# A weighted fractional problem involving a singular nonlinearity and an
$L^{1}$ datum

Akasmika Panda1, Debajyoti Choudhuri2,111Corresponding author and Leandro S.
Tavares3
Department of Mathematics, National Institute of Technology Rourkela, India1,2
Center of Science and Technology, Federal University of Cariri, Brazil3
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this article, we show the existence of a unique entropy solution to the
following problem:

$\begin{split}(-\Delta)_{p,\alpha}^{s}u&=f(x)h(u)+g(x)~{}\text{in}~{}\Omega,\\\
u&>0~{}\text{in}~{}\Omega,\\\
u&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\end{split}$

where the domain $\Omega\subset\mathbb{R}^{N}$ is bounded and contains the
origin, $\alpha\in[0,\frac{N-ps}{2})$, $s\in(0,1)$, $2-\frac{s}{N}<p<\infty$,
$sp<N$, $g\in L^{1}(\Omega)$, $f\in L^{q}(\Omega)$ for $q>1$ and $h$ is a
general singular function with singularity at 0. Further, the fractional
$p$-Laplacian with weight $\alpha$ is given by

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

Keywords: Weighted fractional Sobolev spaces, singular nonlinearity, entropy
solution.
AMS Classification: 35J60, 35R11, 35A15.

##  1 Introduction

We consider the following problem

$\begin{split}(-\Delta)_{p,\alpha}^{s}u&=f(x)h(u)+g(x)~{}\text{in}~{}\Omega,\\\
u&>0~{}\text{in}~{}\Omega,\\\
u&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\end{split}$ (1.1)

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $s\in(0,1)$,
$2-\frac{s}{N}<p<\infty$, $sp<N$, $0\leq\alpha<\frac{N-ps}{2}$ with $g\in
L^{1}(\Omega)$ and $f\in L^{q}(\Omega)$, for some $q>1$ are two positive
functions. The singularity $h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a
non-increasing and continuous function such that

$\underset{v\rightarrow
0^{+}}{\lim}~{}h(v)\in(0,\infty]~{}\text{and}~{}\underset{v\rightarrow\infty}{\lim}h(v):=h(\infty)<\infty.$
(1.2)

Regarding the growth conditions on $h$ near zero and infinity it will be
considered that there are constants $K_{1},K_{2},M,N>0$ in such a way that

$h(v)\leq\frac{K_{1}}{v^{\gamma}}~{}\text{if}~{}v<{M},~{}\text{for}~{}0<\gamma<1,$
(1.3)
$h(v)\leq\frac{K_{2}}{v^{\theta}}~{}\text{if}~{}v>N,~{}\text{for}~{}\theta>0.$
(1.4)

Note that in the case (1.4) we have $h(\infty)=0$. The weighted fractional
$p$-Laplacian is defined as

$(-\Delta)_{p,\alpha}^{s}u(x)=\text{P
V}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\frac{dy}{|x|^{\alpha}|y|^{\alpha}},~{}\forall
x\in\mathbb{R}^{N}.$ (1.5)

This operator $(-\Delta)^{s}_{p,\alpha}$ is the nonlocal version of the local
operator the operator
$-\operatorname{div}(|x|^{-\alpha}|\nabla\cdot|^{p-2}\nabla\cdot)$. The
Caffarelli$-$Kohn$-$Nirenberg inequality, derived in [7], is strongly related
to the problem

$-\operatorname{div}(|x|^{-\alpha}|\nabla u|^{p-2}\nabla
u)=0,~{}\text{in}~{}\mathbb{R}^{N}$ (1.6)

and each weak supersolution to the above problem $\eqref{CKN}$ satisfies the
weak Harnack inequality, for every $\alpha<N-p$. Thus, in this sense, we say
that $|x|^{-\alpha}$ is an admissible weight. A nonlocal version of the
Caffarelli$-$Kohn$-$Nirenberg inequality, in a bounded domain, has been
introduced by Abdellaoui & Bentifour in [2] with $\alpha=\frac{N-sp}{2}$. For
the case $\Omega=\mathbb{R}^{N}$, the generalization of the classical
Caffarelli$-$Kohn$-$Nirenberg inequality is given in [2] for
$\alpha<\frac{N-sp}{2}$ and is known as the weighted fractional Sobolev
inequality. This condition on $\alpha$ is related in some sense to the
admissible weight described in [13]. The derivation of the weighted fractional
Sobolev inequality is based on the following improved Hardy inequality, i.e.
for every $u\in C_{0}^{\infty}(\Omega)$,

$\displaystyle\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy$
$\displaystyle-\Lambda_{N,s,p}\int_{\mathbb{R^{N}}}\frac{|u|^{p}}{|x|^{sp}}dx$
$\displaystyle\geq
C\int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{N+sp}}\frac{dxdy}{|x|^{\frac{N-sp}{2}}|y|^{\frac{N-sp}{2}}}.$
(1.7)

Here, $\Lambda_{N,s,p}$ is the best Hardy constant and
$v(x)=|x|^{\frac{N-sp}{p}}u(x)$. Therefore, the class of operators like
$(-\Delta)^{s}_{p,\alpha}$ appear naturally when one deals with the inequality
(1). We quote [2, 3, 11] for further references.
The main goal of this manuscriptis to obtain the existence and uniqueness of
an entropy solution to $\eqref{ent}$. The concept of entropy solution is very
useful when technical difficulties arise in proving the uniqueness of the weak
solution. Regarding such approach in problems involving an $L^{1}$ datum we
point out [1, 5, 4] and its references. Such notion was considered for the
first time by Boccardo et al. [5] and Bénilan et al [4], where the authors
considered the following problem with an $L^{1}$ datum $g$:

$\begin{split}A(u)&=g(x)~{}\text{in}~{}\Omega,\\\
u&=0~{}\text{in}~{}\Omega,\end{split}$ (1.8)

where $u\mapsto A(u)$ is a monotone operator acting on $W_{0}^{1,p}(\Omega)$
for $1<p<N$. The above mentioned problem $\eqref{boccardo problem}$ is a local
sub-case of the problem $\eqref{ent}$ with $s=1,\alpha=0$ and $f=0$. Although
there is a good amount of literature pertaining to the case $f\neq 0$, the
notion of entropy solution was not previously applied in the literature to
guarantee the existence and uniqueness of solution.
The case of measure data was considered in [9], where the authors have shown
the existence of a renormalized solution to the following problem:

$\begin{split}-\operatorname{div}(a(x,\nabla u))&=\mu~{}\text{in}~{}\Omega,\\\
u&=0~{}\text{in}~{}\Omega.\end{split}$ (1.9)

Here, $\mu$ is a bounded Radon measure and
$u\mapsto-\operatorname{div}(a(x,\nabla u))$ is a monotone operator acting on
$W_{0}^{1,p}(\Omega)$. Kuusi et al. in [16], dealt with a Dirichlet problem
involving an $L^{1}$ datum and the fractional $p$-Laplacian. The authors used
the basic and optimal nonlinear Wolff potential estimates to obtain the
existence of solution in an appropriate fractional Sobolev space. An important
piece of work which is worth mentioning here is due to Abdellaoui et al. in
[1]. The authors have considered the case $f\equiv 0$ of the problem
$\eqref{ent}$ and obtained a unique entropy solution using some algebraic
inequalities.
An improvement of the problems above was considered by Panda et al. [23],
where the authors studied a local singular problem involving a measure datum.
The problem $\eqref{ent}$ with $\alpha=0,s=1,p=2$ and with
$\alpha=0,s\in(0,1),p=2$ has been analysed by Panda et al. [23] and by Ghosh
et al. [12] respectively. The authors guaranteed the existence of a weak
solution as well as a very weak solution to the problem using an approximation
argument. The case of purely singular problem, i.e. $\eqref{ent}$ with
$g\equiv 0$, involving the fractional $p$-Laplace operator was treated in [8]
for $\gamma>0$. Some equally important works in the literature on these type
of problems that involves a singular term and a source term in the form of a
measure can be found in [15, 20, 21, 22] and the references therein.
In this manuscript, by using adapting the ideas of [1, 23], we extend the work
of Abdellaoui [1] by considering a singular function $h$, having a singularity
at 0, in the right hand side of the problem. Due to the irregularity near the
boundary, singular problems admits solutions in a weak distributional sense,
for compactly supported test functions. See Section 2 for the notations.

######  Theorem 1.1.

There exists a positive nontrivial weak solution $u$ to the problem
$\eqref{ent}$ in $X_{0}^{t,m,\alpha}(\Omega)$, for every $1\leq
m<\frac{N(p-1)}{N-s}$ and $0<t<s$, in the sense of Definition 2.8. More
precisely,

$\int_{Q}\frac{|u(x)-u(y)|^{m}}{|x-y|^{N+tm}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}\leq
C,$ (1.10)

for every $1\leq m<\frac{N(p-1)}{N-s}$ and every $0<t<s$. Moreover,
$u\in\mathcal{T}_{0}^{s,p,\alpha}(\Omega)$.

The weak solution space do not lie in the natural energy space corresponding
to the operator $(-\Delta)_{p,\alpha}^{s}$, i.e. $X_{0}^{s,p,\alpha}(\Omega)$,
but it has a lower degree of differentiability and integrability. The next
theorem proves the uniqueness result.

######  Theorem 1.2.

Let $g\in L^{1}(\Omega)$. Then, the problem $\eqref{ent}$ admits a unique
entropy solution in $\mathcal{T}_{0}^{s,p,\alpha}(\Omega)$ in the sense of
Definition 2.9.

The paper is organized as follows: In Section 2, we introduce the weighted
fractional Sobolev space which is the natural solution space to the given
problem $\eqref{ent}$. Further, we also define some useful function spaces and
provide some auxiliary results. The existence of a positive weak solution to
the problem $\eqref{ent}$ is proved in Section 3 by an approximation argument
and using some apriori estimates. Section 4 is all about showing the existence
and uniqueness of a positive entropy solution to $\eqref{ent}$. In this
process, we show the equivalence between the weak solution and the entropy
solution to $\eqref{ent}$. Additionally, we prove the existence of an entropy
solution to the problem $\eqref{ent}$ with $h(u)=\frac{1}{u^{\gamma}}$ and
with a general function $g(x,w)=w^{r}+g(x)$ for $r<p-1$.

##  2 Properties of weighted fractional Sobolev space and some preliminary
definitions & results

We now introduce the weighted fractional Sobolev space which is a natural
solution space to the problem $\eqref{ent}$. Let us consider $\Omega$ to be a
bounded domain in $\mathbb{R}^{N}$, $s\in(0,1)$, $1<p<\infty$ and
$\alpha\in[0,\frac{N-sp}{2})$. Denote

$d\nu=\frac{dx}{|x|^{2\alpha}},~{}~{}d\mu=\frac{dxdy}{|x-y|^{N+sp}|x|^{\alpha}|y|^{\alpha}}.$

The fractional Sobolev space with weight $\alpha$ (refer [2]) is defined by

$W^{s,p,\alpha}(\mathbb{R}^{N})=\left\\{u\in
L^{p}(\mathbb{R}^{N},d\nu):\iint_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}d\mu<\infty\right\\}.$

Further, denote $Q=\mathbb{R}^{2N}\setminus(\Omega^{c}\times\Omega^{c})$ with
$\Omega^{c}=\mathbb{R}^{N}\setminus\Omega$. Then, the space
$X^{s,p,\alpha}(\Omega)$ is defined by

$\displaystyle X^{s,p,\alpha}(\Omega)$
$\displaystyle=\left\\{u:\mathbb{R}^{N}\rightarrow\mathbb{R}~{}\text{is
measurable},u|_{\Omega}\in
L^{p}(\Omega,d\nu)~{}\text{and}~{}\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}<\infty\right\\}$

and this space is a Banach space endowed with the norm:

$\|u\|_{X^{s,p,\alpha}(\Omega)}=\|u\|_{L^{p}(\Omega,d\nu)}+\left(\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}\right)^{\frac{1}{p}}.$

Define a space $X_{0}^{s,p,\alpha}(\Omega)$ by

$X_{0}^{s,p,\alpha}(\Omega)=\\{u\in X^{s,p,\alpha}(\Omega):u=0\text{ in
}\mathbb{R}^{N}\setminus\Omega\\},$

which is a Banach space when endowed with the following norm:

$\|u\|_{X_{0}^{s,p,\alpha}(\Omega)}^{p}=\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}.$

If $\alpha=0$, denote
$W^{s,p,\alpha}(\mathbb{R}^{N})=W^{s,p}(\mathbb{R}^{N}),~{}X^{s,p,\alpha}(\Omega)=X^{s,p}(\Omega),~{}X_{0}^{s,p,\alpha}(\Omega)=X_{0}^{s,p}(\Omega)$
which are the usual well-known fractional Sobolev spaces. Since the domain
$\Omega$ is bounded, the embedding $L^{1}(\Omega,d\nu)\hookrightarrow
L^{1}(\Omega)$ is continuous and we will use this embedding throughout the
article.
The following is the weighted fractional Sobolev inequality which is an
improvement of the fractional Sobolev inequality and is obtained by the help
of weighted Hardy inequality (see (1)).

######  Theorem 2.1 ([2], Theorem 1.4, Theorem 1.5).

Let $s\in(0,1)$ and $1<p<\frac{N}{s}$. If $\alpha<\frac{N-sp}{2}$, then there
exists constant $S=S(s,N,p,\alpha)>0$ such that

$\iint_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}d\mu\geq
S\left(\int_{\mathbb{R}^{N}}\frac{|u(x)|^{p_{s}^{*}}}{|x|^{\frac{2\alpha
p_{s}^{*}}{p}}}\right)^{\frac{p}{p_{s}^{*}}}$

for every $u\in C_{c}^{\infty}(\mathbb{R}^{N})$. Here, the constant
$p_{s}^{*}=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent.

The next theorem is known as the fractional Caffarelli$-$Kohn$-$Nirenberg
inequality in a bounded domain, proved in [2].

######  Theorem 2.2.

If $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ and
$\alpha=\frac{N-sp}{2}$, then for every $q<p$ there exists constant
$C=C(\Omega,s,q,N)>0$ such that

$\iint_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}d\mu\geq
C\left(\int_{\mathbb{R}^{N}}\frac{|u(x)|^{p_{q,s}^{*}}}{|x|^{\frac{2\alpha
p_{q,s}^{*}}{p}}}\right)^{\frac{p}{p_{q,s}^{*}}}$

for every $u\in C_{c}^{\infty}(\mathbb{R}^{N})$, where
$p^{*}_{q,s}=\frac{Np}{N-sq}$.

The truncation function $T_{k}:\mathbb{R}\rightarrow\mathbb{R}$, for a fixed
$k>0$, is defined by

$\displaystyle T_{k}(s)=\max\\{-k,\min\\{k,s\\}\\}$

and define

$G_{k}(v)=(|v|-k)^{+}sign(v).$

Clearly, $T_{k}(v)+G_{k}(v)=v$ for every $v\in\mathbb{R}$.

######  Definition 2.3.

We define $\mathcal{T}^{s,p,\alpha}_{0}(\Omega)$ to be the class of measurable
functions $u$ on $\Omega$ such that $T_{k}(u)\in X^{s,p,\alpha}_{0}(\Omega)$
for all $k>0$.

The following well-known algebraic inequalities will be frequently used
throughout this article.

######  Remark 2.4.

Let $p\geq 1$, $\beta>0$ and $a_{1},a_{2}\in[0,\infty]$. Then, there exist
constants $c_{1},c_{2},c_{3}>0$ such that

  1. 1.

$(a_{1}+a_{2})^{\beta}\leq c_{1}a_{1}^{\beta}+c_{2}a_{2}^{\beta}$,

  2. 2.

$|a_{1}-a_{2}|^{p-2}(a_{1}-a_{2})(a_{1}^{\beta}-a_{2}^{\beta})\geq
c_{3}|a_{1}^{\frac{\beta+p-1}{p}}-a_{2}^{\frac{\beta+p-1}{p}}|^{p}$,

  3. 3.

$|a_{1}+a_{2}|^{\beta-1}|a_{1}-a_{2}|^{p}\leq
c_{3}|a_{1}^{\frac{\beta+p-1}{p}}-a_{2}^{\frac{\beta+p-1}{p}}|^{p}$ for
$\beta\geq 1$,

  4. 4.

for $a_{1},a_{2}\in\mathbb{R}$,

$|a_{1}-a_{2}|^{p-2}(a_{1}-a_{2})(T_{k}(a_{1})-T_{k}(a_{2}))\geq|T_{k}(a_{1})-T_{k}(a_{2})|^{p},$
$|a_{1}-a_{2}|^{p-2}(a_{1}-a_{2})(G_{k}(a_{1})-G_{k}(a_{2}))\geq|G_{k}(a_{1})-G_{k}(a_{2})|^{p},$

  5. 5.

$\left(a_{1}^{p-2}a_{1}-a_{2}^{p-2}a_{2}\right)\cdot(a_{1}-a_{2})\geq
2^{2-p}|a_{1}-a_{2}|^{p}$, when $p\geq 2,$

  6. 6.

$\left(a_{1}^{p-2}a_{1}-a_{2}^{p-2}a_{2}\right)\cdot(a_{1}-a_{2})\geq\frac{1}{2}\left(|a_{1}|+|a_{2}|\right)^{p-2}|a_{1}-a_{2}|^{2}$,
when $p<2$.

######  Definition 2.5 ([1]).

A measurable function $u:\Omega\rightarrow\mathbb{R}$ is said to be in the
weighted Marcinkiewicz space, denoted by $M^{q}(\Omega,d\nu)$ for
$0<q<\infty$, if

$\nu(\\{x\in\Omega:|u(x)|>t\\})\leq\frac{C}{t^{q}},\text{ for }~{}t>0\text{
and }0<C<\infty.$

Assume $\Omega\subset\mathbb{R}^{N}$ to be bounded, then

  1. 1.

$M^{q_{1}}(\Omega,d\nu)\subset M^{q_{2}}(\Omega,d\nu)$, $\forall q_{1}\geq
q_{2}>0$,

  2. 2.

for $1\leq q<\infty$ and $0<\epsilon<q-1$, we have the following continuous
embedding

$L^{q}(\Omega,d\nu)\subset M^{q}(\Omega,d\nu)\subset
L^{q-\epsilon}(\Omega,d\nu).$ (2.11)

Abdellaoui & Bentifour [2] provided a useful comparison principle which is
stated in the following lemma.

######  Lemma 2.6.

Assume that $\Omega\subset\mathbb{R^{N}}$ is a bounded domain, $p\geq 1$,
$H(\cdot,\cdot)$ is a positive continuous function and
$\frac{H(x,v)}{v^{p-1}}$ is decreasing for $v>0$. Furthermore, let $v,w\in
X^{s,p,\alpha}_{0}(\Omega)$ satisfy

$\displaystyle\begin{split}&\begin{cases}(-\Delta)^{s}_{p,\alpha}v\geq
H(x,v)&\text{in}~{}\Omega,\\\ (-\Delta)^{s}_{p,\alpha}w\leq
H(x,w)&\text{in}~{}\Omega,\\\ v,w>0&\text{in}~{}\Omega.\end{cases}\end{split}$

Then, $w\leq v$ in $\Omega$.

We now provide different notions of solutions to problem $\eqref{ent}$.

######  Definition 2.7.

Let $u$ be a measurable function. Then
$u\in\mathcal{T}_{0}^{s,p,\alpha}(\Omega)$ if for every $k>0$, $T_{k}(u)\in
X_{0}^{s,p,\alpha}(\Omega)$.

######  Definition 2.8 (Weak solution).

Let $g\in L^{1}(\Omega)$. A function $u\in X_{0}^{t,m,\alpha}(\Omega)$, for
$0<t<s$ and $1\leq m<p$, is said to be a positive weak solution to
$\eqref{ent}$ if

$\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))d\mu=\int_{\Omega}fh(u)\phi+\int_{\Omega}g\phi$
(2.12)

for every $\phi\in C_{c}^{\infty}(\Omega)$ and for every
$\omega\subset\subset\Omega$, there exists $C_{\omega}>0$ such that $u\geq
C_{\omega}>0$.

In many cases, it is very tough to guarantee the uniqueness of weak solutions
and to overcome this difficulty, the concept of entropy solution comes into
picture to prove the uniqueness of solutions.

######  Definition 2.9 (Entropy solution).

Let $g\in L^{1}(\Omega)$. A function
$u\in\mathcal{T}_{0}^{s,p,\alpha}(\Omega)$ is said to be an entropy solution
to $\eqref{ent}$ if it satisfies the following:

  1. 1.

$\int_{D_{l}}|u(x)-u(y)|^{p-1}d\mu\rightarrow 0~{}~{}\text{ as
}l\rightarrow\infty,$ (2.13)

where

$D_{l}=\\{(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}:l+1\leq\max\\{|u(x)|,|u(y)|\\}\text{
with }\min\\{|u(x)|,|u(y)|\\}\leq l\\},$

  2. 2.

for every $\varphi\in X^{s,p,\alpha}_{0}(\Omega)\cap L^{\infty}(\Omega)$,
$k>0$,

$\displaystyle\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))$
$\displaystyle[T_{k}(u(x)-\varphi(x))-T_{k}(u(y)-\varphi(y))]d\mu$
$\displaystyle\leq\int_{\Omega}fh(u)T_{k}(u-\varphi)+\int_{\Omega}gT_{k}(u-\varphi).$
(2.14)

The solution $u$ is positive if for every $\omega\subset\subset\Omega$, there
exists $C_{\omega}>0$ such that $u\geq C_{\omega}>0$.

######  Remark 2.10.

It is easy to show that every entropy solution of $\eqref{ent}$ is a weak
solution of $\eqref{ent}$.

##  3 Existence of weak solution - Proof of Theorem 1.1

In this section, we prove the existence of a positive weak solution to the
problem $\eqref{ent}$ in the sense of Definition 2.8. The proof is through
approximation, since the problem involves an $L^{1}$ function $g$. Let us
consider the following approximating problem:

$\begin{split}(-\Delta)_{p,\alpha}^{s}u_{n}&=f_{n}h_{n}(u_{n}+1/n)+g_{n}~{}\text{in}~{}\Omega,\\\
u_{n}&>0~{}\text{in}~{}\Omega,\\\
u_{n}&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.\end{split}$ (3.15)

Here, $f_{n}=T_{n}(f)$, $h_{n}=T_{n}(h)$ and $\\{g_{n}\\}\subset
L^{\infty}(\Omega)$ is an increasing sequence such that $g_{n}\rightarrow g$
strongly in $L^{1}(\Omega)$. We say $u_{n}\in X_{0}^{s,p.\alpha}(\Omega)$ is a
weak solution to $\eqref{ent approx}$, if $u_{n}$ satisfies

$\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\phi(x)-\phi(y))d\mu=\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)\phi+\int_{\Omega}g_{n}\phi,$
(3.16)

for every $\phi\in C_{c}^{\infty}(\Omega)$.

######  Lemma 3.1.

Let $h$ satisfies $\eqref{eqq1}$ and $\eqref{eqqq}$ with
$0<\gamma,\theta<\infty$. Then, for a fixed $n\in\mathbb{N}$, there exists a
unique weak solution $u_{n}\in X_{0}^{s,p,\alpha}(\Omega)\cap
L^{\infty}(\Omega)$ to $\eqref{ent approx}$.

###### Proof.

The proof follows the ideas of Panda et al. [23] using Schauder’s fixed point
argument. Let us define a map,

$G:L^{p}(\Omega,d\nu)\rightarrow X_{0}^{s,p,\alpha}(\Omega)\subset
L^{p}(\Omega,d\nu)$

such that for any $w\in L^{p}(\Omega,d\nu)$ we obtain a unique weak solution
$v\in X_{0}^{s,p,\alpha}(\Omega)$ to the following problem by Lax-Milgram
theorem.

$\displaystyle(-\Delta)^{s}_{p,\alpha}v$ $\displaystyle=$ $\displaystyle
h_{n}\left(|w|+\frac{1}{n}\right)f_{n}+g_{n}~{}\text{in}~{}\Omega,$
$\displaystyle v$ $\displaystyle=$ $\displaystyle
0~{}\text{on}~{}\partial\Omega.$ (3.17)

Further, $C_{c}^{\infty}(\Omega)$ is dense in $X_{0}^{s,p,\alpha}(\Omega)$
with respect to the norm topology of $X_{0}^{s,p,\alpha}(\Omega)$. Hence, we
are allowed to choose $v$ as a test function in the weak formulation of
$\eqref{helping}$. Thus, we get

$\displaystyle\int_{Q}|v(x)-v(y)|^{p}d\mu$
$\displaystyle={\int_{\Omega}h_{n}\left(|w|+\frac{1}{n}\right)f_{n}v}+\int_{\Omega}g_{n}v$
$\displaystyle\leq{K_{1}\int_{(|w|+\frac{1}{n}<M)}\frac{f_{n}v}{(|w|+\frac{1}{n})^{\gamma}}}+\max_{[M,N]}\\{h_{n}\\}\int_{(M\leq(|w|+\frac{1}{n})\leq
N)}f_{n}v$
$\displaystyle~{}~{}~{}~{}~{}~{}+K_{2}\int_{(|w|+\frac{1}{n}>N)}\frac{f_{n}v}{(|w|+\frac{1}{n})^{\theta}}+C(n)\int_{\Omega}|v|$
$\displaystyle\leq
K_{1}n^{1+\gamma}\int_{(|w|+\frac{1}{n}<M)}|v|+n\max_{[M,N]}\\{h_{n}\\}\int_{(M\leq(|w|+\frac{1}{n})\leq
N)}|v|$
$\displaystyle~{}~{}~{}~{}~{}~{}+K_{2}n^{1+\theta}\int_{(|w|+\frac{1}{n}>N)}|v|+C(n)\int_{\Omega}|v|$
$\displaystyle\leq
C^{\prime}(n,\gamma,\theta)\int_{\Omega}|v|d\nu=C^{\prime}(n,\gamma,\theta)\|v\|_{L^{1}(\Omega,d\nu)}.$

Now by applying the weighted Sobolev embedding theorem, i.e. Theorem 2.1,
there exists a constant $C(n,\gamma,\theta)>0$ such that

$\|v\|_{X_{0}^{s,p,\alpha}(\Omega)}\leq C(n,\gamma,\theta),$ (3.18)

where $C(n,\gamma,\theta)$ is independent of $w$. We will now show the
continuity and compactness of the operator $G$ to apply the Schauder fixed
point theorem.
Claim 1: The map $G$ is continuous over $X_{0}^{s,p,\alpha}(\Omega)$.
Let us consider a sequence $\\{w_{k}\\}\subset X_{0}^{s,p,\alpha}(\Omega)$
that converges strongly to $w$ with respect to the $X_{0}^{s,p,\alpha}$-norm.
Thus, by the uniqueness of the weak solution, we denote $v_{k}=G(w_{k})$,
$v=G(w)$ and let $\bar{v_{k}}=v_{k}-v$. According to the Remark 2.4-(5) and
(6),

$\displaystyle\text{if }1<p<2,\int_{Q}$
$\displaystyle\left(|v_{k}(x)-v_{k}(y)|+|v(x)-v(y)|\right)^{p-2}|\bar{v_{k}}(x)-\bar{v_{k}}(y)|^{2}d\mu$
$\displaystyle\leq
2\int_{\Omega}\left(h_{n}\left(|w_{k}|+\frac{1}{n}\right)f_{n}-h_{n}\left(|w|+\frac{1}{n}\right)f_{n}\right)\bar{v_{k}}$
(3.19)

and

$\displaystyle\text{if }p\geq
2,\int_{Q}|\bar{v_{k}}(x)-\bar{v_{k}}(y)|^{p}d\mu\leq
2^{p-2}\int_{\Omega}\left(h_{n}\left(|w_{k}|+\frac{1}{n}\right)f_{n}-h_{n}\left(|w|+\frac{1}{n}\right)f_{n}\right)\bar{v_{k}}.$
(3.20)

Hence, by Theorem 2.1 and Hölder’s inequality we establish the following for
the case of $p\geq 2$.

$\displaystyle\|\bar{v_{k}}\|^{p}_{X_{0}^{s,p,\alpha}(\Omega)}$
$\displaystyle\leq
2^{p-2}\left\|h_{n}\left(|w_{k}|+\frac{1}{n}\right)f_{n}-h_{n}\left(|w|+\frac{1}{n}\right)f_{n}\right\|_{L^{(p_{s}^{*})^{\prime}}(\Omega)}\|\bar{v_{k}}\|_{L^{p_{s}^{*}}(\Omega)}$
$\displaystyle\leq
C_{1}(N,p,\alpha,\Omega)\left\|h_{n}\left(|w_{k}|+\frac{1}{n}\right)f_{n}-h_{n}\left(|w|+\frac{1}{n}\right)f_{n}\right\|_{L^{(p_{s}^{*})^{\prime}}(\Omega)}\left\|\frac{\bar{v_{k}}}{|x|^{2\alpha/p}}\right\|_{L^{p_{s}^{*}}(\Omega)}$
$\displaystyle\leq
C_{2}\left\|h_{n}\left(|w_{k}|+\frac{1}{n}\right)f_{n}-h_{n}\left(|w|+\frac{1}{n}\right)f_{n}\right\|_{L^{(p_{s}^{*})^{\prime}}(\Omega)}\left\|\bar{v_{k}}\right\|_{X_{0}^{s,p,\alpha}(\Omega)},$

where $(p_{s}^{*})^{\prime}$ is the Hölder conjugate of $p_{s}^{*}$. This
implies

$\|\bar{v_{k}}\|_{X_{0}^{s,p,\alpha}(\Omega)}\leq
C\left\|h_{n}\left(|w_{k}|+\frac{1}{n}\right)f_{n}-h_{n}\left(|w|+\frac{1}{n}\right)f_{n}\right\|_{L^{(p_{s}^{*})^{\prime}}(\Omega)}^{1/{p-1}}.$

Now on applying the dominated convergence theorem we prove that
$\lim\limits_{k\rightarrow\infty}\|\bar{v_{k}}\|_{X_{0}^{s,p,\alpha}(\Omega)}=0$.
This gives
$\lim\limits_{k\rightarrow\infty}\|v_{k}-v\|_{X_{0}^{s,p,\alpha}(\Omega)}=0$.
In this similar manner, for the case $1<p<2$, we can prove the strong
convergence $v_{k}\rightarrow v$ in $X_{0}^{s,p,\alpha}(\Omega)$. Thus, $G$ is
continuous from $X_{0}^{s,p,\alpha}(\Omega)$ to $X_{0}^{s,p,\alpha}(\Omega)$.
Claim 2: $G(X_{0}^{s,p,\alpha}(\Omega))$ is relatively compact in
$X_{0}^{s,p,\alpha}(\Omega)$.
Let $\\{w_{k}\\}$ is a bounded sequence in $X_{0}^{s,p,\alpha}(\Omega)$. Then,
there exists $w\in X_{0}^{s,p,\alpha}(\Omega)$ and a subsequence of
$\\{w_{k}\\}$ (still denoted as $\\{w_{k}\\}$) such that $w_{k}\rightarrow w$
weakly in $X_{0}^{s,p,\alpha}(\Omega)$. We have already proved in $\eqref{eq}$
that

$\displaystyle\|G(w_{k})\|_{X_{0}^{s,p,\alpha}(\Omega)}\leq C,$

for $C>0$ independent of $k$. Thus, there exists $v\in
X_{0}^{s,p,\alpha}(\Omega)$ such that $G(w_{k})\rightarrow v$ weakly in
$X_{0}^{s,p,\alpha}(\Omega)$. Now by passing the limit $k\rightarrow\infty$ in
the weak formulation and using some basic computation we prove that $G(w)=v$.
Following the similar lines used in the proof of Claim 1, we finally show that
$\lim\limits_{k\rightarrow\infty}\|G(w_{k})-G(w)\|_{X_{0}^{s,p,\alpha}(\Omega)}=0$.
Hence the claim.
Thus, on using the Schauder fixed point theorem to $G$, we obtain a fixed
point $u_{n}\in X_{0}^{s,p,\alpha}(\Omega)$ that is also a weak solution to
the problem $\eqref{ent approx}$ in $X_{0}^{s,p,\alpha}(\Omega)$. Further, by
the strong maximum principle [Lemma 2.3, [19]], $u_{n}>0$ in $\Omega$.
Claim 3: Uniqueness of weak solution.
To prove this claim, suppose the problem $\eqref{ent approx}$ admits two
different weak solutions $u_{n}$ and $v_{n}$. Let us take
$\phi=(u_{n}-v_{n})^{+}$ as a test function in the weak formulation
$\eqref{weak formula approx}$. Thus, for $p\geq 2$, with the consideration of
Lemma 9 in [18], we obtain the following.

$\displaystyle 0$
$\displaystyle\leq\int_{Q}|(u_{n}(x)-v_{n}(x))^{+}-(u_{n}(y)-v_{n}(y))^{+}|^{p}d\mu$
$\displaystyle\leq\int_{Q}\left[|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))-|v_{n}(x)-v_{n}(y)|^{p-2}(v_{n}(x)-v_{n}(y))\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times[(u_{n}(x)-v_{n}(x))^{+}-(u_{n}(y)-v_{n}(y))^{+}]d\mu$
$\displaystyle=\int_{\Omega}\left(f_{n}h_{n}(u_{n}+1/n)-f_{n}h_{n}(v_{n}+1/n)\right)(u_{n}-v_{n})^{+}$
$\displaystyle\leq 0.$

This implies $(u_{n}-v_{n})^{+}=0$ a.e in $\Omega$, since $u_{n}=v_{n}=0$ in
$\mathbb{R}^{N}\setminus\Omega$ and thus $u_{n}\leq v_{n}$ a.e. in $\Omega$.
The same proof holds for $1<p<2$. Proceeding similarly with
$\phi=(v_{n}-u_{n})^{+}$ as a test function, we can show that $u_{n}\geq
v_{n}$ a.e. in $\Omega$. This proves the claim.
Since the right hand side of $\eqref{ent approx}$ belongs to
$L^{\infty}(\Omega)$, by virtue of Lemma 4.3 of [1], we conclude that
$u_{n}\in L^{\infty}(\Omega)$. ∎

######  Lemma 3.2.

Let $u_{n}$ be a unique positive weak solution to $\eqref{ent approx}$. Then,

  1. 1.

the sequence $\\{u_{n}\\}$ is an increasing sequence w.r.t $n$ and for every
set $\omega\subset\subset\Omega$ there exists constant $C_{\omega}>0$,
independent of $n$, such that

$u_{n}\geq C_{\omega}>0,\text{ for every }n\in\mathbb{N}.$ (3.21)

  2. 2.

$h(u_{n})f\in L^{1}(\Omega)$.

###### Proof.

Let us consider the problems satisfied by $u_{n}$ and $u_{n+1}$. Then,
subtracting these two problems and taking the test function
$(u_{n}-u_{n+1})^{+}$ in its weak formulation we get

$\displaystyle 0$
$\displaystyle\leq\int_{Q}|(u_{n}(x)-u_{n+1}(y))^{+}-(u_{n}(x)-u_{n+1}(y))^{+}|^{p}d\mu$
$\displaystyle\leq\int_{Q}\left(I_{p}(u_{n})-I_{p}(u_{n+1}\right)((u_{n}(x)-u_{n+1}(x))^{+}-(u_{n}(y)-u_{n+1}(y))^{+})d\mu$
$\displaystyle=\int_{\Omega}\left(f_{n}h_{n}(u_{n}+1/n)-f_{n+1}h_{n+1}(u_{n+1}+1/\\{n+1\\})\right)(u_{n}-u_{n+1})^{+}+\int_{\Omega}(g_{n}-g_{n+1})(u_{n}-u_{n+1})^{+}$
$\displaystyle\leq\int_{\Omega}\left(f_{n+1}h_{n+1}(u_{n}+1/\\{n+1\\})-f_{n+1}h_{n+1}(u_{n+1}+1/\\{n+1\\})\right)(u_{n}-u_{n+1})^{+}$
$\displaystyle\leq 0,$

where $I_{p}(v)=|v(x)-v(y)|^{p-2}(v(x)-v(y))$. This implies
$(u_{n}-u_{n+1})^{+}=0$ a.e. in $\Omega$ and hence $u_{n}\leq u_{n+1}$ a.e. in
$\Omega$.
Further, from the definition of the problem (3.15), $u_{1}>0$ in $\Omega$ and
belongs to $L^{\infty}(\Omega)$, i.e. there exists $M>0$ such that
$\|u_{1}\|_{L^{\infty}(\Omega)}\leq M$. Since $\\{u_{n}\\}$ is an increasing
sequence, $u_{n}$ verifies $\eqref{positive}$ for all $n\geq 1$. This proves
$(1)$.
Consider the following nonlinear eigenvalue problem:

$\begin{split}(-\Delta)_{p,\alpha}^{s}v&=\lambda|v|^{p-2}v~{}\text{in}~{}\Omega,\\\
v&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.\end{split}$ (3.22)

Moreover, from the weighted fractional Sobolev embedding, we have the Rayleigh
quotient as follows.

$\lambda_{1}=\underset{v\in X_{0}^{s,p,\alpha}(\Omega),v\neq
0}{\inf}\left\\{\frac{\|v\|_{X_{0}^{s,p,\alpha}(\Omega)}^{p}}{\|v\|^{p}_{L^{p}(\Omega)}}\right\\}$

and $\lambda_{1}\in(0,\infty)$. Let $\phi_{1}$ be the eigenfunction
corresponding to the eigenvalue $\lambda_{1}$. Then, it is not difficult to
prove that either $\phi_{1}>0$ a.e. in $\Omega$ or $\phi_{1}<0$ a.e. in
$\Omega$, refer Proposition 2.3 of [14]. Indeed, if $\psi=\phi_{1}^{-}$ is
chosen as a test function to the following weak formulation

$\int_{Q}|\phi_{1}(x)-\phi_{1}(y)|^{p-2}(\phi_{1}(x)-\phi_{1}(y))(\psi(x)-\psi(y))d\mu=\lambda_{1}\int_{\Omega}|\phi_{1}|^{p-2}\phi_{1}\psi,$

we get $\phi_{1}^{-}\equiv 0$ a.e. in $\Omega$. So, $\phi_{1}>0$ a.e. in
$\Omega$. We already know that $u_{1}\in X_{0}^{s,p,\alpha}(\Omega)$ is a weak
solution to $\eqref{ent approx}$ with $n=1$ and $h$ is a nonincreasing
function. Thus, we may choose $c>0$ sufficiently small such that

$\displaystyle(-\Delta)_{p,\alpha}^{s}(c\phi_{1})-f_{1}h_{1}(c\phi_{1}+1)-g_{1}$
$\displaystyle<0$
$\displaystyle=(-\Delta)_{p,\alpha}^{s}u_{1}-f_{1}h_{1}(u_{1}+1)-g_{1},~{}\text{in
}\Omega.$

By using the comparison principle, stated in Lemma 2.6, we observe
$c\phi_{1}\leq u_{1}$ in $\Omega$. Following the arguments as in Lazer and
McKenna [17], for $\gamma<1$, we get $h(\phi_{1})\in L^{1}(\Omega)$. This
implies $h(u_{1})f\leq h(c\phi_{1})f\in L^{1}(\Omega)$. Therefore,
$h(u_{n})f\in L^{1}(\Omega)$ for each $n$ and this concludes the proof of
$(2)$. ∎

In the proceeding lemma we find some apriori estimates so that we can pass the
limit $n\rightarrow\infty$ in $\eqref{weak formula approx}$.

######  Lemma 3.3.

Let $h$ satisfies the growth conditions given in $\eqref{eqq1}$,
$\eqref{eqqq}$ for $0<\gamma<1$, $\theta>0$ and let $u_{n}\in
X_{0}^{s,p,\alpha}(\Omega)$ be a positive weak solution to $\eqref{ent
approx}$. Then, the sequence $\\{u_{n}\\}$ is bounded in
$L^{m_{1}}(\Omega,d\nu)$ for every $m_{1}<\frac{N(p-1)}{N-sp}$. Furthermore,

$\int_{Q}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+tm}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}\leq
C,$ (3.23)

for every $1\leq m<\frac{N(p-1)}{N-s}$ and $0<t<s$.

###### Proof.

Let $u_{n}\in X_{0}^{s,p,\alpha}(\Omega)$ be a unique positive weak solution
to $\eqref{ent approx}$. Then, for any $k\geq 1$, using $\phi=T_{k}(u_{n})$ as
a test function in $\eqref{weak formula approx}$, we have

$\displaystyle\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}$
$\displaystyle(u_{n}(x)-u_{n}(y))(T_{k}(u_{n}(x)-T_{k}(u_{n}(y))d\mu$
$\displaystyle=\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)T_{k}(u_{n})+\int_{\Omega}g_{n}T_{k}(u_{n})$
$\displaystyle\leq{K_{1}\int_{(u_{n}+\frac{1}{n}<M)}\frac{f_{n}T_{k}(u_{n})}{(u_{n}+\frac{1}{n})^{\gamma}}}+\max_{[M,N]}\\{h(s)\\}\int_{(M\leq
u_{n}+\frac{1}{n}\leq N)}f_{n}T_{k}(u_{n})$
$\displaystyle~{}~{}~{}~{}+K_{2}\int_{(u_{n}+\frac{1}{n}>N)}\frac{f_{n}T_{k}(u_{n})}{(u_{n}+\frac{1}{n})^{\theta}}+k\int_{\Omega}g_{n}$
$\displaystyle\leq
K_{1}M^{1-\gamma}\int_{(u_{n}+\frac{1}{n}<M)}f+k\max_{[M,N]}\\{h(s)\\}\int_{(M\leq
u_{n}+\frac{1}{n}\leq N)}f$
$\displaystyle~{}~{}~{}~{}+\frac{K_{2}k}{N^{\theta}}\int_{(u_{n}+\frac{1}{n}>N)}f+C_{1}k$
$\displaystyle\leq C_{1}k$ (3.24)

On using Theorem 2.1, Remark 2.4 and the above inequality $\eqref{entropy1}$,
we have

$\displaystyle
S\left(\int_{\Omega}\frac{|T_{k}(u_{n})(x)|^{p_{s}^{*}}}{|x|^{\frac{2\alpha
p_{s}^{*}}{p}}}dx\right)^{\frac{p}{p_{s}^{*}}}$
$\displaystyle\leq\int_{Q}|T_{k}(u_{n})(x)-T_{k}(u_{n})(y)|^{p}d\mu$
$\displaystyle\leq\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(T_{k}(u_{n}(x)-T_{k}(u_{n}(y))d\mu$
$\displaystyle\leq C_{1}k,~{}\forall k>0.$ (3.25)

Hence,

$\displaystyle\nu(\\{x\in\Omega:|u_{n}(x)|\geq k\\})$
$\displaystyle\leq\nu(\\{x\in\Omega:|T_{k}(u_{n}(x))|=k\\})$
$\displaystyle\leq
C_{2}\int_{\Omega}\frac{|T_{k}(u_{n})(x)|^{p_{s}^{*}}}{k^{p_{s}^{*}}|x|^{\frac{2\alpha
p_{s}^{*}}{p}}}dx$ $\displaystyle\leq\frac{C}{k^{\frac{N(p-1)}{N-sp}}}.$
(3.26)

Therefore, by Definition 2.5, $\\{u_{n}\\}$ is bounded in
$M^{\frac{N(p-1)}{N-sp}}(\Omega,d\nu)$ and hence bounded in
$L^{m_{1}}(\Omega,d\nu)$ for every $m_{1}<\frac{N(p-1)}{N-sp}$. Consequently,
the sequence $\\{|u_{n}|^{p-2}u_{n}\\}$ is bounded in $L^{m_{2}}(\Omega,d\nu)$
for every $m_{2}<\frac{N}{N-sp}$.
Let us choose $m<\frac{N(p-1)}{N-s}$ and $t<s$ very close to $s$ such that
$\frac{mp(s-t)}{p-m}<\alpha$. Denote

$w_{n}(x)=T_{1}(u_{n}(x))-\frac{1}{(u_{n}(x)+1)^{\beta}},\text{ for some
}\beta>0\text{ which will be chosen later}.$

On using $w_{n}$ as a test function in $\eqref{weak formula approx}$, we
establish the following.

$\displaystyle\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(T_{1}(u_{n}(x))-T_{1}(u_{n}(y)))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))\left(\frac{1}{(u_{n}(y)+1)^{\beta}}-\frac{1}{(u_{n}(x)+1)^{\beta}}\right)$
$\displaystyle=\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)\left(T_{1}(u_{n}(x))-\frac{1}{(u_{n}(x)+1)^{\beta}}\right)+\int_{\Omega}g_{n}\left(T_{1}(u_{n}(x))-\frac{1}{(u_{n}(x)+1)^{\beta}}\right)$
$\displaystyle\leq{K_{1}\int_{(u_{n}+\frac{1}{n}<M)}\frac{f_{n}T_{1}(u_{n})}{(u_{n}+\frac{1}{n})^{\gamma}}}+\max_{[M,N]}\\{h(s)\\}\int_{(M\leq
u_{n}+\frac{1}{n}\leq
N)}f_{n}T_{1}(u_{n})+K_{2}\int_{(u_{n}+\frac{1}{n}>N)}\frac{f_{n}T_{1}(u_{n})}{(u_{n}+\frac{1}{n})^{\theta}}+C_{1}$
$\displaystyle\leq
K_{1}M^{1-\gamma}\int_{(u_{n}+\frac{1}{n}<M)}f+\max_{[M,N]}\\{h(s)\\}\int_{(M\leq
u_{n}+\frac{1}{n}\leq
N)}f+\frac{K_{2}}{N^{\theta}}\int_{(u_{n}+\frac{1}{n}>N)}f+C_{1}$
$\displaystyle\leq C_{2}.$ (3.27)

On the other hand, the use of Remark 2.4 implies that

$\displaystyle\int_{Q}\frac{|(u_{n}+1)^{\frac{p+\beta-1}{p}}(x)-(u_{n}+1)^{\frac{p+\beta-1}{p}}(y)|^{p}}{(u_{n}(x)+1)^{\beta}(u_{n}(y)+1)^{\beta}}d\mu$
$\displaystyle\leq\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))\frac{(u_{n}(x)+1)^{\beta}-(u_{n}(y)+1)^{\beta}}{(u_{n}(x)+1)^{\beta}(u_{n}(y)+1)^{\beta}}d\mu$
$\displaystyle\leq\int_{Q}|T_{1}(u_{n}(x))-T_{1}(u_{n}(y))|^{p}d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))\left(\frac{1}{(u_{n}(y)+1)^{\beta}}-\frac{1}{(u_{n}(x)+1)^{\beta}}\right)$
$\displaystyle\leq\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(T_{1}(u_{n}(x))-T_{1}(u_{n}(y)))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))\left(\frac{1}{(u_{n}(y)+1)^{\beta}}-\frac{1}{(u_{n}(x)+1)^{\beta}}\right).$
(3.28)

Thus, the inequalities $\eqref{3.2 refer}$ and $\eqref{3.2 refer 1}$ together
yields

$\int_{Q}\frac{|(u_{n}+1)^{\frac{p+\beta-1}{p}}(x)-(u_{n}+1)^{\frac{p+\beta-1}{p}}(y)|^{p}}{(u_{n}(x)+1)^{\beta}(u_{n}(y)+1)^{\beta}}d\mu\leq
C.$ (3.29)

Abdellaoui et al. [Lemma 3.2, [1]] established the following inequality for
certain range of $\beta>0$.

$\int_{\Omega}\int_{\Omega}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+tm}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}\leq
C_{4}\left(\int_{Q}\frac{|(u_{n}+1)^{\frac{p+\beta-1}{p}}(x)-(u_{n}+1)^{\frac{p+\beta-1}{p}}(y)|^{p}}{(u_{n}(x)+1)^{\beta}(u_{n}(y)+1)^{\beta}}d\mu\right)^{\frac{m}{p}},$
(3.30)

for $m<\frac{N(p-1)}{N-s}$ and $0<t<s$. Then, by combining $\eqref{1st 1}$ and
$\eqref{1st 2}$, the following is obtained.

$\int_{\Omega}\int_{\Omega}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+tm}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}\leq
C.$ (3.31)

We can always choose a $R>0$ such that $|x-y|\sim|y|$ for
$y\in\mathbb{R^{N}}\setminus B_{R}$, where
$B_{R}=\\{x\in\mathbb{R}^{N}:|x|<R\\}$. Assume that
$(x,y)\in\Omega\times(B_{R}\setminus\Omega)$, then

$\underset{(x,y)\in\Omega\times(B_{R}\setminus\Omega)}{\sup}\left\\{\frac{1}{|x-y|^{N+mt}}\right\\}\leq
C<\infty.$

Since $\\{u_{n}\\}$ is bounded in $L^{m_{1}}(\Omega,d\nu)$ for every
$m_{1}<\frac{N(p-1)}{N-sp}$, the following is obtained for
$m<\frac{N(p-1)}{N-s}$.

$\displaystyle\int_{\Omega}\int_{B_{R}\setminus\Omega}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+mt}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}$
$\displaystyle=\int_{\Omega}\int_{B_{R}\setminus\Omega}\frac{|u_{n}(x)|^{m}}{|x-y|^{N+mt}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}$
$\displaystyle\leq
C\int_{\Omega}\int_{B_{R}\setminus\Omega}|u_{n}(x)|^{m}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}$
$\displaystyle\leq C_{3}$ (3.32)

and similarly

$\displaystyle\int_{\Omega}\int_{\mathbb{R^{N}}\setminus
B_{R}}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+mt}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}$
$\displaystyle=\int_{\Omega}\int_{\mathbb{R^{N}}\setminus
B_{R}}\frac{|u_{n}(x)|^{m}}{|x|^{\alpha}}\frac{dxdy}{|y|^{N+mt+\alpha}}$
$\displaystyle\leq C_{4}\int_{\mathbb{R^{N}}\setminus
B_{R}}\frac{dy}{|y|^{N+mt+\alpha}}$ $\displaystyle\leq C_{5}.$ (3.33)

On combining $\eqref{ball R}$ and $\eqref{outside ball R}$ we obtain

$\int_{\Omega}\int_{\mathbb{R^{N}}\setminus\Omega}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+mt}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}.$
(3.34)

In a similar way we can prove

$\int_{\mathbb{R^{N}}\setminus\Omega}\int_{\Omega}\frac{|u_{n}(x)-u_{n}(y)|^{m}}{|x-y|^{N+mt}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}.$
(3.35)

With the consideration of $\eqref{1st}$, $\eqref{2nd}$ and $\eqref{3rd}$, the
claim given in $\eqref{1st 1st}$ is proved which concludes the proof. ∎

We now prove our first main result, i.e. the existence of positive weak
solution to $\eqref{ent}$ stated in Theorem 1.1.

###### Proof of Theorem 1.1.

According to Lemma 3.3, the sequence $\\{T_{k}(u_{n})\\}$ is bounded in
$X_{0}^{s,p,\alpha}(\Omega)$ and $\\{|u_{n}|^{p-2}u_{n}\\}$ is bounded in
$L^{m_{2}}(\Omega,d\nu)$, for every $m_{2}<\frac{N}{N-sp}$. Thus, there exists
a measurable function $u$ such that, upto a subsequential level,
$T_{k}(u_{n})\rightarrow T_{k}(u)$ weakly in $X_{0}^{s,p,\alpha}(\Omega)$,
$|u|^{p-2}u\in L^{m_{2}}(\Omega,d\nu)$ and $T_{k}(u)\in
X_{0}^{s,p,\alpha}(\Omega)$. Hence, $u_{n}\rightarrow u$ a.e. in $\Omega$ and
$u\equiv 0$ in $\mathbb{R}^{N}\setminus\Omega$.
Let us fix the following notations:

$\Phi(x,y)=\phi(x)-\phi(y),~{}I_{n}(x,y)=|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y)),$
$I(x,y)=|u(x)-u(y)|^{p-2}(u(x)-u(y)),~{}d\mu=\frac{1}{|x-y|^{N+sp}}\frac{dxdy}{|x|^{\alpha}|y|^{\alpha}}.$

Then, from $\eqref{weak formula approx}$ we have

$\displaystyle\int_{Q}I_{n}(x,y)\Phi(x,y)d\mu$
$\displaystyle=\int_{\Omega}f_{n}h_{n}(u_{n}+1/{n})\phi+\int_{\Omega}g_{n}\phi,~{}\forall\phi\in
C_{c}^{\infty}(\Omega).$

On rewriting the above equation, we obtain

$\displaystyle\int_{Q}I(x,y)\Phi(x,y)d\mu+\int_{Q}(I_{n}(x,y)-I(x,y))\Phi(x,y)d\mu=\int_{\Omega}f_{n}h_{n}(u_{n}+1/{n})\phi+\int_{\Omega}g_{n}\phi.$
(3.36)

Since $g_{n}\rightarrow g$ strongly in $L^{1}(\Omega)$, we have

$\lim\limits_{n\rightarrow\infty}\int_{\Omega}g_{n}\phi=\int_{\Omega}g\phi,~{}\forall\phi\in
C_{c}^{\infty}(\Omega).$

Now with the help of the dominated convergence theorem and $\eqref{positive}$,
we are able to pass the limit $n\rightarrow\infty$ in the following integral.

$\lim\limits_{n\rightarrow\infty}\int_{\Omega}f_{n}h_{n}(u_{n}+1/{n})\phi=\int_{\Omega}h(u)f\phi.$

We express

$\displaystyle\int_{Q}(I_{n}(x,y)-I(x,y))\Phi(x,y)d\mu$
$\displaystyle=\int_{\Omega\times\Omega}(I_{n}(x,y)-I(x,y))\Phi(x,y)d\mu$
$\displaystyle~{}~{}+\int_{(\mathbb{R}^{N}\setminus\Omega)\times\Omega}(I_{n}(x,y)-I(x,y))\Phi(x,y)d\mu$
$\displaystyle~{}~{}+\int_{\Omega\times(\mathbb{R}^{N}\setminus\Omega)}(I_{n}(x,y)-I(x,y))\Phi(x,y)d\mu$
$\displaystyle=T_{1,n}+T_{2,n}+T_{3,n}.$

Clearly, $I_{n}\rightarrow I$ a.e. in $\mathbb{R}^{N}\times\mathbb{R}^{N}$. By
the Vitali’s lemma and Lemma 3.2, $I_{n}\rightarrow I$ strongly in
$L^{1}(\Omega\times\Omega,d\mu)$. Therefore, $T_{1,n}\rightarrow 0$ as
$n\rightarrow\infty$.
Consider $(x,y)\in(B_{R}\setminus\Omega)\times\Omega$, then

$\underset{(x,y)\in(B_{R}\setminus\Omega)\times\Omega}{\sup}\left\\{\frac{1}{|x-y|^{N+sp}}\right\\}\leq
C<\infty$

and

$\left|\frac{(I_{n}(x,y)-I(x,y))\Phi(x,y)}{|x-y|^{N+sp}|x|^{\alpha}|y|^{\alpha}}\right|\leq
C\frac{(|u_{n}(y)|^{p-1}+|u(y)|^{p-1})|\phi(y)|}{|x|^{\alpha}|y|^{\alpha}}=CJ_{n}(x,y).$

Denote $J(x,y)=\frac{2|u(y)|^{p-1}|\phi(y)|}{|x|^{\alpha}|y|^{\alpha}}$. Then,
$J_{n}\rightarrow J$ strongly in $L^{1}((B_{R}\setminus\Omega)\times\Omega)$.
Hence, by the dominated convergence theorem, $T_{2,n}\rightarrow 0$ as
$n\rightarrow\infty$ and similarly we can also prove that $T_{3,n}\rightarrow
0$ as $n\rightarrow\infty$. Hence, on passing the limit $n\rightarrow\infty$
in $\eqref{entropy2}$, we have

$\displaystyle\int_{Q}I(x,y)\Phi(x,y)d\mu=\int_{\Omega}h(u)f\phi+\int_{\Omega}g\phi,~{}\forall\phi\in
C_{c}^{\infty}(\Omega).$ (3.37)

Further, $\eqref{2nd 2nd}$ follows from $\eqref{1st 1st}$ and the Fatou’s
lemma. Finally, with the consideration of Lemma 3.2 and $\eqref{3rd 3rd}$, it
is proved that $u$ is a weak solution to $\eqref{ent}$ in the sense of
Definition 2.8. Hence the proof. ∎

######  Remark 3.4.

If we assume $g$ to be a bounded Radon measure, then the above existence
result, i.e. Theorem 1.1, holds.

##  4 Existence of entropy solution - Proof of Theorem 1.2

The problem $\eqref{ent}$ admits a positive weak solution by Theorem 1.1.
According to Remark 2.10, every entropy solution of $\eqref{ent}$ is also a
weak solution to $\eqref{ent}$, i.e. the entropy solution satisfies
$\eqref{weak form}$. The following lemma is a consequence of Lemma 3.8 of [1],
which says that every entropy solution to $\eqref{ent}$ satisfies $\eqref{weak
form}$ for a larger class of test function space.

######  Lemma 4.1 (Lemma 3.8, [1]).

Suppose $u$ is a positive entropy solution to $\eqref{ent}$. Then, $u$
satisfies

$\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(\psi(x)-\psi(y))d\mu=\int_{\Omega}fh(u)\psi+\int_{\Omega}g\psi,$

for every $\psi\in X_{0}^{s,p,\alpha}(\Omega)\cap L^{\infty}(\Omega)$ s.t.
$\psi\equiv 0$ in the set $\\{u>l\\}$ for some $l>0$.

Let $u_{n}$ be a weak solution to $\eqref{ent approx}$. Hence, the sequence
$\\{u_{n}\\}$ is an increasing sequence w.r.t $n$, by Lemma 3.2. Further,
$\eqref{truncation bound}$ implies that $\\{T_{k}(u_{n})\\}$ is a uniformly
bounded sequence in $X_{0}^{s,p,\alpha}(\Omega)$ for every $k>0$. Before
proving the existence of an entropy solution, we provide the following
compactness result.

######  Lemma 4.2 (Lemma 3.6, [1]).

Let the sequence $\\{u_{n}\\}\subset X_{0}^{s,p,\alpha}(\Omega)$ be a positive
increasing sequence w.r.t $n$ and $(-\Delta)^{s}_{p,\alpha}u_{n}\geq 0$ in
$\Omega$. Furthermore, assume the sequence $\\{T_{k}(u_{n})\\}$ is uniformly
bounded in $X_{0}^{s,p,\alpha}(\Omega)$ for every $k>0$. Then, there exists
$u$ such that $u_{n}\uparrow u$ a.e. in $\Omega$, $T_{k}(u)\in
X_{0}^{s,p,\alpha}(\Omega)$ and

$T_{k}(u_{n})\rightarrow T_{k}(u)\text{ strongly in
}X_{0}^{s,p,\alpha}(\Omega).$

We now prove Theorem 1.2 and show the equivalence between the weak solution
and the entropy solution to $\eqref{ent}$.

###### Proof of Theorem 1.2..

The proof of this theorem is divided into the following two parts.
Part 1: There exists a positive entropy solution to $\eqref{ent}$ in the sense
of Definition 2.9.
By Theorem 1.1, we guarantee the existence of a weak solution $u$ to
$\eqref{ent}$. Then, $\eqref{truncation bound}$ and Lemma 4.2 imply that
$T_{k}(u_{n})\rightarrow T_{k}(u)$ strongly in $X_{0}^{s,p,\alpha}(\Omega)$.
Let us take $\phi=T_{1}(G_{l}(u_{n}))$, for a fixed $l>0$, in $\eqref{weak
formula approx}$ and we get

$\displaystyle\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}$
$\displaystyle(u_{n}(x)-u_{n}(y))(T_{1}(G_{l}(u_{n}))(x)-T_{1}(G_{l}(u_{n}))(y))d\mu$
$\displaystyle=\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)T_{1}(G_{l}(u_{n}))+\int_{\Omega}g_{n}T_{1}(G_{l}(u_{n}))$
$\displaystyle\leq\int_{\\{u_{n}\geq l\\}}f_{n}h_{n}(l+1/n)+\int_{\\{u_{n}\geq
l\\}}g_{n}.$ (4.38)

Consider the set $D_{l}$ as given in Definition 2.9. Thus, for $(x,y)\in
D_{l}$ it is easy to show that

$\displaystyle|u_{n}(x)-u_{n}(y)|^{p-1}\leq|u_{n}(x)-u_{n}(y)|^{p-2}$
$\displaystyle(u_{n}(x)-u_{n}(y))(T_{1}(G_{l}(u_{n}))(x)-T_{1}(G_{l}(u_{n}))(y)).$
(4.39)

By considering $\eqref{entropy 3}$, $\eqref{D_l}$ and using the Fatou’s lemma,
we have

$\displaystyle\int_{D_{l}}|u(x)-u(y)|^{p-1}d\mu$
$\displaystyle\leq\lim\limits_{n\rightarrow\infty}\inf\left\\{\int_{D_{l}}|u_{n}(x)-u_{n}(y)|^{p-1}d\mu\right\\}$
$\displaystyle\leq\int_{\\{u\geq l\\}}fh(l)+\int_{\\{u\geq l\\}}g.$

In fact,

$\int_{\\{u\geq l\\}}fh(l)+\int_{\\{u\geq l\\}}g\rightarrow 0~{}\text{ as
}~{}l\rightarrow\infty.$

Hence, we establish $\eqref{condition 1}$, i.e. we have

$\int_{D_{l}}|u(x)-u(y)|^{p-1}d\mu\rightarrow 0~{}\text{ as
}~{}l\rightarrow\infty.$

Now it remains to prove $\eqref{condition 2}$. For this, we consider
$\varphi\in X_{0}^{s,p,\alpha}(\Omega)\cap L^{\infty}(\Omega)$. Using
$T_{k}(u_{n}-\varphi)$ as a test function in $\eqref{weak formula approx}$ and
following the notations used in the proof of Theorem 1.1, we get

$\int_{Q}I_{n}(x,y)(T_{k}(u_{n}-\varphi)(x)-T_{k}(u_{n}-\varphi)(y))d\mu=\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)T_{k}(u_{n}-\varphi)+\int_{\Omega}g_{n}T_{k}(u_{n}-\varphi).$
(4.40)

The integrand in the first term of the above equation $\eqref{entropy 5}$ can
be decomposed as

$I_{n}(x,y)(T_{k}(u_{n}-\varphi)(x)-T_{k}(u_{n}-\varphi)(y))=I_{1,n}(x,y)+I_{2,n}(x,y),$
(4.41)

where

$\displaystyle
I_{1,n}(x,y)=|(u_{n}(x)-\varphi(x))-(u_{n}(y)-\varphi(y))|^{p-2}$
$\displaystyle\left((u_{n}(x)-\varphi(x))-(u_{n}(y)-\varphi(y))\right)$
$\displaystyle\times\left[T_{k}(u_{n}-\varphi)(x)-T_{k}(u_{n}-\varphi)(y)\right]$

and

$\displaystyle I_{2,n}(x,y)=$
$\displaystyle\left[I_{n}(x,y)-|(u_{n}(x)-\varphi(x))-(u_{n}(y)-\varphi(y))|^{p-2}\left((u_{n}(x)-\varphi(x))-(u_{n}(y)-\varphi(y))\right)\right]$
$\displaystyle~{}~{}~{}~{}\times\left[T_{k}(u_{n}-\varphi)(x)-T_{k}(u_{n}-\varphi)(y)\right].$

Clearly, $I_{1,n}(x,y)\geq 0$ a.e. in $Q$ and

$\displaystyle
I_{1,n}(x,y)\rightarrow|(u(x)-\varphi(x))-(u(y)-\varphi(y))|^{p-2}$
$\displaystyle\left((u(x)-\varphi(x))-(u(y)-\varphi(y))\right)$
$\displaystyle\times\left[T_{k}(u-\varphi)(x)-T_{k}(u-\varphi)(y)\right]~{}\text{
a.e. in}~{}Q.$

Thus, on using the Fatou’s lemma we establish

$\displaystyle\lim\limits_{n\rightarrow\infty}\inf\left\\{\int_{Q}I_{1,n}(x,y)d\mu\right\\}\geq$
$\displaystyle\int_{Q}|(u(x)-\varphi(x))-(u(y)-\varphi(y))|^{p-2}$
$\displaystyle~{}\left((u(x)-\varphi(x))-(u(y)-\varphi(y))\right)\times\left[T_{k}(u-\varphi)(x)-T_{k}(u-\varphi)(y)\right]d\mu.$
(4.42)

According to Abdellaoui et al. [1],

$\displaystyle\lim\limits_{n\rightarrow\infty}$
$\displaystyle\int_{Q}I_{2,n}(x,y)d\mu$
$\displaystyle=\int_{Q}\left[I(x,y)-|(u(x)-\varphi(x))-(u(y)-\varphi(y))|^{p-2}\left((u(x)-\varphi(x))-(u(y)-\varphi(y))\right)\right]$
$\displaystyle~{}~{}~{}~{}~{}\times\left[T_{k}(u-\varphi)(x)-T_{k}(u-\varphi)(y)\right]d\mu.$
(4.43)

On the other hand, by using the dominated convergence theorem, Lemma 3.2 and
the strong convergence $T_{k}(u_{n})\rightarrow T_{k}(u)$ in
$X_{0}^{s,p,\alpha}(\Omega)$, we observe that

$\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)T_{k}(u_{n}-\varphi)+\int_{\Omega}gT_{k}(u_{n}-\varphi)\rightarrow\int_{\Omega}fh(u)T_{k}(u-\varphi)+\int_{\Omega}gT_{k}(u-\varphi),~{}\text{as}~{}n\rightarrow\infty.$
(4.44)

Therefore, on combining the results from $\eqref{entropy 6}-\eqref{entropy 9}$
and then passing the limit $n\rightarrow\infty$ in $\eqref{entropy 5}$, we
conclude

$\int_{Q}I(x,y)(T_{k}(u-\varphi)(x)-T_{k}(u-\varphi)(y))d\mu\leq\int_{\Omega}fh(u)T_{k}(u-\varphi)+\int_{\Omega}gT_{k}(u-\varphi),$
(4.45)

for every $\varphi\in X_{0}^{s,p,\alpha}(\Omega)\cap L^{\infty}(\Omega)$.
Thus, $\eqref{condition 2}$ holds.
Part 2: Uniqueness of entropy solutions.
Let $u$ be the positive entropy solution obtained from Part 1 of this proof
and $u_{n}$ be the unique solution to $\eqref{ent approx}$. Then, from Lemma
4.2, $u=\lim\limits_{n\rightarrow\infty}\sup\\{u_{n}\\}$.
We prove this theorem by method of contradiction. For that, suppose $\bar{u}$
is another entropy solution to $\eqref{ent}$. Let us fix $n$ and define
$\psi_{n}=(u_{n}-\bar{u})^{+}$. Thus, for
$k\gg\|u_{n}\|_{L^{\infty}(\Omega)}$, $\psi_{n}=(u_{n}-T_{k}(\bar{u}))^{+}$.
This implies $\psi_{n}\in X_{0}^{s,p,\alpha}(\Omega)\cap L^{\infty}(\Omega)$
and $\psi_{n}\equiv 0$ in the set
$\\{\bar{u}>\|u_{n}\|_{L^{\infty}(\Omega)}\\}$. Hence, on choosing
$\phi=\psi_{n}$ in $\eqref{weak formula approx}$, we have

$\displaystyle\int_{Q}|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\psi_{n}(x)-\psi_{n}(y))d\mu$
$\displaystyle=\int_{\Omega}f_{n}h_{n}(u_{n}+1/n)\psi_{n}+\int_{\Omega}g_{n}\psi_{n}$
$\displaystyle\leq\int_{\Omega}fh(u_{n})\psi_{n}+\int_{\Omega}g\psi_{n}.$
(4.46)

Since $\bar{u}$ is an entropy solution to $\eqref{ent}$, by Lemma 4.1, we get

$\displaystyle\int_{Q}|\bar{u}(x)-\bar{u}(y)|^{p-2}(\bar{u}(x)-\bar{u}(y))(\psi_{n}(x)-\psi_{n}(y))d\mu$
$\displaystyle=\int_{\Omega}fh(\bar{u})\psi_{n}+\int_{\Omega}g\psi_{n}.$
(4.47)

Subtracting $\eqref{entropy 11}$ from $\eqref{entropy 10}$ we reach that

$\displaystyle 0$ $\displaystyle\leq C\int_{Q}|\psi_{n}(x)-\psi_{n}(y)|^{p}$
$\displaystyle\leq\int_{Q}\left(|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))-|\bar{u}(x)-\bar{u}(y)|^{p-2}(\bar{u}(x)-\bar{u}(y))\right)(\psi_{n}(x)-\psi_{n}(y))d\mu$
$\displaystyle\leq\int_{\Omega}f(h(u_{n})-h(\bar{u}))\psi_{n}$
$\displaystyle\leq 0.$

Therefore, $\psi_{n}=0$ a.e. in $Q$ and $u_{n}\leq\bar{u}$ a.e. for every $n$.
Use the fact $u=\lim\limits_{n\rightarrow\infty}\sup\\{u_{n}\\}$ to obtain

$u\leq\bar{u}~{}\text{a.e in}~{}Q.$ (4.48)

Further, choose $h\gg k$. Then, the two entropy solutions $u$ and $\bar{u}$
satisfy

$\int_{Q}I(x,y)[T_{k}(u(x)-T_{h}(\bar{u}(x)))-T_{k}(u_{n}(y)-T_{h}(\bar{u}(y)))]d\mu\leq\int_{\Omega}(fh(u)+g)T_{k}(u-T_{h}(\bar{u}))$
(4.49)

and

$\int_{Q}\bar{I}(x,y)[T_{k}(\bar{u}(x)-T_{h}(u(x)))-T_{k}(\bar{u}(y)-T_{h}(u(y)))]d\mu\leq\int_{\Omega}(fh(\bar{u})+g)T_{k}(\bar{u}-T_{h}(u)),$
(4.50)

where

$\bar{I}(x,y)=|\bar{u}(x)-\bar{u}(y)|^{p-2}(\bar{u}(x)-\bar{u}(y))~{}\text{
and }~{}I(x,y)=|u(x)-u(y)|^{p-2}(u(x)-u(y)).$

Clearly, as $h\rightarrow\infty$,

$\int_{\Omega}gT_{k}(u-T_{h}(\bar{u}))+\int_{\Omega}gT_{k}(\bar{u}-T_{h}(u))\rightarrow
0$ (4.51)

and

$\int_{\Omega}fh(u)T_{k}(u-T_{h}(\bar{u}))+\int_{\Omega}fh(\bar{u})T_{k}(\bar{u}-T_{h}(u))\rightarrow\int_{\Omega}fh(u)T_{k}(u-\bar{u})+\int_{\Omega}fh(\bar{u})T_{k}(\bar{u}-u).$
(4.52)

From $\eqref{one side}$, we already have $u\leq\bar{u}$ a.e. in $Q$. Thus, the
right hand side of $\eqref{entropy 15}$ becomes

$\displaystyle\int_{\Omega}fh(u)T_{k}(u-\bar{u})+\int_{\Omega}fh(\bar{u})T_{k}(\bar{u}-u)$
$\displaystyle=\int_{\Omega}f(h(u)-h(\bar{u}))T_{k}(u-\bar{u})$
$\displaystyle\leq 0.$ (4.53)

Let us define the following two sets:

$D(h)=\\{(x,y)\in Q:u(x)<h,u(y)<h\\}\text{ and }\bar{D}(h)=\\{(x,y)\in
Q:\bar{u}(x)<h,\bar{u}(y)<h\\}.$

Since $u\leq\bar{u}$, we get $u<h$ in the set $\\{\bar{u}<h\\}$. On combining
the inequalities $\eqref{entropy 12}-\eqref{entropy 16}$, we obtain

$\displaystyle o(h)$
$\displaystyle\geq\int_{Q}I(x,y)(T_{k}(u(x)-T_{h}(\bar{u}(x)))-T_{k}(u_{n}(y)-T_{h}(\bar{u}(y))))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q}\bar{I}(x,y)(T_{k}(\bar{u}(x)-T_{h}(u(x)))-T_{k}(\bar{u}(y)-T_{h}(u(y))))d\mu$
$\displaystyle=\int_{D(h)}I(x,y)(T_{k}(u(x)-T_{h}(\bar{u}(x)))-T_{k}(u_{n}(y)-T_{h}(\bar{u}(y))))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q\setminus
D(h)}I(x,y)(T_{k}(u(x)-T_{h}(\bar{u}(x)))-T_{k}(u_{n}(y)-T_{h}(\bar{u}(y))))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{\bar{D}(h)}\bar{I}(x,y)(T_{k}(\bar{u}(x)-u(x))-T_{k}(\bar{u}(y)-u(y)))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q\setminus\bar{D}(h)}\bar{I}(x,y)(T_{k}(\bar{u}(x)-T_{h}(u(x)))-T_{k}(\bar{u}(y)-T_{h}(u(y))))d\mu$
$\displaystyle=\int_{\bar{D}(h)}[\bar{I}(x,y)-I(x,y)](T_{k}(\bar{u}(x)-u(x))-T_{k}(\bar{u}(y)-u(y)))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{D(h)\setminus\bar{D}(h)}I(x,y)(T_{k}(u(x)-T_{h}(\bar{u}(x)))-T_{k}(u_{n}(y)-T_{h}(\bar{u}(y))))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q\setminus
D(h)}I(x,y)(T_{k}(u(x)-h)-T_{k}(u_{n}(y)-h))d\mu$
$\displaystyle~{}~{}~{}~{}+\int_{Q\setminus\bar{D}(h)}\bar{I}(x,y)(T_{k}(\bar{u}(x)-T_{h}(u(x)))-T_{k}(\bar{u}(y)-T_{h}(u(y))))d\mu$
$\displaystyle=J_{1}(h)+J_{2}(h)+J_{3}(h)+J_{4}(h).$ (4.54)

It is easy to check that

$\displaystyle J_{1}(h)$
$\displaystyle=\int_{\bar{D}(h)}[\bar{I}(x,y)-I(x,y)](T_{k}(\bar{u}(x)-u(x))-T_{k}(\bar{u}(y)-u(y)))d\mu$
$\displaystyle\geq
C\int_{\bar{D}(h)}|T_{k}(\bar{u}(x)-u(x))-T_{k}(\bar{u}(y)-u(y))|^{p}d\mu.$
(4.55)

According to Abdellaoui et al. [1],

$J_{2}(h)+J_{3}(h)+J_{4}(h)\geq o(h).$ (4.56)

Hence, by the inequalities $\eqref{entropy 17}-\eqref{entropy 19}$, we
establish the following.

$o(h)+C\int_{\bar{D}(h)}|T_{k}(\bar{u}(x)-u(x))-T_{k}(\bar{u}(y)-u(y))|^{p}d\mu\leq
o(h).$

Now on passing the limit $h\rightarrow\infty$, we conclude that

$\int_{\mathbb{R}^{N}}|T_{k}(\bar{u}(x)-u(x))-T_{k}(\bar{u}(y)-u(y))|^{p}d\mu=0.$

This implies $T_{k}(\bar{u}-u)$ is a constant function. Since $u=\bar{u}=0$ in
$\mathbb{R}^{N}\setminus\Omega$, we reach at the conclusion that $u=\bar{u}$
a.e. in $Q$.
Thus, from Part 1 and Part 2 of this proof we conclude that
$u\in\mathcal{T}^{s,p,\alpha}_{0}(\Omega)$ is a unique positive entropy
solution to $\eqref{ent}$ in the sense of Definition 2.9 and is equivalent to
the weak solution as obtained in Theorem 1.1. ∎

###  4.1 Problem $\eqref{ent}$ with a power nonlinearity.

We consider the following problem with a singularity, power nonlinearity and
an $L^{1}$ datum. We show that this problem possesses an entropy solution.

$\begin{split}(-\Delta)_{p,\alpha}^{s}w&=\frac{f(x)}{w^{\gamma}}+w^{r}+g(x)~{}\text{in}~{}\Omega,~{}r<p-1,\gamma<1,\\\
w&>0~{}\text{in}~{}\Omega,\\\
w&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.\end{split}$ (4.57)

The following is the definition of entropy solution to the problem $\eqref{ent
special}$.

######  Definition 4.3.

A function $w\in\mathcal{T}^{s,p,\alpha}_{0}(\Omega)$ is said to be an entropy
solution of $\eqref{ent special}$ if it satisfies $\eqref{condition 1}$ and
for every $\varphi\in W^{s,p,\alpha}_{0}(\Omega)\cap L^{\infty}(\Omega)$ and
every $k>0$,

$\displaystyle\int_{Q}|w(x)-w(y)|^{p-2}(w(x)-w(y))$
$\displaystyle[T_{k}(w(x)-\varphi(x))-T_{k}(w(y)-\varphi(y))]d\mu$
$\displaystyle\leq\int_{\Omega}\frac{f}{w^{\gamma}}T_{k}(w-\varphi)+\int_{\Omega}w^{r}T_{k}(w-\varphi)+\int_{\Omega}gT_{k}(w-\varphi).$
(4.58)

Further, for every $\omega\subset\subset\Omega$, there exists a $C_{\omega}>0$
such that $w\geq C_{\omega}>0$.

######  Theorem 4.4.

Let $r<p-1$ and $0<\gamma<1$. Then, the problem $\eqref{ent special}$ admits a
positive entropy solution in $\mathcal{T}_{0}^{s,p,\alpha}(\Omega)$ in the
sense of Definition 4.3.

###### Proof.

Denote $f_{n}=T_{n}(f)$ and let $\\{g_{n}\\}\subset L^{\infty}(\Omega)$ be a
nonnegative, increasing sequence converging strongly to $g$ in
$L^{1}(\Omega)$. Consider the following approximating problem:

$\begin{split}(-\Delta)_{p,\alpha}^{s}w_{n}&=\frac{f_{n}(x)}{(w_{n}+1/n)^{\gamma}}+w_{n}^{r}+g_{n}(x)~{}\text{in}~{}\Omega,\\\
w_{n}&>0~{}\text{in}~{}\Omega,\\\
w_{n}&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.\end{split}$ (4.59)

The corresponding energy functional is given by

$\displaystyle\Psi(w_{n})$
$\displaystyle=\frac{1}{p}\int_{Q}|w_{n}(x)-w_{n}(y)|^{p}d\mu-\frac{1}{1-\gamma}\int_{\Omega}f_{n}\left((w_{n}^{+}+1/n)^{1-\gamma}-(1/n)^{1-\gamma}\right)$
$\displaystyle~{}~{}~{}~{}-\frac{1}{r+1}\int_{\Omega}(w_{n}^{+})^{r+1}-\int_{\Omega}g_{n}w_{n}^{+}.$
(4.60)

The functional $\Psi$ is $C^{1}$, coercive and weakly lower semi-continuous in
$X_{0}^{s,p,\alpha}(\Omega)$. Thus, there exists a critical point of $\Psi$
and hence a weak solution to the problem $\eqref{ent special}$, denoted by
$w_{n}\in X_{0}^{s,p,\alpha}(\Omega)$. The strong maximum principle [Lemma
2.3, [19]] and the comparison principle, i.e. Lemma 2.6, imply that the
solution $w_{n}>0$ is unique and the sequence $(w_{n})$ is an increasing
sequence with respect to $n$.
Clearly, by Lemma 2.6, we obtain $u_{n}\leq w_{n}$ in $\Omega$ for every $n$,
where $u_{n}$ is the positive weak solution to $\eqref{ent approx}$ with
$h(t)=t^{-\gamma}$. According to Lemma 3.2, $u_{n}^{-\gamma}f\in
L^{1}(\Omega)$ and hence $w_{n}^{-\gamma}f\in L^{1}(\Omega)$.
Claim: the sequence $\\{w_{n}\\}$ is uniformly bounded in $L^{p-1}(\Omega)$.
We will use the method of contradiction to prove our claim. Assume that
$\\{w_{n}\\}$ is an unbounded sequence in $L^{p-1}(\Omega)$. Hence, there
exists a subsequence of $\\{w_{n}\\}$, still denoted as $\\{w_{n}\\}$, such
that $M_{n}=\|w_{n}^{p-1}\|_{L^{1}(\Omega)}\rightarrow\infty$ as
$n\rightarrow\infty$. Let us define

$\bar{w}_{n}=\frac{w_{n}}{M_{n}^{\frac{1}{p-1}}}.$

This implies $\|\bar{w}_{n}\|_{L^{p-1}(\Omega)}=1.$ Since $w_{n}$ is a
solution to $\eqref{ent special approx}$, $\bar{w}_{n}$ satisfies

$\begin{split}(-\Delta)_{p,\alpha}^{s}\bar{w}_{n}&=\frac{f_{n}(x)}{M_{n}({w_{n}}+1/n)^{\gamma}}+M_{n}^{\frac{r-p+1}{p-1}}\bar{w}_{n}^{r}+M_{n}^{-1}g_{n}(x)~{}\text{in}~{}\Omega,\\\
\bar{w}_{n}&>0~{}\text{in}~{}\Omega,\\\
\bar{w}_{n}&=0~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega.\end{split}$ (4.61)

Denote
$H_{n}=\frac{f_{n}}{M_{n}({w_{n}}+1/n)^{\gamma}}+M_{n}^{\frac{r-p+1}{p-1}}\bar{w}_{n}^{r}+M_{n}^{-1}g_{n}$.
Then, it is simple to prove that $\|H_{n}\|_{L^{1}(\Omega)}\rightarrow 0$ as
$n\rightarrow\infty$. Following the proof of Lemma 3.3, we establish that
$\\{\bar{w}_{n}\\}$ is bounded in $L^{m_{1}}(\Omega,d\nu)$ for every
$m_{1}<\frac{N(p-1)}{N-sp}$ and also bounded in $X_{0}^{t,m,\alpha}(\Omega)$
for every $1\leq m<\frac{N(p-1)}{N-s}$, $0<t<s$. Thus, there exists
$\bar{w}\in X_{0}^{t,m,\alpha}(\Omega)\cap L^{m_{1}}(\Omega,d\nu)$ such that
$T_{k}(\bar{w})\in X_{0}^{s,p,\alpha}(\Omega)$ and
$T_{k}(\bar{w}_{n})\rightarrow T_{k}(\bar{w})$ weakly in
$X_{0}^{s,p,\alpha}(\Omega)$. We now use the Vitali’s lemma to prove
$\bar{w}_{n}^{p-1}\rightarrow\bar{w}^{p-1}$ strongly in $L^{1}(\Omega)$ and we
obtain $\|\bar{w}\|_{L^{p-1}}(\Omega)=1$.
On readopting the proof of Lemma 3.3 with the test function
$T_{k}(\bar{w}_{n})$ we obtain

$\|T_{k}(\bar{w}_{n})\|_{X_{0}^{s,p,\alpha}(\Omega)}\rightarrow
0,~{}\text{as}~{}n\rightarrow\infty.$

Thus, $T_{k}(\bar{w})=0$ for every $k$ and this contradicts to
$\|\bar{w}\|_{L^{p-1}(\Omega)}=1$. This proves the claim and $\\{w_{n}\\}$ is
uniformly bounded in $L^{r}(\Omega)$.
We can now guarantee the existence of a weak solution $w\in
X_{0}^{t,m,\alpha}(\Omega)$ to the problem $\eqref{ent special}$ for every
$1\leq m<\frac{N(p-1)}{N-s}$, $0<t<s$. Further, $T_{k}({w_{n}})\rightarrow
T_{k}({w})$ weakly in $X_{0}^{s,p,\alpha}(\Omega)$. The proof follows
verbation of the proofs in Lemma 3.3 and Theorem 1.1.
We have already shown that $\\{w_{n}\\}$ is an increasing sequence. Hence, by
Lemma 4.2, $T_{k}({w_{n}})$ converges strongly to $T_{k}({w})$ in
$X_{0}^{s,p,\alpha}(\Omega)$. Proceeding on the similar lines used in the
proof of Theorem 1.2, we get the existence of an entropy solution $w$ to
$\eqref{ent special}$ in the sense of Definition 4.3. Hence the proof. ∎

## Acknowledgement

The author Akasmika Panda thanks the financial assistantship received from the
Ministry of Human Resource Development (M.H.R.D.), Govt. of India. The author
Debajyoti Choudhuri thanks SERB for the financial assistanceship received to
carry out this research vide MTR/2018/000525. Both the authors also
acknowledge the facilities received from the Department of Mathematics,
National Institute of Technology Rourkela.

## Data Availability statement

The article has no data associated to it whatsoever.

## Conflict of interest statement

There is no conflict of interest whatsoever.

## References

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