y}\psi_{R}[1,y]\right):y\in\Sigma\right\\}<C_{0,\varepsilon}.$

###### Proof.

For simplicity denote $t_{R}=t_{R,1,y}$ and $\psi_{R}=\psi_{R}[1,y]$. We argue
by contradiction so let us assume that there exist $R_{n}\rightarrow\infty$
and $y_{n}\in\Sigma$ such that
$E_{\varepsilon}\left(t_{R_{n}}\psi_{R_{n}}\right)\geq C_{0,\varepsilon}$ for
every $n\in\mathbb{N}$. Since
$t_{R_{n}}\psi_{R_{n}}\in\mathcal{N}_{\varepsilon}$ we can write

$\displaystyle
E_{\varepsilon}\left(t_{R_{n}}\psi_{R_{n}}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\left\|t_{R_{n}}\psi_{R_{n}}\right\|_{\lambda}^{2}+\varepsilon\left(\frac{1}{p}-\frac{1}{2^{*}}\right)\left|t_{R_{n}}\psi_{R_{n}}\right|_{2^{*}}^{2^{*}}$
(4.13)
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p}\right)t_{R_{n}}^{2}\left\|w\left(\tau_{-y_{n}}(\cdot)\right)\right\|_{\lambda}^{2}+\varepsilon\left(\frac{1}{p}-\frac{1}{2^{*}}\right)t_{R_{n}}^{2^{*}}\left|w\left(\tau_{-y_{n}}(\cdot)\right)\right|_{2^{*}}^{2^{*}}.$

We can notice that in our setting $0<m\leq C_{0,\varepsilon}\leq
E_{\varepsilon}\left(t_{R_{n}}\psi_{R_{n}}\right)\leq\mathcal{A}_{\varepsilon,R}<2m$
and that
$0<c\leq\left\|w\left(\tau_{-y_{n}}(\cdot)\right)\right\|_{\lambda}^{2}\leq
C<\infty,\forall n\in\mathbb{N}$. Thus $0<c_{1}\leq t_{R_{n}}\leq
C_{1}<\infty$ follows from (4.13). So, up to a subsequence, we can assume
$t_{R_{n}}\rightarrow t>0$. Since $R_{n}\rightarrow\infty$, the same estimates
provided in the energy estimates of [28] help us prove
$E_{\varepsilon}\left(t_{R_{n}}\psi_{R_{n}}\right)\rightarrow
E_{\varepsilon,\infty}(tw)$, and we get

$\displaystyle C_{0,\varepsilon}$ $\displaystyle\leq
E_{\varepsilon,\infty}(tw)=E_{\infty}(tw)-\frac{\varepsilon}{2^{*}}|tw|_{2^{*}}^{2^{*}}$
$\displaystyle\leq
E_{\infty}(w)-\frac{\varepsilon}{2^{*}}|tw|_{2^{*}}^{2^{*}}$
$\displaystyle=m-\frac{\varepsilon}{2^{*}}|tw|_{2^{*}}^{2^{*}}<m$

which gives rises to a contradiction taking into account Corollary 4.8 and
Lemma 4.7$(a)$. ∎

Proof of Theorem 1.3 Firstly, we recollect all the values that have been
stated and used in obtaining the previous few results:

$\begin{gathered}\mathcal{A}_{\varepsilon,R}=\max\left\\{E_{\varepsilon}\left(t_{R,s,y}\psi_{R}[s,y]\right):s\in[0,1],y\in\Sigma\right\\},\\\
\mathcal{B}_{\varepsilon,R}=\max\left\\{E_{\varepsilon}\left(t_{R,1,y}\psi_{R}[1,y]\right):y\in\Sigma\right\\},\\\
C_{0,\varepsilon}=\inf\left\\{E_{\varepsilon}(u):u\in\mathcal{N}_{\varepsilon},\beta(u)=0\right\\}.\end{gathered}$

By Corollaries 4.6 and 4.8, and Lemmas 4.5, 4.7, 4.9 and 4.10, the following
inequalities

$\begin{cases}(a)&\mathcal{B}_{\varepsilon,R}<C_{0,\varepsilon}\leq\mathcal{A}_{\varepsilon,R}\\\
(b)&m<\frac{c_{0}+m}{2}<C_{0,\varepsilon}\leq\mathcal{A}_{\varepsilon,R}\leq\mathcal{A}<2m\\\
(c)&\mathcal{A}_{\varepsilon,R}<2m_{\varepsilon}\end{cases}$ (4.14)

hold true for all $R>\max\\{\bar{R},\widetilde{R},\widehat{R}\\}$ and for all
$0<\varepsilon<\min\\{\bar{\varepsilon},\widetilde{\varepsilon}\\}$. Assume
$\delta$ such that $0<$
$\delta<\min\left\\{\frac{m}{2},2m-\mathcal{A},\frac{C_{0}-m}{2}\right\\}$.
Further, we take $\varepsilon_{\delta}$ according to Proposition 4.3.

Finally, to prove Theorem 1.3, we claim that $E_{\varepsilon}$ constrained on
$\mathcal{N}_{\varepsilon}$ has a (PS)-sequence in
$\left[C_{0,\varepsilon},\mathcal{A}_{\varepsilon,R}\right]$ for every
$0<\varepsilon<\widehat{\varepsilon}:=\min\left\\{\varepsilon_{\delta},\bar{\varepsilon},\widetilde{\varepsilon}\right\\}$.
Having done this, the Proposition 4.3 will guarantee the existence of a non-
zero critical point $\bar{u}$ with
$E_{\varepsilon}(\bar{u})\leq\mathcal{A}_{\varepsilon,R}$.

We argue by contradiction, so let us suppose that there is no (PS)-sequence in
the interval $\left[C_{0,\varepsilon},\mathcal{A}_{\varepsilon,R}\right]$.
Then, standard deformation arguments give the existence of $\eta>0$ such that
the sublevel
$E_{\varepsilon}^{C_{0,\varepsilon}-\eta}:=\left\\{u\in\mathcal{N}_{\varepsilon}:E_{\varepsilon}(u)\leq
C_{0,\varepsilon}-\eta\right\\}$ is a deformation retract of the sublevel
$E_{\varepsilon}^{\mathcal{A}_{\varepsilon,R}}:=\left\\{u\in\mathcal{N}_{\varepsilon}:E_{\varepsilon}(u)\leq\mathcal{A}_{\varepsilon,R}\right\\}$,
i.e., there exists a continuous function
$\sigma:E_{\varepsilon}^{\mathcal{A}_{\varepsilon,R}}\rightarrow
E_{\varepsilon}^{C_{0,\varepsilon}-\eta}$ such that

$\sigma(u)=u\quad\text{ for any }u\in
E_{\varepsilon}^{C_{0,\varepsilon}-\eta}.$ (4.15)

Moreover, taking into account the inequality 4.14 $(a)$, $\eta$ can be chosen
so small that

$C_{0,\varepsilon}-\eta>\mathcal{B}_{\varepsilon,R}.$ (4.16)

Let us define the map $\mathcal{H}:[0,1]\times\Sigma\rightarrow\mathbb{R}^{N}$
by

$\mathcal{H}(s,y)=\beta\left(\sigma\left(t_{R,s,y}\psi_{R}[s,y]\right)\right).$
(4.17)

Applying (4.16), (4.17), and the arguments similar to that established in
Lemma 4.9, we can find a point $(\tilde{s},\tilde{y})\in[0,1]\times\Sigma$ for
which

$0=\mathcal{H}(\tilde{s},\tilde{y})=\beta\left(\sigma\left(t_{R,\tilde{s},\tilde{y}}\psi_{R}[\tilde{s},\tilde{y}]\right)\right).$

Then,
$E_{\varepsilon}\left(\sigma\left(t_{R,\tilde{s},\tilde{y}}\psi_{R}[\tilde{s},\tilde{y}]\right)\right)\geq
C_{0,\varepsilon}$ in contrast to
$\sigma\left(t_{R,s,y}\psi_{R}[s,y]\right)\in
E_{\varepsilon}^{C_{0,\varepsilon}-\eta}$ for every
$(s,y)\in[0,1]\times\Sigma$, so the claim must hold true.

Let the critical point, which we have found out, be $\bar{u}\in
E_{\varepsilon}^{\mathcal{A}_{\varepsilon,R}}$. In order to prove that
$\bar{u}$ is a constant sign function, assume, by contradiction, that
$\bar{u}=\bar{u}^{+}-\bar{u}^{-}$, with $\bar{u}^{\pm}\not\equiv 0$. We
determine that $\bar{u}^{\pm}\in\mathcal{N}_{\varepsilon}$ by multiplying the
equation in ($P_{\varepsilon}$) by $\bar{u}^{\pm}$, so

$E_{\varepsilon}(\bar{u})=E_{\varepsilon}\left(\bar{u}^{+}\right)+E_{\varepsilon}\left(\bar{u}^{-}\right)\geq
2m_{\varepsilon},$

contrary to 4.14(c).

Acknowledgments. D. Ganguly is partially supported by the INSPIRE faculty
fellowship (IFA17-MA98). D. Gupta is supported by the PMRF.

Conflict of interest: All authors certify that there is no actual or potential
conflict of interest about this article.

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