# Algebraic topological techniques for elliptic problems involving fractional
Laplacian

A. Panda111University of Monastir<EMAIL_ADDRESS>D.
Choudhuri222University of Monastir<EMAIL_ADDRESS>A. Bahrouni333Department
of Mathematics, National Institute of Technology Rourkela
<EMAIL_ADDRESS>
###### Abstract

We prove the existence of infinitely many solutions to an elliptic problem by
borrowing the techniques from algebraic topology. The solution(s) thus
obtained will also be proved to be bounded.

Keywords: Fractional Laplacian, Fractional Sobolev Space, homology group.

AMS Classification: 35R11, 35J75, 35J60, 46E35.

##  1 Introduction

We propose to study the following singular problem with a mixed operator:

$\displaystyle\begin{split}a(-\Delta)^{s}u+b(-\Delta)u&=\lambda|u|^{-\gamma-1}u+\mu|u|^{2_{s}^{*}-2}u,~{}\text{in}~{}\Omega,\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\end{split}$ (1.1)

where $\Omega\subset\mathbb{R}^{N}(N\geq 2)$ is a bounded domain, $a,b\geq 0$,
$0<s<1<2<2_{s}^{*}<2^{*}$, $\lambda>0$, $\mu\in\mathbb{R}$, $\gamma\in(0,1)$,
and

$(-\Delta)^{s}u(x)=P.V.\int_{\mathbb{R}^{N}}\frac{(u(x)-u(y))}{|x-y|^{N+2s}}dy,~{}\forall~{}x\in\Omega,$

is the fractional Laplacian. We refer the operator
$``a(-\Delta)^{s}+b(-\Delta)"$ as a mixed operator since it possesses both
local as well as nonlocal features, and thus we refer the problems of kind
$\eqref{main prob}$ as “nonlocal-local” elliptic problems. For a detailed
study on this operator refer [6] and [14]. The mixed operators with different
order are nowadays gaining popularity in applied sciences, in theoretical
studies and also in real world applications. The development of the literature
includes viscosity solution methods [4], Cahn-Hilliard equations [9], Aubry-
Mather theory [12], phase transitions [8], probability and stochastics [29],
fractional damping effects [13], decay estimates for parabolic equations [15],
population dynamics [14], Bernstein-type regularity results [7].
We begin by considering the following two spaces $X$ and $Y$ which are the
closures of $C_{c}^{\infty}(\Omega)$ in $H^{s}(\Omega)$ and $H^{1}(\Omega)$,
respectively (refer Section 2 for these notations). Let
$Q=\mathbb{R}^{2N}\setminus((\mathbb{R}^{N}\setminus\Omega)\times(\mathbb{R}^{N}\setminus\Omega))$
and define the spaces as follows:

$\displaystyle X$ $\displaystyle=$
$\displaystyle\left\\{u:\Omega\rightarrow\mathbb{R}:u~{}\text{is measurable
and}~{}\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy<\infty\right\\}$
$\displaystyle Y$ $\displaystyle=$ $\displaystyle\left\\{u\in
L^{2}(\Omega):|\nabla u|\in L^{2}(\Omega)\right\\}.$

The spaces $X,Y$ are Banach spaces with respect to the following norms:

$\displaystyle\begin{split}\|u\|_{X}&=\|u\|_{2}+\left(\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dydx\right)^{\frac{1}{2}}\\\
&=\|u\|_{2}+[u]_{s,2},\end{split}$ (1.2)

and

$\displaystyle\begin{split}\|u\|_{Y}&=\|u\|_{2}+\left(\int_{\Omega}|\nabla
u|^{2}dx\right)^{\frac{1}{2}}\\\ &=\|u\|_{2}+\|u\|_{1,2},\end{split}$ (1.3)

respectively. The norm $[\cdot]$ defined in (1.2) is the Gagliardo norm. Note
that here $\|u\|_{\alpha}=(\int_{\Omega}|u|^{\alpha}dx)^{\frac{1}{\alpha}}$
for $1<\alpha<\infty$.
The study of the nonlocal-local elliptic problem in (1.1), firstly directed
our attention to fix a function space in which the solution(s) will be seeked
for. Define the space $Z=\\{u:a[u]_{s,2}^{2}+b\|u\|_{1,2}^{2}<\infty\\}$
equipped with the following norm:

$\displaystyle\|u\|$ $\displaystyle=$
$\displaystyle(a[u]_{s,2}^{2}+b\|u\|_{1,2}^{2})^{\frac{1}{2}}.$ (1.4)

It is easy to see that the space $Z$ is a Banach space and also reflexive with
respect to the norm $\|\cdot\|$, defined in $\eqref{norm}$.
The use of algebraic topological techniques to study problems having a
singular nonlinearity is a rarity in the literature. Thus, the problem
discussed here is new as the consideration of a singularity with a critical
exponent with $\mu\in\mathbb{R}$ handled with Morse theoretic approach is not
found anywhere in the literature to our knowledge. The question about the
existence and multiplicity of positive weak solutions to problem $\eqref{main
prob}$ with $\mu>0$ and with either $a=0$ or $b=0$ has been answered in [17,
18, 19, 22, 23, 31] and the references therein. The authors of these works
followed different tools such as variational method, concentration compactness
method, Nehari manifold method etc. but none of them used the techniques from
algebraic topology to study $\eqref{main prob}$ with $a,b>0$ and
$\mu\in\mathbb{R}$.
With the help of algebraic topological techniques, the existence and
multiplicity results for the following problem have been established by many
researchers under different growth conditions on the reaction term $f$ (either
subcritical or critical growth conditions):

$\displaystyle\begin{split}\mathcal{L}u&=f(x,u),~{}\text{in}~{}\Omega,\\\
u&=0,~{}\text{on}~{}\partial\Omega.\end{split}$ (1.5)

Problems of type $\eqref{prob s=1}$ were treated recently by Iannizzotto et
al. in [24] (finite multiplicity with $\mathcal{L}$ being the fractional
$p$-Laplacian), Ferrara et al. in [16] (at least one non-trivial solution with
$\mathcal{L}$ being a fractional integro-differential operator), D. Choudhuri
in [11] (infinitely many solutions with $\mathcal{L}$ being the
$p(x)$-Laplacian) and the references therein. The paper by Papageorgiou &
Rădulescu [32], dealt with a nonlinear Robin problem and proved the
multiplicity by producing three nontrivial solutions. The techniques thus
differed from problem-to-problem addressed.
The double phase problems of type $\eqref{prob s=1}$ with $\mathcal{L}$ being
a $(p,q)$-Laplacian or a fractional $(p,q)$-Laplacian have been widely studied
by many authors with different techniques. For instance, when
$\mathcal{L}=(-\Delta_{p}-\Delta_{q})$ with $p,q>1$, Gongbao & Gao [21], Yin &
Yang [36] used variational method, Liang et al. [27] used Morse theoretical
technique for the case $q=2\neq p$, and Marano et al. [28], Mugani &
Papageorgiou [30] used the variational method with Morse theory and truncation
comparison techniques. When
$\mathcal{L}=((-\Delta_{p})^{s}+(-\Delta_{q})^{s})$ with $p,q>1$ and
$s\in(0,1)$, the problem $\eqref{prob s=1}$ has been discussed in [1, 5, 20,
25] using variational methods, and in [10] using the Morse theory. With a
combination of variational technique and Morse technique, N. S. Papageorgiou,
V. D. Rădulescu, D. D. Repovš in [34] established the existence (for
$(p-1)$-superlinear case) and multiplicity (for $(p-1)$-linear resonant case)
for Robin problems with $(p,q)$-Laplacian. For more details on double phase
problems one can refer the recent piece of works by A. Bahrouni, V. D.
Rǎdulescu & D. D. Repovš [2, 3] and the bibliography therein.
Motivated by the former works, in this article, we study the singular problem
$\eqref{main prob}$ using variational techniques and algebraic topological
methods, specifically the Morse theory and the critical groups (refer Section
2). We establish the existence of infinitely many solutions to $\eqref{main
prob}$ in Section 3 followed by two subsections. Subsection 3.1 and Subsection
3.2 deal with the existence part and the multiplicity part, respectively. In
the Appendix, we will establish the boundedness result of these weak
solutions. The books by Perera et al. [35] and Papageorgiou et al. [33] are
strongly recommended for a better understanding on the usage of Morse theory
in the study of various elliptic PDEs.

##  2 Mathematical preliminaries

A quintessential condition which the functional requires to satisfy is the the
Palais-Smale condition (denoted by $(PS)$-condition) which is as follows.

######  Definition 2.1.

Let $X$ be a Banach space, and $I:X\rightarrow\mathbb{R}$ be a
$C^{1}(X,\mathbb{R})$ functional. Given $c\in\mathbb{R}$ we say that the
functional $I$ satisfies the Palais-Smale condition (or the
$(PS)_{c}$-condition) at level $c$ if any bounded sequence $(u_{n})\subset X$
such that $I(u_{n})\rightarrow c$, and $I^{\prime}(u_{n})\rightarrow 0$ as
$n\rightarrow\infty$ has a convergent subsequence in $X$.

Below are the Sobolev embedding results that will be used throughout the
article.

######  Theorem 2.2.

[[26]] Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain, $0<s<1$, and
$2s<N$. Further, assume that $r\leq 2_{s}^{*}=\frac{2N}{N-2s}$. Then, there
exists $C=C(r,s,N,\Omega)>0$ such that

$\|u\|_{L^{r}(\Omega)}\leq C\|u\|_{X},~{}\forall u\in X.$

Moreover, this embedding is continuous for any $r\in[1,2_{s}^{*}]$, and
compact for any $r\in[1,2_{s}^{*})$. The above embedding holds also for $Z$.

######  Theorem 2.3 ([26]).

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain, and $N>2$. Then, for
every $\bar{r}\leq 2^{*}=\frac{2N}{N-2}$ there exists
$\bar{C}=\bar{C}(r,N,\Omega)>0$ such that

$\|u\|_{L^{\bar{r}}(\Omega)}\leq\bar{C}\|u\|_{Y},~{}\forall u\in Y.$

Moreover, this embedding is continuous for any $r\in[1,2^{*}]$, and compact
for any $r\in[1,2^{*})$. The above embedding holds also for $Z$.

We now present the fundamental tool that will be used to work with, namely the
homology theory [33], which will be followed by the definition of deformation
(see [33]).

######  Definition 2.4.

A “homology theory” on a family of pairs of spaces $(X,A)$ consists of:

  1. 1.

A sequence $\\{H_{k}(X,A)\\}_{k\in\mathbb{N}_{0}}$ of abelian groups known as
“homology group” for the pair $(X,A)$ (note that for the pair $(X,\phi)$, we
write $H_{k}(X),k\in\mathbb{N}_{0}$).

  2. 2.

To every map of pairs $\varphi:(X,A)\rightarrow(Y,B)$ is associated a
homomorphism $\varphi^{*}:H_{k}(X,A)\rightarrow H_{k}(Y,B)$ for all
$k\in\mathbb{N}_{0}$.

  3. 3.

To every $k\in\mathbb{N}_{0}$ and every pair $(X,A)$ is associated a
homomorphism $\partial:H_{k}(X,A)\rightarrow H_{k-1}(A)$ for all
$k\in\mathbb{N}_{0}$.

Here, $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. These objects satisfy the
following axioms:

($A_{1}$)

If $\varphi=id_{X}$, then $\varphi_{*}=id|_{H_{k}(X,A)}$.

($A_{2}$)

If $\varphi:(X,A)\rightarrow(Y,B)$, and $\psi:(Y,B)\rightarrow(Z,C)$ are maps
of pairs, then $(\psi\circ\varphi)_{*}=\psi_{*}\circ\varphi_{*}$.

($A_{3}$)

If $\varphi:(X,A)\rightarrow(Y,B)$ is a map of pairs, then
$\partial\circ\varphi_{*}=(\varphi|_{A})_{*}\circ\partial$.

($A_{4}$)

If $i:A\rightarrow X$ and $j:(X,\phi)\rightarrow(X,A)$ are inclusion maps,
then the following sequence is exact

$...\xrightarrow[]{\partial}H_{k}(A)\xrightarrow[]{i_{*}}H_{k}(X)\xrightarrow[]{j_{*}}H_{k}(X,A)\xrightarrow[]{\partial}H_{k-1}(A)\rightarrow...$

Recall that a chain
$...\xrightarrow[]{\partial_{K+1}}C_{k}(X)\xrightarrow[]{\partial_{k}}C_{K-1}(X)\xrightarrow[]{\partial_{k-1}}C_{k-2}(X)\xrightarrow[]{\partial_{k-2}}...$
is said to be exact if $im(\partial_{k+1})=ker(\partial_{k})$ for each
$k\in\mathbb{N}_{0}$.

($A_{5}$)

If $\varphi,\psi:(X,A)\rightarrow(Y,B)$ are homotopic maps of pairs, then
$\varphi_{*}=\psi_{*}$.

($A_{6}$)

(Excision): If $U\subseteq X$ is an open set with
$\bar{U}\subseteq\text{int}(A)$, and $i:(X\setminus U,A\setminus
U)\rightarrow(X,A)$ is the inclusion map, then $i_{*}:H_{k}(X\setminus
U,A\setminus U)\rightarrow H_{k}(X,A)$ is an isomorphism.

($A_{7}$)

If $X=\\{*\\}$, then $H_{k}({*})=0$ for all $k\in\mathbb{N}$.

######  Definition 2.5.

A continuous map $F:X\times[0,1]\to X$ is a deformation retraction of a space
$X$ onto a subspace $A$ if, for every $x\in X$ and $a\in A$, $F(x,0)=x$,
$F(x,1)\in A$, and $F(a,1)=a$.

An important result in Morse theory is stated below.

######  Theorem 2.6.

Let $I\in C^{2}(X)$ satisfy the Palais-Smale condition, and let ‘$a$’ be a
regular value of $I$. Then, $H_{*}(X,I^{a})\neq 0$, implies that $K_{I}\cap
I^{a}\neq\emptyset$ where

$K_{I}=\\{u\in X:I^{\prime}(u)=0\\},$

and

$I^{a}=\\{u\in X:I(u)\leq a\\}.$

######  Remark 2.7.

Another notation which will be used in the article is

$K_{I,D}=\\{u\in X:I(u)\in D\\}$

where $D$ is a connected subset of $\mathbb{R}$.

######  Remark 2.8.

Prior to applying the Morse lemma we recall that for a Morse function the
following holds:

  1. 1.

$H_{*}(I^{c},I^{c}\setminus\text{Crit}(I,c))=\oplus_{j}H_{*}(I^{c}\cap
N^{j},(I^{c}\setminus\\{x^{j}\\})\cap N^{j}).$

  2. 2.

$H_{k}(I^{c}\cap N,I^{c}\setminus\\{x\\}\cap
N)=\begin{cases}\mathbb{R},&k=m(x)\\\ 0,&\text{otherwise}\end{cases}$

where $m(x)$ is a Morse index of $x$, and $x$ is a critical point of $f$.

  3. 3.

Further

$H_{k}(I^{a},I^{b})=\oplus_{\\{i:m(x_{i})=k\\}}\mathbb{Z}=\mathbb{Z}^{m_{k}(a,b)}$

where $m_{k}(a,b)=n(\\{i:m(x_{i})=k,x_{i}\in K_{I,(a,b)}\\})$. Here, $n(S)$ is
the number of elements in the set $S$.

  4. 4.

Morse relation

$\sum_{u\in K_{J,[a,b]}}\sum_{k\geq 0}dimC_{k}(I,u)t^{k}=\sum_{k\geq
0}dimH_{k}(J^{a},J^{b})t^{k}+(1+t)Q_{t}$

for all $t\in\mathbb{R}$.

In this paper, we use the notion of local $(m,n)$-linking $(m,n\in\mathbb{N})$
(see Definition 2.3, [34]) to prove the existence of solution.

######  Definition 2.9.

Let $W$ be a Banach space, $I\in C^{1}(W,\mathbb{R})$, and $0$ an isolated
critical point of $I$ with $I(0)=0$. Further, assume that $m,n\in\mathbb{N}$.
We say that $I$ has a “local $(m,n)$-linking” near the origin if there exist a
neighborhood U of 0, $E_{0}\neq\emptyset$, $E\subseteq U$, and $D\subseteq W$
such that $0\notin E_{0}\subseteq E$, $E_{0}\cap D=\emptyset$, and

  1. 1.

0 is the only critical point of $I$ in $I^{0}\cap U$, where $I^{0}=\\{u\in
W:I(u)\leq 0\\}$,

  2. 2.

$Dim~{}im(i^{*})-Dim~{}im(j^{*})\geq n$, where

$i^{*}:H_{m-1}(E_{0})\rightarrow H_{m-1}(W\setminus
D)~{}\text{and}~{}j^{*}:H_{m-1}(E_{0})\rightarrow H_{m-1}(E)$

are the homomorphisms induced by the inclusion maps $i:E_{0}\rightarrow
W\setminus D$ and $j:E_{0}\rightarrow E$,

  3. 3.

$I|_{E}\leq 0\leq I|_{U\cap D\setminus\\{0\\}}$.

##  3 Proof of the main results

Let $F(x,t)=\int_{0}^{t}f(s,x)ds$ be the primitive of
$f(x,t)=\lambda|t|^{-\gamma-1}t+\mu|t|^{2_{s}^{*}-2}t$.
We say $u\in Z$ to be a weak solution of $\eqref{main prob}$ if for every
$\varphi\in Z$, we have

$\displaystyle
a\iint_{Q}\frac{(u(x)-u(y))}{|x-y|^{N+2s}}(\varphi(x)-\varphi(y))dxdy+b\int_{\Omega\Omega}\nabla
u\cdot\nabla\varphi dx=$
$\displaystyle\int_{\Omega}\lambda|u|^{-\gamma-1}u\varphi(x)dx$
$\displaystyle+\int_{\Omega}\mu|u|^{2_{s}^{*}-2}u\varphi(x)dx.$

Clearly, a weak solution to problem $\eqref{main prob}$ is a critical point of
the corresponding energy functional

$\displaystyle\begin{split}I(u)=&\frac{a}{2}\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy+\frac{b}{2}\int_{\Omega}|\nabla
u|^{2}dx-\int_{\Omega}F(x,u)dx.\end{split}$ (3.1)

However, it is easy to see that the functional $I$ is not $C^{1}(Z)$ due to
the presence of the singular term. Therefore, instead of working with the
original functional $I$, we will use a cut-off functional, $\bar{I}$.

###  3.1 Existence of solution using a local $(1,1)$ linking at 0

We first define the functional $\bar{I}$. Let us consider the following cut-
off function.

$\xi(t)=\begin{cases}1,~{}\text{if}~{}|t|\leq l\\\ \xi~{}\text{is decreassing,
if}~{}l\leq t\leq 2l\\\ 0,~{}\text{if}~{}|t|\geq 2l.\end{cases}$

Let us now consider the following cut-off problem:

$\displaystyle\begin{split}a(-\Delta)^{s}u-b\Delta
u&=\lambda\cfrac{u}{|u|^{\gamma+1}}+\tilde{g}(u)~{}\text{in}~{}\Omega,\\\
u&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\end{split}$ (3.2)

where,

$\tilde{g}(u)=\mu|u|^{2_{s}^{*}-2}u\xi(\|u\|).$

Let $\tilde{F}(x,t)=\int_{0}^{t}\tilde{f}(s,x)ds$ be the primitive of
$\tilde{f}(x,t)=\lambda|t|^{-\gamma-1}t+\tilde{g}(t)$. We say $\tilde{u}\in Z$
to be a weak solution of $\eqref{main3}$ if for every $\varphi\in Z$, we have

$\displaystyle
a\iint_{Q}\frac{(\tilde{u}(x)-\tilde{u}(y))}{|x-y|^{N+2s}}(\varphi(x)-\varphi(y))dxdy+b\int_{\Omega\Omega}\nabla\tilde{u}\cdot\nabla\varphi
dx=$
$\displaystyle\int_{\Omega}\lambda|\tilde{u}|^{-\gamma-1}\tilde{u}\varphi(x)dx$
$\displaystyle+\int_{\Omega}\mu\xi(\|\tilde{u}\|)|\tilde{u}|^{2_{s}^{*}-2}\tilde{u}\varphi(x)dx.$

Consequently, a weak solution to problem $\eqref{main3}$ is a critical point
of the corresponding energy functional

$\displaystyle\begin{split}\bar{I}(u)=&\frac{a}{2}\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy+\frac{b}{2}\int_{\Omega}|\nabla
u|^{2}dx-\int_{\Omega}\tilde{F}(x,u)dx.\end{split}$ (3.3)

######  Remark 3.1.

One can easily see that if $u$ is a weak solution to (3.2) with $\|u\|\leq l$,
then $u$ is also a weak solution to (1.1).

######  Remark 3.2.

It is easy to observe that $I(u)=I(|u|)$. Hence, if a solution exists it has
to be nonnegative. Furthermore, due to the presence of the singular term the
solution is forced to be positive a.e. in $\Omega$.

We first verify that that functional $\bar{I}$ satisfies the (PS)- condition.
Let us denote

$S=\underset{u\in
X\setminus\\{0\\}}{\inf}\frac{\|u\|_{X}^{2}}{\|u\|_{L^{2_{s}^{*}}(\Omega)}^{2}}$
(3.4)

which is the best Sobolev constant in the Sobolev embedding (Theorem 2.2).

######  Theorem 3.3.

The functional $\bar{I}$ satisfies the $(PS)_{c}$-condition for

$c<c_{*}=\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)\mu^{\frac{-2}{2_{s}^{*}-2}}S^{\frac{2_{s}^{*}}{2_{s}^{*}-2}}-\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)^{-\frac{\gamma-1}{\gamma+1}}\left[|\Omega|^{\frac{2_{s}^{*}-1+\gamma}{2_{s}^{*}}}S^{-\frac{1-\gamma}{2}}\frac{\lambda}{1-\gamma}\right]^{\frac{2}{1+\gamma}},$

where $c_{*}>0$ for sufficiently small $\lambda,|\mu|>0$.

###### Proof.

We see that although the functional has been modified with a cut-off, yet the
functional is not $C^{1}$ which is not in the premise of the Palais-Smale
condition. However, we will devise a scheme to tackle this situation by
picking the sequence in a way that the singularity at $0$ gets avoided since
the functional is $C^{1}$ in $Z\setminus\\{0\\}$. Suppose $(u_{n})\subset Z$
is an eventually zero sequence, then apparently it converges to $0$, and we
discard the sequence immediately. Suppose, $(u_{n})\subset Z$ is a sequence
with infinitely many terms of the sequence equal to $0$, then we will hand-
pick a subsequence of $(u_{n})$ with all non-zero terms. Thus, without loss of
generality we will let $(u_{n})$ such that $u_{n}\neq 0$ for every
$n\in\mathbb{N}$. Let this sequence $(u_{n})$ be such that

$\displaystyle\bar{I}(u_{n})\rightarrow
c,~{}\text{and}~{}\bar{I}^{\prime}(u_{n})\rightarrow 0$ (3.5)

as $n\rightarrow\infty$. Therefore, let us consider a sequence $(u_{n})\subset
Z$ such that $\bar{I}(u_{n})\rightarrow c$ for some $c\in\mathbb{R}$, and
$\bar{I}^{\prime}(u_{n})\rightarrow 0$ as $n\rightarrow\infty$. It is not
difficult to see that the subsequence is bounded in $Z$. This has the
following consequences:

$\displaystyle\begin{split}u_{n}&\rightharpoonup
u_{0}~{}\text{in}~{}Z;\|u_{n}\|\rightarrow M,\\\ u_{n}&\rightarrow
u_{0}~{}\text{in}~{}L^{r}(\Omega)~{}\text{for any}~{}1\leq r<2_{s}^{*},\\\
u_{n}&\rightarrow u_{0}\neq 0~{}\text{a.e. in}~{}\Omega,\end{split}$ (3.6)

as $n\rightarrow\infty$. Now, consider

$\displaystyle\begin{split}o(1)=&\langle\bar{I}^{\prime}(u_{n}),u_{n}-u\rangle\\\
=&(\|u_{n}\|^{2}-\|u\|^{2})-\lambda\int_{\Omega}|u_{n}|^{-\gamma-1}u_{n}(u_{n}-u)dx-\mu\int_{\Omega}\xi(\|u_{n}\|)|u_{n}|^{2_{s}^{*}-2}u_{n}(u_{n}-u)dx\\\
=&(\|u_{n}\|^{2}-\|u\|^{2})-\mu\xi(M)\int_{\Omega}|u_{n}|^{2_{s}^{*}}dx+\mu\xi(M)\int_{\Omega}|u|^{2_{s}^{*}}dx+o(1)\\\
=&\|u_{n}-u\|^{2}-\mu\xi(M)\int_{\Omega}|u_{n}-u|^{2_{s}^{*}}dx+o(1).\end{split}$
(3.7)

We obtain

$\displaystyle\begin{split}\underset{n\rightarrow\infty}{\lim}\|u_{n}-u\|^{2}&=\mu\xi(M)\underset{n\rightarrow\infty}{\lim}\int_{\Omega}|u_{n}-u|^{2_{s}^{*}}dx=\mu\xi(M)N^{2_{s}^{*}}.\end{split}$
(3.8)

If $\mu\leq 0$, we produce a contradiction from $\eqref{PS_2_AUX}$, and we
guarantee that $u_{n}\rightarrow u$ strongly in $Z$. Therefore, we now proceed
for $\mu>0$.
Further, if $N=0$, then we obtain $u_{n}\rightarrow u$ in $Z$ as
$n\rightarrow\infty$, since $M>0$. Therefore, we will show that $N=0$. On the
contrary let us assume that $N>0$. Thus, we get

$\displaystyle\begin{split}0\leq\underset{n\rightarrow\infty}{\lim}\|u_{n}-u\|^{2}&=\mu\xi(M)N^{2_{s}^{*}}.\end{split}$
(3.9)

Next, from $\eqref{conv1}$, (3.9), and $\eqref{best}$ we have

$\displaystyle\begin{split}SN^{2}&\leq\mu\xi(M)N^{2_{s}^{*}}\leq\mu
N^{2_{s}^{*}}\\\ M^{2}-\|u\|^{2}&=\mu\xi(M)N^{2_{s}^{*}}.\end{split}$ (3.10)

It also follows from (3.10) that

$\displaystyle\begin{split}N&\geq\left(\frac{S}{\mu}\right)^{\frac{1}{2_{s}^{*}-2}},\end{split}$
(3.11)

and

$\displaystyle\begin{split}M^{2}\geq
SN^{2}&\geq\mu^{\frac{-2}{2_{s}^{*}-2}}S(S^{\frac{2}{2_{s}^{*}-2}})\\\
&=\mu^{\frac{-2}{2_{s}^{*}-2}}S^{\frac{2_{s}^{*}}{2_{s}^{*}-2}}.\end{split}$
(3.12)

Consider

$\displaystyle\begin{split}c+o(1)=&\bar{I}(u_{n})-\frac{1}{2_{s}^{*}}\langle\bar{I}^{\prime}(u_{n}),u_{n}\rangle\\\
=&a\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)[u_{n}]_{s,2}^{2}+b\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)\|\nabla
u_{n}\|_{2}^{2}+\lambda\left(\frac{1}{2_{s}^{*}}-\frac{1}{1-\gamma}\right)\int_{\Omega}|u_{n}|^{1-\gamma}dx\\\
\geq&\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)\|u_{n}\|^{2}-\frac{\lambda}{1-\gamma}\int_{\Omega}|u_{n}|^{1-\gamma}dx.\end{split}$
(3.13)

Therefore, by the Brezis-Lieb theorem, the Young’s inequality, and the results
derived in this theorem, we pass the limit $n\rightarrow\infty$ in
$\eqref{contradiction}$ to obtain the following:

$\displaystyle\begin{split}c\geq&\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)(M^{2}+\|u\|^{2})-\frac{\lambda}{1-\gamma}\int_{\Omega}|u|^{1-\gamma}dx\\\
\geq&\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)(M^{2}+\|u\|^{2})-|\Omega|^{\frac{2_{s}^{*}-1+\gamma}{2_{s}^{*}}}S^{-\frac{1-\gamma}{2}}\frac{\lambda}{1-\gamma}\|u\|^{1-\gamma}\\\
\geq&\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)(M^{2}+\|u\|^{2})-\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)\|u\|^{2}-\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)^{-\frac{\gamma-1}{\gamma+1}}\left[|\Omega|^{\frac{2_{s}^{*}-1+\gamma}{2_{s}^{*}}}S^{-\frac{1-\gamma}{2}}\frac{\lambda}{1-\gamma}\right]^{\frac{2}{1+\gamma}}\\\
\geq&\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)M^{2}-\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)^{-\frac{\gamma-1}{\gamma+1}}\left[|\Omega|^{\frac{2_{s}^{*}-1+\gamma}{2_{s}^{*}}}S^{-\frac{1-\gamma}{2}}\frac{\lambda}{1-\gamma}\right]^{\frac{2}{1+\gamma}}\\\
\geq&\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)\mu^{\frac{-2}{2_{s}^{*}-2}}S^{\frac{2_{s}^{*}}{2_{s}^{*}-2}}-\left(\frac{1}{2}-\frac{1}{2_{s}^{*}}\right)^{-\frac{\gamma-1}{\gamma+1}}\left[|\Omega|^{\frac{2_{s}^{*}-1+\gamma}{2_{s}^{*}}}S^{-\frac{1-\gamma}{2}}\frac{\lambda}{1-\gamma}\right]^{\frac{2}{1+\gamma}}\\\
=&c_{*}\end{split}$ (3.14)

for sufficiently small $\lambda>0$ and $|\mu|>0$. This is a contradiction. ∎

######  Theorem 3.4.

The functional $\bar{I}$ has a local $(1,1)$ \- linking at the origin.

###### Proof.

We define $V=\mathbb{R}$. Clearly $V$ is a one dimensional vector subspace of
$Z$. We now choose $\nu\in(0,1)$ small enough so that
$K_{\bar{I}}\cap\overline{B_{\nu}(0)}=\\{0\\}$ where $B_{\nu}(0)=\\{u\in
Z:\|u\|<\nu\\}$, and $K_{\bar{I}}=\\{u\in Z:\bar{I}^{\prime}(u)=0\\}$. Define

$E=V\cap\overline{B_{\nu}(0)}$

for small enough $\nu\in(0,1)$. Note that any two norms are equivalent over
$E$. Therefore, using this to our advantage, for a sufficiently small $\nu>0$
there exists $\delta>0$, we have

$\|u\|\leq\nu\Rightarrow\|u\|_{\infty}\leq\delta~{}\text{for all}~{}u\in E.$

Therefore,

$\displaystyle\begin{split}\bar{I}(u)=&\frac{a}{2}\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy+\frac{b}{2}\int_{\Omega}|\nabla
u|^{2}dx-\frac{\lambda}{1-\gamma}\int_{\Omega}|u|^{1-\gamma}dx-\frac{\mu}{2_{s}^{*}}\int_{\Omega}\xi(\|u\|)|u|^{2_{s}^{*}}dx\\\
=&\frac{1}{2}\|u\|^{2}-\frac{\lambda}{1-\gamma}\int_{\Omega}|u|^{1-\gamma}dx-\frac{\mu}{2_{s}^{*}}\int_{\Omega}\xi(\|u\|)|u|^{2_{s}^{*}}dx\\\
\leq&\frac{1}{2}\|u\|^{2}-\frac{\lambda}{1-\gamma}C\|u\|^{1-\gamma}-\frac{\mu}{2_{s}^{*}}C^{\prime}\|u\|^{2_{s}^{*}}\leq
0\\\ \end{split}$ (3.15)

where equivalence of norms followed from the set $E$ being in a finite
dimensional space $V$. Further, define

$D^{\prime}=\left\\{u\in
Z:\|u\|^{2}\geq\frac{4\lambda}{1-\gamma}\|u\|_{1-\gamma}^{1-\gamma}\right\\}.$

Thus, for any $u\in D^{\prime}$,

$\displaystyle\begin{split}\bar{I}(u)=&\frac{a}{2}\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy+\frac{b}{2}\int_{\Omega}|\nabla
u|^{2}dx-\frac{\lambda}{1-\gamma}\int_{\Omega}|u|^{1-\gamma}dx-\frac{\mu}{2_{s}^{*}}\int_{\Omega}\xi(\|u\|)|u|^{2_{s}^{*}}dx\\\
=&\frac{1}{2}\|u\|^{2}-\frac{\lambda}{1-\gamma}\int_{\Omega}|u|^{1-\gamma}dx-\frac{\mu}{2_{s}^{*}}\int_{\Omega}|u|^{2_{s}^{*}}dx\\\
\geq&\frac{1}{4}\|u\|^{2}-\frac{|\mu|}{2_{s}^{*}}C\|u\|^{2_{s}^{*}}>0\end{split}$
(3.16)

holds for small enough $\|u\|<\nu$. This holds for all
$D=D^{\prime}\cap(B_{\nu}(0)\setminus\\{0\\})$. Define

$E_{0}=V\cap\partial B_{\nu}(0).$

Clearly $E_{0}\cap D=\emptyset$. Therefore, we arrive at the following

$I|_{E}\leq 0<I|_{D}.$

Further, define

$h:[0,1]\times Z\setminus D^{\prime}\rightarrow Z\setminus D^{\prime}$

as

$h(t,u)=(1-t)u+t\cdot\nu\frac{v}{\|v\|}$

where $v=\alpha\in V$. Clearly $0\in D^{\prime}$ which makes the definition of
$h(\cdot,\cdot)$ valid. Observe that

$\displaystyle\begin{split}h(0,u)&=u,~{}\text{in}~{}Z\setminus D^{\prime}\\\
h(1,u)&=\nu\frac{v}{\|v\|},~{}\text{in}~{}V\cap\partial
B_{\nu}(0)=E_{0}.\end{split}$ (3.17)

Thus, $E_{0}$ is a retract of $Z\setminus D^{\prime}$. Therefore,

$i^{*}:H_{0}(E_{0})\rightarrow H_{0}(Z\setminus D)$

is an isomorphism. Note that $E_{0}=\\{v,-v\\}$ for some $v\neq 0$. Therefore,
from $DimH_{0}(E_{0})=2$ since $H_{0}(E_{0})=\mathbb{R}\oplus\mathbb{R}$.
Thus, $Dim~{}im(i^{*})=2$.
Further, $E$ is an interval $[-\alpha,\alpha]$ which is contractible to a
point. Thus, $H_{0}(E)=\mathbb{R}$. Hence, if

$j^{*}:H_{0}(E_{0})\rightarrow H_{0}(E),$

then $Dim~{}im(i^{*})-Dim~{}im(j^{*})=2-1=1$. Thus, the hypothesis of the
Definition 2.9 is satisfied, and hence there exists a local $(1,1)$ linking at
0. ∎

######  Proposition 3.5.

$C_{k}(\bar{I},\infty)=0$ for all $k\in\mathbb{N}$.

###### Proof.

Let $t>0$, and consider

$\displaystyle\begin{split}\frac{d}{dt}\bar{I}(tu)=&\langle\bar{I}^{\prime}(tu),u\rangle\\\
=&\frac{1}{t}\langle\bar{I}^{\prime}(tu),tu\rangle\\\
=&\frac{1}{t}\left[a\iint_{Q}\frac{|tu(x)-tu(y)|^{2}}{|x-y|^{N+2s}}dxdy+b\int_{\Omega}|\nabla
tu|^{2}dx\right.\\\ &\left.-\int_{\Omega}\tilde{f}(x,tu)tudx\right]\\\
=&\frac{1}{t}\left[a\iint_{Q}\frac{|tu(x)-tu(y)|^{2}}{|x-y|^{N+2s}}dxdy+b\int_{\Omega}|\nabla
tu|^{2}dx\right.\\\
&\left.-\lambda\int_{\Omega}|tu|^{1-\gamma}dx-\mu\int_{\Omega}\xi(\|tu\|)|tu|^{2_{s}^{*}}dx\right]\end{split}$
(3.18)

Observe that for large $t>0$ we have $\bar{I}(tu)\geq\mathbb{J}_{0}>c$, for
some $c>0$, since $\bar{I}(tu)\rightarrow\infty$ as $t\rightarrow\infty$.
Therefore,

$\displaystyle\frac{d}{dt}\bar{I}(tu)$ $\displaystyle>$ $\displaystyle 0$
(3.19)

for sufficiently large $t>0$. Thus, there exists a unique $h(u)>0$ such that
$\bar{I}(h(u)u)=\mathbb{J}_{0}$. This actually implies that $h\in C(\partial
B_{1})$. On extending $h(\cdot)$ to $Z\setminus\\{0\\}$ by defining

$\tilde{h}(u)=\frac{1}{\|u\|}h\left(\frac{u}{\|u\|}\right)$

for all $u\in Z\setminus\\{0\\}$, we obtain $\tilde{h}\in Z\setminus\\{0\\}$,
and $\bar{I}(\tilde{h}(u)u)=\mathbb{J}_{0}$. Also if
$\bar{I}(u)=\mathbb{J}_{0}$ then $\tilde{h}(u)=1$. Therefore, we define

$\displaystyle\hat{h}(u)=$ $\displaystyle 1,$ if
$\bar{I}(u)\geq\mathbb{J}_{0}$ $\displaystyle\hat{h}(u)=$
$\displaystyle\tilde{h}(u),$ if $\bar{I}(u)\leq\mathbb{J}_{0}$

which makes $\hat{h}$ continuous. Further define

$g(t,u)=(1-s)u+s\hat{h}(u)u$

for all $(t,u)\in[0,1]\times(Z\setminus\\{0\\})$. Thus, we have

$g(0,u)=u,~{}g(1,u)=\tilde{h}(u)u\in\bar{I}^{\mathbb{J}_{0}},$

and

$g(t,.)|_{\bar{I}^{\mathbb{J}_{0}}}=id|_{\bar{I}^{\mathbb{J}_{0}}}$

for all $t\in[0,1]$. What we conclude from here is that
$\bar{I}^{\mathbb{J}_{0}}$ is a deformation of $Z\setminus\\{0\\}$. By
standard definition of a homotopy, it is easy to see that $\partial
B_{1}=\\{u\in\bar{Z}:\|u\|=1\\}$ is a deformation of $Z\setminus\\{0\\}$.
Thus, on choosing $\mathbb{J}_{0}$ sufficiently negative we get

$C_{k}(\bar{I},\infty)=H_{k}(\bar{Z},\partial B_{1})=0$

for all $k\in\mathbb{N}$. ∎

######  Remark 3.6.

We use a variant of the result - If $X$ is a Banach space, $\bar{I}\in
C^{1}(X)$, $0\in K_{I}$ is isolated, and $\bar{I}$ has a local $(m,n)$-linking
near the origin, then rank $C_{m}(\bar{I},0)\geq n$. Since we obtained a
$(1,1)$ linking we conclude that rank of $C_{1}(\bar{I},0)\geq 1$. In our case
$0$ is not a critical point of $\bar{I}$, however one can still construct
$C_{1}(\bar{I},0)$ owing to not only the fact that $\bar{I}$ is well-defined
at $0$ but also having a ‘sharp’ isolated non regularity there.

######  Theorem 3.7.

There exists a solution, say $u_{0}$, to the problem (3.2).

###### Proof.

As seen in the Remark 3.6 we have that rank $C_{1}(\bar{I},0)\geq 1$. In
tandem with this and Proposition 3.5, we are in a position to apply the
Proposition $6.2.42$ (refer Theorem 4.1 in Appendix) of [33] that guarantees
the existence of a $u_{0}\in Z$ such that

$u_{0}\in K_{\bar{I}}\setminus\\{0\\},$

which further implies that $u_{0}\in Z\cap L^{\infty}(\Omega)$ is a solution
of (3.2). For the boundedness of $u_{0}$, please refer the Theorem 4.2 in the
Appendix. Thus, the existence of a nontrivial, bounded solution to (3.2) has
been established. ∎

###  3.2 Multiplicity results

######  Theorem 3.8.

The problem (3.2) has at least two nontrivial solutions in $Z\cap
L^{\infty}(\Omega).$

###### Proof.

From the Theorem 3.7 it can be assumed that there is one nontrivial solution
to (3.2). Let us further assume that there is exactly one nontrivial solution
to (3.2). We at first show that $H_{k}(Z,\bar{I}^{-a})$ for all $k\geq 0$.
Pick a $u\in\\{v\in Z:\|v\|=1\\}=\partial B^{\infty}$, where
$B^{\infty}=\\{v\in Z:\|v\|\leq 1\\}$. We make use of the equation (3.16)
which explains that for small enough $t>0$ we have $\bar{I}(tu)>0$. The
functional $\bar{I}$ being $C^{1}$ in $Z\setminus\\{0\\}$ indicates that
$\bar{I}^{\prime}(tu)>0$ for this small $t>0$. Also from (3.19) and in
combination with the fact that the functional is superlinear, we have that for
large enough $t>0$, $\bar{I}^{\prime}(tu)<0$. Thus, there exists a unique
$t(u)$ such that $\bar{I}^{\prime}(t(u)u)=0$ since due to our assumption that
there exists exactly one nontrivial solution. We can thus say that there
exists a $C^{1}$-function $T:Z\setminus\\{0\\}\rightarrow\mathbb{R}^{+}$
defined by $u\mapsto t(u)$. We now define a standard deformation retract
$\eta$ of $Z\setminus B_{r}(0)$ into $\bar{I}^{-a}$ as follows (refer
Definition 2.5).

$\eta(s,u)=\begin{cases}(1-s)u+sT\left(\frac{u}{\|u\|}\right)\frac{u}{\|u\|},&\|u\|\geq\nu,\bar{I}(u)\geq-a\\\
u,&\bar{I}(u)\leq-a.\end{cases}$

It is not difficult to see that $\eta$ is a $C^{1}$ function over $[0,1]\times
Z\setminus B_{r}(0)$. On using the map $\delta(s,u)=\frac{u}{\|u\|}$, for
$u\in Z\setminus B_{r}(0)$ we claim that $H_{k}(Z,Z\setminus
B_{r}(0))=H_{k}(B^{\infty},S^{\infty})$ for all $k\geq 0$. This is because,
$H_{k}(B^{\infty},S^{\infty})\cong H_{k}(*,0)$. From elementary computation of
homology groups with two $0$-dimensional simplices it is easy to see that
$H_{k}(*,0)=\\{0\\}$ for each $k\geq 0$. A result in [35] tells us that

$C_{n}(\bar{I},0)=\begin{cases}\mathbb{R},&\text{if}~{}m(0)=N\\\
0,&\text{otherwise}.\end{cases}$

Clearly, $m(0)\geq 2$. Therefore, from the Morse relation in the Remark
$\ref{rem1}-4$ and the result above taken from [35], we have for $a>0$

$\displaystyle\sum_{u\in K_{\bar{I},[-a,\infty)}}\sum_{k\geq
0}dimC_{k}(\bar{I},u)t^{k}$ $\displaystyle=t^{m(0)}+p(t)$ (3.20)

where $m(0)$ is the Morse index of $0$, and $p(t)$ contains the rest of the
powers of $t$ corresponding to the other critical points, if any. On further
using the the Morse relation we obtain

$\displaystyle t^{m(0)}+p(t)$ $\displaystyle=(1+t)Q_{t}.$ (3.21)

This is because the $H_{k}$s are all trivial groups. Hence, $Q_{t}$ either
contains $t^{m(0)}$ or $t^{m(0)-1}$ or both. Thus, there exists at least two
nontrivial $u\in K_{\bar{I},[-a,\infty)}$ with $2\leq m(0)\leq N+1$. ∎

######  Theorem 3.9.

Let $\Omega$ be as above.Then, $\bar{I}$ has infinitely many critical points
in $Z$.

###### Proof.

We appeal to the Morse theory again from which we obtain the following.

$\displaystyle\sum_{u\in K_{\bar{I},[-a,\infty)}}\sum_{k\geq
0}dimC_{k}(\bar{I},u)t^{k}$ $\displaystyle=t^{m(0)}+2\sum_{k\geq
0}\alpha_{k}t^{k}.$ (3.22)

The factor $2$ is due to the fact that if $u$ is a critical point then $(-u)$
is also a critical point. $\alpha_{k}$s are nonnegative integers. As in the
proof of the Theorem 3.8, we have $H_{k}(Z,\bar{I}_{-a}^{\infty})=0$.
Therefore, we have the following identity over $\mathbb{R}$.

$\displaystyle t^{m(0)}+2\sum_{k\geq 0}\alpha_{k}t^{k}$
$\displaystyle=(1+t)Q_{t}.$ (3.23)

In particular for $t=1$ we have $1+2A=2B$, where the series on the left of the
identity in (3.23) for $t=1$ is denoted by $A$, and the term on the right of
the same for $t=1$ is denoted by $B$. This is possible only when the sum is
infinite as finite sum leads to a contradiction that there exists an odd and
an even number that agree. Thus, there exists infinitely many solutions. ∎

The following is the main existence result for the problem $\eqref{main
prob}$.

######  Theorem 3.10.

The problem $\eqref{main prob}$ admits infinitely many nontrivial solutions in
$Z\cap L^{\infty}(\Omega)$.

###### Proof.

According to Theorem $\ref{existence_one_soln}-\ref{subcrit}$, the problem
$\eqref{main3}$ has infinitely many solutions in $Z\cap L^{\infty}(\Omega)$.
With a suitable choice of $l$ and by Remark 3.2, we conclude our proof. ∎

##  4 Appendix

######  Theorem 4.1.

(Theorem $6.2.42$ of Papagiorgiou et al. [33]) If $X$ is a Banach space, $I\in
C^{1}(X)$, $I$ satisfies the $Ce$-condition, $K_{I}$ is finite with $0\in
K_{I}$, and for some $k\in\mathbb{N}$ we have $C_{k}(I,0)\neq 0$,
$C_{k}(I,\infty)=0$, then there exists a $u\in K_{I}$ such that $\bar{(}u)<0$,
$C_{k-1}(I,u)\neq 0$ or $I(u)>0$, and $C_{k+1}(I,u)\neq 0$.

######  Theorem 4.2.

Any solution to (3.2) is in $L^{\infty}(\Omega)$.

###### Proof.

The argument sketched here is a standard one, and hence we shall only show
that an improvement in integrability is possible up to $L^{\infty}$ assuming
an integrability of certain order, say $p>1$. The boundedness follows from a
bootstrap argument. Without loss of generality we can consider the set
$\Omega^{\prime}=\\{x\in\Omega:u(x)>1\\}$, and thus by the positivity of a
fixed solution (refer Remark 3.2), say, $u$ we have $u=u^{+}>0$ a.e. in
$\Omega$. Let $u\in L^{p}(\Omega)$ for $p>1$. Let $a=\max\\{\lambda,|\mu|\\}$.
On testing with $u^{p}$ to obtain the following:

$\displaystyle\begin{split}C\|u^{\frac{p+1}{2}}\|_{2_{s}^{*}}^{2}&\leq\|u^{\frac{p+1}{2}}\|\\\
&\leq\left(\lambda\int_{\Omega^{\prime}}|u|^{p-\gamma}dx+\mu\int_{\Omega^{\prime}}|u|^{2_{s}^{*}-1+p}dx\right)\frac{(p+1)^{2}}{4p}\\\
&\leq
a\left(\int_{\Omega^{\prime}}|u|^{p}(1+|u|^{2_{s}^{*}-1})dx\right)\frac{(p+1)^{2}}{4p};~{}\text{since
in}~{}\Omega^{\prime}~{}\text{we have}~{}u>1\\\ &\leq
2a\left(\int_{\Omega^{\prime}}|u|^{p}|u|^{2_{s}^{*}}dx\right)\frac{(p+1)^{2}}{4p}\\\
&\leq
2aC^{\prime\prime}\|u\|_{\beta^{*}}^{2_{s}^{*}}\|u^{p}\|_{\mathfrak{t}};~{}\text{since
by using H\"{o}lder's inequality}.\end{split}$ (4.1)

Here, $\mathfrak{t}=\frac{\beta^{*}}{\beta^{*}-2_{s}^{*}}$ for some
$\beta^{*}>1$, $\mathfrak{t}^{*}=\frac{tN}{N-ts}<2_{s}^{*}$. Thus, we also
have

$\displaystyle\begin{split}C^{\prime}\|u^{\frac{p}{2}}\|_{\beta^{*}}^{2}&\leq
C^{\prime}\|u^{\frac{p+1}{2}}\|_{\beta^{*}}^{2}\leq\|u^{\frac{p+1}{2}}\|_{2_{s}^{*}}^{2}.\end{split}$
(4.2)

From the story so far, we know the following

$\displaystyle C^{\prime}\|u^{\frac{p}{2}}\|_{\beta^{*}}^{2}$
$\displaystyle\leq
2aC^{\prime\prime}\|u\|_{\beta^{*}}^{2_{s}^{*}}\|u^{p}\|_{\mathfrak{t}}.$
(4.3)

For the fixed $\beta^{*}>1$, we set $\eta=\frac{\beta^{*}}{2\mathfrak{t}}>1$
for a suitable choice of $t$, and $\tau=\mathfrak{t}p$ to get

$\displaystyle\|u\|_{\eta\tau}$ $\displaystyle\leq
C^{\frac{\mathfrak{t}}{\tau}}\|u\|_{\tau};~{}\text{where}~{}C=2aC^{\prime\prime}\|u\|_{\beta^{*}}^{\alpha^{+}}~{}\text{is
a fixed quantity for a fixed solution}~{}u.$ (4.4)

Let us now iterate with $\tau_{0}=\mathfrak{t}$,
$\tau_{n+1}=\eta\tau_{n}=\eta^{n+1}\mathfrak{t}$. After $n$ iterations, the
inequality $\eqref{moser1}$ yields

$\displaystyle\|u\|_{\tau_{n+1}}$ $\displaystyle\leq
C^{\sum\limits_{i=0}^{n}\frac{\mathfrak{t}}{\tau_{i}}}\prod\limits_{i=0}^{n}\left(\frac{\tau_{i}}{\mathfrak{t}}\right)^{\frac{\mathfrak{t}}{\tau_{i}}}\|u\|_{\mathfrak{t}}.$
(4.6)

By using $\eta>1$ and the method of iteration, i.e. $\tau_{0}=\mathfrak{t}$,
$\tau_{n+1}=\eta\tau_{n}=\eta^{n+1}\mathfrak{t}$, we have

$\sum\limits_{i=0}^{\infty}\frac{\mathfrak{t}}{\tau_{i}}=\sum\limits_{i=0}^{\infty}\frac{1}{\eta^{i}}=\frac{\eta}{\eta-1},$

and

$\prod\limits_{i=0}^{\infty}\left(\frac{\tau_{i}}{\mathfrak{t}}\right)^{\frac{\mathfrak{t}}{\tau_{i}}}=\eta^{\frac{\eta^{2}}{(\eta-1)^{2}}}.$

Hence, on passing the limit $n\rightarrow\infty$ in (4.6), we end up getting

$\displaystyle\|u\|_{\infty}$ $\displaystyle\leq
C^{\frac{\eta}{\eta-1}}\eta^{\frac{\eta^{2}}{(\eta-1)^{2}}}\|u\|_{\mathfrak{t}}.$
(4.7)

Thus, $u\in L^{\infty}(\Omega)$. ∎

##  5 References

## References

  * [1] V. Ambrosio and T. Isernia. On a fractional $p\&q$ Laplacian problem with critical Sobolev-Hardy exponents, Mediterranean Journal of Mathematics, 15(6):219, 2018.
  * [2] A. Bahrouni, V. D. Rǎdulescu, D. D. Repovš, A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications, Nonlinearity, 31 (4), 1516-1534, 2018.
  * [3] A. Bahrouni, V. D. Rǎdulescu, D.D. Repovš, Double phase transonic flow problems with variable growth: non-linear patterns and stationary waves, Nonlinearity, 32 (7), 2481-2495, 2019.
  * [4] G. Barles, C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (3), 567-585, 2008.
  * [5] M. Bhakta, D. Mukherjee, et al. Multiplicity results for $(p;q)$ fractional elliptic equations involving critical nonlinearities, Advances in Differential Equations, 24(3/4), 185-228, 2019.
  * [6] S. Biagi, S. Dipierro, E. Valdinoci and E. Vecchi, Mixed local and nonlocal elliptic operators: regularity and maximum principles, arxiv:2005.06907v2 [math.AP] 20 May 2020.
  * [7] X. Cabré, S. Dipierro, E. Valdinoci, The Bernstein technique for integro-differential equations, arXiv:2010.00376v1 [math.AP] 1 Oct 2020.
  * [8] X. Cabré, J. Serra, An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions, Nonlinear Anal. 137, 246-265, 2016.
  * [9] L. Caffarelli, E. Valdinoci, A priori bounds for solutions of a nonlocal evolution PDE, Analysis and numerics of partial differential equations, 141–163, Springer INdAM Ser., 4, Springer, Milan, 2013.
  * [10] F. Chen, Y. Yang, Existence results for fractional $(p,q)$-Laplacian equations via Morse theory, DOI- 10.13140/RG.2.2.30120.80642, 2020.
  * [11] D. Choudhuri, Understanding the existence of multiple solutions using the Morse theory – a short note, arXiv:2003.04239 [math.AP], 9 Mar 2020.
  * [12] R. de la Llave, E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (4), 1309-1344, 2009.
  * [13] F. Dell’Oro, V. Pata, Second order linear evolution equations with general dissipation, arxiv:1811.07667 [math.AP] 19 Nov 2018.
  * [14] S. Dipierro, E. Proietti Lippi and E. Valdinoci, (Non)local logistic equations with Neumann conditions, arXiv:2101.02315v1 [math.AP] 7 Jan 2021.
  * [15] S. Dipierro, E. Valdinoci, V. Vespri, Decay estimates for evolutionary equations with fractional time-diffusion, J. Evol. Equ., 19 (2), 435-462, 2019.
  * [16] M. Ferrara, G. M. Bisci and B. Zhang, Existence of weak solutions for non-local fractional problems via Morse theory,  Discrete Contin. Dyn. Syst. Ser. B, 19 (8), 2493-2499, 2014.
  * [17] A. Ghanmi and K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fractional Differential Calculus, 6(2), 201-217, 2016.
  * [18] K. Saoudi, S. Ghosh and D. Choudhuri, Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity, J. Math. Phys., 60, 101509 1-28, 2019.
  * [19] J. Giacomoni, T. Mukherjee and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal., 6 (3), 327-354, 2017\.
  * [20] D. Goel, D. Kumar, and K. Sreenadh. Regularity and multiplicity results for fractional $(p;q)$-Laplacian equations. Communications In Contemporary Mathematics, 2019\. https://doi.org/10.1142/S0219199719500652
  * [21] L. Gongbao and Z. Guo. Multiple solutions for the $p\&q$-Laplacian problem with critical exponent, Acta Mathematica Scientia, 29(4), 903-918, 2009.
  * [22] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations, 189, 487-512, 2003.
  * [23] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 9, 12, 197-220, 2004.
  * [24] A. Iannizzotto, S. Liu, K. Perera, M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, arXiv:1403.5388v1 [math.AP] 21 Mar 2014.
  * [25] T. Isernia. Fractional $p\&q$-Laplacian problems with potentials vanishing at infinity. Opuscula Mathematica, 40(1), 93-110, 2020.
  * [26] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des sciences mathématiques, 136(5), 521-573, 2012.
  * [27] Z. Liang, Y. Song, and J. Su, Existence of solutions to $(2;p)$-Laplacian equations by Morse theory, Electronic Journal of Differential Equations, 2017, 1-9, 2017.
  * [28] S. A. Marano, S. J. N. Mosconi, and N. S. Papageorgiou. Multiple solutions to $(p;q)$\- Laplacian problems with resonant concave nonlinearity, Advanced Nonlinear Studies, 2015\. https://doi.org/10.1515/ans-2015-5011.
  * [29] A. Mimica, Heat kernel estimates for subordinate Brownian motions, Proc. Lond. Math. Soc. (3) 113 (5), 627–648, 2016.
  * [30] D. Mugnai and N. S. Papageorgiou, Wang’s multiplicity result for superlinear $(p;q)$\- equations without the Ambrosetti-Rabinowitz condition. Transactions of the American Mathematical Society, 366(9), 2014.
  * [31] T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electronic Journal of Differential Equations, 2016(54), 1-23, 2016.
  * [32] N. S. Papageorgiou, V. D. Rădulescu,  Nonlinear Nonhomogeneous Robin Problems with Superlinear Reaction Term,  Adv. Nonlinear Stud., 16 (4), 737-764, 2016.
  * [33] N. S. Papageorgiou, V. D. Rădulescu, D. D. Repovš,  Nonlinear Analysis-Theory and Methods,  Springer Monographs in Mathematics, Springer nature, Cham, 2019.
  * [34] N. S. Papageorgiou, V. D. Rădulescu, D. D. Repovš,  Existence and multiplicity of solutions for double-phase Robin problems,  arXiv:2006.01454v1[math.AP], 2020.
  * [35] K. Perera, R. P. Agarwal and D. O’Regan,  Morse Theoretic Aspects of $p$-Laplacian Type Operators,  Mathematical surveys and Monographs, Amer. Math. Soc., 161, 2010.
  * [36] H. Yin and Z. Yang. Multiplicity of positive solutions to a $p-q$-Laplacian equation involving critical nonlinearity, Nonlinear Analysis: Theory, Methods & Applications, 75(6), 3021-3035, 2012.