# A Detailed Study of Kirchhoff-type Critical Elliptic Equations and $p$-Sub-
Laplacian Operators within the Heisenberg Group $\mathcal{H}_{n}$ Framework

SUBHAM DE  Department of Mathematics, Indian Institute of Technology, Delhi,
India.

(Dated: )

###### Abstract

This article presents a comprehensive study of Kirchhoff-type Critical
Elliptic Equations involving $p$-sub-Laplacian Operators on the Heisenberg
Group $\mathcal{H}_{n}$. It delves into the mathematical framework of
Heisenberg Group, and explores their Spectral Properties. A significant focus
is on the existence and multiplicity of solutions under various conditions,
leveraging concepts like the Mountain Pass Theorem. This work not only
contributes to the theoretical understanding of such groups but also has
implications in fields like Quantum Mechanics and Geometric Group Theory.

Keywords and Phrases:  Heisenberg Group, sub-Laplacian, Twisted laplacian,
Essential Self-Adjointness, Spectrum, Essential Spectra, $p$-sub-Laplacian,
Kirchhoff-type Critical Elliptic Equations, Palais-Smale Condition.

2020 MSC: Primary 35A15, 35A20, 35B38, 35B65, 35J35, 35D30, 35H10, 35J70 .
Secondary 35-02, 35A01, 35A09 .

###### Contents

  1. 1 Sub-Laplacian On The Heisenberg Group
    1. 1.1 Definition $\&$ Construction
    2. 1.2 Twisted Laplacians
    3. 1.3 Essential Self-Adjointness Property
  2. 2 The Spectrum of the sub-Laplacian
  3. 3 Essential Spectra of sub-Laplacian
  4. 4 Kirchhoff-type Critical Elliptic Equations involving $p$-sub-Laplacians on $\mathcal{H}_{n}$
    1. 4.1 Some Important Concepts
    2. 4.2 Introduction to Critical Kirchhoff Equations
  5. 5 Existence and Multiplicity of Solutions
    1. 5.1 Some Important Assumptions
    2. 5.2 Palais-Smale Condition $\left(PS\right)_{c}$
    3. 5.3 The Non-Degenerate Case
      1. 5.3.1 Proof of Theorem (5.3.1)
      2. 5.3.2 Proof of Theorem (5.3.2)
    4. 5.4 The Degenerate Case
      1. 5.4.1 Proof of Theorem (5.4.1)
      2. 5.4.2 Proof of Theorem (5.4.2)

##  1 Sub-Laplacian On The Heisenberg Group

###  1.1 Definition $\&$ Construction

Suppose, we consider the identification of $\mathbb{R}^{2}$ with $\mathbb{C}$
via the map,

$\displaystyle\mathbb{R}^{2}\longrightarrow\mathbb{C}$
$\displaystyle(x,y)\mapsto z=x+iy$

Then, we can interpret, $\mathcal{H}_{3}=\mathbb{C}\times\mathbb{R}$, where,
$\mathcal{H}_{3}$ is the Heisenberg Group defined on $3$ parameters. As
observed before, $\mathcal{H}_{3}$ is a non-commutative and a Unimodular Lie
Group on which the Haar Measure is equal to the usual Lebesgue Measure $dzdt$.

A priori denoting the Lie Algebra associated to $\mathcal{H}_{3}$ as
$\mathfrak{h}$, consisting of all left invariant vector fields on the same, we
can in fact opt for a basis of $\mathfrak{h}$ as $\\{X,Y,T\\}$, where,

$\displaystyle X=\partial_{y_{1}}-2y_{2}\partial_{\tau}\mbox{ , \hskip
20.0pt}Y=\partial_{y_{2}}+2y_{1}\partial_{\tau}\mbox{ , \hskip
20.0pt}T=4\partial_{\tau}$ (1.1)

######  Definition 1.1.1.

(Sub-Laplacian) The sub-Laplacian $\mathcal{L}$ on $\mathcal{H}_{3}$ is
defined by,

$\displaystyle\mathcal{L}=-\left(X^{2}+Y^{2}\right)$ (1.2)

We further introduce the following notations corresponding to the partial
differential operators on $\mathbb{C}$ as,

$\displaystyle\frac{\partial}{\partial z}=\frac{\partial}{\partial
y_{1}}-i\frac{\partial}{\partial y_{2}}$
$\displaystyle\frac{\partial}{\partial\overline{z}}=\frac{\partial}{\partial
y_{1}}+i\frac{\partial}{\partial y_{2}}$

Thus it only suffices to study the vector fields $Z$ and $\overline{Z}$ on
$\mathcal{H}_{3}$ given by,

$\displaystyle Z=X-iY=\frac{\partial}{\partial
z}-2i\overline{z}\frac{\partial}{\partial\tau}$ (1.3)
$\displaystyle\overline{Z}=X+iY=\frac{\partial}{\partial\overline{z}}+2iz\frac{\partial}{\partial\tau}$
(1.4)

Important to note that, $\overline{Z}$ is also well-known as the Hans Lewy
Operator [19], which eventually defies local solvability on $\mathbb{R}^{3}$,
and,

$\displaystyle\mathcal{L}=-\frac{1}{2}\left(Z\overline{Z}+\overline{Z}Z\right)$
(1.5)

We can further compute,

$\displaystyle\mathcal{L}=-\left(\left(\frac{\partial}{\partial
y_{1}}-2y_{2}\frac{\partial}{\partial\tau}\right)^{2}+\left(\frac{\partial}{\partial
y_{2}}+2y_{1}\frac{\partial}{\partial\tau}\right)^{2}\right)$
$\displaystyle=-\Delta-4\left(y_{1}^{2}+y_{2}^{2}\right)\frac{\partial^{2}}{\partial\tau^{2}}+4\left(y_{2}\frac{\partial}{\partial
y_{1}}-y_{1}\frac{\partial}{\partial
y_{2}}\right)\frac{\partial}{\partial\tau}$

provided, $\Delta=\frac{\partial^{2}}{\partial
y_{1}^{2}}+\frac{\partial^{2}}{\partial y_{2}^{2}}$.

Subsequently, the symbol $\sigma(\mathcal{L})$ of $\mathcal{L}$ can be derived
as follows,

$\displaystyle\sigma(\mathcal{L})(y_{1},y_{2},\tau;\xi,\eta,\gamma)=\left(\xi-2y_{2}\gamma\right)^{2}+\left(\eta+2y_{1}\gamma\right)^{2}$
(1.6)

For every $(y_{1},y_{2},\tau)\mbox{ },\mbox{
}(\xi,\eta,\gamma)\in\mathcal{H}_{3}$.

######  Remark 1.1.1.

$\mathcal{L}$ is Nowhere Elliptic on $\mathbb{R}^{3}$.

A priori from the fact that, $[X,Y]=T$, a theorem by H$\ddot{o}$rmander [18,
Theorem 1.1] enables us to conclude that, $\mathcal{L}$ is indeed
Hypoelliptic.

###  1.2 Twisted Laplacians

For $\tau\in\mathbb{R}\setminus\\{0\\}$, let, $Z_{\tau}$ and
$\overline{Z}_{\tau}$ be partial differential operators given by,

$\displaystyle Z_{\tau}=\frac{\partial}{\partial z}-2\overline{z}\tau$
$\displaystyle\overline{Z}_{\tau}=\frac{\partial}{\partial\overline{z}}+2z\tau$

Subsequently, the Twisted Laplacian $L_{\tau}$ is defined as,

$\displaystyle
L_{\tau}=-\frac{1}{2}\left(Z_{\tau}\overline{Z}_{\tau}+\overline{Z}_{\tau}Z_{\tau}\right)$
(1.7)

To be more explicit, we can write,

$\displaystyle L_{\tau}=-\frac{1}{2}\left(\left(\frac{\partial}{\partial
z}-2\overline{z}\tau\right)\left(\frac{\partial}{\partial\overline{z}}+2z\tau\right)+\left(\frac{\partial}{\partial\overline{z}}+2z\tau\right)\left(\frac{\partial}{\partial
z}-2\overline{z}\tau\right)\right)$
$\displaystyle=-\Delta+4\left(y^{2}_{1}+y^{2}_{2}\right)\tau^{2}+4i\left(y_{1}\frac{\partial}{\partial
y_{2}}-y_{2}\frac{\partial}{\partial y_{1}}\right)\tau$ (1.8)

######  Remark 1.2.1.

The fundamental connection between the sub-laplacian and the twisted laplacian
is given by the following result. ( ref. [20], [21], [22] )

######  Theorem 1.2.2.

Suppose, $u\in\mathcal{S}^{\prime}(\mathcal{H}_{3})\cap
C^{\infty}(\mathcal{H}_{3})$ be such that, $\check{u}(z,\tau)$ is a tempered
distribution of $\tau$ on $\mathbb{R}$, $\forall$ $z\in\mathbb{C}$, where,
$\check{u}$ denotes the Inverse Fourier Transform of $u$ with respect to time
$t$. Then, for almost every $\tau\in\mathbb{R}\setminus\\{0\\}$,

$\displaystyle\left(\mathcal{L}u\right)^{\tau}=L_{\tau}u^{\tau}$

where,

$\displaystyle\left(\mathcal{L}u\right)^{\tau}(z)=\left(\mathcal{L}u\right)^{\check{}}(z)\mbox{
, \hskip 20.0pt}z\in\mathbb{C}$

and,

$\displaystyle u^{\tau}(z)=\check{u}(z,\tau)\mbox{ , \hskip
20.0pt}z\in\mathbb{C}$

We recall the definition of Fourier Transform $\hat{f}$ of a function $f\in
L^{1}(\mathbb{R})$ as,

$\displaystyle\hat{f}(\xi):=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}e^{-ix.\xi}f(x)dx\mbox{
, \hskip 20.0pt}\xi\in\mathbb{R}.$ (1.9)

Since, our primary intention is to study the spectral properties of
$L_{\tau}$, we introduce the following.

######  Definition 1.2.1.

The Fourier-Wigner Transform $V_{\tau}(f,g)$ of the functions
$f,g\in\mathcal{S}(\mathbb{R})$ is defined by,

$\displaystyle
V_{\tau}(f,g)(q,p)=\frac{1}{\sqrt{2\pi}}|\tau|^{1/2}\int\limits_{-\infty}^{\infty}e^{i\tau
q.y}f(y-2p)\overline{g(y+2p)}dy\mbox{ , \hskip 20.0pt}\forall\hskip
10.0ptq,p\in\mathbb{R}$

If, $\tau=1$, then, $V_{1}(f,g)=V(f,g)$ , which in fact, defines the Classical
Fourier-Wigner Transform. ( ref. [25],[26], [27] )

It can be further established that,

$\displaystyle V_{\tau}(f,g)(q,p)=|\tau|^{1/2}V(f,g)(\tau q,p)$

For $\tau\in\mathbb{R}\setminus\\{0\\}$ and, $k=0,1,2,\cdots$, we define the
function $e_{k,\tau}$ on $\mathbb{R}$ by,

$\displaystyle e_{k,\tau}(x)=|\tau|^{1/4}e_{k}(\sqrt{|\tau|}x)\mbox{ , \hskip
20.0pt}x\in\mathbb{R}$

Where, $e_{k}$ denotes the Hermite Function defined as follows,

$\displaystyle
e_{k}(x):=\frac{1}{\left(2^{k}k!\sqrt{\pi}\right)^{1/2}}e^{-\frac{x^{2}}{2}}H_{k}(x)\mbox{
, \hskip 20.0pt}x\in\mathbb{R}.$ (1.10)

Such that,

$\displaystyle
H_{k}(x):=(-1)^{k}e^{x^{2}}\left(\frac{d}{dx}\right)^{k}\left(e^{-x^{2}}\right)\mbox{
, \hskip 20.0pt}x\in\mathbb{R}.$ (1.11)

Given $j,k=0,1,2,...$, we define the function $e_{j,k,\tau}$ on $\mathbb{C}$
as follows,

$\displaystyle e_{j,k,\tau}=V_{\tau}(e_{j,\tau}\mbox{ , }e_{k,\tau})$ (1.12)

Further computation yields,

$\displaystyle e_{j,k,1}=V_{1}(e_{j,1}\mbox{ , }e_{k,1})=V(e_{j}\mbox{ ,
}e_{k})$

Where, $V(e_{j}\mbox{ , }e_{k})$ is the Classical Hermite Function on
$\mathbb{C}$. [ ref. [25] ]

######  Remark 1.2.3.

The above result can also be interpreted as an analogue of [25, Proposition
21.1].

######  Proposition 1.2.4.

The set, $\\{e_{j,k,\tau}\mbox{ }|\mbox{ }j,k=0,1,2,....\\}$ is an orthonormal
basis for $L^{2}(\mathbb{C})$.

Indeed we can learn about the respective spectral properties of
$L_{\tau}\mbox{ , }\tau\in\mathbb{R}\setminus\\{0\\}$.

######  Theorem 1.2.5.

For $j,k=0,1,2,....$, the following holds true,

$\displaystyle L_{\tau}e_{j,k,\tau}=(2k+1)|\tau|e_{j,k,\tau}$

###### Proof.

Proof is similar to the derivation cited in [25, Theorem 22.2].

A priori from the statement of [25, Theorem 22.1], it follows that, for
$j=0,1,2,...$ and $k=0,1,2,.....$;

$\displaystyle Z\overline{Z}e_{j,k}=i(2k+2)^{1/2}Ze_{j,k-1}=-(2k+2)e_{j,k}$

and,

$\displaystyle\overline{Z}Ze_{j,k}=i(2k)^{1/2}Ze_{j,k-1}=-(2k)e_{j,k}$

Thus,

$\displaystyle
L_{\tau}e_{j,k}=-\frac{1}{2}\left(Z\overline{Z}+\overline{Z}Z\right)|\tau|e_{j,k,\tau}$

Important to observe that, the above identity also holds for $e_{j,0,\tau}$,
where, $j=0,1,2,.....$ due to the fact that,

$\displaystyle Ze_{j,0}=0\mbox{ , \hskip 20.0pt}\forall\hskip
10.0ptj=0,1,2,....$

∎

###  1.3 Essential Self-Adjointness Property

Our aim in this section is to study the Sub-Laplacian $\mathcal{L}$ as an
unbounded linear operator from $L^{2}(\mathcal{H}_{3})$ to
$L^{2}(\mathcal{H}_{3})$ with dense domain denoted as,
$\mathcal{S}(\mathcal{H}_{3})$.

######  Proposition 1.3.1.

$\mathcal{L}$ is an injective symmetric operator from $L^{2}(\mathcal{H}_{3})$
to $L^{2}(\mathcal{H}_{3})$ with dense domain $\mathcal{S}(\mathcal{H}_{3})$.
Furthermore, it is strictly positive.

Proof follows from integration by parts.

######  Remark 1.3.2.

The above proposition implies that, $\mathcal{L}$ is closed. Suppose we denote
its closure by $\mathcal{L}_{0}$. Hence, $\mathcal{L}_{0}$ is closed,
symmetric and positive operator from $L^{2}(\mathcal{H}_{3})$ to
$L^{2}(\mathcal{H}_{3})$.

In fact, $\mathcal{L}$ is Essentially Self-Adjoint ( ref. [23, Section 4, pp.
1603]) in the following sense that, it has a unique self-adjoint extension,
which, subsequently equals to $\mathcal{L}_{0}$. Further details on essential
self-adjointness can be found in [24, Theorem X.23].

######  Remark 1.3.3.

The results and derivations in this article are also valid for the Sub-
Laplacian on the $n$-dimensional Heisenberg Group $\mathcal{H}_{n}$, $n>1$,
having an underlying space as $\mathbb{C}^{n}\times\mathbb{R}$, although, we
have only explored the case for $n=1$, which is $\mathcal{H}_{3}$ for the sake
of lucidity.

##  2 The Spectrum of the sub-Laplacian

A priori given a closed linear operator $\mathcal{T}$ from a complex Banach
Space $X$ with dense domain $\mathcal{D}(\mathcal{T})$, we provide the
following definitions,

######  Definition 2.0.1.

(Spectrum) The Resolvent Set $\rho(\mathcal{T})$ of $\mathcal{T}$ is defined
as follows,

$\displaystyle\rho(\mathcal{T})\mbox{ }:=\mbox{
}\left\\{\lambda\in\mathbb{C}\mbox{ }|\mbox{ }\mathcal{T}-\lambda I\mbox{
}:\mbox{ }\mathcal{D}(\mathcal{T})\longrightarrow X\mbox{ is
bijective}\right\\}$

Where, $I$ denotes the identity operator on $X$.
The Spectrum, denoted by $\Sigma(\mathcal{T})$ is defined to be the complement
of $\rho(\mathcal{T})$ in $\mathbb{C}$.

######  Definition 2.0.2.

(Point Spectrum) The point spectrum [28] of $\mathcal{T}$, denoted by
$\Sigma_{p}(\mathcal{T})$ is defined as,

$\displaystyle\Sigma_{p}(\mathcal{T})\mbox{ }:=\mbox{
}\left\\{\lambda\in\mathbb{C}\mbox{ }|\mbox{ }\mathcal{T}-\lambda I\mbox{
}:\mbox{ }\mathcal{D}(\mathcal{T})\longrightarrow X\mbox{ is not
injective}\right\\}$

######  Definition 2.0.3.

(Continuous Spectrum) The Continuous Spectrum of $\mathcal{T}$, denoted by
$\Sigma_{c}(\mathcal{T})$, is defined as,

$\displaystyle\Sigma_{c}(\mathcal{T})\mbox{ }:=\mbox{
}\left\\{\lambda\in\mathbb{C}\mbox{ }|\mbox{ }Range(\mathcal{T}-\lambda
I)\mbox{ }\mbox{ is dense in }X\mbox{ , }(\mathcal{T}-\lambda I)^{-1}\mbox{
exists, but is unbounded}\right\\}$

######  Definition 2.0.4.

(Residual Spectrum) The Residual Spectrum of $\mathcal{T}$, denoted by
$\Sigma_{r}(\mathcal{T})$, is defined as,

$\displaystyle\Sigma_{r}(\mathcal{T})\mbox{ }:=\mbox{
}\left\\{\lambda\in\mathbb{C}\mbox{ }|\mbox{ }Range(\mathcal{T}-\lambda
I)\mbox{ }\mbox{ is not dense in }X\mbox{ , }(\mathcal{T}-\lambda
I)^{-1}\mbox{ exists and is bounded}\right\\}$

We can indeed deduce that, $\Sigma_{p}(\mathcal{T})$,
$\Sigma_{c}(\mathcal{T})$ and $\Sigma_{r}(\mathcal{T})$ are mutually disjoint.
Furthermore,

$\displaystyle\Sigma(\mathcal{T})=\Sigma_{p}(\mathcal{T})+\Sigma_{c}(\mathcal{T})+\Sigma_{r}(\mathcal{T})$

######  Proposition 2.0.1.

A priori given a complex and separable Hilbert Space $X$, if $\mathcal{T}$ is
a self-adjoint operator , then,

$\displaystyle\Sigma_{r}(\mathcal{T})=\phi$

With the above notations and concepts, we thus delineate a more precise
illustration of the Spectrum of the sub-Laplacian on the Heisenberg Group.

######  Theorem 2.0.2.

We have,

$\displaystyle\Sigma(\mathcal{L}_{0})=\Sigma_{c}(\mathcal{L}_{0})=\left[0,\infty\right)$

###### Proof.

We first intend to show that, no eigenvalue of $\mathcal{L}_{0}$ lies in the
interval $\left[0,\infty\right)$. We know for a fact that, $0$ is not an
eigenvalue of $\mathcal{L}_{0}$ ( Proof mentioned in [22]). Suppose, $\lambda$
be a positive number such that, $\exists$ a function $u\in
L^{2}(\mathcal{H}_{3})$ satisfying,

$\displaystyle\mathcal{L}_{0}u=\lambda u$

Consequently,

$\displaystyle L_{\tau}u^{\tau}=\lambda u^{\tau}$

Although, the above relation implies that, $u^{\tau}=0$, $\forall$
$\tau\in\mathbb{R}\setminus\\{0\\}$ and,

$\displaystyle|\tau|\neq\frac{\lambda}{(2k+1)}\mbox{ , }\hskip
20.0ptk=0,1,2,\cdots$ (2.1)

This helps us conclude that, $u=0$, a contradiction. Moreover,
$\mathcal{L}_{0}$ being self-adjoint, it implies,

$\displaystyle\Sigma(\mathcal{L}_{0})=\Sigma_{c}(\mathcal{L}_{0})$

Thus, it only suffices to establish that, $(\mathcal{L}_{0}-\lambda I)$ is not
surjective $\forall$ $\lambda\in\left[0,\infty\right)$.

Assuming $(\mathcal{L}_{0}-\lambda I)$ to be surjective for some
$\lambda_{0}\in\left[0,\infty\right)$, we can infer that,
$\lambda_{0}\in\rho(\mathcal{L}_{0})$. Hence, $\exists$ an open interval
$I_{\lambda_{0}}\subset\rho(\mathcal{L}_{0})$ containing $\lambda_{0}$.

Define $f$ on $\mathcal{H}$ as,

$\displaystyle f(x,y,t)=h(x,y)e^{-\frac{t^{2}}{2}}\mbox{ , }\hskip
10.0ptx,y,t\in\mathbb{R}$

Where, $h\in L^{2}(\mathbb{R}^{2})$.

Therefore, for every $\lambda\in I_{\lambda_{0}}$, $\exists$ a function
$u_{\lambda}\in L^{2}(\mathcal{H}_{3})$, such that,

$\displaystyle(\mathcal{L}_{0}-\lambda I)u_{\lambda}=f$

Computing the Inverse Transform with respect to $t$ helps us conclude,

$\displaystyle(L_{\tau}-\lambda
I)u^{\tau}_{\lambda}=he^{-\frac{\tau^{2}}{2}}\mbox{ , }\hskip 20.0pt\mbox{ for
almost every }\tau\in\mathbb{R}\setminus\\{0\\}$ (2.2)

As a consequence, $(L_{\tau}-\lambda I)$ is surjective $\forall$ $\tau\in
S_{\lambda}$ for which the lebesgue measure,

$\displaystyle m(\mathbb{R}\setminus S_{\lambda})=0$

Consider, $\tau\in\bigcap\limits_{r\in I_{\lambda_{0}}\cap\mathbb{Q}}S_{r}$ ,
$\mathbb{Q}$ being the set of all rationals. Hence, $(L_{\tau}-\lambda I)$ is
surjective, and subsequently injective for every $\lambda\in
I_{\lambda_{0}}\cap\mathbb{Q}$.

Thus, $L_{\tau}-\lambda I$ is bijective, $\forall$ $\lambda\in
I_{\lambda_{0}}$. ( Use the fact that, the Resolvent Set of $L_{\tau}$ is open
) Furthermore, we can also observe that, $(L_{\tau}-\lambda I)$ is injective
iff,

$\displaystyle\lambda\neq(2k+1)|\tau|\mbox{ , }\hskip 20.0ptk=0,1,2,\cdots$

A contradiction to the fact that, if we assume $\tau\in\bigcap\limits_{r\in
I_{\lambda_{0}}}S_{r}$ to be sufficiently small, such that, $(2k+1)|\tau|\in
I_{\lambda_{0}}$ for some $k=0,1,2,\cdots$. Hence, the proof is done. ∎

######  Remark 2.0.3.

As an application to Theorem (2.0.2), in the next section, we shall introoduce
various Essential Spectra which’ll be extremely helpful to us.

##  3 Essential Spectra of sub-Laplacian

Given a closed linear operator $\mathcal{T}$ densly defined on a complex
Banach Space $X$.

######  Definition 3.0.1.

The Essential Spectrum of $\mathcal{T}$, denoted as $\Sigma_{DS}(\mathcal{T})$
[Dunford and Schwartz [29] ] is defined as follows,

$\displaystyle\Sigma_{DS}(\mathcal{T})\mbox{ }:=\mbox{
}\left\\{\lambda\in\mathbb{C}\mbox{ }|\mbox{ }Range(\mathcal{T}-\lambda
I)\mbox{ }\mbox{ is not closed in }X\right\\}$

Let us recall the following concept from Functional Analysis.

######  Definition 3.0.2.

(Fredholm Operator) Given any two Banach Spaces $X$ and $Y$, and a bounded
linear operator $\mathcal{T}:X\longrightarrow Y$, we define $\mathcal{T}$ to
be Fredholm if, the following conditions hold true:

  1. 1.

$ker(\mathcal{T})$ is of finite dimension.

  2. 2.

$Range(\mathcal{T})$ is closed.

  3. 3.

$Coker(\mathcal{T})$ is of finite dimension.

If $\mathcal{T}$ is Fredholm, then, Index of $\mathcal{T}$ is defined to be
equal to $\\{dim(ker(\mathcal{T}))-dim(Coker(\mathcal{T}))\\}$.

Denote $\Phi_{W}(\mathcal{T})$ to be the set of all $\lambda\in\mathbb{C}$,
such that, $\mathcal{T}-\lambda I$ is a Fredholm Operator. Furthermore,
suppose, $\Phi_{S}(\mathcal{T})$ be the set of all complex numbers $\lambda$
satisfying, $\mathcal{T}-\lambda I$ is Fredholm with index $0$.

Then, the essential spectrums $\Sigma_{W}(\mathcal{T})$ [Wolf [30] [31]] and,
$\Sigma_{S}(\mathcal{T})$ [Schechter [32]] of $\mathcal{T}$ having the
following definitions,

$\displaystyle\Sigma_{W}(\mathcal{T})=\mathbb{C}\setminus\Phi_{W}(\mathcal{T})$

and,

$\displaystyle\Sigma_{S}(\mathcal{T})=\mathbb{C}\setminus\Phi_{S}(\mathcal{T})$

It is obvious from the respective definitions that,

$\displaystyle\Sigma_{DS}(\mathcal{T})\subseteq\Sigma_{W}(\mathcal{T})\subseteq\Sigma_{S}(\mathcal{T})$
(3.1)

In the particular case when, $\mathcal{T}$ denotes the sub-laplacian on the
Heisenberg Group $\mathcal{H}_{3}$, we can indeed deduce the following
important result.

######  Theorem 3.0.1.

We shall have,

$\displaystyle\Sigma_{DS}(\mathcal{L}_{0})=\Sigma_{W}(\mathcal{L}_{0})=\Sigma_{S}(\mathcal{L}_{0})=\left[0,\infty\right)$
(3.2)

###### Proof.

It only requires us to verify that,

$\displaystyle\left[0,\infty\right)\subseteq\Sigma_{DS}(\mathcal{L}_{0})$

Assume, $\lambda\in\left[0,\infty\right)$, although,
$\lambda\notin\Sigma_{DS}(\mathcal{L}_{0})$. Thus,
$Range(\mathcal{L}_{0}-\lambda I)$ is closed in $L^{2}(\mathcal{H}_{3})$. This
in turn helps us conclude that, $\mathcal{L}_{0}-\lambda I$ is bijective,
i.e., $\lambda\in\mathcal{L}_{0}$, a contradiction. Hence the proof is
complete. ∎

######  Remark 3.0.2.

Similar technique can be implemented to compute the Spectrum of the unique
self-adjoint extension $\Delta_{\mathcal{H}_{3},0}$ of the Laplacian
$\Delta_{\mathcal{H}_{3}}$ on the Heisenberg Group $\mathcal{H}_{3}$ defined
as,

$\displaystyle\Delta_{\mathcal{H}_{3}}=-\left(X^{2}+Y^{2}+T^{2}\right)$

In fact, we have [33],

$\displaystyle\Sigma\left(\Delta_{\mathcal{H}_{3},0}\right)=\Sigma_{c}\left(\Delta_{\mathcal{H}_{3},0}\right)=\left[0,\infty\right)$

Therefore, we can infer,

$\displaystyle\Sigma_{DS}(\Delta_{\mathcal{H}_{3},0})=\Sigma_{W}(\Delta_{\mathcal{H}_{3},0})=\Sigma_{S}(\Delta_{\mathcal{H}_{3},0})=\left[0,\infty\right)$
(3.3)

##  4 Kirchhoff-type Critical Elliptic Equations involving $p$-sub-Laplacians
on $\mathcal{H}_{n}$

###  4.1 Some Important Concepts

A priori given a generalized Heisenberg Group $\mathcal{H}_{n}$, a lie group
of topological dimension $(2n+1)$, having $\mathbb{R}^{2n+1}$ as a background
manifold, endowd with the non-Abelian group law,

$\displaystyle\tau\mbox{ }:\mbox{
}\mathcal{H}_{n}\longrightarrow\mathcal{H}_{n}\mbox{ , }\hskip
10.0pt\tau_{\xi}(\xi^{\prime})=\xi\circ\xi^{\prime}$

where,

$\displaystyle\xi\circ\xi^{\prime}=\left(x+x^{\prime},y+y^{\prime},t+t^{\prime}+2\sum\limits_{i=1}^{n}(y_{i}x_{i}^{\prime}-x_{i}y_{i}^{\prime})\right)\mbox{
, }\hskip 10.0pt\forall\hskip 10.0pt\xi,\xi^{\prime}\in\mathcal{H}_{n}$

Subsequently, inverse of this operation can be deduced as, $\xi^{-1}=-\xi$,
thus,

$\displaystyle(\xi\circ\xi^{\prime})^{-1}=(\xi^{\prime})^{-1}\circ(\xi)^{-1}$

Applying similar concepts as in (1.1) the corresponding Lie Algebra of left-
invariant vector fields is generated by,

$\displaystyle X_{j}=\partial_{x_{j}}+2y_{j}\partial_{t}\mbox{ , \hskip
20.0pt}Y_{j}=\partial_{y_{j}}-2x_{j}\partial_{t}\mbox{ , \hskip
20.0pt}T=4\partial_{t}$ (4.1)

For every $j=1,2,3,\cdots,n$. As a consequence, the basis
$\beta=\\{X_{j},Y_{j},T\\}_{j=1(1)n}$ satisfies the Heisenberg Canonical
Communication Relations for position and momentum,

$\displaystyle[X_{j},Y_{j}]=-\delta_{jk}T$

And all other commutators are zero.

######  Remark 4.1.1.

A vector field in the span of $\beta$ is called Horizontal.

######  Definition 4.1.1.

(Kor$\acute{a}$nyi Norm) It can be observed that, the anisotropic dilation
structure on the Heisenberg Group $\mathcal{H}_{n}$ induces the
Kor$\acute{a}$nyi Norm defined as follows:

$\displaystyle r(\xi):=r(z,t)=(|z|^{4}+t^{2})^{\frac{1}{4}}\mbox{ , }\hskip
10.0pt\forall\hskip 5.0pt\xi=(z,t)\in\mathcal{H}_{n}$

Some properties of Kor$\acute{a}$nyi Norm include that, its homogeneous degree
with resopect to dilations is equal to $1$.

Subsequently, the Kor$\acute{a}$nyi Distance is defined as :

$\displaystyle d_{H}(\xi,\xi^{\prime})=r(\xi^{-1}\circ\xi^{\prime})\mbox{ ,
}\hskip 10.0pt\forall\hskip
5.0pt(\xi,\xi^{\prime})\in\mathcal{H}_{n}\times\mathcal{H}_{n}$

And, the Kor$\acute{a}$nyi Open Ball of radius $R$ centered at $\xi_{0}$ is,

$\displaystyle B_{R}(\xi_{0})=\\{\xi\in\mathcal{H}_{n}\mbox{ }|\mbox{
}d_{H}(\xi,\xi_{0})<R\\}$

######  Remark 4.1.2.

We can indeed infer that, the Haar Measure on $\mathcal{H}_{n}$ is consistent
with the Lebesgue Measure on $\mathbb{R}^{2n+1}$, and is invariant under the
left translations of $\mathcal{H}_{n}$. Moreover, it is $Q$-Homogeneous with
respect to dilations ($Q$ denotes the Hausdorff Dimension).

Thus, the topological dimension of $\mathcal{H}_{n}$ (is equal to $2n+1$) is
strictly less than $Q=2n+2$.

######  Definition 4.1.2.

We define the Horizontal Gradient of a $C^{1}$-function
$u:\mathcal{H}_{n}\longrightarrow\mathbb{R}$ as:

$\displaystyle
D_{H}(u)=\sum\limits_{j=1}^{n}\left((X_{j}u)X_{j}+(Y_{j}u)Y_{j}\right)$ (4.2)

An important observation is that, $D_{H}u$ is in fact an element of
$span(\beta)$. Thus, we can define the natural inner product in $span(\beta)$
as :

$\displaystyle(X,Y)_{H}:=\sum\limits_{j=1}^{n}\left(x^{j}y^{j}+\tilde{x}^{j}\tilde{y}^{j}\right)$

For every $X=\\{x^{j}X_{j}+\tilde{x}^{j}Y_{j}\\}_{j=1(1)n}$ ,
$Y=\\{y^{j}X_{j}+\tilde{y}^{j}Y_{j}\\}_{j=1(1)n}$. This eventually helps us
define the Hilbertian Norm,

$\displaystyle|D_{H}(u)|:=\sqrt{\left(D_{H}(u),D_{H}(u)\right)_{H}}$ (4.3)

for any horizontal vector field $D_{H}(u)$.

######  Definition 4.1.3.

Given any horizontal vector field function, $X=X(\xi)$,
$X=\\{x^{j}X_{j}+\tilde{x}^{j}Y_{j}\\}_{j=1(1)n}$ of class
$C^{1}(\mathcal{H}_{n},\mathbb{R}^{2n})$, the Horizontal Divergence of $X$ is
defined as,

$\displaystyle
div_{H}X=\sum\limits_{j=1}^{n}\left(X_{j}(x^{j})+Y_{j}(y^{j})\right)$

We can generalize the notion of sub-Laplacians in the case for
$\mathcal{H}_{3}$ to a generalized Heisenberg Group $\mathcal{H}_{n}$.

######  Definition 4.1.4.

(sub-Laplacian) For every $u\in C^{2}(\mathcal{H}_{n})$, the sub-Laplacian or,
Kohn-Spencer Laplacian of $u$ is defined as:

$\displaystyle\Delta_{H}(u)=\sum\limits_{j=1}^{n}\left(X^{2}_{j}+Y^{2}_{j}\right)u$
$\displaystyle=\sum\limits_{j=1}^{n}\left(\frac{\partial^{2}}{\partial
x^{2}_{j}}+\frac{\partial^{2}}{\partial
y^{2}_{j}}+4y_{j}\frac{\partial^{2}}{\partial x_{j}\partial
t}-4x_{j}\frac{\partial^{2}}{\partial y_{j}\partial
t}\right)u+4|z|^{2}\frac{\partial^{2}u}{\partial^{2}t}$ (4.4)

H$\ddot{o}$rmander [18] established the fact that, $\Delta_{H}$ is
Hypoelliptic. To be more precise,

$\displaystyle\Delta_{H}(u)=div_{H}D_{H}(u)\mbox{ , }\hskip
10.0pt\forall\hskip 10.0ptu\in C^{2}(\mathcal{H}_{n})$

We can in fact further generalize the Kohn-Spencer Laplacian to obtain the so
called p-Laplacian on $\mathcal{H}_{n}$, having the following expression:

$\displaystyle\Delta_{H,p}(\varphi)=div_{H}\left(|D_{H}(\varphi)|^{p-2}_{H}D_{H}(\varphi)\right)$
(4.5)

for every $\varphi\in C^{\infty}_{c}(\mathcal{H}_{n})$.For further study,
interested readers can refer to [34], [35], [36], [37].

let us recall some significant properties of classic Sobolev Spaces on
$\mathcal{H}_{n}$.

######  Definition 4.1.5.

The standard $L^{p}$-norm is defined as,

$\displaystyle||u||^{p}_{p}=\int\limits_{\omega}|u|^{p}d\xi\mbox{ , }\hskip
10.0pt\forall\hskip 5.0ptu\in\Omega$

A priori given $\Omega$ to be a bounded Lipschitz Domain in $\mathcal{H}_{n}$
or, $\Omega=\mathcal{H}_{n}$. Then we denote $W^{1,p}(\Omega)$ as the
Horizontal Sobolev Space of the functions $u\in L^{p}(\Omega)$, provided,
$D_{H}(u)$ exists in the sense of distributions, and furthermore,
$|D_{H}(u)|\in L^{p}(\Omega)$, endowed with the norm,

$\displaystyle||u||_{W^{1,p}(\Omega)}=\left(||u||^{p}_{p}+||D_{H}(u)||^{p}_{p}\right)^{\frac{1}{p}}$

Consider the function space,

$\displaystyle HW^{1,p}_{V}(\mathcal{H}_{n}):=\left\\{u\in
W^{1,p}(\mathcal{H}_{n}):\int\limits_{\mathcal{H}_{n}}V(\xi)|u(\xi)|^{p}d\xi<\infty\right\\}$

with the followig norm defined on it,

$\displaystyle||u||=||u||_{HW^{1,p}_{V}(\mathcal{H}_{n})}:=\left(||D_{H}(u)||^{p}_{p}+||u||^{p}_{p,V}\right)^{\frac{1}{p}}$
(4.6)

and,

$\displaystyle||u||^{p}_{p,V}=\int\limits_{\mathcal{H}_{n}}V(\xi)|u(\xi)|^{p}d\xi$
(4.7)

Where, $V$ denotes the potenial function. Furthermore, under the assumption
that, $V(\xi)\geq V_{0}>0$, we can in fact conclude that,
$HW^{1,p}_{V}(\mathcal{H}_{n})$ is a reflexive Banach Space. For proof
involving Euclidean setting, it can be found in [38, Lemma 10], whereas, in
case for $\mathcal{H}_{n}$, we shall be needing few minor alterations. The
continuous embedding of $HW^{1,p}_{V}(\mathcal{H}_{n})\hookrightarrow
W^{1,p}(\mathcal{H}_{n})\hookrightarrow L^{t}(\mathcal{H}_{n})$ $\forall$
$p\leq t<p^{*}$, where, $p^{*}:=\frac{Qp}{Q-p}$ is the Critical Sobolev
Exponent on $\mathcal{H}_{n}$.

######  Remark 4.1.3.

In fact, one can establish that, the best value of the constant $V_{0}$,
denoted by $C_{p^{*}}$ is attained in the Folland-Stein Spaces
$S^{1,p}(\mathcal{H}_{n})$, which, also can be interpreted as the completion
of $C^{\infty}_{c}(\mathcal{H}_{n})$ in terms of the norm,

$\displaystyle||D_{H}(u)||_{p}=\left(\hskip
5.0pt\int\limits_{\mathcal{H}_{n}}|D_{H}(u)|^{p}_{H}d\xi\hskip
5.0pt\right)^{\frac{1}{p}}$

Thus, we can obtain the following estimate of $C_{p^{*}}$ of the Folland-Stein
Inequality as,

$\displaystyle C_{p^{*}}=\inf\limits_{u\in S^{1,p}(\mathcal{H}_{n}),u\neq
0}\frac{||D_{H}(u)||^{p}_{p}}{||u||^{p}_{p^{*}}}$

For further details, see [39].

###  4.2 Introduction to Critical Kirchhoff Equations

In this section, we shall deal with a class of Kirchhoff-type Critical
Elliptic Equations (Kirchhoff, $1883$) as a generalization of D’Alembert’s
Wave Equation for free vibrations of elastic strings, involving $p$-sub-
Laplacians, having the following representation,

$\displaystyle
M\left(||D_{H}(u)||^{p}_{p}+||u||^{p}_{p,V}\right)\left\\{-\Delta_{H,p}(u)+V(\xi)|u|^{p-2}u\right\\}=\lambda
f(\xi,u)+|u|^{p^{*}-2}u$ (4.8) $\displaystyle\hskip
100.0pt\xi\in\mathcal{H}_{n}\mbox{ , }\hskip 20.0ptu\in
HW^{1,p}_{V}(\mathcal{H}_{n})$

in both non-degenerate and degenerate cases separately.

######  Remark 4.2.1.

In (4.8), $\lambda$ is a real parameter, and, $M$ denotes the Kirchhoff
Function.

A priori, for pre-determined constants $\rho,P_{0},h,E,L$ having some physical
interpretation, Kirchhoff thus established a model given by the equation,

$\displaystyle\rho\frac{\partial^{2}u}{\partial
t^{2}}-\left(\frac{P_{0}}{h}+\frac{E}{2L}\int\limits_{0}^{L}\left|\frac{\partial
u}{\partial x}\right|^{2}dx\right)\frac{\partial^{2}u}{\partial x^{2}}=0$

In particular, the study of critical Kirchhoff-type problems were first
initially studied in the seminal paper of Br$\acute{e}$zis & Nirenberg [ref.
[40]], in which their intention was to study the Laplacian equations.

Over the years, there have been many generalizations of [40] in various
directions. For instance, Liao et al. [41] studied the following non-local
problem with Critical Sobolev Exponent of the form,

$\displaystyle-\left\\{a+b\int\limits_{\Omega}\left|\nabla
u\right|^{2}dx\right\\}\Delta u=\mu|u|^{2^{*}-2}u+\lambda|u|^{q-2}u\mbox{ ,
}\hskip 20.0ptx\in\Omega$ (4.9) $\displaystyle u=0\mbox{ , }\hskip
10.0ptx\in\partial\Omega$

where, $\Omega\subseteq\mathbb{R}^{N}$ ($N\geq 4$) is a smooth bounded domain,
$2^{*}=\frac{2N}{N-2}$ is the Critical Sobolev Exponent. The existence an
multiplicity of the solutions of (4.9) are obtained by applying the
Variational Methods and the Critical Point Theorem. Liang et al. [42]
effectively followed the similar approach to derive the solutions to the
fractional Schr$\ddot{o}$dinger-Kirchhoff equations with electro-magnetic
fields and critical non-linearity in the no-degenerate Kirchhoff case by using
the fractional versions of the concentration compactness principle
andvariational methods. Results related to the existence of solution in case
of non-degenerate Kirchhoff problems are illustrated, for example, in [43],
[44], [45] and [38].

Furthermore, there are extensive amount of research which are currently going
on for the degenerate case. Interested readers can look at the findings of
wang et al. [46] and even Song & Shi [47] in this regard.

The motivation behind studying the problem (4.8) primarily originates from the
significant applications of the Heisenberg Group. Liang & Pucci [48]
considered a class of critical Kirchhoff-Poisson systems in the Heisenberg
group under suitable assumptions. On the contrary, the existence of multiple
solutions is obtained by using the symmetric Mountain Pass Theorem. A priori
applying this result along with Singular Trudinger-Moser Inequality, Deng &
Tian [49] discussed the existence of solutions for Kirchhoff-type systems
involving $Q$-laplacian operator in the Heisenberg Group,

$\displaystyle-K\left(\int\limits_{\Omega}\left|\nabla_{\mathcal{H}_{n}}u\right|^{Q}d\xi\right)\Delta_{Q}(u)=\lambda\frac{G_{u}(\xi,u,v)}{\rho(\xi)^{\wp}}\hskip
20.0pt\mbox{in }\Omega$
$\displaystyle-K\left(\int\limits_{\Omega}\left|\nabla_{\mathcal{H}_{n}}v\right|^{Q}d\xi\right)\Delta_{Q}(v)=\lambda\frac{G_{v}(\xi,u,v)}{\rho(\xi)^{\wp}}\hskip
20.0pt\mbox{in }\Omega$ $\displaystyle u=v=0\hskip 20.0pt\mbox{ on
}\partial\Omega$

Where, $\Omega$ is an open, smooth and bounded subset of $\mathcal{H}_{n}$,
$K$ is Kirchhoff-type function & non-linear terms : $G_{u}$ and $G_{v}$ have
critical exponent growth.

Whereas, Pucci & Temeprini [50] studied the $(p,q)$ critical systems on the
Heisenberg Group,

$\displaystyle-
div_{H}\left(A(\left|D_{H}(u)\right|_{H})\right)+B\left(|u|\right)u=\lambda
H_{u}(u,v)+\frac{\alpha}{\wp^{*}}|v|^{\beta}|u|^{\alpha-2}u$ $\displaystyle-
div_{H}\left(A(\left|D_{H}(v)\right|_{H})\right)+B\left(|v|\right)v=\lambda
H_{v}(u,v)+\frac{\beta}{\wp^{*}}|u|^{\alpha}|v|^{\beta-2}v$

Subsequently, the existence of entire non-trivial solutions are obtained by
applying the concentration-compactness principle in the vectorial Heisenberg
context and variational methods.

######  Remark 4.2.2.

Further details on these results can be explored from [51].

##  5 Existence and Multiplicity of Solutions

A priori having the concepts discussed in detail in the previous section, we
now proceed towards proving the existence and multiplicity for a special class
of Kirchhoff-type critical elliptic equations as mentioned in (4.8) involving
the $p$-sub-Laplacian operators on $\mathcal{H}_{n}$ for both the degenerate
and the non-degenerate case separately.

###  5.1 Some Important Assumptions

In order to establish our main results, a priori we shall assume that,
$M:\mathbb{R}^{+}_{0}\longrightarrow\mathbb{R}^{+}_{0}$ is a continuous and
non-decreasing function, the potential function $V$ and $M$. Therefore, they
will satisfy the following properties :

  * •

$V:\mathcal{H}_{n}\longrightarrow\mathbb{R}_{+}$ is continuous, and $\exists$
$V_{0}>0$ such that, $V\geq V_{0}>0$ in $\mathcal{H}_{n}$.

  1. 1.

$\exists$ $m_{0}>0$ such that, $\inf\limits_{t\geq 0}M(t)=m_{0}$.

  2. 2.

$\exists$ $\tau\in\left[1,\frac{p^{*}}{p}\right)$ satisfying,

$\displaystyle\tau\mathcal{M}(t)\geq M(t)t\mbox{ , }\hskip 20.0pt\forall\hskip
10.0ptt\geq 0$

Where,

$\displaystyle\mathcal{M}(t):=\int\limits_{0}^{t}M(s)ds.$

  3. 3.

$\exists$ $m_{1}>0$ such that, $M(t)\geq m_{1}t^{\tau-1}$ , $\forall$ $t\geq
0$ and $M(0)=0$.

Furthermore, we impose the following hypotheses on the non-linearity of $f$:

  * •

$f:\mathcal{H}_{n}\times\mathbb{R}\longrightarrow\mathbb{R}$is a
Carath$\acute{e}$odory Function such that, $f$ is odd with respect to the
second variable.

  * •

$\exists$ a constant $r$ satisfying, $p^{*}>r>p\tau$ such that,

$\displaystyle\left|f(\xi,t)\right|\leq a(\xi)\left|t\right|^{r-2}t\mbox{ ,
}\hskip 10.0pt\mbox{ for a.e. }\xi\in\mathcal{H}_{n}\mbox{ and
}t\in\mathbb{R}$

where, $0\leq a(\xi)\in L^{\eta}(\mathcal{H}_{n})\cap
L^{\infty}(\mathcal{H}_{n})$, and, $\eta:=\frac{p^{*}}{p^{*}-r}$,
$\xi\in\mathcal{H}_{n}$.

  * •

$\exists$ a constant $\theta$ satisfying $p\tau<\theta<p^{*}$ such that,
$0<\theta F(\xi,t)\leq f(\xi,t)t$ , $\forall$ $t\in\mathbb{R}_{+}$. We define,

$\displaystyle F(\xi,t):=\int\limits_{0}^{t}f(\xi,s)ds.$

###  5.2 Palais-Smale Condition $\left(PS\right)_{c}$

A priori we adhere to some standard notations, where,
$\mathcal{N}(\mathcal{H}_{n})$ denotes the space of all signed finite Radon
Measures on $\mathcal{H}_{n}$ equipped with the norm. In other words, we
identify $\mathcal{N}(\mathcal{H}_{n})$ with the dual of
$C_{0}(\mathcal{H}_{n})$, the completion of all continuous functions
$u:\mathcal{H}_{n}\longrightarrow\mathbb{R}$, having con=mpact support, and
also is connected to the supremum norm $||.||_{\infty}$.

Important observation one can make here is that, the problem (4.8) has a
variational structure. The Euler-Lagrange Functional,
$\mathcal{J}_{\lambda}:HW^{1,p}_{V}(\mathcal{H}_{n})\longrightarrow\mathbb{R}$
associated to this problem is defined as follows :

$\displaystyle\mathcal{J}_{\lambda}(u)=\frac{1}{p}\mathcal{M}\left(\left|\left|D_{H}(u)\right|\right|^{p}_{p}+\left|\left|u\right|\right|^{p}_{p,V}\right)-\lambda\int\limits_{\mathcal{H}_{n}}F(\xi,u)d\xi-\frac{1}{p^{*}}\int\limits_{\mathcal{H}_{n}}|u|^{p^{*}}d\xi$
(5.1)

It implies that, under the conditions described in (5.1),
$\mathcal{J}_{\lambda}$ is of class
$C^{1}\left(HW^{1,p}_{V}(\mathcal{H}_{n})\right)$. Moreover, for every $u,v\in
HW^{1,p}_{V}(\mathcal{H}_{n})$, we define the Fr$\acute{e}$chet Derivative of
$\mathcal{J}_{\lambda}$ is given as:

$\displaystyle\langle\mathcal{J}^{\prime}_{\lambda}(u),v\rangle=M\left(\left|\left|D_{H}(u)\right|\right|^{p}_{p}+\left|\left|u\right|\right|^{p}_{p,V}\right)\left(\langle\mathcal{A}(u),v\rangle+\int\limits_{\mathcal{H}_{n}}V(\xi)|u|^{p-2}uvd\xi\right)$
$\displaystyle\hskip
50.0pt-\lambda\int\limits_{\mathcal{H}_{n}}f(\xi,u)uvd\xi-\int\limits_{\mathcal{H}_{n}}|u|^{p^{*}-2}uvd\xi$

Where,

$\displaystyle\langle\mathcal{A}_{p}(u),v\rangle=\int\limits_{\mathcal{H}_{n}}\left|D_{H}(u)\right|^{p-2}_{H}.D_{H}(u).D_{H}(v)d\xi$

######  Remark 5.2.1.

It can be verified that the weak solutions for problem (4.8) indeed coincide
with the critical points of $\mathcal{J}_{\lambda}$.

With all these notations and definitions, we define,

######  Definition 5.2.1.

A sequence $\\{u_{n}\\}_{n\in\mathbb{N}}\subset X$ is termed as a Palais-Smale
Sequence for the functional $\mathcal{J}_{\lambda}$ at level $c$ if,

$\displaystyle\mathcal{J}_{\lambda}(u_{n})\longrightarrow c\mbox{ , }\hskip
20.0pt\mathcal{J}^{\prime}_{\lambda}(u_{n})\longrightarrow 0\hskip
10.0pt\mbox{ in }X^{\prime}\mbox{ as }n\rightarrow+\infty$ (5.2)

If (5.2) implies the existence of a subsequence of $\\{u_{n}\\}_{n}$ which
converges in $X$, we assert that, $\mathcal{J}_{\lambda}$ satisfies the
Palais-Smale Condition $(PS)_{c}$. Moreover, if this strongly convergent
subsequence exists only for some $c$ values, we comment that,
$\mathcal{J}_{\lambda}$ satisfies a Local Palais-Smale Condition.

###  5.3 The Non-Degenerate Case

In this section, we shall state and prove two theorems which best illustrates
our purpose of analyzing existence and multiplicity of solutions for the
problem (4.8) in the Non-Degenerate case.

######  Theorem 5.3.1.

A priori we assume that, (5.1) holds true. If $M$ satisfies conditions $(1)$
and $(2)$ of (5.1), and $f$ verifies (5.1), then $\exists$ $\lambda_{1}>0$
such that, for any $\lambda\geq\lambda_{1}$, the problem (4.8) has a non-
trivial solution in $HW^{1,p}_{V}(\mathcal{H}_{n})$.

######  Theorem 5.3.2.

A priori we assume that, (5.1) holds true. If $M$ satisfies conditions $(1)$
and $(2)$ of (5.1), and $f$ verifies (5.1). Additionally, suppose we consider
that, one of the following condition also holds :

  1. 1.

$\exists$ a positive constant $m^{*}>0$ for every $m_{0}>m^{*}$ and
$\lambda>0$.

  2. 2.

$\exists$ a positive constant $\lambda_{2}>0$ for every $\lambda>\lambda_{2}$
and, $m_{0}>0$.

Then, the problem (4.8) admits of at least $n$ pairs of non-trivial weak
solutions in $HW^{1,p}_{V}(\mathcal{H}_{n})$.

Before we indulge ourselves into the proof of these statements, it is
imperative to study all the necessary results required for justification of
the same.

######  Theorem 5.3.3.

(Mountain Pass Theorem) For a given real Banach Space $E$ and $\mathcal{J}\in
C^{1}(E)$, satisfying $\mathcal{J}(0)=0$. We further assume that,

  1. 1.

$\exists$ $\rho,\alpha>0$ such that, $\mathcal{J}(u)\geq\alpha$ $\forall$
$u\in E$, with $||u||_{E}=\rho$;

  2. 2.

$\exists$ $e\in E$ satisfying $||e||_{E}>\rho$ such that, $\mathcal{J}(e)<0$

If we denote, $\gamma:=\left\\{\gamma\in C([0,1],E)\mbox{ }|\mbox{
}\gamma(0)=1\mbox{ , }\gamma(1)=e\right\\}$. Then,

$\displaystyle c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq
1}\mathcal{J}(\gamma(t))\geq\alpha$ (5.3)

and there exists a $(PS)_{c}$ sequence $\\{u_{n}\\}_{n}\subset E$.

One can in fact verify that, $\mathcal{J}_{\lambda}$ satisfies the geometric
properties $(1)$ and $(2)$ of Mountain Pass Theorem (5.3.3).

####  5.3.1 Proof of Theorem (5.3.1)

First, we intend to establish the following:

######  Claim 5.3.4.

We shall have,

$\displaystyle 0<c_{\lambda}=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq
t\leq
1}\mathcal{J}_{\lambda}(\gamma(t))<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}$
(5.4)

for large enough $\lambda$.

###### Proof.

Choose $v_{0}\in HW^{1,p}_{V}(\mathcal{H}_{n})$ such that,

$\displaystyle||v_{0}||=1\mbox{ , and, }\hskip
20.0pt\lim\limits_{t\rightarrow\infty}\mathcal{J}_{\lambda}(tv_{0})=-\infty$

Therefore, $\sup\limits_{t\geq
0}\mathcal{J}_{\lambda}(tv_{0})=\mathcal{J}_{\lambda}(t_{\lambda}v_{0})$ for
some $t_{\lambda}>0$. Thus, $t_{\lambda}$ satisfies,

$\displaystyle
M\left(\left|\left|D_{H}(t_{\lambda}v_{0})\right|\right|^{p}_{p}+\left|\left|t_{\lambda}v_{0}\right|\right|^{p}_{p,V}\right).\left(\left|\left|D_{H}(t_{\lambda}v_{0})\right|\right|^{p}_{p}+\left|\left|t_{\lambda}v_{0}\right|\right|^{p}_{p,V}\right)$
$\displaystyle=\lambda\int\limits_{\mathcal{H}_{n}}f(\xi,t_{\lambda}v_{0})|t_{\lambda}v_{0}|^{2}d\xi+\int\limits_{\mathcal{H}_{n}}|t_{\lambda}v_{0}|^{p^{*}}d\xi$
(5.5)

It suffices to prove that, $\\{t_{\lambda}\\}_{\lambda>0}$ is bounded. Since,
$t_{\lambda}\geq 1$ $\forall$ $\lambda>0$. Thus, by property $(2)$ of (5.1)
and (5.5), we deuce that,

$\tau\mathcal{M}(1)t^{2p\tau}_{\lambda}\geq\tau\mathcal{M}(1)(||t_{\lambda}v_{0}||^{p})^{\tau}$

$\displaystyle\geq
M\left(\left|\left|D_{H}(t_{\lambda}v_{0})\right|\right|^{p}_{p}+\left|\left|t_{\lambda}v_{0}\right|\right|^{p}_{p,V}\right).\left(\left|\left|D_{H}(t_{\lambda}v_{0})\right|\right|^{p}_{p}+\left|\left|t_{\lambda}v_{0}\right|\right|^{p}_{p,V}\right)$
$\displaystyle=\lambda\int\limits_{\mathcal{H}_{n}}f(\xi,t_{\lambda}v_{0})|t_{\lambda}v_{0}|^{2}d\xi+\int\limits_{\mathcal{H}_{n}}|t_{\lambda}v_{0}|^{p^{*}}d\xi$
$\displaystyle\geq
t^{p^{*}}_{\lambda}\int\limits_{\mathcal{H}_{n}}|v_{0}|^{p^{*}}d\xi$ (5.6)

It follows from (5.6) that, $\\{t_{\lambda}\\}_{\lambda>0}$ is bounded, since,
$p\tau<p^{*}$. Next, we shall prove that, $t_{\lambda}\longrightarrow 0$ as,
$\lambda\longrightarrow\infty$. By contradiction, suppose, $\exists$ $t_{0}>0$
and a sequence $\\{\lambda_{n}\\}_{n}$ with $\lambda_{n}\longrightarrow\infty$
as $n\rightarrow\infty$, and consequently, $t_{\lambda_{n}}\longrightarrow
t_{0}$ as $n\rightarrow\infty$.

A simple application of the Lebesgue Dominated Convergence Theorem yields,

$\displaystyle\int\limits_{\mathcal{H}_{n}}f(\xi,t_{\lambda_{n}}v_{0})\left|t_{\lambda_{n}}v_{0}\right|^{2}d\xi\longrightarrow\int\limits_{\mathcal{H}_{n}}f(\xi,t_{\lambda}v_{0})\left|t_{\lambda}v_{0}\right|^{2}d\xi$

as $n\rightarrow\infty$. Thus, we can conclude that,

$\displaystyle\lambda_{n}\int\limits_{\mathcal{H}_{n}}f(\xi,t_{\lambda}v_{0})\left|t_{\lambda}v_{0}\right|^{2}d\xi\longrightarrow\infty\mbox{
, as, }\hskip 10.0ptn\rightarrow\infty$

A contradiction to the fact that, (5.5) holds true. Thus,
$t_{\lambda}\longrightarrow 0$ as, $\lambda\rightarrow\infty$. Another
deduction which can indeed be made from (5.5) is,

$\displaystyle\lim\limits_{\lambda\rightarrow\infty}\lambda\int\limits_{\mathcal{H}_{n}}f(\xi,t_{\lambda}v_{0})\left|t_{\lambda}v_{0}\right|^{2}d\xi=0$

and,

$\displaystyle\lim\limits_{\lambda\rightarrow\infty}\int\limits_{\mathcal{H}_{n}}\left|t_{\lambda}v_{0}\right|^{p^{*}}d\xi=0$

A priori, using the deduction, $t_{\lambda}\longrightarrow 0$ as,
$\lambda\rightarrow\infty$ and the definition of $\mathcal{J}_{\lambda}$, we
obtain,

$\displaystyle\lim\limits_{\lambda\rightarrow\infty}\left(\sup\limits_{t\geq
0}\mathcal{J}_{\lambda}(tv_{0})\right)=\lim\limits_{\lambda\rightarrow\infty}\mathcal{J}_{\lambda}(tv_{0})=0$

Hence, $\exists$ $\lambda_{1}>0$ satisfying for every
$\lambda\geq\lambda_{1}$,

$\displaystyle\sup\limits_{t\geq
0}\mathcal{J}_{\lambda}(tv_{0})<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}$

Choosing $e=t_{1}v_{0}$, for large enough $t_{1}$, it can be verified that,

$\mathcal{J}_{\lambda}(e)<0$

$\displaystyle\implies
0<c_{\lambda}\leq\max\limits_{t\in[0,1]}\mathcal{J}_{\lambda}(\gamma(t))\hskip
40.0pt\mbox{, by setting, }\gamma(t)=tt_{1}v_{0}$

Therefore, we conclude,

$\displaystyle 0<c_{\lambda}=\sup\limits_{t\geq
0}\mathcal{J}_{\lambda}(tv_{0})<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}$

for $\lambda$ large enough, which completes the proof. ∎

####  5.3.2 Proof of Theorem (5.3.2)

To prove the theorem, we shall in fact use the concept related to
Krasnoselskii’s Genus Theory [52]. Given a Banach Space $X$, and let,
$\Lambda$ denotes the class of all closed subsets $A\subset X\setminus\\{0\\}$
that are symmetric with respect to the origin ( in other words, $u\in A$
implies, $-u\in A$ ).

######  Theorem 5.3.5.

For an infinite dimensional Banach Space $X$ and $\mathcal{J}\in C^{1}(X)$ be
an even functional, such that, $\mathcal{J}(0)=0$. Further, we assume that,
$X=Y\oplus Z$, $Y$ being finite dimensional. Then, $\mathcal{J}$ satisfies the
following conditions,

  1. 1.

$\exists$ constants $\rho,\alpha>0$ satisfying, $\mathcal{J}(u)\geq\alpha$ for
every $u\in\partial B_{\rho}\cap Z$.

  2. 2.

$\exists$ $\Theta>0$ such that, $\mathcal{J}$ satisfies the $(PS)_{c}$
condition $\forall$ $c$, with $c\in(0,\Theta)$.

  3. 3.

Given any finite dimensional subspace $\tilde{X}\subset X$, $\exists$
$R=R(\tilde{X})>0$ satisfying $\mathcal{J}(u)\leq 0$ on $\tilde{X}\setminus
B_{R}$

In addition to the above, we further consider, $dim(Y)=k$ and that,
$Y=Span\\{v_{1},v_{2},\cdots,v_{k}\\}$. For $n\geq k$, we inductively choose
that, $v_{n+1}\notin E_{n}=Span\\{v_{1},v_{2},\cdots,v_{n}\\}$. Suppose,
$R_{n}=R(E_{n})$, and, $\Omega_{n}=B_{R_{n}}\cap E_{n}$. Define,

$\displaystyle G_{n}:=\left\\{\psi\in C(\Omega_{n},X)\mbox{ }|\mbox{
}\psi|_{\partial B_{R_{n}\cap E_{n}}}=id\mbox{ and, }\psi\mbox{ is
odd}\right\\}$

and,

$\displaystyle G_{n}:=\left\\{\psi(\overline{\Omega_{n}\setminus V})\mbox{
}|\mbox{ }\psi\in G_{n}\mbox{ , }n\geq j\mbox{ , }V\in\Lambda\mbox{ ,
}\gamma(V)\leq n-j\right\\}$

Where, $0\leq c_{j}\leq c_{j+1}$ and, $c_{j}<\Theta$ for $j>k$, then, we
assert that, $c_{j}$ is a critical value of $\mathcal{J}$.

Furthermore, if $c_{j}=c_{j+l}=c<\Theta$ for every $1\leq l\leq m$ and for
$j>k$. Thus, $\gamma(K_{c})\geq m+1$, provided,

$\displaystyle K_{c}:=\left\\{u\in X\mbox{ }|\mbox{ }\mathcal{J}(u)=c\mbox{
and, }\mathcal{J}^{\prime}(u)=0\right\\}$

Now that, we have introduced all the terminologies required for the proof of
Theorem (5.3.2), we proceed to the proof of the statement of the same.

###### Proof.

We shall apply Theorem (5.3.5) to $\mathcal{J}_{\lambda}$. A priori using the
fact that, $HW^{1,p}_{V}(\mathcal{H}_{n})$ is indeed a reflexive Banach Space
and, $\mathcal{J}_{\lambda}\in
C^{1}\left(HW^{1,p}_{V}(\mathcal{H}_{n})\right)$.

Applying (5.1), we can infer that, the functional $\mathcal{J}_{\lambda}$
satisfies, $\mathcal{J}_{\lambda}(0)=0$. We shall thus prove the theorem in
three steps.
Step 1 :

We can in fact obtain that, $\mathcal{J}_{\lambda}$ indeed satisfies $(1)$ and
$(3)$ of Theorem (5.3.5).
Step 2 :

We claim that, $\exists$ a monotone increasing sequeence $\\{\Phi_{n}\\}_{n}$
in $\mathbb{R}_{+}$, such that,

$\displaystyle c^{\lambda}_{n}=\inf\limits_{E\in\Gamma_{n}}\max\limits_{u\in
E}\mathcal{J}_{\lambda}(u)<\Phi_{n}$

The above definition of $c^{\lambda}_{n}$ allows us to infer,

$\displaystyle c^{\lambda}_{n}=\inf\limits_{E\in\Gamma_{n}}\max\limits_{u\in
E}\mathcal{J}_{\lambda}(u)\leq\inf\limits_{E\in\Gamma_{n}}\max\limits_{u\in
E}\left\\{\mathcal{M}\left(\left|\left|D_{H}(u)\right|\right|^{p}_{p}+||u||^{p}_{p,V}\right)-\frac{1}{p^{*}}\int\limits_{\mathcal{H}_{n}}|u|^{p^{*}}d\xi\right\\}$

Set,

$\displaystyle\Phi_{n}=\inf\limits_{E\in\Gamma_{n}}\max\limits_{u\in
E}\left\\{\mathcal{M}\left(\left|\left|D_{H}(u)\right|\right|^{p}_{p}+||u||^{p}_{p,V}\right)-\frac{1}{p^{*}}\int\limits_{\mathcal{H}_{n}}|u|^{p^{*}}d\xi\right\\}$

Therefore, $\Phi_{n}<\infty$ and, $\Phi_{n}\leq\Phi_{n+1}$ $\forall$
$n\in\mathbb{N}$ ( Follows from the definition of $\Gamma_{n}$ ).
Step 3 :

We claim that, the problem (4.8) has at least $k$ pairs of weak solutions. To
this end, we distingish two cases:
Case I :

Fix $\lambda>0$. We set $m_{0}$ very large such that,

$\displaystyle\sup\limits_{n}\Phi_{n}<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}$

Case II :

using similar discussions as in Theorem (5.3.5), $\exists$ $\lambda_{2}>0$
satisfyng,

$\displaystyle
c^{\lambda}_{n}\leq\Phi_{n}<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}\hskip
40.0pt\forall\mbox{ }\lambda>\lambda_{2}.$

Thus, in any case, we must obtain,

$\displaystyle 0<c^{\lambda}_{1}\leq c^{\lambda}_{2}\leq\cdots\leq
c^{\lambda}_{n}<\Phi_{n}<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}$

Applying similar arguments as described in [52], we can in fact guarantee
that, the levels, $c^{\lambda}_{1}\leq c^{\lambda}_{2}\leq\cdots\leq
c^{\lambda}_{n}$ are indeed the critical values of $\mathcal{J}_{\lambda}$.

If $c^{\lambda}_{j}=c^{\lambda}_{j+1}$ for some $j=1,2,\cdots k-1$, therefore,
by [53, Theorem $4.2$, Remark $2.12$], the set $K_{c^{\lambda}_{j}}$ contains
infinitely many distinct points, and hence, the problem (4.8) has infinitely
many weak solutions.

Finally thus we conclude that, the problem (4.8) has at least $k$ pairs of
solutions, and the proof is done.

∎

###  5.4 The Degenerate Case

In this section, we shall investigate the degradation of the problem (4.8). We
shall always assume that, $M$ satisfies conditions $(2)$ and $(3)$ of (5.1),
and $f$ verifies the coonditions mentioned in (5.1). We state the main results
which helps us comment on the existence and the multiplicity of the solutions
to problem (4.8) in the Degenerate Case.

######  Theorem 5.4.1.

A priori we assume that, (5.1) holds true. If $M$ satisfies conditions $(2)$
and $(3)$ of (5.1), and $f$ verifies (5.1), then $\exists$ $\lambda_{3}>0$
such that, for any $\lambda\geq\lambda_{3}$, the problem (4.8) has a non-
trivial solution in $HW^{1,p}_{V}(\mathcal{H}_{n})$.

######  Theorem 5.4.2.

A priori we assume that, (5.1) holds true. If $M$ satisfies conditions $(2)$
and $(3)$ of (5.1), and $f$ verifies (5.1). Additionally, suppose we consider
that, one of the following condition also holds :

  1. 1.

$\exists$ a positive constant $m_{*}>0$ for every $m_{1}>m_{*}$ and
$\lambda>0$.

  2. 2.

$\exists$ a positive constant $\lambda_{4}>0$ for every $\lambda>\lambda_{4}$
and, $m_{1}>0$.

Then, the problem (4.8) admits of at least $n$ pairs of non-trivial weak
solutions in $HW^{1,p}_{V}(\mathcal{H}_{n})$.

Before indulging ourselves into the proofs of these statements above, we must
state the following lemma which apparently plays a major role in determining
the existence of solution to problem (4.8).

######  Lemma 5.4.3.

Under the assumptions on $M$ and $f$ mentioned above, if suppose,
$\\{u_{n}\\}_{n}\subset HW^{1,p}_{V}(\mathcal{H}_{n})$ be a Palais-Smale
Sequence of functional $\mathcal{J}_{\lambda}$ ( as defined in (5.2) ).

Furthermore, if,

$\displaystyle
0<c_{\lambda}<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p}}$

Then, $\exists$ a subsequence of $\\{u_{n}\\}_{n}$ Strongly Convergent in
$HW^{1,p}_{V}(\mathcal{H}_{n})$.

For the proof of the Lemma, interested readers can look up [51].

######  Remark 5.4.4.

In fact, one can further infer that, the functional $\mathcal{J}_{\lambda}$
satisfies all the assumptions of the Mountain Pass Theorem ( Theorem (5.3.3)
).

####  5.4.1 Proof of Theorem (5.4.1)

###### Proof.

A priori using similar argments as in the proof of the Mountain Pass Theorem
(Theorem (5.3.3)), we get,

$\displaystyle 0<c_{\lambda}=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq
t\leq
1}\mathcal{J}_{\lambda}(\gamma(t))<\left(\frac{1}{\theta}-\frac{1}{p^{*}}\right)\left(m_{0}C^{\tau}_{p^{*}}\right)^{\frac{p^{*}}{p^{*}-p\tau}}$

The rest of the proof follows in a similar manner to that described in Theorem
(5.3.1). ∎

####  5.4.2 Proof of Theorem (5.4.2)

###### Proof.

The proof of the theorem is similar to that of Theorem (5.3.2). ∎

## Statements and Declarations

### Conflicts of Interest Statement

I as the sole author of this article certify that I have no affiliations with
or involvement in any organization or entity with any financial interest (such
as honoraria; educational grants; participation in speakers’ bureaus;
membership, employment, consultancies, stock ownership, or other equity
interest; and expert testimony or patent-licensing arrangements), or non-
financial interest (such as personal or professional relationships, affi
liations, knowledge or beliefs) in the subject matter or materials discussed
in this manuscript.

### Data Availability Statement

I as the sole author of this aarticle confirm that the data supporting the
findings of this study are available within the article [and/or] its
supplementary materials.

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