# Multiplicity of positive solutions for a class of nonhomogeneous 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.

The paper is concerned with positive solutions to problems of the type

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

where $\mathbb{B}^{N}$ denotes the hyperbolic space,
$1<p<2^{*}-1:=\frac{N+2}{N-2}$, $\;\lambda<\frac{(N-1)^{2}}{4}$, and $f\in
H^{-1}(\mathbb{B}^{N})$ ($f\not\equiv 0$) is a non-negative functional. The
potential $a\in L^{\infty}(\mathbb{B}^{N})$ is assumed to be strictly
positive, such that $\lim_{d(x,0)\rightarrow\infty}a(x)\rightarrow 1,$ where
$d(x,0)$ denotes the geodesic distance. First, the existence of three positive
solutions is proved under the assumption that $a(x)\leq 1$. Then the case
$a(x)\geq 1$ is considered, and the existence of two positive solutions is
proved. In both cases, it is assumed that $\mu(\\{x:a(x)\neq 1\\})>0.$
Subsequently, we establish the existence of two positive solutions for
$a(x)\equiv 1$ and prove asymptotic estimates for positive solutions using
barrier-type arguments. The proofs for existence combine variational
arguments, key energy estimates involving hyperbolic bubbles.

###### Key words and phrases:

Hyperbolic space, hyperbolic bubbles, Palais-Smale decomposition, Mountain
pass geometry, Lusternik-Schnirelman Category theory, energy estimate, min-max
method.

###### 2010 Mathematics Subject Classification:

Primary: 35J20, 35J60, 58E30

##  1\. Introduction

In this paper, we aim to study the existence, multiplicity and asymptotic
estimates of solutions to the following elliptic problem on the hyperbolic
space $\mathbb{B}^{N}$

$\left.\begin{aligned} -\Delta_{\mathbb{B}^{N}}u-\lambda
u&=a(x)|u|^{p-1}u+f(x)\text{ in }\mathbb{B}^{N},\\\ u&>0\;\text{ in
}\mathbb{B}^{N},\\\ u&\in
H^{1}\left(\mathbb{B}^{N}\right),\end{aligned}\right\\}$ ($\mathcal{P}$)

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 $a(x)\in
L^{\infty}(\mathbb{B}^{N}).$ Further, $0<a\in
L^{\infty}\left(\mathbb{B}^{N}\right),$ and $0\not\equiv f\in
H^{-1}\left(\mathbb{B}^{N}\right)$ is a non-negative functional i.e.,
$f(u)\geq 0$ whenever $u\geq 0$. Let us postpone the discussion on the
technical assumptions of function $a(x)$ for a while.

If the hyperbolic space $\mathbb{B}^{N}$ is replaced with the Euclidean space
$\mathbb{R}^{N},$ i.e., when the equation ($\mathcal{P}$) is posed on
$\mathbb{R}^{N}$ with $f\equiv 0,$ has been investigated widely in the last
few decades, and several seminal results have been obtained, we name a few,
e.g., [5, 6, 7, 8, 15, 16, 22, 23], and this list is far from being complete.
The difficulty in treating this problem arises because the domain
$\mathbb{R}^{N}$ is unbounded, and standard variational methods would fail due
to the lack of compactness of Sobolev embedding even in the subcritical
regime. So to tackle such issues, several authors have introduced new tools,
particularly the papers mentioned above. Firstly, the existence of Ground
state is established by using delicate energy estimates and carefully
analysing the breaking levels of Palais-Smale sequences (see [5]); we also
refer to [10] for a comprehensive treatment of the problem in the last thirty
years. Then onwards, the question of the multiplicity of solutions came into
prominence for slightly modified problems in the Euclidean space
$\mathbb{R}^{N},$

$\left.\begin{aligned} -\Delta u+a(x)u&=|u|^{p-1}u\text{ in
}\mathbb{R}^{N},\\\ u&\in
H^{1}\left(\mathbb{R}^{N}\right),\end{aligned}\right\\}$ ($\mathcal{EP}$)

where the potential $a(x)\rightarrow a_{\infty}>0$ as $|x|\rightarrow\infty.$
Under the radially symmetric assumption on $a(x),$ existence of infinitely
many solutions was obtained by Berestycki-Lions in [8]. Moreover, the question
is even more interesting when the symmetric assumption on the potential $a(x)$
is dropped. However, considerable progress has also been made in the case in
which $a(x)$ is not radially symmetric. In fact, the existence of infinitely
many positive solutions is obtained in [12]. Also, see [11, 13, 14, 25, 26].

Adachi-Tanaka [2] considered Eq. ($\mathcal{P}$) in the whole Euclidean space,
with $\lambda=-1$ , and studied the multiplicity results. In fact, the problem
($\mathcal{P}$) is considered as a perturbation of the classical scalar field
equation. From the mathematical point of view, it is natural to ask whether
the problem ($\mathcal{P}$) admits a positive solution and if yes, then its
multiplicity/uniqueness, i.e., whether the positive solutions are stable after
the perturbation of type ($\mathcal{P}$) is studied. These questions were
quite comprehensively studied by Adachi-Tanaka [2]. Also, refer to [1, 3]. In
[2], the existence of four solutions has been obtained under the hypothesis
$({\mathbf{A}_{1}})$ below. Moreover, in [9, 21], the existence of two
positive solutions is established when the potential $a$ satisfies
$({\mathbf{A}_{2}})$, and $f\not\equiv 0$ (but small). Although, the cases
$({\mathbf{A}_{1}})$ and $({\mathbf{A}_{2}})$ do not cover the case
$a(x)\equiv 1,$ Zhu treated this case in [29], where he proved existence of
two positive solutions. The papers mentioned above employ topological
arguments, like Lusternik-Schnirelmn (L-S) category and the Min-Max arguments,
to obtain their multiplicity results. But for such arguments to work, precise
energy estimates of solutions to the “limiting problem” are required so that
we are away from the critical level (breaking level) of the Palais-Smale
sequences. By the “limiting problem,” we mean the following problem

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

It is well-known that the above problem admits unique radially symmetric
solutions $W\in C^{\infty}(\mathbb{R}^{N})$ up to translations. Furthermore,
it satisfies

$W(x)\sim|x|^{\frac{-(N-1)}{2}}\,e^{-|x|}\quad\mbox{as}\
|x|\rightarrow\infty.$

In particular, $W\in L^{p}(\mathbb{R}^{N})$ for all $p\geq 1.$ As described,
the energy estimates in the papers mentioned earlier were involved with
integrals of $W$, and this decay estimate plays a pivotal role in it.

Now coming back to our problem ($\mathcal{P}$) in the hyperbolic space
setting, even if it seems that the equation is a generalization of problems in
the Euclidean space, it has many fascinating phenomena. Let us start with the
seminal result of Sandeep-Mancini [24], where the author showed the
existence/uniqueness of positive solutions to the problem

$-\Delta_{\mathbb{B}^{N}}u\,-\,\lambda u\,=\,|u|^{p-1}u,\quad u\in
H^{1}\left(\mathbb{B}^{N}\right),$ (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 $N=2.$ They established in the subcritical case, i.e., $p>1$
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 radially symmetric with respect to some point
and unique up to hyperbolic isometries, except possibly for $N=2$ and
$\lambda>\frac{2(p+1)}{(p+3)^{2}}.$ Furthermore, the radially symmetric
solution $\mathcal{V}$ satisfies the following asymptotic estimates

$\lim_{r\rightarrow\infty}\dfrac{\log\mathcal{V}^{2}}{r}\;=\;-{(N-1)+\sqrt{(N-1)^{2}-4\lambda}},$

where $r:=d(x,0)$ denotes the geodesic distance (we refer Section 2 for more
details).

Concerning the multiplicity of (1.2), the existence of infinitely many radial
sign-changing solutions, compactness, and non-degeneracy was studied in ([17,
19, 20]). In this article, we are interested in whether the positive solutions
still exist under the perturbation of type ($\mathcal{P}$). If it exists, then
study its asymptotic estimates and multiplicity. In our previous article [18],
we showed the existence of a positive solution with high energy when $f\equiv
0.$ Here we considered a multiplicity of solutions along the line of previous
authors. As one anticipates, we follow the topological /variational arguments
to obtain multiple solutions. Still, the major hurdle lies in the energy
estimates involving solutions to (1.2) since one could see easily that
$\mathcal{V}\notin L^{p}(\mathbb{B}^{N})$ for $p\in[1,2).$ This step is quite
delicately handled in Section 6. Moreover, we also studied asymptotic
estimates of solutions to ($\mathcal{P}$) for $a(x)\equiv 1$ and $f$ satisfies
some decay estimates. Indeed the ode approach won’t work in this case, as
apriori $f$ is not given to be a radial function, and hence we tackle this
problem using the barrier argument. (See Section 5)

Now let us describe all the necessary assumptions before stating our main
theorems. We investigate the solutions of ($\mathcal{P}$) under the following
cases separately:

$\displaystyle\left(\mathbf{A}_{1}\right):$ $\displaystyle\quad
a(x)\in(0,1]\;\;\forall x\in\mathbb{B}^{N},\;\;\mu(\\{x:a(x)\neq
1\\})>0,\;\;\inf_{x\in\mathbb{B}^{N}}a(x)>0,\text{ and }$ $\displaystyle
a(x)\rightarrow 1\text{ as }d(x,0)\rightarrow\infty,\ \mbox{where}\ \mu\
\mbox{denotes the hyperbolic measure}.$
$\displaystyle\left(\mathbf{A}_{2}\right):$ $\displaystyle\;\;a(x)\geq
1\;\;\forall x\in\mathbb{B}^{N},\;\;\mu(\\{x:a(x)\neq 1\\})>0,\;\;a\in
L^{\infty}(\mathbb{B}^{N})\text{ and }a(x)\rightarrow 1$ $\displaystyle\text{
as }d(x,0)\rightarrow\infty.$ $\displaystyle\left(\mathbf{A}_{3}\right):$
$\displaystyle\;\;a(x)\equiv 1\;\;\forall x\in\mathbb{B}^{N}.$

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

$\lambda\in\begin{cases}\left(-\infty,\frac{2(p+1)}{(p+3)^{2}}\right],&N=2,\\\
\left(-\infty,\frac{(N-1)^{2}}{4}\right),&N\geq 3.\end{cases}$ (1.3)

We are now in a position to state this article’s main theorems. Let us begin
with the Adachi-Tanaka [2] type result in the hyperbolic space setting :

######  Theorem 1.1.

Let $a\in C\left(\mathbb{B}^{N}\right)$ satisfies
$\left(\mathbf{A}_{1}\right).$ In addition, assume that $a$ also satisfies

$a(x)\geqslant 1-\operatorname{C\,exp}(-\delta\,d(x,0))\quad\forall
x\in\mathbb{B}^{N},$ (1.4)

for some positive constants $C$ and $\delta.$ Then there exists $\delta_{0}>0$
such that the equation ($\mathcal{P}$) has at least three positive solutions
for any non-negative $f\in H^{-1}\left(\mathbb{B}^{N}\right)$ with
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq\delta_{0}$ and for $\lambda$
satisfying (1.3).

######  Remark 1.1.

In contrast with Adachi-Tanaka [2], here we obtain the existence of at least
three solutions instead of four. This is purely a technical reason for not
getting the fourth solution, which can be attributed to the new energy
estimates phenomenon in the hyperbolic space.

Next, we assume $a(x)\geq 1,$ and we prove the following result :

######  Theorem 1.2.

Let $a$ satisfies $\left(\mathbf{A}_{2}\right),0\not\equiv f\in
H^{-1}\left(\mathbb{B}^{N}\right)$ is a non-negative functional and
$S_{1,\lambda}$ be defined as in (3.10). Furthermore, if

$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}<C_{p}S_{1,\lambda}^{\frac{p+1}{2(p-1)}}\text{
where
}C_{p}:=\left(p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\right)^{-\frac{1}{p-1}}\left(\frac{p-1}{p}\right).$

Then ($\mathcal{P}$) admits at least two positive solutions for $\lambda$
satisfying (1.3).

Further, if $a$ satisfies $\left(\mathbf{A}_{3}\right)$, i.e., ($\mathcal{P}$)
becomes the following

$\left.\begin{aligned} -\Delta_{\mathbb{B}^{N}}u-\lambda
u&=|u|^{p-1}u+f(x)\text{ in }\mathbb{B}^{N},\\\ u&>0\;\text{ in
}\mathbb{B}^{N},\\\ u&\in
H^{1}\left(\mathbb{B}^{N}\right),\end{aligned}\right\\}$
($\mathcal{P^{{}^{\prime}}}$)

where all the notations are the same as for the problem ($\mathcal{P}$) then
we have the following theorem.

######  Theorem 1.3.

Assume that $a$ satisfies $\left(\mathbf{A}_{3}\right)$. Then there exists
$\delta_{0}^{{}^{\prime}}>0$ such that the problem $\eqref{4.aaa}$ has at
least two positive solutions any non-negative $f\in
H^{-1}\left(\mathbb{B}^{N}\right)$ with
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq\delta_{0}^{{}^{\prime}}$ and
for $\lambda$ satisfying (1.3).

The paper is organized as follows: In Section 2, we introduce some of the
notations, geometric definitions, and preliminaries concerning the hyperbolic
space. Section 3 describes the energy functional, setting up the problem, and
associated auxiliary lemmas involving functionals. In Section 4, we state and
prove the Palais-Smale decomposition theorem as Proposition 4.1 and 4.2.
Whereas in Section 5, we obtain asymptotic estimates for the solution of
$\eqref{4.aaa}.$ Section 6 is devoted to the key energy estimates involving
the solutions of (1.2). The proof of Theorem 1.1 and Theorem 1.2 are given in
Section 7. Finally, Section 8 is devoted to the proof of Theorem 1.3.

##  2\. 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 $\mathbb{B}^{N}$ 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 $a$ centred 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 [27].

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

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

Laplace-Beltrami operator on $\mathbb{B}^{N}$. It is well known that the
$N$-dimensional hyperbolic space $\mathbb{B}^{N}$ admits a polar coordinate
decomposition structure. Namely, for $x\in\mathbb{B}^{N}$ we can write
$x=(r,\Theta)=(r,\theta_{1},\ldots,\theta_{N-1})\in(0,\infty)\times\mathbb{S}^{N-1}$,
where $r$ denotes the geodesic distance between the point $x$ and a fixed pole
$0$ in $\mathbb{B}^{N}$ and $\mathbb{S}^{N-1}$ is the unit sphere in the
$N$-dimensional euclidean space $\mathbb{R}^{N}$. Recall that the Riemannian
Laplacian of a scalar function $u$ on $\mathbb{B}^{N}$ is given by

$\Delta_{\mathbb{B}^{N}}u(r,\Theta)=\frac{1}{(\sinh
r)^{N-1}}\frac{\partial}{\partial r}\left[(\sinh r)^{N-1}\frac{\partial
u}{\partial r}(r,\Theta)\right]\\\
+\frac{1}{\sinh^{2}r}\Delta_{\mathbb{S}^{N-1}}u(r,\Theta),$ (2.2)

where $\Delta_{\mathbb{S}^{N-1}}$ is the Riemannian Laplacian on the unit
sphere $\mathbb{S}^{N-1}$.

A sharp Poincaré-Sobolev inequality. (see [24])

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

$\frac{(N-1)^{2}}{4}=\inf_{u\in
H^{1}(\mathbb{B}^{N})\setminus\\{0\\}}\dfrac{\int_{\mathbb{B}^{N}}|\nabla_{\mathbb{B}^{N}}u|^{2}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}}{\int_{\mathbb{B}^{N}}|u|^{2}\,\mathrm{~{}d}V_{\mathbb{B}^{N}}}.$
(2.3)

######  Remark 2.1.

A consequence of (2.3) 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\. Energy functional and preliminary lemmas

###  3.1. Unperturbed equation

Firstly, let us recall the asymptotic estimates of positive solutions to the
following homogeneous problem

$\begin{gathered}-\Delta_{\mathbb{B}^{N}}w-\lambda w=|w|^{p-1}w,\;w>0\;\text{
in }\mathbb{B}^{N},w\in H^{1}\left(\mathbb{B}^{N}\right).\end{gathered}$ (3.1)

Then by elliptic regularity, any solution, $w\in H^{1}(\mathbb{B}^{N}),$ is
also in $C^{\infty}$ and satisfies the decay property (See [24, Lemma 3.4]):
for every $\varepsilon>0,$ there exist positive constants
$C_{1}^{\varepsilon}$ and $C_{2}^{\varepsilon}$ such that there holds

$C_{1}^{\varepsilon}{\text{e}}^{-(c(N,\lambda)+\varepsilon)\,d(x,0)}\leq
w(x)\leq
C_{2}^{\varepsilon}{\text{e}}^{-(c(N,\lambda)-\varepsilon)\,d(x,0)},\quad\hbox{for
all}\ x\in\mathbb{B}^{N},$ (3.2)

where $c(N,\lambda)\,=\,\frac{1}{2}(N-1+\sqrt{(N-1)^{2}-4\lambda}).$

###  3.2. Energy functional

For given $a(x)$ and $f(x)$, we define
$I_{\lambda,a,f}(u):H^{1}\left(\mathbb{B}^{N}\right)\rightarrow\mathbb{R}$ by

$I_{\lambda,a,f}(u)=\frac{1}{2}\|u\|_{H_{\lambda}}^{2}-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)u_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)-\int_{\mathbb{B}^{N}}f(x)u(x)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
(3.3)

It is obvious that if $u$ is a critical point of $I_{\lambda,a,f}$, then $u$
is the solution to the following problem

$\displaystyle-\Delta_{\mathbb{B}^{N}}u-\lambda u$
$\displaystyle=a(x)u_{+}^{p}+f(x)\text{ in }\mathbb{B}^{N},$ (3.4)
$\displaystyle u$ $\displaystyle\in H^{1}\left(\mathbb{B}^{N}\right).$

######  Remark 3.1.

If we take $v=u_{-}$ as a test function in (3.4) where $u$ is a weak solution
of (3.4) and $f$ is a non-negative functional, we obtain $u_{-}=0$, i.e.,
$u\geq 0$. Thus $u>0$ follows from the maximum principle, and hence $u$ is a
solution to ($\mathcal{P}$).

Define

$J_{\lambda,a,f}(v)=\max_{t>0}I_{\lambda,a,f}(tv):\tilde{\Sigma}_{+}\rightarrow\mathbb{R},$
(3.5)

where

$\displaystyle\Sigma:=\left\\{v\in
H^{1}\left(\mathbb{B}^{N}\right);\;\|v\|_{H_{\lambda}}=1\right\\},$
$\displaystyle\tilde{\Sigma}_{+}:=\left\\{v\in\Sigma:\;v_{+}\not\equiv
0\right\\}.$

In the subsequent sections, we will establish that the positive solutions of
($\mathcal{P}$) correspond to the critical points of
$I_{\lambda,a,f}(u):H^{1}\left(\mathbb{B}^{N}\right)\rightarrow\mathbb{R}$ or
$J_{\lambda,a,f}(v)$ : $\tilde{\Sigma}_{+}\rightarrow\mathbb{R}$. To this end
we set

$\displaystyle\underline{\mathrm{a}}:=\inf_{x\in\mathbb{B}^{N}}a(x)>0,$
$\displaystyle\bar{a}:=\sup_{x\in\mathbb{B}^{N}}a(x)=1.$

Using the definition of $J_{\lambda,a,f},$ and carrying out some easy
calculations we obtain

$\displaystyle J_{\lambda,a,0}(v)$
$\displaystyle=I_{\lambda,a,0}\left(\left(\int_{\mathbb{B}^{N}}a(x)v_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{-\frac{1}{p-1}}v\right)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p+1}\right)\left(\int_{\mathbb{B}^{N}}a(x)v_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{-\frac{2}{p-1}}.$
(3.6)

Therefore

$\bar{a}^{-\frac{2}{p-1}}J_{\lambda,1,0}(v)=J_{\lambda,\bar{a},0}(v)\leq
J_{\lambda,a,0}(v)\leq
J_{\lambda,\underline{\mathrm{a}},0}(v)=\underline{\mathrm{a}}^{-\frac{2}{p-1}}J_{\lambda,1,0}(v).$

Further, since $w$ is the unique radial solution of (3.1), we have

$\max_{t\in[0,1]}I_{\lambda,1,0}(tw)=I_{\lambda,1,0}(w).$ (3.7)

Moreover,

$\bar{a}^{-\frac{2}{p-1}}I_{\lambda,1,0}(w)\leq\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,0}(v)\leq\underline{\mathrm{a}}^{-\frac{2}{p-1}}I_{\lambda,1,0}(w).\quad$
(3.8)

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}}}\\\
$ (3.9)

and the energy levels

$S_{1,\lambda}:=\inf_{u\in
H^{1}\left(\mathbb{B}^{N}\right)\backslash\\{0\\}}J_{\infty}(u),\quad
S_{m,\lambda}:=m^{\frac{p-1}{p+1}}S_{1,\lambda},\quad m=2,3,4,\cdots$ (3.10)

###  3.3. Auxliary Lemmas

We require the following auxiliary lemmas to prove Theorem 1.1.

The subsequent lemmas give us the inequalities involving
$I_{\lambda,a,f}\;(J_{\lambda,a,f})$ and
$I_{\lambda,a(\varepsilon),0}\;(J_{\lambda,a(\varepsilon),0})$ for
$\varepsilon\in(0,1).$

######  Lemma 3.1.

  1. (i)

The following inequality holds for $u\in H^{1}\left(\mathbb{B}^{N}\right)$ and
$\varepsilon\in(0,1)$

$(1-\varepsilon)I_{\lambda,\frac{a}{1-\varepsilon},0}(u)-\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}\leq
I_{\lambda,a,f}(u)\leq(1+\varepsilon)I_{\lambda,\frac{a}{1+\varepsilon},0}(u)+\frac{1}{2\varepsilon}\left\|f\right\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}.$
(3.11)

  2. (ii)

Suppose $v\in\tilde{\Sigma}_{+}$and $\varepsilon\in(0,1)$. Then there holds

$(1-\varepsilon)^{\frac{p+1}{p-1}}J_{\lambda,a,0}(v)-\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}\leq
J_{\lambda,a,f}(v)\leq(1+\varepsilon)^{\frac{p+1}{p-1}}J_{\lambda,a,0}(v)+\frac{1}{2\varepsilon}\left\|f\right\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}.$
(3.12)

  3. (iii)

In particular, there exists $d_{0}>0$ such that if
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{0}$, then,

$\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,f}(v)>0.$

In the next lemma, for $v\in\tilde{\Sigma}_{+}$, we analyse the function
$\tilde{g}(t):[0,\infty)\rightarrow\mathbb{R}$ defined by

$\tilde{g}(t):=I_{\lambda,a,f}(tv).$

######  Lemma 3.2.

  1. (i)

The function $\tilde{g}$ has at most two critical points in $[0,\infty)$ for
every $v\in\tilde{\Sigma}_{+}$.

  2. (ii)

If $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{0}\left(d_{0}\right.$ as
chosen in Lemma 3.1$\left.\right)$, then for any $v\in\tilde{\Sigma}_{+}$,
there exists a unique $t_{a,f}(v)>0$ such that
$I_{\lambda,a,f}\left(t_{a,f}(v)v\right)=J_{\lambda,a,f}(v)$, where
$J_{\lambda,a,f}$ is defined as in (3.6). Moreover, $t_{a,f}(v)>0$ satisfies

$t_{a,f}(v)>\left(p\int_{\mathbb{B}^{N}}a(x)v_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{-\frac{1}{p-1}}\geq\left(pS_{1,\lambda}^{-\frac{(p+1)}{2}}\right)^{-\frac{1}{p-1}}.$
(3.13)

Additionally, we also have

$I_{\lambda,a,f}^{\prime\prime}\left(t_{a,f}(v)v\right)(v,v)<0.$ (3.14)

(iii) Any critical point of $\tilde{g}$ distinct from $t_{a,f}(v)$ lies in
$\left[0,\left(1-\frac{1}{p}\right)^{-1}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\right]$.

We omit the details of the proof of the above two lemmas. They can be proved
exactly in the spirit of [2]. The following proposition characterises all the
critical points of the functional $I_{\lambda,a,f}$ in terms of the functional
$J_{\lambda,a,f}$.

######  Proposition 3.3.

Assume $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{2}$ where
$d_{2}=\min\left\\{d_{1},\left(1-\frac{1}{p}\right)r_{1}\right\\}>0$ and
$d_{1},r_{1}$ as chosen in Proposition 7.1. Then the following holds

  1. (i)

$J_{\lambda,a,f}\in C^{1}\left(\tilde{\Sigma}_{+},\mathbb{R}\right)$ and

$J_{\lambda,a,f}^{\prime}(v)h=t_{a,f}(v)I_{\lambda,a,f}^{\prime}\left(t_{a,f}(v)v\right)h,$
(3.15)

for all $h\in T_{v}\tilde{\Sigma}_{+}=\left\\{h\in
H^{1}\left(\mathbb{B}^{N}\right)\mid\langle
h,v\rangle_{H_{\lambda}}=0\right\\}$.

  2. (ii)

$v\in\tilde{\Sigma}_{+}$is a critical point of $J_{\lambda,a,f}(v)$ iff
$t_{a,f}(v)v\in H^{1}\left(\mathbb{B}^{N}\right)$ is a critical point of
$I_{\lambda,a,f}(u)$.

  3. (iii)

In addition, the set containing all the critical points of
$I_{\lambda,a,f}(u)$ can be written as

$\left\\{t_{a,f}(v)v\mid
v\in\tilde{\Sigma}_{+},J_{\lambda,a,f}^{\prime}(v)=0\right\\}\cup\left\\{\mathcal{U}_{a,f}(x)\right\\},$
(3.16)

where $\mathcal{U}_{a,f}$ is a critical point of $I_{\lambda,a,f}$ obtained in
Proposition 7.1.

###### Proof.

We skip the proof for brevity. The proof can be concluded with the necessary
modifications for the hyperbolic space. For details, we refer [2]. ∎

##  4\. Palais-Smale Characterization

In this section, we study the Palais-Smale sequences (PS sequences)
corresponding to the problem ($\mathcal{P}$). We say a sequence $u_{n}\in
H^{1}\left(\mathbb{B}^{N}\right)$ is a Palais-Smale sequence for
$I_{\lambda,a,f}$ at a level $d$ if
$I_{\lambda,a,f}\left(u_{n}\right)\rightarrow d$ and
$I_{\lambda,a,f}^{\prime}\left(u_{n}\right)\rightarrow 0$ 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
$d(x,0)\rightarrow\infty.$

In the subsequent propositions, we examine the Palais-Smale condition for
$I_{\lambda,a,f}(u)$ and $J_{\lambda,a,f}(v)$. In particular, we prove the
following proposition :

######  Proposition 4.1.

Assume $0<a\in L^{\infty}\left(\mathbb{B}^{N}\right),a(x)\rightarrow 1$ as
$d(x,0)\rightarrow\infty$ and $0\not\equiv f\in
H^{-1}\left(\mathbb{B}^{N}\right)$ is a non-negative functional and suppose
that a sequence $\left\\{u_{j}\right\\}_{j=1}^{\infty}\subset
H^{1}\left(\mathbb{B}^{N}\right)$ satisfies

$\displaystyle I_{\lambda,a,f}^{\prime}\left(u_{j}\right)\rightarrow
0\quad\text{ in }H^{-1}\left(\mathbb{B}^{N}\right),$ $\displaystyle
I_{\lambda,a,f}\left(u_{j}\right)\rightarrow c\in\mathbb{R}$

as $j\rightarrow\infty$. Then there exists a subsequence - still denoted by
$\left\\{u_{j}\right\\}_{j=1}^{\infty}$, a critical point $u_{0}(x)$ of
$I_{\lambda,a,f}(u)$, an integer $\ell\in\mathbb{N}\cup\\{0\\}$, and $\ell$
sequences of points
$\left\\{y_{j}^{1}\right\\}_{j=1}^{\infty},\ldots,\left\\{y_{j}^{\ell}\right\\}_{j=1}^{\infty}\subset\mathbb{B}^{N}$
such that

  1. (1)

$d(y_{j}^{k},0)\rightarrow\infty\text{ as }j\rightarrow\infty\;\;\forall
k=1,2,\ldots,\ell,$

  2. (2)

$d(y_{j}^{k},y_{j}^{k^{\prime}})\rightarrow\infty\text{ as
}j\rightarrow\infty\text{ for }k\neq k^{\prime},$

  3. (3)

$\left\|u_{j}(x)-\left(u_{0}(x)+\sum_{k=1}^{\ell}w(\tau_{-y_{j}^{k}}(x))\right)\right\|_{{H_{\lambda}}}\rightarrow
0$ as $j\rightarrow\infty$,

  4. (4)

$I_{\lambda,a,f}\left(u_{j}\right)\rightarrow
I_{\lambda,a,f}\left(u_{0}\right)+\ell I_{\lambda,1,0}(w)$ as
$j\rightarrow\infty,$

where $\tau_{a},$ $a\in\mathbb{B}^{N}$ denotes the hyperbolic translation, and
$w$ is the unique positive radial solution to the unperturbed equation.

###### Proof.

The proof is a straightforward adaption of [18, Proposition 3.1] in the case
$f\not\equiv 0.$ We also refer ([22], [23] and [28]) for the Euclidean case. ∎

Next, we study the Palais-Smale condition for $J_{\lambda,a,f}.$

######  Proposition 4.2.

Suppose $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{2}$ for $d_{2}>0$ as
given in Proposition 3.3. Then,

  1. (a)

As the $\operatorname{dist}$
${}_{H_{\lambda}\left(\mathbb{B}^{N}\right)}\left(v_{j},\partial\tilde{\Sigma}_{+}\right)=\inf\left\\{\left\|v_{j}-u\right\|_{H_{\lambda}}:u\in\Sigma,u_{+}\equiv
0\right\\}{\xrightarrow{j}}\;\;0$
implies $J_{\lambda,a,f}\left(v_{j}\right)\rightarrow\infty$.

  2. (b)

Suppose that $\left\\{v_{j}\right\\}_{j=1}^{\infty}\subset\tilde{\Sigma}_{+}$
satisfies as $j\rightarrow\infty$

$\displaystyle J_{\lambda,a,f}\left(v_{j}\right)\rightarrow c\text{ for some
}c>0,$ (4.1)
$\displaystyle\left\|J_{\lambda,a,f}^{\prime}\left(v_{j}\right)\right\|_{T_{v_{j}}^{*}\tilde{\Sigma}_{+}}\equiv\sup\left\\{J_{\lambda,a,f}^{\prime}\left(v_{j}\right)h;\;h\in
T_{v_{j}}\tilde{\Sigma}_{+},\;\|h\|_{H_{\lambda}}=1\right\\}\rightarrow 0.$

Then there exists a subsequence - still denoted by
$\left\\{v_{j}\right\\}_{j=1}^{\infty}$, a critical point $u_{0}(x)\in
H^{1}\left(\mathbb{B}^{N}\right)$ of $I_{\lambda,a,f}(u)$, an integer
$\ell\in\mathbb{N}\cup\\{0\\}$ and $\ell$ sequences of points
$\left\\{y_{j}^{1}\right\\}_{j=1}^{\infty},\ldots,\left\\{y_{j}^{\ell}\right\\}_{j=1}^{\infty}\subset\mathbb{B}^{N}$
such that

  1. (1)

$d(y_{j}^{k},0)\rightarrow\infty\text{ as }j\rightarrow\infty\;\;\forall
k=1,2,\ldots,\ell,$

  2. (2)

$d(y_{j}^{k},y_{j}^{k^{\prime}})\rightarrow\infty\text{ as
}j\rightarrow\infty\text{ for }k\neq k^{\prime},$

  3. (3)

$\left\|v_{j}(x)-\frac{u_{0}(x)+\sum_{k=1}^{\ell}w(\tau_{-y_{j}^{k}}(x))}{\left\|u_{0}(x)+\sum_{k=1}^{\ell}w(\tau_{-y_{j}^{k}}(x))\right\|_{{H_{\lambda}}}}\right\|_{{H_{\lambda}}}\rightarrow
0\text{ as }j\rightarrow\infty,\\\ \text{ where
}\tau_{a},a\in\mathbb{B}^{N}\text{ denotes the hyperbolic translation,}$

  4. (4)

$J_{\lambda,a,f}\left(v_{j}\right)\rightarrow
I_{\lambda,a,f}\left(u_{0}\right)+\ell I_{\lambda,1,0}(w)\text{ as
}j\rightarrow\infty.$

###### Proof.

For any $\varepsilon\in(0,1)$ and using (3.12) and (3.6), we obtain,

$\displaystyle J_{\lambda,a,f}\left(v_{j}\right)$
$\displaystyle\geq(1-\varepsilon)^{\frac{p+1}{p-1}}J_{\lambda,a,0}\left(v\right)-\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}$
$\displaystyle\geq(1-\varepsilon)^{\frac{p+1}{p-1}}\left(\frac{1}{2}-\frac{1}{p+1}\right)\left(\int_{\mathbb{B}^{N}}a(x)v_{j+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right)^{-\frac{2}{p-1}}-\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}.$

As
$\operatorname{dist}\left(v_{j},\partial\tilde{\Sigma}_{+}\right)\rightarrow
0$ gives

$\displaystyle\left(v_{j}\right)_{+}\rightarrow 0\text{ in
}H^{1}\left(\mathbb{B}^{N}\right),$
$\displaystyle\left(v_{j}\right)_{+}\rightarrow 0\text{ in
}L^{p+1}\left(\mathbb{B}^{N}\right).$

Therefore,

$\left|\int_{\mathbb{B}^{N}}a(x)v_{j}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\right|\leq\left.\|a\|\right._{L^{\infty}\left(\mathbb{B}^{N}\right)}\int_{\mathbb{B}^{N}}\left|v_{j+}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\xrightarrow{j}0.$

Hence $J_{\lambda,a,f}\left(v_{j}\right)\rightarrow\infty$ as dist
${}_{H^{1}\left(\mathbb{B}^{N}\right)}\left(v_{j},\partial\tilde{\Sigma}_{+}\right)\rightarrow
0.$ This proves part $(a).$

For part $(b),$ using (3.13) and (3.15), we get

$\displaystyle\left\|I_{\lambda,a,f}^{\prime}\left(t_{a,f}\left(v_{j}\right)v_{j}\right)\right\|_{H^{-1}\left(\mathbb{B}^{N}\right)}$
$\displaystyle=\frac{1}{t_{a,f}\left(v_{j}\right)}\left\|J_{\lambda,a,f}^{\prime}\left(v_{j}\right)\right\|_{T_{v_{j}}^{*}\tilde{\Sigma}_{+}}$
$\displaystyle\leq\left(pS_{1,\lambda}^{-\frac{p+1}{2}}\right)^{\frac{1}{p-1}}\left\|J_{\lambda,a,f}^{\prime}\left(v_{j}\right)\right\|_{T_{v_{j}}\tilde{\Sigma}_{+}}\stackrel{{\scriptstyle
j}}{{\rightarrow}}0.$

Further, we also have
$I_{\lambda,a,f}\left(t_{a,f}(v_{j})v_{j}\right)=J_{\lambda,a,f}\left(v_{j}\right)\rightarrow
c$ as $j\rightarrow\infty$. Applying Palais-Smale lemma for
$I_{\lambda,a,f}(u)$ (Proposition 4.1), the rest follows.

∎

The subsequent corollary is an outcome of the above Proposition 4.2. Before
moving to the corollary, note that we say $J_{\lambda,a,f}(v)$ satisfies
$(\mathrm{PS})_{c}$ if and only if any sequence
$\left(v_{j}\right)_{j=1}^{\infty}\subseteq$ $\tilde{\Sigma}_{+}$satisfying
(4.1) has a strongly convergent subsequence in
$H^{1}\left(\mathbb{B}^{N}\right)$.

######  Corollary 4.3.

Suppose that $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{2}$ for $d_{2}$
as in Proposition 3.3. Then $J_{\lambda,a,f}(v)$ satisfies the condition
$(\mathrm{PS})_{c}$ for
$c<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)$ where
$w$ is the unique radial solution of (3.1) and $\mathcal{U}_{a,f}$ is the
critical point of $I_{\lambda,a,f}$ obtained in Proposition 7.1.

###### Proof.

Proposition 4.2 suggests that the condition (PS)c breaks down only at levels

$c=I_{\lambda,a,f}\left(u_{0}\right)+\ell I_{\lambda,1,0}(w),$

where $\ell\in\mathbb{N}$ and $u_{0}\in H^{1}\left(\mathbb{B}^{N}\right)$ is a
critical point of $I_{\lambda,a,f}(u)$ .
From Proposition 7.1, we have

$I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)=\inf_{u\in
B\left(r_{1}\right)}I_{\lambda,a,f}(u)\leq I_{\lambda,a,f}(0)=0,$ (4.2)

Furthermore, all the critical points of $I_{\lambda,a,f}(u)$ except
$\mathcal{U}_{a,f}(x)$ corresponds to a critical point $J_{\lambda,a,f}(v)$,
which follows from (3.16). Thus there exists $v_{1}\in\tilde{\Sigma}_{+}$ for
a critical point $u_{1}$ of $I_{\lambda,a,f}(u)$ such that
$I_{\lambda,a,f}\left(u_{1}\right)=J_{\lambda,a,f}\left(v_{1}\right)>0$ by
using $(iii)$ of Lemma 3.1. Consequently,

$I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)=\inf\left\\{I_{\lambda,a,f}\left(u_{0}\right)\mid
u_{0}\in H^{1}\left(\mathbb{B}^{N}\right)\text{ is a critical point of
}I_{\lambda,a,f}(u)\right\\}.$

Hence $I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)$ is
the lowest level where $(PS)_{c}$ breaks. ∎

##  5\. Asymptotic estimates for solutions of ($\mathcal{P^{{}^{\prime}}}$)

This section is devoted to deriving asymptotic estimates for positive
solutions to ($\mathcal{P^{{}^{\prime}}}$) for $\lambda\leq 0$. It is worth
noting that when $f\equiv 0,$ the precise estimates were obtained by Sandeep-
Mancini in their seminal paper (See [24, Lemma 3.4]). Indeed they showed using
the moving plane method that all positive solutions to the homogeneous
equation are radial with respect to a point. Further, asymptotic was obtained
by analysing the corresponding ode. On the other hand, when dealing with
$f\not\equiv 0$ and non-radial, the solution $u$ need not be radial; hence,
this approach does not help us obtain asymptotic estimates for solutions of
$\eqref{4.aaa}.$ Thus we follow the approach of constructing suitable barriers
as sub and super solutions to obtain the desired asymptotic estimates. When
$f\equiv 0,$ we recover the optimal estimates obtained by Sandeep-Mancini for
radial solutions. In particular, we prove the following theorem :

######  Theorem 5.1.

Let $u$ be a positive solution of ($\mathcal{P^{{}^{\prime}}}$) and $f\in
L^{2}(\mathbb{B}^{N}),$ non-negative and assume

$f(x)\leq\,C\;\exp{-(k+\varepsilon)p\,d(x,0)},$

for all $x\in\mathbb{B}^{N}$ and for some positive constants $k,C,$ and
$\varepsilon.$

Then, for any $\delta>0$, there exist positive constants $C_{1},C_{2}$ such
that

$C_{1}\exp(-((N-1)+\delta)d(x,0))\leqslant u(x)\leqslant
C_{2}\exp(-((N-1)-\delta)d(x,0))$

for all $x\in\mathbb{B}^{N},$ and $\lambda=0.$ Furthermore, for $\lambda<0$,
there exist positive constants $C^{{}^{\prime}}_{1},C^{{}^{\prime}}_{2}$ such
that

$C^{{}^{\prime}}_{1}\exp(-(c^{{}^{\prime}}(n,\lambda)+\delta)|\lambda|d(x,0))\leqslant
u(x)\leqslant
C^{{}^{\prime}}_{2}\exp(-(c^{{}^{\prime}}(n,\lambda)-\delta)|\lambda|d(x,0))$

for all $x\in\mathbb{B}^{N}$ and
$c^{{}^{\prime}}(n,\lambda)=\frac{(N-1)+\sqrt{(N-1)^{2}-4\lambda}}{2|\lambda|}$.

###### Proof.

The solution $u\in H^{1}(\mathbb{B}^{N}),$ this immediately implies
$\lim_{d(x,0)\rightarrow\infty}u(x)=0\text{ a.e.}$ Furthermore, using the
Calderon-Zygmund estimate and elliptic regularity, we have $u\in
C^{2}\left(\mathbb{B}^{N}\right)$; thus,
$\lim_{d(x,0)\rightarrow\infty}u(x)=0$ for all $x\in\mathbb{B}^{N}.$ The proof
is divided into two cases: $\lambda<0$ and $\lambda=0.$

Case 1: $\lambda<0$

Choose $\alpha>0$ such that $\frac{\alpha^{2}|\lambda|-1}{\alpha(N-1)}\geq 1$.
To be precise, $\alpha\in[c^{{}^{\prime}}(N,\lambda),\infty)$ where

$c^{{}^{\prime}}(N,\lambda)=\frac{(N-1)+\sqrt{(N-1)^{2}-4\lambda}}{2|\lambda|}.$

Thus we can choose $R_{1}>0$ large enough such that

$\alpha^{2}|\lambda|-\alpha(N-1)\coth{d(x,0)}\geq 1,\;\;\;\;\forall d(x,0)\geq
R_{1}.$ (5.1)

For $m=\min\left\\{\frac{1}{|\lambda|}u(x)\mid d(x,0)=R_{1}\right\\}>0$, set
$v_{1}(x):=v_{1}(r)\,=me^{-\alpha|\lambda|\left(d(x,0)-R_{1}\right)},$ where
$r:=d(x,0).$ Now for any $L>R_{1}$, denote

$\Omega(L)=\left\\{x\in\mathbb{B}^{N}\mid R_{1}<d(x,0)<L\quad\text{ and
}\quad|\lambda|v_{1}(x)>u(x)\right\\}.$

Then $\Omega(L)$ is open. Moreover, for $x\in\Omega(L)$ and using (5.1) we
have

$\displaystyle\Delta_{\mathbb{B}^{N}}\left(u-|\lambda|v_{1}\right)(x)$
$\displaystyle=\Delta_{\mathbb{B}^{N}}u(x)\,\,-|\lambda|\,\Delta_{\mathbb{B}^{N}}(v_{1}(x))$
$\displaystyle=-\lambda
u-\,u^{p}\,-\,f(x)-|\lambda|\,\left(\frac{\partial^{2}}{\partial
r^{2}}v_{1}(r)+(N-1)\coth r\frac{\partial}{\partial r}v_{1}(r)\right)$
$\displaystyle=-\lambda
u-\,u^{p}\,-\,f(x)-|\lambda|\left[\alpha^{2}|\lambda|^{2}-\alpha|\lambda|(N-1)\coth{r}\right]v_{1}(x)$
$\displaystyle\leqslant|\lambda|u(x)-|\lambda|^{2}\left[\alpha^{2}|\lambda|-\alpha(N-1)\coth{r}\right]v_{1}(x)$
$\displaystyle\leqslant|\lambda|\left(u-|\lambda|v_{1}\right)(x)$
$\displaystyle<0$

Applying the maximum principle, for $x\in\Omega(L)$ will result in

$\displaystyle u(x)-|\lambda|v_{1}(x)$
$\displaystyle\geqslant\min\left\\{\left(u-|\lambda|v_{1}\right)(x)\mid
x\in\partial\Omega(L)\right\\}$
$\displaystyle=\min\left\\{0,\min_{d(x,0)=L}\left(u-|\lambda|v_{1}\right)(x)\right\\}.$

Since
$\displaystyle\lim_{d(x,0)\rightarrow+\infty}u(x)=\lim_{d(x,0)\rightarrow+\infty}v_{2}(x)=0$,
by letting $L\rightarrow\infty$, we see that $\Omega(L)$ is empty and hence

$u(x)\geqslant|\lambda|v_{1}(x)\text{ for all }d(x,0)\geqslant R_{1},$ (5.2)

By the supposition on $f(x)$ there exists some $\varepsilon,\text{ and }C>0$
such that

$f(x)\leqslant
Ce^{-(c^{{}^{\prime}}(N,\lambda)+\varepsilon)|\lambda|p\,d(x,0)}\text{ for all
}x\in\mathbb{B}^{N}\text{. }$ (5.3)

(5.2) will imply the existence of a $C_{1}>0$

$u(x)\geqslant
C_{1}e^{(c^{{}^{\prime}}(N,\lambda)+\delta)|\lambda|d(x,0)}\quad\text{ for all
}x\in\mathbb{B}^{N},\quad\mbox{and for any }\ \delta>0.$ (5.4)

Choosing $\varepsilon$ appropriately, and using (5.3), (5.4) together will
provide $R_{2}>0$ such that

$(u(x))^{p}\geqslant f(x)\quad\text{ for }\quad d(x,0)\geqslant R_{2}.$

Moreover, since $p>1,$ there holds

$u^{p}\,=\,\circ(u)\text{ for }d(x,0)\rightarrow\infty.$

Let $\beta>0$ be such that $\beta^{2}|\lambda|-(N-1)\beta\leq 1$, i.e.,
$\beta\leq c(n,\lambda)^{{}^{\prime}}.$

Define $v_{2}(x)=Me^{-\beta|\lambda|\left(d(x,0)-R_{4}\right)}$, where

$M=\max\left\\{u(x)\mid d(x,0)=R_{2}\right\\}>0.$

Further, for any $L>R_{4}$, denote

$\tilde{\Omega}(L)=\left\\{x\in\mathbb{B}^{N}\mid R_{4}<d(x,0)<L\quad\text{
and }\quad u(x)>v_{2}(x)\right\\}.$

Then $\tilde{\Omega}(L)$ is open and, for $x\in\tilde{\Omega}(L)$,

$\displaystyle\Delta_{\mathbb{B}^{N}}\left(v_{2}-u\right)(x)$
$\displaystyle=\left[\beta^{2}|\lambda|^{2}-\beta|\lambda|(N-1)\coth
r\right]v_{2}(x)+\lambda u\,+\,u^{p}+f(x)$ $\displaystyle\leqslant-\lambda
v_{2}+\lambda u\,+\,2u^{p}$ $\displaystyle\leqslant-\lambda v_{2}+\lambda
u\,+\,\circ(u)$ $\displaystyle=-\lambda(v_{2}-u)(x)+\circ(u)$
$\displaystyle<0.$

By the maximum principle, for $x\in\widetilde{\Omega}(L)$,

$\displaystyle v_{2}(x)-u(x)$
$\displaystyle\geqslant\min\left\\{\left(v_{2}-u\right)(x)\mid
x\in\partial\tilde{\Omega}(L)\right\\}$
$\displaystyle=\min\left\\{0,\min_{d(x,0)=L}\left(v_{2}-u\right)(x)\right\\}.$

Since
$\displaystyle\lim_{d(x,0)\rightarrow+\infty}u(x)=\lim_{d(x,0)\rightarrow+\infty}v_{2}(x)=0$,
by letting $L\rightarrow\infty$, we see again that $\tilde{\Omega}(L)$ is
empty and hence

$v_{2}(x)\geqslant u(x)\text{ for all }d(x,0)\geqslant R_{4}.$

Now by choosing $\alpha=\beta=c^{{}^{\prime}}(N,\lambda),$ the proof is
complete.

Case 2: $\lambda=0$

This case can also be tackled similarly by appropriately choosing the
functions $v_{1}$ and $v_{2}$.
To be precise, let

$v_{1}=me^{-\gamma\left(d(x,0)-R_{1}^{{}^{\prime}}\right)}\text{ and
}v_{2}=Me^{-\eta\left(d(x,0)-R_{2}^{{}^{\prime}}\right)}\text{ for some
}\gamma,R_{1}^{{}^{\prime}},\eta,R_{2}^{{}^{\prime}}>0$

where $m=\min\left\\{u(x)\mid d(x,0)=R_{1}^{{}^{\prime}}\right\\}>0$ and
$M=\max\left\\{u(x)\mid d(x,0)=R_{2}^{{}^{\prime}}\right\\}>0.$
Indeed $\gamma>0$ satisfies $\gamma>N-1$, and thus $R_{1}^{{}^{\prime}}$ is
chosen such that $\gamma-(N-1)\coth{r}>0$ for all $r>R_{1}^{{}^{\prime}}.$
Also, $R_{2}^{{}^{\prime}}$ is chosen similarly as $R_{3}$ mentioned above.
Further, we can conclude the lemma by applying the maximum principle in the
hyperbolic balls of radius $R_{1}^{{}^{\prime}}$ and $R_{2}^{{}^{\prime}}$,
and proceeding as in the previous case. ∎

##  6\. Key Energy Estimates

This section is devoted to deriving key energy estimates for the functional
$I_{\lambda,a,f}$ with $a(x)\leq 1$. The subsequent energy estimates will play
a pivotal role in the existence of solutions. In fact with the help of the
proposition 6.1, we shall show that the energy of the functional is below the
critical level given in the Palais-Smale decomposition.

######  Proposition 6.1.

Let $a$ satisfies $0<a\in
L^{\infty}\left(\mathbb{B}^{N}\right),a(x)\rightarrow 1$ as
$d(x,0)\rightarrow\infty$ and (1.4). Further, assume that
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}{\leq}\;d_{2},\;f\geq
0,\;f\not\equiv 0$ and $\tilde{\mathcal{U}}_{a,f}$ is any critical point of
$I_{\lambda,a,f}$. Then there exists $R>0$ such that

$I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)<I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)\right)+I_{\lambda,1,0}(w),\;\;\;$
(6.1)

for all $d(y,0)\geq R$ and $t>0$.
Moreover, if $a$ satisfies $\left(\mathbf{A}_{3}\right),$ i.e., $a\equiv 1$,
we have

$\sup_{t\geqslant
0}I_{\lambda,1,f}\left(\tilde{\mathcal{U}}_{1,f}+tw\left(\tau_{y}(x)\right)\right)<I_{\lambda,1,f}\left(\tilde{\mathcal{U}}_{1,f}\right)+I_{\lambda,1,0}(w),$
(6.2)

for all $d(y,0)\geq R.$

###### Proof.

Performing straightforward calculations implies

$\displaystyle
I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)$
$\displaystyle=\frac{1}{2}\left\|\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right\|_{H_{\lambda}}^{2}$
(6.3)
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle-\int_{\mathbb{B}^{N}}f(x)\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=\frac{1}{2}\left\|\tilde{\mathcal{U}}_{a,f}(x)\right\|_{H_{\lambda}}^{2}+\frac{t^{2}}{2}\|w\|_{H^{1}\left(\mathbb{B}^{N}\right)}^{2}$
$\displaystyle+t\left\langle\tilde{\mathcal{U}}_{a,f}(x),w(\tau_{-y}(x))\right\rangle_{H_{\lambda}}$
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle-\frac{t^{p+1}}{p+1}\int_{\mathbb{B}^{N}}\;a(x)\;(w(\tau_{-y}(x))^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left\\{\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)^{p+1}\right.$
$\displaystyle-\left.\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p+1}-t^{p+1}w(\tau_{-y}(x))^{p+1}\right\\}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle-\int_{\mathbb{B}^{N}}f(x)\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

Now for all $h\in H^{1}\left(\mathbb{B}^{N}\right)$, we have

$\displaystyle 0$
$\displaystyle=I_{\lambda,a,f}^{\prime}\left(\tilde{\mathcal{U}}_{a,f}(x)\right)(h)$
$\displaystyle=\left\langle\tilde{\mathcal{U}}_{a,f}(x),h\right\rangle_{H_{\lambda}}-\int_{\mathbb{B}^{N}}a(x)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p}h\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)-\int_{\mathbb{B}^{N}}fh\mathrm{~{}d}V_{\mathbb{B}^{N}}(x),$

i.e.,

$\left\langle\tilde{\mathcal{U}}_{a,f}(x),h\right\rangle_{H_{\lambda}}=\int_{\mathbb{B}^{N}}a(x)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p}h\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)+\int_{\mathbb{B}^{N}}fh\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

In particular, for $h=tw(\tau_{-y}(x))$ in the above yields

$\displaystyle
t\left\langle\tilde{\mathcal{U}}_{a,f}(x),w(\tau_{-y}(x))\right\rangle_{H_{\lambda}}$
$\displaystyle=t\int_{\mathbb{B}^{N}}a(x)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p}w(\tau_{-y}(x))\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)+t\int_{\mathbb{B}^{N}}fw(\tau_{-y}(x))\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

Hence utilizing the above equation and appropriately rearranging the terms in
(6.3) will result in

$\displaystyle
I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)=I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)\right)+I_{\lambda,1,0}(tw)$
$\displaystyle+\frac{t^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))w(\tau_{-y}(x))^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left\\{\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)^{p+1}-\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p+1}\right.$
$\displaystyle\left.-t(p+1)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p}w(\tau_{-y}(x))-t^{p+1}w(\tau_{-y}(x))^{p+1}\right\\}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)\right)+I_{\lambda,1,0}(tw)\;+\;\underbrace{(I)-(II)}.$

where

$I:=\frac{t^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))w(\tau_{-y}(x))^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x),$
(6.4)

and

$\displaystyle II$
$\displaystyle:=\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left\\{\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)^{p+1}-\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p+1}\right.$
(6.5)
$\displaystyle\left.-t(p+1)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p}w(\tau_{-y}(x))-t^{p+1}w(\tau_{-y}(x))^{p+1}\right\\}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

To complete the proof of the proposition, we need to show that
$(I)\;-\;(II)<0,$ for suitably chosen $R>0.$

Using the continuity, we easily get

$I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)\rightarrow
I_{\lambda,a,f}(\tilde{\mathcal{U}}_{a,f}(x))$

as $t\rightarrow 0$. In addition, we also have

$I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)\rightarrow-\infty\text{
as }t\rightarrow\infty.$

Thus using the above two facts, we can find $m,M$ with $0<m<M$ such that

$I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x)\right)<I_{\lambda,a,f}\left(\tilde{\mathcal{U}}_{a,f}(x)\right)+I_{\lambda,1,0}(w)\;\text{
for all }t\in(0,m)\cup(M,\infty).$

As a result, to prove the proposition at hand, it suffices to show (6.1) for
$t\in[m,M]$. Hence to finish the proof, we need to show $I<II$. To this end,
let us recall the following standard $p$-th inequalities from calculus.

  1. (1)

$(s+t)^{p+1}-s^{p+1}-t^{p+1}-(p+1)s^{p}t\geq 0$ for all
$(s,t)\in[0,\infty)\times[0,\infty)$.

  2. (2)

For any $r>0$ we can find a constant $A(r)>0$ such that

$(s+t)^{p+1}-s^{p+1}-t^{p+1}-(p+1)s^{p}t\geq A(r)t^{2},$

for all $(s,t)\in[r,\infty)\times[0,\infty)$.

We can estimate $II$ with the help of the above inequality as follows:

Set $A:=A(r):=A\left(\min_{d(x,0)\leq 1}\tilde{\mathcal{U}}_{a,f}(x)\right)>$
0, then

$\displaystyle II$
$\displaystyle:=\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left\\{\left(\tilde{\mathcal{U}}_{a,f}(x)+tw(\tau_{-y}(x))\right)^{p+1}-\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p+1}\right.$
$\displaystyle\left.-t(p+1)\left(\tilde{\mathcal{U}}_{a,f}(x)\right)^{p}w(\tau_{-y}(x))-t^{p+1}w(\tau_{-y}(x))^{p+1}\right\\}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\geq\frac{1}{p+1}\int_{d(x,0)\leq
1}a(x)A(r)t^{2}w^{2}(\tau_{-y}(x))\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\geq\frac{m^{2}\underline{\mathrm{a}}A(r)}{p+1}\underbrace{\int_{d(x,0)\leq
1}w^{2}(\tau_{-y}(x))\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)}_{E_{1}}$

Estimate of $E_{1}:$ We shall estimate $E_{1}$ in the domain $d(x,0)\leq 1.$
Using traingle inequality we have

$1-\frac{d(x,0)}{d(y,0)}\leq\frac{d(x,y)}{d(y,0)}\leq
1+\frac{d(x,0)}{d(y,0)}.$

Since, $d(x,0)\leq 1,$ there exist $R>0$ and $\varepsilon_{R}>0$ such that
whenver $d(y,0)>R,$ there holds

$1-\varepsilon_{R}\leq\frac{d(x,y)}{d(y,0)}\leq 1+\varepsilon_{R},$

where $\varepsilon_{R}\rightarrow 0$ as $R\rightarrow\infty.$ Thus using above
and (3.2) we conclude for any $\varepsilon>0,$

$\displaystyle E_{1}$ $\displaystyle:=\int_{d(x,0)\leq
1}w^{2}(\tau_{-y}(x))\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\geq
C_{\varepsilon}\int_{d(x,0)\leq
1}e^{-2(c(N,\lambda)+\varepsilon)d(x,y)}\;\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\geq
C_{\varepsilon}\;e^{-2(c(N,\lambda)+\varepsilon)(1+\varepsilon_{R})d(y,0)}\underbrace{\int_{d(x,0)\leq
1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)}_{:=C}$
$\displaystyle=\tilde{C_{\varepsilon}}\;e^{-2(c(N,\lambda)+\varepsilon)(1+\varepsilon_{R})d(y,0)}.$

Therefore we have

$II\geq\frac{\tilde{C_{\varepsilon}}m^{2}\underline{\mathrm{a}}A(r)}{p+1}\;e^{-2(c(N,\lambda)+\varepsilon)(1+\varepsilon_{R})d(y,0)}.$
(6.6)

Estimate of I : Let us now compute an estimate on $I$ for
$\delta>c(n,\lambda)(p+1)+(N-1)$, then for every $\varepsilon^{\prime}>0,\\\
\delta>(c(n,\lambda)-\varepsilon^{\prime})(p+1)+(N-1).$ We shall estimate $I$
as follows:

$\displaystyle I$
$\displaystyle=\frac{t^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))w(\tau_{-y}(x))^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
(6.7) $\displaystyle\leq
C_{\varepsilon^{\prime}}\frac{t^{p+1}}{p+1}\int_{\mathbb{B}^{N}}\;(1-a(x))e^{-(c(n,\lambda)-\varepsilon^{\prime})(p+1)d(x,y)}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq
C_{\varepsilon^{\prime}}\frac{t^{p+1}}{p+1}\int_{\mathbb{B}^{N}}e^{-\delta
d(x,0)}e^{(c(n,\lambda)-\varepsilon^{\prime})(p+1)(d(x,0)-d(y,0))}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq
C_{\varepsilon^{\prime}}\frac{t^{p+1}}{p+1}e^{-(c(n,\lambda)-\varepsilon^{\prime})(p+1)d(y,0)}\int_{\mathbb{B}^{N}}e^{-\delta
d(x,0)+(c(n,\lambda)-\varepsilon^{\prime})(p+1)d(x,0)}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\leq
C_{\varepsilon^{\prime}}\frac{t^{p+1}}{p+1}e^{-(c(n,\lambda)-\varepsilon^{\prime})(p+1)d(y,0)}\int_{0}^{\infty}e^{-\delta
r+(c(n,\lambda)-\varepsilon^{\prime})(p+1)r+(N-1)r}\mathrm{~{}d}r$
$\displaystyle\leq
C_{\varepsilon^{\prime}}\frac{M^{p+1}}{p+1}e^{-(c(n,\lambda)-\varepsilon^{\prime})(p+1)d(y,0)}.$

Thus we have deduced

$I\leq
C_{\varepsilon^{\prime}}\frac{M^{p+1}}{p+1}e^{-(c(n,\lambda)-\varepsilon^{\prime})(p+1)d(y,0)}.$
(6.8)

Now applying(6.6) and (6.8), we can choose $R_{0}>R>0$ large enough and also
choose ${\varepsilon}$ and $\varepsilon^{\prime}$ appropriately such that

$(I)<(II)\text{ for }d(y,0)\geq R_{0}.$

As a result, (6.1) is proved. This completes the proof (6.1). Now the proof of
(6.2) can be concluded in a similar line by noting that $(I)$ is zero and
$\underline{a}=1.$

∎

##  7\. Proof of Theorem 1.1 and Theorem 1.3

###  7.1. Existence of the first solution of ($\mathcal{P}$) for $a(x)$
satisfying $\left(\mathbf{A}_{1}\right)$ or $\left(\mathbf{A}_{3}\right)$

The below-mentioned proposition helps us establish the existence of the first
positive solution in the neighbourhood of 0.

######  Proposition 7.1.

For $d_{0}$ as chosen in Lemma 3.1 and $a(x)$ satisfying
$\left(\mathbf{A}_{1}\right)$ or $\left(\mathbf{A}_{3}\right)$, there exists
$r_{1}>0$ and $d_{1}\in(0,d_{0}]$ such that

  1. (i)

$I_{\lambda,a,f}(u)$ is strictly convex in $B\left(r_{1}\right)=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right)\right.$ :
$\left.\|u\|_{H_{\lambda}}<r_{1}\right\\}$.

  2. (ii)

If $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{1}$, then

$\inf_{\|u\|_{H_{\lambda}}=r_{1}}I_{\lambda,a,f}(u)>0.$

Moreover, there exists a unique critical point $\mathcal{U}_{a,f}(x)$ of
$I_{\lambda,a,f}(u)$ in $B\left(r_{1}\right)$. Furthermore,
$\mathcal{U}_{a,f}(x)$ satisfies

$\mathcal{U}_{a,f}(x)\in B\left(r_{1}\right)\text{ and
}I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)=\inf_{u\in
B\left(r_{1}\right)}I_{\lambda,a,f}(u).$

###### Proof.

We proceed to prove part $(i)$ as follows:

$I_{\lambda,a,f}^{\prime\prime}(u)(h,h)=\|h\|_{H_{\lambda}}^{2}-p\int_{\mathbb{B}^{N}}a(x)u_{+}^{p-1}h^{2}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$
(7.1)

Applying Hölder inequality, Sobolev inequality and the fact that $a\leq 1$ or
$a\equiv 1$, we get an estimate on the second term of RHS of (7.1) as follows

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

Thus using this above estimate in (7.1) yields

$I_{\lambda,a,f}^{\prime\prime}(u)(h,h)\geq\left(1-pS_{1,\lambda}^{-\frac{p+1}{2}}\|u\|_{H_{\lambda}}^{p-1}\right)\|h\|_{H_{\lambda}}^{2}.$

Defining $r_{1}=p^{-\frac{1}{p-1}}S_{1,\lambda}^{\frac{p+1}{2(p-1)}}$ results
in $I_{\lambda,a,f}^{\prime\prime}(u)$ being positive definite for $u\in
B\left(r_{1}\right)$. Therefore, $I_{\lambda,a,f}(u)$ is strictly convex in
$B\left(r_{1}\right)$. With this, we are done with the proof of part $(i)$.

$(ii)$ Assuming $\|u\|_{H_{\lambda}}=r_{1}$ gives

$\displaystyle I_{\lambda,a,f}(u)$
$\displaystyle=\frac{1}{2}\|u\|_{H_{\lambda}}^{2}-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)u_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)-\langle
f,u\rangle\geq\frac{1}{2}r_{1}^{2}-\frac{1}{p+1}S_{1,\lambda}^{-\frac{p+1}{2}}r_{1}^{p+1}$
$\displaystyle-r_{1}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p+1}S_{1,\lambda}^{-\frac{p+1}{2}}r_{1}^{p-1}\right)r_{1}^{2}-r_{1}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}.$

Further,

$I_{\lambda,a,f}(u)\geq\left(\frac{1}{2}-\frac{1}{p(p+1)}\right)r_{1}^{2}-r_{1}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)},$

where we have used $r_{1}^{p-1}=\frac{1}{p}S_{1,\lambda}^{\frac{p+1}{2}}$.
Thus there exists $d_{1}\in\left(0,d_{0}\right]$ such that

$\inf_{\|u\|_{H_{\lambda}}=r_{1}}I_{\lambda,a,f}(u)>0\quad\text{ for
}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{1}.$

Moreover, there exists a unique critical point $\mathcal{U}_{a,f}(x)$ of
$I_{\lambda,a,f}(u)$ in $B\left(r_{1}\right)$ because $I_{\lambda,a,f}(u)$ is
strictly convex in $B\left(r_{1}\right)$ and
$\inf_{\|u\|_{H_{\lambda}}=r_{1}}I_{\lambda,a,f}(u)>0=I_{\lambda,a,f}(0)$.
Furthermore, this critical point satisfies

$I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)=\inf_{\|u\|_{H_{\lambda}}<r_{1}}I_{\lambda,a,f}(u).$

This completes the proof of the proposition.
∎

###  7.2.  The case $a(x)\leq 1,$ $\mu\\{x:a(x)\neq 1\\}>0:$ Existence of
second and third solutions.

We now aim to prove the existence of the second and third positive solutions.
To fulfil this aim, we will utilize the Lusternik-Schnirelman Category theory,
a careful investigation of Palais-Smale characterization, and energy estimates
involving hyperbolic bubbles to prove the multiplicity result. The following
notation will be used to define level sets in the subsequent sections.

$\left[J_{\lambda,a,f}\leq c\right]=\left\\{v\in\tilde{\Sigma}_{+}\mid
J_{\lambda,a,f}(v)\leq c\right\\}$

for $c\in\mathbb{R}$. To compute the critical points of $J_{\lambda,a,f}(v)$,
we will show for a sufficiently small $\varepsilon>0$,

$\operatorname{cat}\left(\left[J_{\lambda,a,f}\leq
I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)-\varepsilon\right]\right)\geq
2$

where cat denotes Lusternik-Schnirelman Category.

We now study the properties of the functional $J_{\lambda,a,0}$ under the
condition $\mathbf{A}_{1}$.

######  Lemma 7.2.

Assume $a$ satisfies $0<a\in
L^{\infty}\left(\mathbb{B}^{N}\right),a(x)\rightarrow 1$ as
$d(x,0)\rightarrow\infty$, (1.4) and $\mathbf{A}_{1}$. Then there holds

  1. (i)

$\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,0}(v)=I_{\lambda,1,0}(w)$.

  2. (ii)

$\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,0}(v)$ is not attained.

  3. (iii)

$J_{\lambda,a,0}(v)$ satisfies $(PS)_{c}$ for
$c\in\left(-\infty,I_{\lambda,1,0}(w)\right)\cup\left(I_{\lambda,1,0}(w),2I_{\lambda,1,0}(w)\right)$.

###### Proof.

Using (3.8) and $\mathbf{A}_{\mathbf{1}}$, we immediately get

$\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,0}(v)\geq I_{\lambda,1,0}(w).$

Now define $w_{l}(x)=w(\tau_{le}(x))$ for a unit vector $e$ in
$\mathbb{R}^{N}$ and $0<l<1$ so that $le\in\mathbb{B}^{N}$. Moreover,
$l\rightarrow\infty$ in the disc model of the hyperbolic space means
$l\rightarrow 1$. Applying Lemma 3.2, corresponding to
$\bar{w}_{l}=\frac{w_{l}}{\left\|w_{l}\right\|}\in\tilde{\Sigma}_{+}$ implies
the existence of a unique $t_{a,0}\left(\bar{w}_{l}\right)$ such that

$J_{\lambda,a,0}\left(\frac{w_{l}}{\left\|w_{l}\right\|}\right)=I_{\lambda,a,0}\left(t_{a,0}\left(\bar{w}_{l}\right)\frac{w_{l}}{\left\|w_{l}\right\|}\right).$

Let us now determine the RHS of the above equation

$I_{\lambda,a,0}\left(t_{a,0}\left(\bar{w}_{l}\right)\frac{w_{l}}{\left\|w_{l}\right\|}\right)=\frac{t_{a,0}^{2}\left(\bar{w}_{l}\right)}{2}\left\|\bar{w}_{l}\right\|_{H_{\lambda}}^{2}-\frac{t_{a,0}^{p+1}\left(\bar{w}_{l}\right)}{p+1}\int_{\mathbb{B}^{N}}a(x)\left(\bar{w}_{l}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

Also, $t_{a,0}\left(\bar{w}_{l}\right)$ can be expressed in an explicit form
that occurs in the proof of Lemma 3.2 which is given by

$t_{a,0}\left(\bar{w}_{l}\right)=\left(\int_{\mathbb{B}^{N}}a(x)\bar{w}_{l}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right)^{-\frac{1}{p-1}}\stackrel{{\scriptstyle
l\rightarrow\infty}}{{\longrightarrow}}\left(\frac{\|w\|_{H_{\lambda}}}{\|w\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}}\right)^{\frac{p+1}{p-1}}.$

Since $w$ is the unique radial solution of (3.1), we further get

$\displaystyle J_{\lambda,a,0}\left(\bar{w}_{l}\right)\stackrel{{\scriptstyle
l\rightarrow\infty}}{{\longrightarrow}}$
$\displaystyle\frac{1}{2}\left\\{\frac{\|w\|_{H_{\lambda}}}{\|w\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}}\right\\}^{\frac{2(p+1)}{(p-1)}}-\frac{1}{p+1}\left(\left\\{\frac{\|w\|_{H_{\lambda}}}{\|w\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}}\right\\}^{\frac{(p+1)^{2}}{(p-1)}}\times\frac{\|w\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}}{\|w\|_{H_{\lambda}}^{p+1}}\right)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p+1}\right)\|w\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}=I_{\lambda,1,0}(w).$

Hence $(i)$ follows.
We will now show $(ii)$ by contradiction, i.e., let us assume that there
exists $v_{0}\in\tilde{\Sigma}_{+}$ such that
$J_{\lambda,a,0}\left(v_{0}\right)=\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,0}(v)=I_{\lambda,1,0}\left(w\right)$.
Define, the Nehari manifold $\mathcal{N}$ as

$\mathcal{N}:=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right):\left(I_{\lambda,1,0}\right)^{\prime}(u)(u)=0\right\\}.$

It is not difficult to find a $t_{v_{0}}>0$ such that
$t_{v_{0}}v_{0}\in\mathcal{N}$. Further, note that that for any
$v\in\mathcal{N}$, we have
$\|v\|_{H_{\lambda}}^{2}=\int_{\mathbb{B}^{N}}(v)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$,
and consequently,

$I_{\lambda,1,0}(v)=\frac{p-1}{2(p+1)}\|v\|_{H_{\lambda}}^{2}=\frac{p-1}{2(p+1)}\int_{\mathbb{B}^{N}}(v)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}\geq\frac{p-1}{2(p+1)}S_{1,\lambda}^{\frac{p+1}{p-1}},$

where $S_{1,\lambda}$ is as defined in (3.10). Thus $I_{\lambda,1,0}(v)\geq
I_{\lambda,1,0}\left(w\right)$ for all $v\in\mathcal{N}$. Moreover,
$w\in\mathcal{N}$, and hence

$\inf_{v\in\mathcal{N}}I_{\lambda,1,0}(v)=I_{\lambda,1,0}\left(w\right).$

Therefore,

$\displaystyle
I_{\lambda,1,0}\left(w\right)=J_{\lambda,a,0}\left(v_{0}\right):=$
$\displaystyle\max_{t>0}I_{\lambda,a,0}\left(tv_{0}\right)\geq
I_{\lambda,a,0}\left(t_{v_{0}}v_{0}\right)$ (7.2) $\displaystyle=$
$\displaystyle\frac{t_{v_{0}}^{2}}{2}\left\|v_{0}\right\|_{H_{\lambda}}^{2}-\frac{t_{v_{0}}^{p+1}}{p+1}\int_{\mathbb{B}^{N}}a(x)\left(v_{0}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=$
$\displaystyle\frac{t_{v_{0}}^{2}}{2}\left\|v_{0}\right\|_{H_{\lambda}}^{2}-\frac{t_{v_{0}}^{p+1}}{p+1}\int_{\mathbb{B}^{N}}\left(v_{0}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle+\frac{t_{v_{0}}^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))\left(v_{0}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle=$ $\displaystyle
I_{\lambda,1,0}\left(t_{v_{0}}v_{0}\right)+\frac{t_{v_{0}}^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))\left(v_{0}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle\geq$ $\displaystyle
I_{\lambda,1,0}\left(w\right)+\frac{t_{v_{0}}^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))\left(v_{0}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x).$

Thus the above inequality and $\mathbf{A}_{\mathbf{1}}$ result in

$\frac{t_{v_{0}}^{p+1}}{p+1}\int_{\mathbb{B}^{N}}(1-a(x))\left(v_{0}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)=0.$
(7.3)

Thus

$\left(v_{0}\right)_{+}\equiv 0\;\text{ in
}\left\\{x\in\mathbb{B}^{N}:a(x)\neq 1\right\\}.$ (7.4)

Moreover, the inequality in (7.2) becomes an equality by substituting (7.3)
into (7.2). Therefore,

$\inf_{\mathcal{N}}I_{\lambda,1,0}(v)=I_{\lambda,1,0}\left(w\right)=I_{\lambda,1,0}\left(t_{v_{0}}v_{0}\right).$

Thus $t_{v_{0}}v_{0}$ is a constraint critical point of $I_{\lambda,1,0}$.
Therefore $t_{v_{0}}v_{0}>0$ follows from the Lagrange multiplier and maximum
principle, which further implies $v_{0}>0$ in $\mathbb{B}^{N}$. This
contradicts (7.4). Hence $(2)$ holds.
The proof of part $(3)$ follows from the Palais-Smale decomposition. ∎

######  Lemma 7.3.

Let $a$ as in Theorem 1.1. Then there exists a constant $\delta_{0}>0$ such
that if $J_{\lambda,a,0}(v)\leq I_{\lambda,1,0}(w)+\delta_{0}$, then

$\int_{\mathbb{B}^{N}}\frac{x}{m(x)}|v(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\neq
0,$ (7.5)

where $m(x)>0$ is defined such that $d(\frac{x}{m},0)=\frac{1}{2}$, i.e.,
$m(x)=\frac{|x|}{\tanh\left(\frac{1}{4}\right)}.$

###### Proof.

Suppose on the contrary that there exists a sequence
$\left\\{v_{n}\right\\}\subset\tilde{\Sigma}_{+}$ such that

$J_{\lambda,a,0}\left(v_{n}\right)\leq I_{\lambda,1,0}(w)+\frac{1}{n}$ and
$\int_{\mathbb{B}^{N}}\frac{x}{m}|v_{n}(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0$ hold.

Then there exists $\tilde{v}_{n}\subset\tilde{\Sigma}_{+}$ by Ekeland’s
variational principle such that

$\displaystyle\left\|v_{n}-\tilde{v}_{n}\right\|_{H_{\lambda}}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0,$ $\displaystyle
J_{\lambda,a,0}\left(\tilde{v}_{n}\right)$ $\displaystyle\leq
J_{\lambda,a,0}\left(v_{n}\right)\leq I_{\lambda,1,0}(w)+\frac{1}{n},$
$\displaystyle J_{\lambda,a,0}^{\prime}\left(\tilde{v}_{n}\right)$
$\displaystyle\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0\text{ in
}H^{-1}\left(\mathbb{B}^{N}\right).$

The above implies $\left\\{\tilde{v}_{n}\right\\}$ is a Palais Smale sequence
for $J_{\lambda,a,0}$ at the level $I_{\lambda,1,0}(w)$.
Further, by Proposition 4.2, we have
$\left\\{y_{n}\right\\}\subset\mathbb{B}^{N}$ such that
$d(y_{n},0)\stackrel{{\scriptstyle n}}{{\rightarrow}}\infty$ and

$\left\|\tilde{v}_{n}-\frac{w\left(\tau_{-y_{n}}(x)\right)}{\left\|w\left(\tau_{-y_{n}}(x)\right)\right\|_{H^{1}\left(\mathbb{B}^{N}\right)}}\right\|_{H^{1}\left(\mathbb{B}^{N}\right)}{\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0}$

Therefore,

$\displaystyle\left\|v_{n}-\frac{w\left(\tau_{-y_{n}}(x)\right)}{\left\|w\left(\tau_{-y_{n}}(x)\right)\right\|_{H_{\lambda}}}\right\|_{H_{\lambda}}\leq$
$\displaystyle\left\|v_{n}-\tilde{v}_{n}\right\|_{H_{\lambda}}$
$\displaystyle+\left\|\tilde{v}_{n}-\frac{w\left(\tau_{-y_{n}}(x)\right)}{\left\|w\left(\tau_{-y_{n}}(x)\right)\right\|_{H_{\lambda}}}\right\|_{H_{\lambda}}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}0.$

Thus we can deduce

$\displaystyle\circ(1)$
$\displaystyle=\int_{\mathbb{B}^{N}}\frac{x}{m}|v_{n}(x)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)=\int_{\mathbb{B}^{N}}\tanh(\frac{1}{4})\frac{x}{|x|}\left(\frac{w\left(\tau_{-y_{n}}(x)\right)}{\left\|w\left(\tau_{-y_{n}}(x)\right)\right\|_{H_{\lambda}}}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}+\circ(1)$
$\displaystyle=\frac{\tanh(\frac{1}{4})}{\|w\|_{H_{\lambda}}^{p+1}}\int_{\mathbb{B}^{N}}\frac{\tau_{y_{n}}(y)}{|\tau_{y_{n}}(y)|}|w(y)|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(y)\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\not\longrightarrow}}\,0,\ \text{upto a subsequence}.$

Hence we have come to a contradiction. ∎

Finally, in this section, we state some refinement of Corollary 4.3.

######  Proposition 7.4.

Assume $a$ as in Lemma 1.1. Then for any $\varepsilon>0$, there exists
$d(\varepsilon)\in\left(0,d_{2}\right]$ such that for
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d(\varepsilon)$, the following
holds

  1. (i)

$\inf_{v\in\tilde{\Sigma}_{+}}J_{\lambda,a,f}(v)\in\left[I_{\lambda,1,0}(\omega)-\varepsilon,I_{\lambda,1,0}(\omega)+\varepsilon\right]$.

  2. (ii)

$J_{\lambda,a,f}(v)$ satisfies $(\mathrm{PS})_{c}$ for

$\displaystyle c\in$
$\displaystyle\left(-\infty,I_{\lambda,a,f}\left(u_{loc\min}(a,f;x)\right)+I_{\lambda,1,0}(\omega)\right)$
$\displaystyle\cup\left(I_{\lambda,a,f}\left(u_{\text{loc
}\min}(a,f;x)\right)+I_{\lambda,1,0}(\omega),2I_{\lambda,1,0}(\omega)-\varepsilon\right).$

Now Lusternik-Schnirelmn (L-S) category theory will help us find the second
and third positive solutions to ($\mathcal{P}$). Note that the $(L-S)$
category of $A$ with respect to $M$ is denoted by cat $(A,M)$. Particularly,
cat $(M)$ denotes cat $(M,M)$.
The following proposition is vital to obtain the second and third solutions to
($\mathcal{P}$).

######  Proposition 7.5.

Suppose $M$ is a Hilbert manifold and $\Psi\in C^{1}(M,\mathbb{R})$. Assume
that for $c_{0}\in\mathbb{R}$ and $k\in\mathbb{N}$

  1. (i)

$\Psi(x)$ satisfies $(PS)_{c}$ for $c\leq c_{0}.$

  2. (ii)

$\operatorname{cat}\left(\left\\{x\in M:\Psi(x)\leq c_{0}\right\\}\right)\geq
k.$

Then $\Psi(x)$ has at least $k$ critical points in $\left\\{x\in M:\Psi(x)\leq
c_{0}\right\\}$.

######  Lemma 7.6.

([2], Lemma 2.5) Let $N\geq 1$ and $M$ be a topological space. Assume that
there exist two continuous mappings

$F:S^{N-1}_{\mathbb{B}^{N}}\left(:=\left\\{x\in\mathbb{B}^{N}:d(x,0)=1\right\\}\right)\rightarrow
M,\quad G:M\rightarrow S^{N-1}_{\mathbb{B}^{N}}$

such that $G\circ F$ is homotopic to the identity map Id:
$S^{N-1}_{\mathbb{B}^{N}}\rightarrow S^{N-1}_{\mathbb{B}^{N}}$, i.e, there is
a continuous $\operatorname{map}\eta:[0,1]\times
S^{N-1}_{\mathbb{B}^{N}}\rightarrow S^{N-1}_{\mathbb{B}^{N}}$ such that

$\displaystyle\eta(0,x)=(G\circ F)(x)\text{ for all }x\in
S^{N-1}_{\mathbb{B}^{N}}$ $\displaystyle\eta(1,x)=x\text{ for all }x\in
S^{N-1}_{\mathbb{B}^{N}}$

Then cat $(M)\geq 2$.

Taking into account the above lemma, our next goal will be to construct two
mappings:

$\displaystyle F:S^{N-1}_{\mathbb{B}^{N}}\rightarrow\left[J_{\lambda,a,f}\leq
I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)-\varepsilon\right],$
$\displaystyle G:\left[J_{\lambda,a,f}\leq
I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)-\varepsilon\right]\rightarrow
S^{N-1}_{\mathbb{B}^{N}}$

such that $G\circ F$ is homotopic to the identity map.

Let us define $F_{R}:S^{N-1}_{\mathbb{B}^{N}}\rightarrow\tilde{\Sigma}_{+}$ as
follows:
For $d(y,0)\geq R_{0}$, where $R_{0}$ is as found in Proposition 6.1, (6.1)
holds for all $t>0$. For $d(y,0)\geq R_{0}$, we will find $s=s(f,y)$ such that

$\displaystyle\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))=$ $\displaystyle
t_{a,f}\left(\frac{\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))}{\left\|\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))\right\|_{H_{\lambda}}}\right)$
$\displaystyle\times\frac{\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))}{\left\|\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))\right\|_{H_{\lambda}}}.$

This implies

$\left\|\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))\right\|_{H_{\lambda}}=t_{a,f}\left(\frac{\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))}{\left\|\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))\right\|_{H_{\lambda}}}\right).$
(7.6)

Therefore,

$\displaystyle
J_{\lambda,a,f}\left(\frac{\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))}{\left\|\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))\right\|_{H_{\lambda}}}\right)$
$\displaystyle=I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)+sw(\tau_{-y}(x))\right)$
$\displaystyle<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w).$

######  Proposition 7.7.

([2], Proposition 2.6) Assume $a$ as in Theorem 1.1. Then there exists
$d_{3}\in\left(0,d_{2}\right]$ and $R_{1}>R_{0}$ such that for any
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{3}$ and any $d(y,0)\geq
R_{1}$, there exists a unique $s=s(f,y)>0$ in a neighbourhood of 1, satisfying
(7.6). In addition,

$\left\\{y\in\mathbb{B}^{N}:d(y,0)>R_{1}\right\\}\rightarrow(0,\infty);\quad
y\mapsto s(f,y)$

is continuous.

Now we define a function
$F_{R}:S^{N-1}_{\mathbb{B}^{N}}\rightarrow\tilde{\Sigma}_{+}$by

$F_{R}(y)=\frac{\mathcal{U}_{a,f}(x)+s(f,\frac{\tanh(\frac{R}{2})}{\tanh\frac{1}{2}}y)w(\tau_{-\frac{\tanh(\frac{R}{2})}{\tanh\frac{1}{2}}y}(x))}{\left\|{\mathcal{U}_{a,f}(x)+s(f,\frac{\tanh(\frac{R}{2})}{\tanh\frac{1}{2}}y)w(\tau_{-\frac{\tanh(\frac{R}{2})}{\tanh\frac{1}{2}}y}(x))}\right\|_{H_{\lambda}}}$

for $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{3}$ and $R\geq R_{1}$.
Then we have,

######  Proposition 7.8.

For $0<\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{3}$ and $R\geq R_{1}$,
there exists $\varepsilon_{0}=\varepsilon_{0}(R)>0$ such that

$F_{R}\left(S^{N-1}_{\mathbb{B}^{N}}\right)\subseteq\left[J_{\lambda,a,f}\leq
I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)-\varepsilon_{0}\right].$

###### Proof.

The following expression follows from the construction of $F_{R}$

$F_{R}\left(S^{N-1}_{\mathbb{B}^{N}}\right)\subseteq\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)\right]$

Hence the proposition follows as $F\left(S^{N-1}_{\mathbb{B}^{N}}\right)$ is
compact. ∎

Thus we construct a mapping

$F_{R}:S^{N-1}_{\mathbb{B}^{N}}\rightarrow\left[J_{\lambda,a,f}\leq
I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)-\varepsilon_{0}(R)\right]$

Now the following lemma is crucial for constructing the mapping $G$.

######  Lemma 7.9.

There exists $d_{4}\in\left(0,d_{3}\right]$ such that if
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq d_{4}$, then

$\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)\right]\subseteq\left[J_{\lambda,a,0}<I_{\lambda,1,0}(w)+\delta_{0}\right]$
(7.7)

where $\delta_{0}>0$ is as found in lemma 7.3.

###### Proof.

For any $\varepsilon\in(0,1)$, the following holds using (3.12)

$J_{\lambda,a,0}(v)\leq(1-\varepsilon)^{-\frac{p+1}{p-1}}\left(J_{\lambda,a,f}(v)+\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}\right)\text{
for all }v\in\tilde{\Sigma}_{+}.$ (7.8)

Now, if

$v\in\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)\right],$

then

$J_{\lambda,a,f}(v)<I_{\lambda,1,0}(w)$

because $I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)\leq 0$.
Therefore, (7.8) implies

$J_{\lambda,a,0}(v)\leq(1-\varepsilon)^{-\frac{p+1}{p-1}}\left(I_{\lambda,1,0}(w)+\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}\right),$

for all $v\in\left[J_{\lambda,a,f}\leq
I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)\right].$

Thus
$v\in\left[J_{\lambda,a,0}\leq(1-\varepsilon)^{-\frac{p+1}{p-1}}\left(I_{\lambda,1,0}(w)+\frac{1}{2\varepsilon}\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}^{2}\right)\right]$.
Since $\varepsilon\in(0,1)$ is arbitrary, we get

$v\in\left[J_{\lambda,a,0}<I_{\lambda,1,0}(w)+\delta_{0}\right]\text{ for
sufficiently small }\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}.$

Hence (7.7) follows. ∎

We are now in a position to define the function $G$ as follows:

$G:\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)\right]\rightarrow
S^{N-1}_{\mathbb{B}^{N}}$
$G(v):=tanh(\frac{1}{2})\frac{\int_{\mathbb{B}^{N}}\frac{x}{m}|v|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)}{\left|\int_{\mathbb{B}^{N}}\frac{x}{m}|v|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)\right|}$

where $m$ as defined in Lemma 7.3, and the above function is well defined
again by Lemma 7.3 and by Lemma 7.9. Besides, we will show that these
developments, i.e., $F$ and $G$ will serve our purpose.

######  Proposition 7.10.

For a sufficiently large $R\geq R_{1}$ and for sufficiently small
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}>0$, we have,

$G\circ F_{R}:S^{N-1}_{\mathbb{B}^{N}}\rightarrow S^{N-1}_{\mathbb{B}^{N}}$

is homotopic to identity.

###### Proof.

The proof follows as in [2]. ∎

We are now in a situation to establish our main results:

######  Proposition 7.11.

For sufficiently large $R\geq R_{1}$,

$\text{ cat
}\left(\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)\right)+I_{\lambda,1,0}(w)-\varepsilon_{0}(R)\right]\right)\geq
2$

###### Proof.

The proof of the proposition follows by combining Lemma 7.6 and Proposition
7.10. ∎

The above proposition led us to the following multiplicity results.

######  Theorem 7.12.

Let $a$ satisfies the assumptions as in Theorem 1.1. Then there exists
$d_{5}>0$ such that if $\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\leq
d_{5},\;f\geq 0,\;f\not\equiv 0$, then $J_{\lambda,a,f}(v)$ has at least two
critical points in

$\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)(a,f;x)\right)+I_{\lambda,1,0}(w)\right]$

###### Proof.

Combining Corollary 4.3, Proposition 7.11, and Proposition 7.5, the theorem
follows. ∎

We can now finish the proof of Theorem 1.1 as follows:
Firstly, set $u^{(1)}(x)=\mathcal{U}_{a,f}(x)$ as found in Proposition 7.1.
Also, using (4.2) $u^{(1)}(x)$ satisfies

$I_{\lambda,a,f}\left(u^{(1)}(x)\right)\leq 0.$

By Theorem 7.12, $J_{\lambda,a,f}(v)$ has at least two critical points
$v^{(2)}(x),v^{(3)}(x)$ in

$\left[J_{\lambda,a,f}<I_{\lambda,a,f}\left(\mathcal{U}_{a,f}(x)(a,f;x)\right)+I_{\lambda,1,0}(\omega)\right].$

Then
$u^{(2)}(x)=t_{a,f}\left(v^{(2)}\right)v^{(2)}(x),u^{(3)}(x)=t_{a,f}\left(v^{(3)}\right)v^{(3)}(x)$
will be the corresponding solutions to ($\mathcal{P}$) using Proposition 3.3.
Moreover, by Lemma 3.1, we get

$\displaystyle 0$
$\displaystyle<I_{\lambda,a,f}\left(u^{(k)}(x)\right)=J_{\lambda,a,f}\left(v^{(k)}(x)\right)$
$\displaystyle<I_{\lambda,a,f}\left(u^{(1)}(x)\right)+I_{\lambda,1,0}(\omega)\quad\text{
for }k=2,3.$

Hence $u^{(1)}(x),u^{(2)}(x),u^{(3)}(x)$ are distinct, and $\mathcal{P}$
possesses at least three positive solutions.

###  7.3.  The case $a(x)\equiv 1:$ Existence of the second solution

The Remark 3.1 suggests that we need to find the critical points of the energy
functional $I_{\lambda,1,f}$ to guarantee the existence of solutions to
($\mathcal{P^{{}^{\prime}}}$).

###### Proof.

There exists $r_{1}>0$ such that

$I_{\lambda,1,f}(u)>0\quad\text{ for }u\in S_{r_{1}}=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right)\mid\|u\|=r_{1}\right\\},$ (7.9)

where $r_{1}$ is as found in Proposition 7.1. Also, using Proposition 7.1 and
(4.2), we found a positive solution $\mathcal{U}_{1,f}(x)$ of
($\mathcal{P^{{}^{\prime}}}$) in $B\left(r_{1}\right)$ with
$I_{\lambda,1,f}\left(\mathcal{U}_{1,f}(x)\right)\leqslant 0$.
Now fix $y$ such that (6.2) holds. Further, it is not difficult to find
$t_{0}>0$ such that
$I_{\lambda,1,f}\left(\mathcal{U}_{1,f}(x)+tw\left(\tau_{y}(x)\right)\right)<0$
and $\|\mathcal{U}_{1,f}(x)+tw\left(\tau_{y}(x)\right)\|_{H_{\lambda}}>r_{1}$
for $t\geqslant t_{0}$.

Set

$\displaystyle\Gamma=\left\\{\gamma\in
C\left([0,1],H^{1}\left(\mathbb{B}^{N}\right)\right)\mid\gamma(0)=\mathcal{U}_{1,f},\;\gamma(1)=\mathcal{U}_{1,f}+t_{0}w\left(\tau_{y}\right)\right\\},$
$\displaystyle c=\inf_{\gamma\in\Gamma}\max_{s\in[0,1]}I(\gamma(s)).$

Moreover, we have

$0<c=\inf_{\gamma\in\Gamma}\max_{s\in[0,1]}I(\gamma(s))<I_{\lambda,1,f}\left(\mathcal{U}_{1,f}(x)\right)+I_{\lambda,1,0}(w),$
(7.10)

which follows from (7.9) and 6.2.
Thus applying the mountain-pass theorem of Ambrosetti and Rabinowitz and then
using PS characterization (4), we get a solution of
($\mathcal{P^{{}^{\prime}}}$), say $\mathcal{V}_{1,f}$, such that

$c=I_{\lambda,1,f}\left(\mathcal{V}_{1,f}(x)\right)+mI_{\lambda,1,0}(w),$
(7.11)

for some non-negative integer $m$. Furthermore, 7.11 and 7.10 imply
$\mathcal{U}_{1,f}\neq\mathcal{V}_{1,f}$.
With this, we have finished the proof of Theorem 1.3. ∎

##  8\. Proof of Theorem 1.2

In this section, we prove Theorem 1.2 by finding two positive critical points
of the functional $I_{\lambda,a,f}$ (as defined in (3.3)). We essentially
follow the approach in the spirit of Jeanjean [21]. Towards that, we partition
$H^{1}\left(\mathbb{B}^{N}\right)$ into the following three disjoint sets:

$\begin{gathered}U_{1}:=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right):u=0\text{ or }g(u)>0\right\\},\quad
U_{2}:=\left\\{u\in H^{1}\left(\mathbb{B}^{N}\right):g(u)<0\right\\},\\\
U:=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right)\backslash\\{0\\}:g(u)=0\right\\}\end{gathered}$

where $g:H^{1}\left(\mathbb{B}^{N}\right)\rightarrow\mathbb{R}$ is defined as

$g(u):=\|u\|_{H_{\lambda}}^{2}-p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}.$

######  Remark 8.1.

Observe that $\|u\|_{H_{\lambda}}$ and
$\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}$ are bounded away from 0 for all
$u\in U$. It follows from the fact that $p>1$ and Poincaré-Sobolev inequality
on the hyperbolic space.

Further, define

$c_{0}:=\inf_{U_{1}}I_{\lambda,a,f}(u)\quad\text{ and }\quad
c_{1}:=\inf_{U}I_{\lambda,a,f}(u).$ (8.1)

######  Remark 8.2.

Clearly,
$g(tu)=t^{2}\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{2}-t^{p+1}p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}$
for any $t>0$. Moreover, for $u\in H^{1}\left(\mathbb{B}^{N}\right)$ with
$\|u\|_{H_{\lambda}}=1$, there exists unique $t=t(u)$ such that $tu\in U$. On
the other hand, $g(tu)=\left(t^{2}-t^{p+1}\right)\|u\|_{H_{\lambda}}^{2}$ for
any $u\in U$. Thus

$tu\in U_{1}\text{ for all }t\in(0,1)\quad\text{ and }\quad tu\in U_{2}\text{
for all }t>1\text{. }$

######  Lemma 8.1.

The following inequality holds $\forall u\in U$,

$\frac{p-1}{p}\|u\|_{H_{\lambda}}\geq
C_{p}S_{1,\lambda}^{\frac{p+1}{2(p-1)}},$

where $S_{1,\lambda}$ as defined in (3.10) and $C_{p}$ as defined in Theorem
1.2.

###### Proof.

As $u\in U$, we get
$\|u\|_{L^{p+1}}=\frac{\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{\frac{2}{p+1}}}{\left(p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\right)^{\frac{1}{p+1}}}.$
This, together with the definition of $S_{1,\lambda}$, gives

$\|u\|_{H_{\lambda}}\geq
S_{1,\lambda}^{\frac{1}{2}}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}=S_{1,\lambda}^{\frac{1}{2}}\frac{\|u\|_{H_{\lambda}}^{\frac{2}{p+1}}}{\left(p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\right)^{\frac{1}{p+1}}}\quad\forall
u\in U.$

Therefore, for all $u\in U$, we have

$\|u\|_{H_{\lambda}}\geq\frac{S_{1,\lambda}^{\frac{p+1}{2(p-1)}}}{\left(p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\right)^{\frac{1}{p-1}}}=\frac{p}{p-1}C_{p}S_{1,\lambda}^{\frac{p+1}{2(p-1)}}.$

Thus the lemma follows. ∎

######  Lemma 8.2.

Suppose

$\inf_{u\in
H^{1}\left(\mathbb{B}^{N}\right),\;\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)=1}}\left\\{C_{p}\|u\|_{H_{\lambda}}^{\frac{2p}{p-1}}-\langle
f,u\rangle\right\\}>0,$ (8.2)

where $C_{p}$ is defined in Theorem 1.2. Then $c_{0}<c_{1}$, where $c_{0}$ and
$c_{1}$ are as defined in (8.1).

###### Proof.

Define,

$\tilde{J}(u):=\frac{1}{2}\|u\|_{H_{\lambda}}^{2}-\frac{\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}}{p+1}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}-\langle
f,u\rangle,\quad u\in H^{1}\left(\mathbb{B}^{N}\right).$ (8.3)

  1. Step 1:

This step aims to show the existence of a constant $\alpha>0$ such that

$\left.\frac{d}{dt}\tilde{J}(tu)\right|_{t=1}\geq\alpha\quad\forall u\in U.$

It directly follows from the definition of $\tilde{J}$ that

$\left.\frac{d}{dt}\tilde{J}(tu)\right|_{t=1}=\|u\|_{H_{\lambda}}^{2}-\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}-{\langle
f,u\rangle}.$

Therefore, from the definition of $U$ and substituting the value of $C_{p}$,
we have for $u\in U$

$\displaystyle\left.\frac{d}{dt}\tilde{J}(tu)\right|_{t=1}=\frac{p-1}{p}\|u\|_{H_{\lambda}}^{2}-{\langle
f,u\rangle}$
$\displaystyle=\left(p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\right)^{\frac{1}{p-1}}C_{p}\|u\|_{H_{\lambda}}^{2}-\langle
f,u\rangle$ (8.4)
$\displaystyle=\left(\frac{\|u\|_{H_{\lambda}}^{2}}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}}\right)^{\frac{1}{p-1}}C_{p}\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{2}-\langle
f,u\rangle$
$\displaystyle=C_{p}\frac{\|u\|_{H_{\lambda}}^{\frac{2p}{p-1}}}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{\frac{p+1}{p-1}}}-\langle
f,u\rangle.$

Furthermore, the given hypothesis, i.e., (8.2) implies there exists $d>0$ such
that

$\inf_{u\in
H^{1}\left(\mathbb{B}^{N}\right),\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)=1}}\left\\{C_{p}\|u\|_{H_{\lambda}}^{\frac{2p}{p-1}}-\langle
f,u\rangle\right\\}\geq d.$ (8.5)

Now,

$\displaystyle\eqref{4bbb}$ $\displaystyle\Longleftrightarrow
C_{p}\frac{\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{\frac{2p}{p-1}}}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{\frac{p+1}{p-1}}}-\langle
f,u\rangle\geq d,\quad\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}=1$
$\displaystyle\Longleftrightarrow
C_{p}\frac{\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{\frac{2p}{p-1}}}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{\frac{p+1}{p-1}}}-\langle
f,u\rangle\geq d\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)},\quad u\in
H^{1}\left(\mathbb{B}^{N}\right)\backslash\\{0\\}.$

Hence, step 1 follows by using the above estimate in (8.4) and by Remark
(8.1).

  2. Step 2:

Let $u_{n}$ be a minimizing sequence for $I_{\lambda,a,f}$ on $U$, i.e.,
$I_{\lambda,a,f}\left(u_{n}\right)\rightarrow c_{1}$ and
$\left\|u_{n}\right\|_{H_{\lambda}}^{2}=p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\left\|u_{n}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}$.
Thus for $n$ large, we get

$c_{1}+o(1)\geq
I_{\lambda,a,f}\left(u_{n}\right)\geq\tilde{J}\left(u_{n}\right)\geq\left(\frac{1}{2}-\frac{1}{p(p+1)}\right)\left\|u_{n}\right\|_{H_{\lambda}}^{2}-\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\left\|u_{n}\right\|_{H_{\lambda}}.$

As a result, $\left\\{\tilde{J}\left(u_{n}\right)\right\\}$ is a bounded
sequence. Also, $\left\|u_{n}\right\|_{H_{\lambda}}$ and
$\left\|u_{n}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}$ are bounded.
Claim: $c_{0}<0$.
To prove the above claim, it suffices to show that there exists $v\in U_{1}$
such that $I_{\lambda,a,f}(v)<0$. Remark (8.2) implies we can choose $u\in U$
such that $\langle f,u\rangle>0$. Therefore,

$I_{\lambda,a,f}(tu)\leq
t^{2}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\left[\frac{p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}}{2}-\frac{t^{p-1}}{p+1}\right]-t\langle
f,u\rangle<0.$

for $t<<1$. Moreover, by Remark (8.2), $tu\in U_{1}$. This proves the claim.

Now $I_{\lambda,a,f}\left(u_{n}\right)<0$ for large $n$ by using the above
claim. Consequently,

$0>I_{\lambda,a,f}\left(u_{n}\right)\geq\left(\frac{1}{2}-\frac{1}{p(p+1)}\right)\left\|u_{n}\right\|_{H_{\lambda}}^{2}-\left\langle
f,u_{n}\right\rangle.$

Therefore, $p>1$ implies $\left\langle f,u_{n}\right\rangle>0$ for all large
$n$ . As a a result, $\frac{d}{dt}\tilde{J}\left(tu_{n}\right)<0$ for $t>0$
small enough. Thus, by Step 1, there exists $t_{n}\in(0,1)$ such that
$\frac{d}{dt}\tilde{J}\left(t_{n}u_{n}\right)=0$. In addition, $t_{n}$ is
unique because

$\frac{d^{2}}{dt^{2}}\tilde{J}(tu)=\|u\|_{H_{\lambda}}^{2}-p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}t^{p-1}\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}=\left(1-t^{p-1}\right)\|u\|_{H_{\lambda}}^{2}>0,\;\forall
u\in U,\;\forall t\in[0,1).$

  3. Step 3:

The goal of this step is to prove the following

$\liminf_{n\rightarrow\infty}\left\\{\tilde{J}\left(u_{n}\right)-\tilde{J}\left(t_{n}u_{n}\right)\right\\}>0.$
(8.6)

We can notice that
$\tilde{J}\left(u_{n}\right)-\tilde{J}\left(t_{n}u_{n}\right)=\int_{t_{n}}^{1}\frac{d}{dt}\left\\{\tilde{J}\left(tu_{n}\right)\right\\}\mathrm{d}t$
and that for all $n\in\mathbb{N}$, there is $\xi_{n}>0$ such that
$t_{n}\in\left(0,1-2\xi_{n}\right)$ and
$\frac{d}{dt}\tilde{J}\left(tu_{n}\right)\geq\alpha$ for
$t\in\left[1-\xi_{n},1\right]$.

To prove (8.6), it is enough to show that $\xi_{n}>0$ can be chosen
independent of $n\in\mathbb{N}$. But this is true because, by step 1, we have
$\left.\frac{d}{dt}\tilde{J}\left(tu_{n}\right)\right|_{t=1}\geq\alpha$.
Moreover, the boundedness of $\left\\{u_{n}\right\\}$ gives

$\left|{\frac{d^{2}}{dt^{2}}\tilde{J}\left(tu_{n}\right)}\right|=\left|\left\|u_{n}\right\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{2}-p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}t^{p-1}\left\|u_{n}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\right|=\left|\left(1-t^{p-1}\right)\left\|u_{n}\right\|_{H_{\lambda}}^{2}\right|\leq
C,$

for all $n\geq 1$ and $t\in[0,1]$.

  4. Step 4:

It straight away follows from the definition of $I_{\lambda,a,f}$ and
$\tilde{J}$ that
$\frac{d}{dt}I_{\lambda,a,f}(tu)\geq\frac{d}{dt}\tilde{J}(tu)$ for all $u\in
H^{1}\left(\mathbb{B}^{N}\right)$ and for all $t>0$. Therefore,

$I_{\lambda,a,f}\left(u_{n}\right)-I_{\lambda,a,f}\left(t_{n}u_{n}\right)=\int_{t_{n}}^{1}\frac{d}{dt}\left(I_{\lambda,a,f}\left(tu_{n}\right)\right)\mathrm{d}t\geq\int_{t_{n}}^{1}\frac{d}{dt}\tilde{J}\left(tu_{n}\right)\mathrm{d}t=\tilde{J}\left(u_{n}\right)-\tilde{J}\left(t_{n}u_{n}\right).$

Since $\left\\{u_{n}\right\\}\in U$ is a minimizing sequence for
$I_{\lambda,a,f}$, and $t_{n}u_{n}\in U_{1}$, we deduce using (8.6) that

$c_{0}=\inf_{u\in U_{1}}I_{\lambda,a,f}(u)<\inf_{u\in
U}I_{\lambda,a,f}(u)=c_{1}$

This completes the proof of the lemma. ∎

It is worth mentioning explicitly the problem at infinity corresponding to
(3.4) :

$\begin{gathered}-\Delta_{\mathbb{B}^{N}}w-\lambda w=w_{+}^{p},\;\text{ in
}\mathbb{B}^{N},\;w\in H^{1}\left(\mathbb{B}^{N}\right).\end{gathered}$ (8.7)

and the associated functional
$I_{\lambda,1,0}:H^{1}\left(\mathbb{B}^{N}\right)\rightarrow\mathbb{R}$
defined by

$I_{\lambda,1,0}(u)=\frac{1}{2}\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}^{2}-\frac{1}{p+1}\int_{\mathbb{B}^{N}}u_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}.$

Define,

$X_{1}:=\left\\{u\in
H^{1}\left(\mathbb{B}^{N}\right)\backslash\\{0\\}:\left(I_{\lambda,1,0}\right)^{\prime}(u)=0\right\\},\quad
S^{\infty}:=\inf_{X_{1}}I_{\lambda,1,0}.$ (8.8)

######  Remark 8.3.

We can easily see
$I_{\lambda,1,0}(u)=\frac{p-1}{2(p+1)}\|u\|_{H_{\lambda}}^{2}$ on $X_{1}$.
Further,(3.10) also gives $\|u\|_{H_{\lambda}}^{2}\geq
S_{1,\lambda}^{\frac{p+1}{p-1}}$ on $X_{1}$. Consequently, $S^{\infty}\geq$
$\frac{p-1}{2(p+1)}S_{1,\lambda}^{\frac{p+1}{p-1}}>0$. Moreover, it is known
from [24] that $S_{1,\lambda}$ is achieved by unique positive radial solution
$w$ of (3.1). Therefore,

$I_{\lambda,1,0}\left(w\right)=\frac{p-1}{2(p+1)}S_{1,\lambda}^{\frac{p+1}{p-1}}.$

Thus $S^{\infty}$ is achieved by $w$.

######  Proposition 8.3.

Suppose (8.2) and all the assumptions in the Theorem 1.2 hold. Then there
exists a critical point $u_{0}\in U_{1}$ of $I_{\lambda,a,f}$ such that
$I_{\lambda,a,f}\left(u_{0}\right)=c_{0}$. In particular, $u_{0}$ is a weak
positive solution to ($\mathcal{P}$).

###### Proof.

We divide the proof into the following few steps.

  1. Step 1:

$c_{0}>-\infty.$
As $I_{\lambda,a,f}(u)\geq\tilde{J}(u)$ so, to prove Step 1, it is enough to
show that $\tilde{J}$ is bounded from below. The definition of $U_{1}$ implies

$\tilde{J}(u)\geq\left[\frac{1}{2}-\frac{1}{p(p+1)}\right]\|u\|_{H_{\lambda}}^{2}-\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\|u\|_{H_{\lambda}}\text{
for all }u\in U_{1}.$ (8.9)

Since the RHS of the above inequality is a quadratic function in
$\|u\|_{H_{\lambda}}$ implies $\tilde{J}$ is bounded from below. Hence Step 1
follows.

  2. Step 2:

We aim to find a bounded PS sequence $\left\\{u_{n}\right\\}\subset U_{1}$ for
$I_{\lambda,a,f}$ at the level $c_{0}$.
Let $\left\\{u_{n}\right\\}\subset\bar{U}_{1}$ such that
$I_{\lambda,a,f}\left(u_{n}\right)\rightarrow c_{0}$. As
$I_{\lambda,a,f}(u)\geq\tilde{J}(u)$ so, from (8.9), we get
$\left\\{u_{n}\right\\}$ is a bounded sequence. Since by Lemma 8.2,
$c_{0}<c_{1}$, without restriction we can assume $u_{n}\in U_{1}$. Therefore,
by Ekeland’s variational principle, we can extract a PS sequence from
$\left\\{u_{n}\right\\}$ in $U_{1}$ for $I_{\lambda,a,f}$ at the level
$c_{0}$. We still denote this PS sequence by $\left\\{u_{n}\right\\}$. Thus
step 2 follows.

  3. Step 3:

In this step, we show that there exists $u_{0}\in U_{1}$ such that
$u_{n}\rightarrow u_{0}$ in $H^{1}\left(\mathbb{B}^{N}\right)$.
Applying PS decomposition (4) gives

$u_{n}-u_{0}-\sum_{i=1}^{m}w^{i}\left(\tau_{n}^{i}(x)\right)\rightarrow
0\text{ in }H^{1}\left(\mathbb{B}^{N}\right)$ (8.10)

for some $u_{0}$ such that
$\left(I_{\lambda,a,f}\right)^{\prime}\left(u_{0}\right)=0$ and some
appropriate $w^{i}$ and $\left\\{\tau_{n}^{i}\right\\}$. We will proceed by
the method of contradiction to show that $m=0$, which in turn will imply step
3. Assume that there is $w^{i}\neq 0$ for $i\in\\{1,2,\cdots,m\\}$ such that
$\left(I_{\lambda,1,0}\right)^{\prime}\left(w^{i}\right)=0$, i.e,
$\left\|w^{i}\right\|_{H_{\lambda}}^{2}=\int_{\mathbb{B}^{N}}\left(w^{i}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$.
Therefore,

$\displaystyle g\left(w^{i}\right)$
$\displaystyle=\left\|w^{i}\right\|_{H_{\lambda}}^{2}-p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\left\|w^{i}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}$
$\displaystyle=\int_{\mathbb{B}^{N}}\left(w^{i}\right)_{+}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\int_{\mathbb{B}^{N}}\left|w^{i}\right|^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}$
$\displaystyle\leq\left\|w^{i}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\left(1-p\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\right)<0,$

where for the last inequality, we have used that $p>1$ and
$\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\geq 1$. Further, using the
Remark 8.3, we get $I_{\lambda,1,0}\left(w^{i}\right)\geq S^{\infty}>0$ for
all $1\leq i\leq m$. Therefore, $I_{\lambda,a,f}\left(u_{n}\right)\rightarrow
I_{\lambda,a,f}\left(u_{0}\right)+\sum_{i=1}^{m}I_{\lambda,1,0}\left(w_{i}\right)$
implies $I_{\lambda,a,f}\left(u_{0}\right)<c_{0}$. Thus $u_{0}\notin U_{1}$,
i.e., $g\left(u_{0}\right)\leq 0.$

We have $g\left(u_{n}\right)\geq 0$ because $u_{n}\in U_{1}$. We now compute
$g\left(u_{0}+\sum_{i=1}^{m}w^{i}\left(\tau_{n}^{i}(x)\right)\right)$. Thus
(8.10) and uniform continuity of $g$ implies

$0\leq\liminf_{n\rightarrow\infty}g\left(u_{n}\right)=\liminf_{n\rightarrow\infty}g\left(u_{0}+\sum_{i=1}^{m}w^{i}\left(\tau_{n}^{i}(x)\right)\right).$
(8.11)

On the other hand, as
$\tau_{n}^{i}(0)\rightarrow\infty,\;d(\tau_{n}^{i}(0),\tau_{n}^{j}(0))\rightarrow\infty$
for $1\leq i\neq j\leq m$ the supports of $u_{0}(\bullet)$ and
$w^{i}\left(\tau_{n}^{i}(\bullet)\right)$ are going increasingly far away as
$n\rightarrow\infty$. Therefore,

$\lim_{n\rightarrow\infty}g\left(u_{0}+\sum_{i=1}^{m}w^{i}\left(\tau_{n}^{i}(x)\right)\right)=g\left(u_{0}\right)+\lim_{n\rightarrow\infty}\sum_{i=1}^{m}g\left(w^{i}\left(\tau_{n}^{i}(x)\right)\right)=g\left(u_{0}\right)+\sum_{i=1}^{m}g\left(w^{i}\right),$

where the last equality follows from the translation invariance of $g$. Now
because $g\left(u_{0}\right)\leq 0$ and $g\left(w^{i}\right)<0$ for $1\leq
i\leq m$, we get a contradiction to (8.11). This proves step 3.

  4. Step 4:

Using the previous steps, we can conclude that
$I_{\lambda,a,f}\left(u_{0}\right)=c_{0}$ and
$\left(I_{\lambda,a,f}\right)^{\prime}\left(u_{0}\right)=0$. Thus, $u_{0}$ is
a weak solution to (3.4); combining this with Remark 3.1, we complete the
proof of the proposition.

∎

######  Proposition 8.4.

Assume (8.2) holds. Then $I_{\lambda,a,f}$ has a second critical point
$v_{0}\neq u_{0}$. In particular, $v_{0}$ is a positive solution to
($\mathcal{P}$).

###### Proof.

For $u_{0}$ to be the critical point found in Proposition 8.3 and $w$ to be as
in Remark 8.3, set $w_{t}(x):=tw\left(x\right).$

Claim 1: $u_{0}+w_{t}\in U_{2}$ for $t>0$ large enough.
As $p>1$ and $\|a\|_{L^{\infty}\left(\mathbb{B}^{N}\right)}\geq 1$, we have

$\displaystyle g\left(u_{0}+w_{t}\right)$
$\displaystyle\leq\left\|u_{0}\right\|_{H_{\lambda}}^{2}+\left\|w_{t}\right\|_{H_{\lambda}}^{2}+2\left\langle
u_{0},w_{t}\right\rangle_{H_{\lambda}}-p\left(\left\|u_{0}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}+\left\|w_{t}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\right)$
$\displaystyle\leq(1+\varepsilon)\left\|w_{t}\right\|_{H_{\lambda}}^{2}+(1+C(\varepsilon))\left\|u_{0}\right\|_{H_{\lambda}}^{2}-p\left(\left\|u_{0}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}+\left\|w_{t}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\right)$
$\displaystyle=t^{2}(1+\varepsilon)\left\|w\right\|_{H_{\lambda}}^{2}+(1+C(\varepsilon))\left\|u_{0}\right\|_{H_{\lambda}}^{2}-p\left(\left\|u_{0}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}+t^{p+1}\left\|w\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\right),$

where the second last step follows from Young’s inequality with
$\varepsilon>0$. Moreover, as $w$ is the solution to (3.1) implies

$\left\|w\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}=\left\|w\right\|_{H_{\lambda}}^{2}.$

Finally,

$\begin{gathered}g\left(u_{0}+w_{t}\right)\leq(1+C(\varepsilon))\left\|u_{0}\right\|_{H_{\lambda}}^{2}-p\left\|u_{0}\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}+\left\|w\right\|_{H_{\lambda}}^{2}\left[(1+\varepsilon)t^{2}-pt^{p+1}\right]\\\
\end{gathered}$

Thus choosing $\varepsilon>0$ such that $1+\varepsilon<p$ gives
$g\left(u_{0}+w_{t}\right)<0$ for $t>0$ large enough. Hence the claim follows.
Claim 2:
$I_{\lambda,a,f}\left(u_{0}+w_{t}\right)<I_{\lambda,a,f}\left(u_{0}\right)+I_{\lambda,1,0}\left(w_{t}\right)\forall
t>0$.
As $u_{0},w_{t}>0$, using $w_{t}$ as the test function for (3.4) yields

$\left\langle
u_{0},w_{t}\right\rangle_{H_{\lambda}}=\int_{\mathbb{B}^{N}}a(x)u_{0}^{p}w_{t}\mathrm{~{}d}V_{\mathbb{B}^{N}}+\left\langle
f,w_{t}\right\rangle.$

Therefore, utilizing the above expression and assumption $a\geq 1$, we compute
the following

$\displaystyle I_{\lambda,a,f}\left(u_{0}+w_{t}\right)=$
$\displaystyle\frac{1}{2}\left\|u_{0}\right\|_{H_{\lambda}}^{2}+\frac{1}{2}\left\|w_{t}\right\|_{H_{\lambda}}^{2}+\left\langle
u_{0},w_{t}\right\rangle_{H_{\lambda}}$
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left(u_{0}+w_{t}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)-\left\langle
f,u_{0}\right\rangle-\left\langle f,w_{t}\right\rangle$ $\displaystyle=$
$\displaystyle
I_{\lambda,a,f}\left(u_{0}\right)+I_{\lambda,1,0}\left(w_{t}\right)+\left\langle
u_{0},w_{t}\right\rangle_{H_{\lambda}}+\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)u_{0}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle+\frac{1}{p+1}\int_{\mathbb{B}^{N}}w_{t}^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left(u_{0}+w_{t}\right)^{p+1}\mathrm{~{}d}V_{\mathbb{B}^{N}}-\left\langle
f,w_{t}\right\rangle$ $\displaystyle\leq$ $\displaystyle
I_{\lambda,a,f}\left(u_{0}\right)+I_{\lambda,1,0}\left(w_{t}\right)$
$\displaystyle+\frac{1}{p+1}\int_{\mathbb{B}^{N}}a(x)\left[(p+1)u_{0}^{p}w_{t}+u_{0}^{p+1}+w_{t}^{p+1}-\left(u_{0}+w_{t}\right)^{p+1}\right]\mathrm{~{}d}V_{\mathbb{B}^{N}}(x)$
$\displaystyle<$ $\displaystyle
I_{\lambda,a,f}\left(u_{0}\right)+I_{\lambda,1,0}\left(w_{t}\right)$

This proves the claim. Further, the straightforward calculation gives

$I_{\lambda,1,0}\left(w_{t}\right)=\frac{t^{2}}{2}\left\|w\right\|_{H_{\lambda}}^{2}-\frac{t^{p+1}}{p+1}\left\|w\right\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{p+1}\rightarrow-\infty\quad\text{
as }\quad t\rightarrow\infty.$ (8.12)

From (8.12) and Remark 8.3, we have

$\sup_{t>0}I_{\lambda,1,0}\left(w_{t}\right)=I_{\lambda,1,0}\left(w_{1}\right)=I_{\lambda,1,0}\left(w\right)=S^{\infty}.$

Combing this with Claim 2 yields

$I_{\lambda,a,f}\left(u_{0}+w_{t}\right)<I_{\lambda,a,f}\left(u_{0}\right)+S^{\infty}\quad\forall
t>0.$ (8.13)

Claim 2, together with (8.12), results in

$I_{\lambda,a,f}\left(u_{0}+w_{t}\right)<I_{\lambda,a,f}\left(u_{0}\right)\quad\text{
for }\quad t\text{ large enough. }$ (8.14)

We now fix $t_{0}>0$ large enough such that (8.14) and Claim 1 are satisfied.
Then set

$\gamma:=\inf_{i\in\Gamma}\max_{t\in[0,1]}I_{\lambda,a,f}(i(t)),$

where

$\Gamma:=\left\\{i\in
C\left([0,1],H^{1}\left(\mathbb{B}^{N}\right)\right):i(0)=u_{0},\;i(1)=u_{0}+w_{t_{0}}\right\\}$

As $u_{0}\in U_{1}$ and $u_{0}+w_{t_{0}}\in U_{2}$, for every $i\in\Gamma$,
there exists $t_{i}\in(0,1)$ such that $i\left(t_{i}\right)\in U$. Therefore,

$\max_{t\in[0,1]}I_{\lambda,a,f}(i(t))\geq
I_{\lambda,a,f}\left(i\left(t_{i}\right)\right)\geq\inf_{U}I_{\lambda,a,f}(u)=c_{1}.$

Thus, using Lemma 8.2, we have $\gamma\geq
c_{1}>c_{0}=I_{\lambda,a,f}\left(u_{0}\right)$.

Claim 3: For $S^{\infty}$, as defined in (8.8),
$\gamma<I_{\lambda,a,f}\left(u_{0}\right)+S^{\infty}$.
Observe that $\lim_{t\rightarrow 0}\left\|w_{t}\right\|_{H_{\lambda}}=0$.
Thus, if we define $\tilde{i}(t)=u_{0}+w_{tt_{0}}$,
then $\lim_{t\rightarrow 0}\left\|\tilde{i}(t)-u_{0}\right\|_{H_{\lambda}}=0$.
As a result, $\tilde{i}\in\Gamma$. Therefore, using (8.13) will give us

$\gamma\leq\max_{t\in[0,1]}I_{\lambda,a,f}(\tilde{i}(t))=\max_{t\in[0,1]}I_{\lambda,a,f}\left(u_{0}+w_{tt_{0}}\right)<I_{\lambda,a,f}\left(u_{0}\right)+S^{\infty}$

Hence the claim follows. Thus

$I_{\lambda,a,f}\left(u_{0}\right)<\gamma<I_{\lambda,a,f}\left(u_{0}\right)+S^{\infty}.$

Applying Ekeland’s variational principle, there exists a PS sequence
$\left\\{u_{n}\right\\}$ for $I_{\lambda,a,f}$ at the level $\gamma$. Also,
note that $\left\\{u_{n}\right\\}$ is a bounded sequence. Further, from PS
decomposition and Remark (8.3), we have
$S^{\infty}=I_{\lambda,1,0}\left(w\right)$ and $u_{n}\rightarrow v_{0}$ for
some $v_{0}\in H^{1}\left(\mathbb{B}^{N}\right)$ such that
$\left(I_{\lambda,a,f}\right)^{\prime}\left(v_{0}\right)=0$ and
$I_{\lambda,a,f}\left(v_{0}\right)=\gamma$. Further, as
$I_{\lambda,a,f}\left(u_{0}\right)<\gamma$, we conclude $v_{0}\neq u_{0}$.
Finally, $\left(I_{\lambda,a,f}\right)^{\prime}\left(v_{0}\right)=0$, along
with the Remark 3.1, completes the proof of the proposition. ∎

######  Lemma 8.5.

If
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}<C_{p}S_{1,\lambda}^{\frac{p+1}{2(p-1)}}$,
then (8.2) holds.

###### Proof.

We can find an $\varepsilon>0$ such that
$\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}<C_{p}S_{1,\lambda}^{\frac{p+1}{2(p-1)}}-\varepsilon$
using the given assumption. Therefore, using Lemma 8.1. we have

$\langle
f,u\rangle\leq\|f\|_{H^{-1}\left(\mathbb{B}^{N}\right)}\|u\|_{H_{\lambda}}<\left[C_{p}S_{1,\lambda}^{\frac{p+1}{2(p-1)}}-\varepsilon\right]\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)}\leq\frac{p-1}{p}\|u\|_{H_{\lambda}}^{2}-\varepsilon\|u\|_{H_{\lambda}\left(\mathbb{B}^{N}\right)},\;\forall
u\in U.$

Thus

$\inf_{U}\left[\frac{p-1}{p}\|u\|_{H_{\lambda}}^{2}-\langle
f,u\rangle\right]\geq\varepsilon\inf_{U}\|u\|_{H_{\lambda}}.$

Moreover, Remark 8.1 gives us that $\|u\|_{H_{\lambda}}$ is bounded away from
0 on $U$, so the above expression yields

$\inf_{U}\left[\frac{p-1}{p}\|u\|_{H_{\lambda}}^{2}-\langle
f,u\rangle\right]>0.$

On the other hand,

$\displaystyle\eqref{2aaa}$ $\displaystyle\Leftrightarrow
C_{p}\frac{\|u\|_{H_{\lambda}}^{\frac{2p}{p-1}}}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{\frac{p+1}{p-1}}}-\langle
f,u\rangle>0\quad\text{ for
}\quad\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}=1$
$\displaystyle\Leftrightarrow\frac{\|u\|_{H_{\lambda}}^{\frac{2p}{p-1}}}{\|u\|_{L^{p+1}\left(\mathbb{B}^{N}\right)}^{\frac{p+1}{p-1}}}-\langle
f,u\rangle>0\quad\text{ for }\quad u\in U$
$\displaystyle\Leftrightarrow\frac{p-1}{p}\|u\|_{H_{\lambda}}^{2}-\langle
f,u\rangle>0\quad\text{ for }\quad u\in U.$

Hence the lemma follows. ∎

Combining Proposition 8.3 and Proposition 8.4 with Lemma 8.5, we conclude the
proof of Theorem 1.2.

##  9\. Acknowledgments.

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

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