lemmatheorem lemma propositiontheorem proposition assumptiontheorem assumption
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#  Positive solutions for fractional p- Laplace semipositone problem with
superlinear growth 111 1Indian Institute of Science Education and Research,
Thiruvananthapuram 695551, India 2Department of Mathematics, Indian Institute
of Technology, Guwahati-781039, India Ae-mail<EMAIL_ADDRESS>Be-
mail<EMAIL_ADDRESS>Ce-mail<EMAIL_ADDRESS>De-mail:
<EMAIL_ADDRESS>Acknowledgments: R. Dhanya was supported by INSPIRE
faculty fellowship with grant number DST/INSPIRE/04/2015/003221 when the work
was being carried out. Ritabrata Jana is currently supported by the Prime
Minister Research Fellowship during the execution of this research.

R. Dhanya1,A ,  Ritabrata Jana1,B ,  Uttam Kumar2,C and  Sweta Tiwari2,D

###### Abstract.

We consider a semipositone problem involving the fractional $p$ Laplace
operator of the form

$\displaystyle(-\Delta)_{p}^{s}u$ $\displaystyle=\mu(u^{r}-1)\text{ in
}\Omega,$ $\displaystyle u$ $\displaystyle>0\hskip 39.83368pt\text{ in
}\Omega,$ $\displaystyle u$ $\displaystyle=0\hskip 39.83368pt\text{ on
}\Omega^{c},$

where $\Omega$ is a smooth bounded convex domain in $\mathbb{R}^{N}$,
$p-1<r<p^{*}_{s}-1$, where $p_{s}^{*}:=\frac{Np}{N-ps}$, and $\mu$ is a
positive parameter. We study the behaviour of the barrier function under the
fractional $p$-Laplacian and use this information to prove the existence of a
positive solution for small $\mu$ using degree theory. Additionally, the paper
explores the existence of a ground state positive solution for a
multiparameter semipositone problem with critical growth using variational
arguments.

###### Key words and phrases:

Key words: Nonlocal problem, fractional p- Laplacian, semipositone problem,
indefinite nonlinearities, critical exponent

##  1\. Introduction

In this paper, we consider a fractional $p$-Laplace equation with a
superlinear nonlinearity that changes its sign. We consider $\Omega$ to be a
bounded convex open set in $\mathbb{R}^{N},$ $N\geq 2$ with $C^{2}$-boundary.
Our primary focus is to obtain a positive solution to the following Dirichlet
boundary value problem

$\displaystyle(P^{\mu})\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=\mu(u^{r}-1)\text{
in }\Omega,\\\ u&>0\hskip 39.83368pt\text{ in }\Omega,\\\ u&=0\hskip
39.83368pt\text{ on }\Omega^{c},\end{array}\right.$ (1.4)

where $p-1<r<p^{*}_{s}-1,N>ps$ with $p^{*}_{s}:=\frac{Np}{N-ps}$ is the
fractional critical Sobolev exponent. Upto a suitable normalization constant,
the operator fractional $p$-Laplacian $(-\Delta)_{p}^{s}u$, is defined as

$(-\Delta)_{p}^{s}u(x)=2\thinspace
PV.\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y)}{|x-y|^{N+sp}}\,dy\qquad
x\in\Omega.$ (1.5)

Motivation to study the semipositone problems stems from pure mathematical
interest and also due to its applications in various other areas such as
mathematical physics, population dynamics, and materials science (see [8] and
references therein). Semipositone problems are characterized by a nonlinearity
that changes sign, making the existence of a positive solution difficult to
establish. In [23], authors observed a striking phenomenon that non-negative
solutions of semipositone problems can have interior zeros. As a result of
this intriguing finding, researchers have focused their attention in proving
the existence of positive solutions to semipositone problems. This topic has
been extensively studied for Laplace and p Laplace operators in [16, 19, 29,
32] and many more. The recent advances in the study of nonlocal operators,
particularly in understanding fine boundary regularity [22] and Hopf-type
lemmas [25], have played a crucial role in exploring nonlocal semipositone
problems. As a result, researchers have been able to use these tools to
establish the positivity of solutions in recent works, including [15] and
[27]. The paper [15] employs a sub-supersolution approach to demonstrate the
existence of a solution, while [27] relies on the mountain pass theorem.

In this article, we utilize degree theoretic arguments to demonstrate the
existence of a positive solution to the superlinear semipositone problem
$(P^{\mu})$. To accomplish this, we first establish apriori $L^{\infty}$
estimates for a family of subcritical problems. This inturn reduces to
determining the pointwise bounds for the behavior of a barrier function near
the boundary of $\Omega$ when acted upon by the fractional p Laplacian. The
novelty of this work is to overcome the difficulty which arises due to the
global integration as well as the nonlinearity of the operator while obtaining
this pointwise bound. We anticipate that this estimate may have additional
applications, such as providing an effective upper bound for solutions by
utilizing relevant comparison principles and thereby establishing regularity
results. We next state our main theorem which is proved in Section 5 of this
paper.

######  Theorem 1.1.

For $p\geq 2$, there exists $\mu_{0}>0$ such that the problem $(P^{\mu})$
admits a positive solution for $\mu\in(0,\mu_{0})$ and $p-1<r<p_{s}^{*}-1$.

The use of apriori estimates for nonlinear equations is known to be a vital
tool in proving the existence of solutions to nonlinear problems. This
estimate is often used to verify assumptions of the Leray Schauder degree
theory about the existence of a fixed point. Chen et al.[10] demonstrated the
effectiveness of this approach in proving the existence of solutions to
fractional Laplace equation. Barrios et al. [2] examined more general nonlocal
operators and considered the viscosity solutions of the problem in the
presence of a gradient term. Both proofs rely on constructing a barrier
function and analyzing its behavior near the boundary. In this article, we use
the idea of Gidas-Spruck translated function to obtain the uniform
$L^{\infty}$ estimate for viscosity solutions by studying a proper barrier
function under the fractional p-Laplacian. It is well known that Gidas-Spruck
type estimate transforms the uniform $L^{\infty}$ bound to that of Liouville
type result. In order to obtain the results outlined in Theorem 1.1, we must
rely on a nonexistence assumption, denoted $(\mathcal{NA})$, which is similar
to the nonexistence assumption made by Brasco et al. in [6]. We consider a
sub-critical exponent problem with $p-1<q<p_{s}^{*}-1$ either when
$\mathcal{H}$ is the entire space $\mathbb{R}^{N}$ or a half-space in
$\mathbb{R}^{N}$ given by

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=u^{q}\text{ in
}\mathcal{H},\\\ u&>0\hskip 5.69046pt\mbox{ in }\mathcal{H};\\\ u&=0\hskip
5.69046pt\text{ on }\mathbb{R}^{N}\setminus\mathcal{H}.\end{array}\right.$
(1.9)

  * $(\mathcal{NA})$

Problem (1.9) does not admit nontrivial viscosity solution in
$C^{\alpha}(\mathcal{H}).$

Although to the best of our knowledge this result has not been proven yet, we
have a strong basis for believing that it is a plausible assumption.
Specifically, we note that analogous results have been established for both
the fractional Laplacian (i.e., for $0<s<1$ and $p=2$) in Theorem 1.1 and 1.2
of Quass and Xia[31], and for the local case ( $-\Delta_{p},\,1<p<\infty$) in
Theorem 1.1 and Lemma 2.8 of Zou [34]. Therefore, since nonexistence results
are already known for these special cases, it is reasonable to assume
$\mathcal{(NA)}.$

In a recent preprint, Lopera et al. [27] investigated a semipositone problem
with superlinear nonlinearity that is similar to $(P^{\mu})$. They utilized
the variational method to establish the existence of a nonnegative solution
when $r$ belongs to the interval $(p-1,p_{s}^{*}-1)$ and $\mu$ is sufficiently
small. However, their findings only guarantee the positivity of the solution
for specific values of $r$ and small $\mu$. In contrast, our solution to
problem $(P^{\mu})$ does not impose any such restrictions and guarantees the
positivity of the solution for all values of $r$.

In the last section of this article, we consider a multiparameter semipositone
problem involving critical Sobolev exponent given as below:

$\displaystyle\left.\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=\lambda
u^{p-1}+\mu(u^{p^{*}_{s}-1}-1)\text{ in }\Omega,\\\ u&>0\hskip
101.00737pt\text{ in }\Omega,\\\ u&=0\hskip 101.00737pt\text{ on
}\Omega^{c},\end{array}\right\\}$ (1.13)

When $\lambda=0$ and $\mu>0,$ the problem (1.13) reduces to (1.4) for
$r=p^{*}_{s}-1.$ We observe that the scaling
$u\mapsto\mu^{\frac{1}{p^{*}_{s}-p}}u$ transforms the equation in the
following critical semipositone nonlocal problem which we call as
$(P_{\lambda}^{\mu})$

$\displaystyle(P^{\mu}_{\lambda})\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=\lambda
u^{p-1}+u^{p^{*}_{s}-1}-\mu\text{ in }\Omega,\\\ u&>0\hskip 85.35826pt\text{
in }\Omega,\\\ u&=0\hskip 85.35826pt\text{ on }\Omega^{c}.\end{array}\right.$
(1.17)

where $\lambda,\mu>0$ are parameters. For a given $\lambda>0$, the solution of
the problem (1.17) are the critical points of the energy functional
$E_{\mu}:D_{0}^{s,p}(\Omega)\rightarrow\mathbb{R}$ defined by

$E_{\mu}(u)=\frac{1}{p}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy-\frac{\lambda}{p}\int_{\Omega}u^{p}\,dx-\frac{1}{p^{*}_{s}}\int_{\Omega}u^{p^{*}_{s}}\,dx+\int_{\Omega}\mu
u\,dx.$

All the weak solutions of problem (1.17) lie on the set

$\mathcal{N}_{\mu}=\left\\{\begin{aligned} u\in D_{0}^{s,p}(\Omega):&\ u>0\
\text{in}\ \Omega\thickspace\thickspace\text{and}\\\
&\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy=\int_{\Omega}(\lambda
u^{p}+u^{p^{*}_{s}}-\mu u)\,dx\end{aligned}\right\\}$

A weak solution that minimizes $E_{\mu}$ on $\mathcal{N}_{\mu}$ is a ground
state solution for (1.17). Let

$\lambda_{1}=\inf_{u\in
D^{s,p}_{0}(\Omega)\setminus{\\{0\\}}}\dfrac{\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy}{\int_{\Omega}|u(x)|^{p}\,dx}$
(1.18)

is the first Dirichlet eigenvalue of the fractional $p$ Laplacian, which is
positive. We prove that,

######  Theorem 1.2.

For $p\geq 2$, $N\geq sp^{2}$ and $\lambda\in(0,\lambda_{1})$, there exists
$\mu^{*}>0$ such that $\mu\in(0,\mu^{*})$, problem (1.17) has a ground state
solution $u_{\mu}\in C_{d}^{0,\alpha}(\overline{\Omega})$ for some
$\alpha\in(0,s]$.

##  2\. Preliminaries

###  2.1. Function spaces

To begin, we will revisit the definitions of several Sobolev spaces with
fractional orders that are utilized in this article. For a smooth, bounded
domain $\Omega\subset\mathbb{R}^{N}$ (where $N\geq 2$) we denote the standard
$L^{p}(\Omega)$ norm by $|\cdot|_{L^{p}(\Omega)}$, where $p\in[1,\infty]$. For
a measurable function $u:\mathbb{R}^{N}\rightarrow\mathbb{R}$ and for
$p\in(1,\infty)$ and $s\in(0,1)$, let

$[u]_{W^{s,p}(\mathbb{R}^{N})}:=\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\right)^{1/p}$

be the Gagliardo seminorm. For $sp<N$, we consider the space

$W^{s,p}(\mathbb{R}^{N}):=\left\\{u\in{L}^{p}(\mathbb{R}^{N}):[u]_{{W^{s,p}(\mathbb{R}^{N})}}<\infty\right\\}.$

Then $W^{s,p}(\mathbb{R}^{N})$ is a Banach space with respect to the norm

$\|u\|_{{W^{s,p}(\mathbb{R}^{N})}}=\left(\|u\|^{p}_{L^{p}(\mathbb{R}^{N})}+[u]^{p}_{{W^{s,p}(\mathbb{R}^{N})}}\right)^{\frac{1}{p}}.$

To address the Dirichlet boundary condition, we consider the space

$W_{0}^{s,p}(\Omega):=\left\\{u\in
W^{s,p}(\mathbb{R}^{N}):u=0\medspace\text{in}\medspace\mathbb{R}^{N}\setminus\Omega\right\\},$

which is a Banach space endowed with the norm
$\|\cdot\|=\|\cdot\|_{{W^{s,p}(\mathbb{R}^{N})}}$. Moreover the embedding
$W^{s,p}_{0}(\Omega)\hookrightarrow L^{r}(\Omega)$ is continuous for $1\leq
r\leq p^{*}_{s}$ and compact for $1\leq r<p^{*}_{s}$. Due to continuous
embedding of $W^{s,p}_{0}(\Omega)\hookrightarrow L^{r}(\Omega)$ for $1\leq
r\leq p^{*}_{s}$, we define the equivalent norm on $W^{s,p}_{0}(\Omega)$ as

$\|u\|_{W^{s,p}_{0}}:=\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\right)^{1/p}.$

In order to handle the critical growth in problem (1.13), we will work with
the following function spaces

$D^{s,p}(\mathbb{R}^{N}):=\left\\{u\in{L}^{p_{s}^{*}}(\mathbb{R}^{N}):[u]_{{W^{s,p}(\mathbb{R}^{N})}}<\infty\right\\},$
$D_{0}^{s,p}(\Omega):=\left\\{u\in
D^{s,p}(\mathbb{R}^{N}):u=0\medspace\text{in}\medspace\mathbb{R}^{N}\setminus\Omega\right\\}.$

Note that for the bounded domain $\Omega$,
$D_{0}^{s,p}(\Omega)=W_{0}^{s,p}(\Omega).$ Next, we recall some weighted
Hölder spaces. Let the distance function
$d:\overline{\Omega}\rightarrow\mathbb{R}_{+}$ be defined by

$d(x):=\text{dist}(x,\partial\Omega),\;x\in\overline{\Omega}.$

The weighted Hölder type spaces are defined as follows:

$\displaystyle C^{0}_{d}(\overline{\Omega}):=\bigg{\\{}u\in
C^{0}(\overline{\Omega}):\frac{u}{d^{s}}\text{ admits a continuous extension
to }\overline{\Omega}\bigg{\\}},$ $\displaystyle
C^{0,\alpha}_{d}(\overline{\Omega}):=\bigg{\\{}u\in
C^{0}(\overline{\Omega}):\frac{u}{d^{s}}\text{ admits a }\alpha\text{
-H\"{o}lder continuous extension to }\overline{\Omega}\bigg{\\}}$

equipped with the norms

$\displaystyle\|u\|_{C_{d}^{0}(\overline{\Omega})}:=\|\frac{u}{d^{s}}\|_{L^{\infty}(\Omega)},$
$\displaystyle\|u\|_{C_{d}^{0,\alpha}(\overline{\Omega})}:=\|u\|_{C_{d}^{0}(\overline{\Omega})}+\sup_{x,y\in\overline{\Omega},\,x\not=y}\frac{|u(x)/d^{s}(x)-u(y)/d^{s}(y)|}{|x-y|^{\alpha}},$

respectively. The embedding
$C^{0,\alpha}_{d}(\overline{\Omega})\hookrightarrow
C^{0}_{d}(\overline{\Omega})$ is compact, for all $\alpha\in(0,1).$

###  2.2. Notions of solutions

Next, we specify the two notions of the solutions, viz. weak and the viscosity
solutions of the following boundary value problem:

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u(x)&=h(x,u)\hskip
14.22636pt\text{ in }\Omega,\\\ u(x)&=0\hskip 44.10185pt\text{ on
}\Omega^{c}.\end{array}\right.$ (2.21)

We also consider the equivalence between these two solutions and refer to
[24], and [3] for the details. Let $p\in(1,\infty),$
$h:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function which
satisfy the growth condition

$|h(x,t)|\leq\gamma(|t|)+\phi(x),$ (2.22)

where $\gamma\geq 0$ is continuous and $\phi\in L^{\infty}_{loc}(\Omega).$ We
define,

$L_{sp}^{p-1}(\mathbb{R}^{N}):=\left\\{u\in
L^{p-1}_{loc}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\dfrac{|u(x)|^{p-1}}{(1+|x|)^{N+sp}}\,dx<\infty\right\\}.$

######  Definition .

A function $u\in W^{s,p}(\mathbb{R}^{N})\cap L_{sp}^{p-1}(\mathbb{R}^{N})$ is
a weak super solution (sub solution) of (2.21) if

$\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{N+sp}}\,dx\,dy\geq(\leq)\int_{\Omega}h(x,u)\varphi\,dx.$

for every $\varphi\in W_{0}^{s,p}(\Omega)$ and $\varphi\geq 0.$

We say that $u$ is a weak solution of (2.21) if it is both a weak
supersolution and subsolution to the problem.

Next, we define a viscosity sub and super solution of (2.21) as given in [3].
Let $D\subset\Omega$ be an open set; we define

$C_{\beta}^{2}(D):=\left\\{u\in C^{2}(D):\sup_{x\in
D}\left(\frac{\min\\{d_{u}(x),1\\}^{\beta-1}}{|\nabla
u(x)|}+\frac{|D^{2}u(x)|}{d_{u}(x)^{\beta-2}}\right)<\infty\right\\},$

where $d_{u}$ is the distance from the set of critical points of $u$ denoted
as $N_{u}$, that is

$d_{u}(x):=dist(x,N_{u}),\qquad N_{u}:=\\{x\in\Omega:\nabla u(x)=0\\}.$

Motivation for the definition of the space $C_{\beta}^{2}(D)$ comes from [24].

######  Definition .

[3, Definition 2.2] A function $u$ is a viscosity super-solution (subsolution)
of (2.21) if

  1. (i)

$u<+\infty\thickspace(u>-\infty)$ a.e. in $\mathbb{R}^{N}$,
$u>-\infty\thickspace(u<+\infty)$ a.e. in $\Omega$.

  2. (ii)

$u$ is lower (upper) semicontinuous in $\Omega$.

  3. (iii)

If $\phi\in C^{2}(B(x_{0},r))\cap L_{sp}^{p-1}(\mathbb{R}^{N})$ for some
$B(x_{0},r)\subset\Omega$ such that $\phi(x_{0})=u(x_{0}),\phi\leq u(\phi\geq
u)$ in $B(x_{0},r),\thickspace\phi=u$ in $\Omega\setminus B(x_{0},r)$ and one
of the following conditions hold:

    1. (a)

$p>\frac{2}{2-s}$ or $\nabla\phi(x_{0})\neq 0$,

    2. (b)

$1<p\leq\frac{2}{2-s}$, $\nabla\phi(x_{0})=0$ such that $x_{0}$ is an isolated
critical point of $\phi$, and $\phi\in C_{\beta}^{2}(B(x_{0},r))$ for some
$\beta>\frac{sp}{p-1}$,

then

$(-\Delta)_{p}^{s}\phi(x_{0})\geq(\leq)h(x_{0},\phi(x_{0}).$

  4. (iv)

$u_{-}:=\max\\{-u,0\\}\thickspace(u_{+}:=\max\\{u,0\\})$ belongs to
$L_{sp}^{p-1}(\mathbb{R}^{N})$.

We say that $u$ is a viscosity solution of (2.21) if it is both a viscosity
supersolution and subsolution to the problem.

Notations: The paper uses various notations, which we will clarify here.
Sections 3 and 4 consider the case where $p>\frac{2}{2-s}$. However, for
sections 5 and 6, the paper restricts $p$ to the interval $(2,\infty)$ due to
the lack of improved weighted regularity results. Unless stated otherwise,
$C$, $C_{i}$, $C_{K,u}$, and other similar notations denote positive generic
constants, which may vary in value within the same line. For simplicity, the
notation $a^{p-1}$ denotes $a^{p-1}=|a|^{p-2}a$ for any real number $a$.

##  3\. Pointwise estimates for a barrier function under s-fractional
p-Laplace operator

In this section, we prove some important estimates which are used in
subsequent sections to prove the main results on a nonlocal superlinear
semipositone problem with subcritical growth.

We begin this section with an important observation that if $u$ is smooth
enough, then the principal value $P.V.$ in the definition of fractional
$p$-Laplacian can be replaced with integral over $\mathbb{R}^{N}$ when
$p>\frac{2}{2-s}.$ We can also define

$(-\Delta)_{p}^{s}u(x)=P.V.\int_{\mathbb{R}^{N}}\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+sp}}\,dz.$
(3.23)

The equivalence of the definitions (1.5) and (3.23) can be proved using a
change of variable.

######  Lemma .

For $\frac{2}{2-s}<p<\infty,$ suppose that $u\in
L^{\infty}(\mathbb{R}^{N})\cap C^{1,1}_{loc}(\Omega),$ then for $x\in\Omega,$

$(-\Delta)_{p}^{s}u(x)=\int_{\mathbb{R}^{N}}\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+sp}}\,dz.$

###### Proof.

Let $p\geq 2$ and $x$ belongs to a compact subset $K$ of $\Omega.$ Using [21,
Lemma 2.11], for proper choice of $\gamma\in(0,1),$ we have

$\displaystyle\int_{B(0,R_{K})}\frac{|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}|}{|z|^{N+sp}}$
(3.24) $\displaystyle\leq
C_{K,u}\int_{B(0,R_{K})}|z|^{\gamma+p-1-N-sp}dz<\infty.$

Since, $u\in L^{\infty}(\mathbb{R}^{N}),$ it can easily be seen that

$\int_{B^{c}(0,R_{K})}\frac{|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}|}{|z|^{N+sp}}<\infty.$

Therefore,
$\displaystyle\frac{|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}|}{|z|^{N+sp}}\in
L^{1}(\mathbb{R}^{N})$ and hence by dominated convergence theorem we can
write,

$\begin{array}[]{llr}\displaystyle\lim_{\epsilon\rightarrow 0}\int_{z\in
B^{c}(0,\epsilon)}\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+sp}}dz=&&\\\\[14.22636pt]
\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\displaystyle\int_{\mathbb{R}^{N}}\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+sp}}dz.\end{array}$

Now if $p\in(\frac{2}{2-s},2)$ then again using the second pointwise estimate
in [21, Lemma 2.11], and arguing as in the above case, for a suitable choice
of $\gamma$, we can prove that

$\displaystyle\frac{|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}|}{|z|^{N+ps}}\in
L^{1}(\mathbb{R}^{N})$

and thus the conclusion follows as before. ∎

###  3.1. Barrier function under fractional p-Laplacian

Our aim in this section is to construct a barrier function and then study the
behavior of the barrier function under the fractional $p$-Laplacian. Later
this result is used to obtain the apriori $L^{\infty}$ estimate in Theorem
4.2. To construct the barrier function, we start by considering a smooth
perturbation of the distance function $d(x):=\text{dist}(x,\partial\Omega)$,
where $x\in\Omega$ is a point in a smooth, bounded domain
$\Omega\subset\mathbb{R}^{N}$ with $N\geq 2$. We define the set
$\Omega_{\delta}=\\{x\in\Omega:d(x,\partial\Omega)<\delta\\}$ for some small
$\delta>0$ such that $d(x)$ is well-defined and $C^{2}$ in $\Omega_{\delta}$.
The barrier function $\xi(x)$ is then constructed as follows:

$\xi(x)=\begin{cases}d^{\beta}(x)&\text{if}\thickspace x\in\Omega_{\delta}\\\
r(x)&\text{if}\thickspace x\in\Omega\setminus\Omega_{\delta}\\\
0&\text{if}\thickspace x\in\Omega^{c}\end{cases}$ (3.25)

for $\beta>0$ and a function $r$ such that $\xi$ is positive and $C^{2}$ in
$\Omega$ with

$r(x)\geq d^{\beta}(x)\text{ for }x\in\Omega\setminus\Omega_{\delta}.$ (3.26)

The proof of the next lemma is inspired by section 3 of [17], where an
estimate of a similar type is proved for the fractional Laplacian. However,
since the operator we deal with is nonlinear, we have to estimate the
integrals differently. We prove the estimate using a different technique that
is tailored to the nonlinear operator.

######  Lemma .

Let $\frac{2}{2-s}<p<\infty$ then there exist $\delta>0$, $C>0$ and
$0<\beta_{0}<\frac{sp}{p-1}<\infty$ such that for all $\beta\in(0,\beta_{0})$

$-(-\Delta)_{p}^{s}\xi(x)\leq-
Cd(x)^{\beta(p-1)-sp}\quad\text{in}\quad\Omega_{\delta}.$

###### Proof.

Throughout the proof, we assume that $x\in\Omega_{\delta}$. For simplicity in
notation, we write fractional $p$-Laplacian as

$-(-\Delta)_{p}^{s}\xi(x)=\int_{\mathbb{R}^{N}}\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy,\quad
x\in\mathbb{R}^{N}.$ (3.27)

where

$\varrho_{+}(\xi,x,y)=(\xi(x+y)-\xi(x))^{p-1}$

and

$\varrho_{-}(\xi,x,y)=(\xi(x-y)-\xi(x))^{p-1}.$

Also

$\displaystyle-(-\Delta)_{p}^{s}\xi(x)=\int_{B(0,\delta)}$
$\displaystyle\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy$
$\displaystyle+\int_{B(0,\delta)^{c}}\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy$

Since $\xi$ is an $L^{\infty}$ function, it is easy to show that there is a
constant $C_{1}$ such that

$\int_{y\in
B^{c}(0,\delta)}\frac{|\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)|}{|y|^{N+sp}}\,dy\leq
C_{1}.$ (3.28)

Now we will estimate the integral over $B(0,\delta)$. We write

$\int_{B(0,\delta)}\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy=I_{1}(x)+I_{2_{+}}(x)+I_{2_{-}}(x)+I_{3}(x),$
(3.29)

where

$\begin{split}I_{1}(x)=&\int_{D_{1}}\frac{-2\xi(x)^{p-1}}{|y|^{N+sp}}\,dy,\\\
I_{2_{+}}(x)=&\int_{D_{2_{+}}}\frac{\varrho_{+}(\xi,x,y)-\xi(x)^{p-1}}{|y|^{N+sp}}\,dy,\\\
I_{2_{-}}(x)=&\int_{D_{2_{-}}}\frac{\varrho_{-}(\xi,x,y)-\xi(x)^{p-1}}{|y|^{N+sp}}\,dy,\\\
I_{3}(x)=&\int_{D_{3}}\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy\end{split}$
(3.30)

with the following domains of integration.

$\begin{split}D_{1}=&\\{y\in B(0,\delta):x+y\notin\Omega\quad\text{and}\quad
x-y\notin\Omega\\},\\\ D_{2_{\pm}}=&\\{y\in B(0,\delta):x\pm
y\in\Omega\quad\text{and}\quad x\mp y\notin\Omega\\},\\\ D_{3}=&\\{y\in
B(0,\delta):x+y\in\Omega\quad\text{and}\quad x-y\in\Omega\\}.\end{split}$
(3.31)

For notational simplicity we denote $d=d(x)$, whenever there is no confusion.
We shall show that each of the integrals $I_{i}$ is bounded above by
$-Cd^{\beta(p-1)-sp}.$ The first integral $I_{1}(x)$ can be estimated similar
to [17, Lemma 3.2] and we obtain

$I_{1}(x)\leq-C_{2}d^{\beta(p-1)-sp}.$ (3.32)

We will estimate $I_{2_{+}}(x)$ in detail and $I_{2_{-}}(x)$ follows in the
similar way. We start the estimation of $I_{2_{+}}(x)$ by making an important
observation that $dist(d^{-1}x+z,d^{-1}\partial\Omega))\leq
d^{-1}(|z|+\inf_{y\in\partial\Omega}|x-y|).$ This implies for
$0<\beta\leq\beta_{0}<\frac{sp}{p-1},$

$dist(d^{-1}x+z,d^{-1}\partial\Omega))^{\beta}-1\leq|z|^{\beta_{0}}.$

The constant $\beta_{0}$ appearing in the above expression will be chosen
later. We split the domain $d^{-1}D_{2_{+}}$ into two parts,

$d^{-1}D_{2_{+}}=(d^{-1}D_{2_{+}}\cap B(0,R))\cup(d^{-1}D_{2_{+}}\cap
B(0,R)^{c})=:B_{1,R}\cup B_{2,R}.$

Then we can write $I_{2_{+}}(x)$ as

$\displaystyle I_{2_{+}}(x)=d^{\beta(p-1)-sp}$
$\displaystyle\int_{B_{1,R}}\frac{(dist(d^{-1}x+z,d^{-1}\partial\Omega)^{\beta}-1)^{p-1}}{|z|^{N+sp}}\,dz$
(3.33) $\displaystyle+d^{\beta(p-1)-sp}$
$\displaystyle\int_{B_{2,R}}\frac{dist(d^{-1}x+z,d^{-1}\partial\Omega)^{\beta}-1)^{p-1}}{|z|^{N+sp}}\,dz$
$\displaystyle-d^{\beta(p-1)-sp}$
$\displaystyle\int_{d^{-1}D_{2_{+}}}\frac{1}{|z|^{N+sp}}\,dz$

The last integral could be easily estimated by $-C_{3}d^{\beta(p-1)-sp}.$ We
will find a proper upper bound for the first two integrals in terms of
$C_{3}$. Observe that,

$\begin{split}\int_{B_{2,R}}\frac{dist(d^{-1}x+z,d^{-1}\partial\Omega)^{\beta}-1)^{p-1}}{|z|^{N+sp}}\,dz\leq\int_{B_{2,R}}\frac{|z|^{\beta_{0}(p-1)}}{|z|^{N+sp}}\,dz\;\;\leq\alpha_{N}R^{\beta_{0}(p-1)-sp}.\end{split}$

where $\alpha_{N}$ denotes the surface measure of unit sphere in
$\mathbb{R}^{N}.$ Now we shall choose $R$ large enough so that
$\alpha_{N}R^{\beta_{0}(p-1)-sp}=\frac{C_{3}}{4}.$ We also have

$\lim_{\beta\rightarrow
0}\int_{B_{1,R}}\frac{dist(d^{-1}x+z,d^{-1}\partial\Omega)^{\beta}-1)^{p-1}}{|z|^{N+sp}}\,dz=0.$

Now we shall choose $\beta_{0}$ small enough so that

$\int_{B_{1,R}}\frac{dist(d^{-1}x+z,d^{-1}\partial\Omega)^{\beta}-1)^{p-1}}{|z|^{N+sp}}\,dz<\frac{C_{3}}{4}\mbox{
for all }0<\beta<\beta_{0}.$

Combining all the above arguments, there exists $\beta_{0}<<1$ and $C_{4}>0$
such that

$\displaystyle I_{2_{+}}(x)\leq-C_{4}d^{\beta(p-1)-sp}\thickspace\text{for}\
\beta\in(0,\beta_{0})$ (3.34)

Finally, we will estimate $I_{3}(x)$ by splitting the domain of its
integration

$D_{3}=B(0,\epsilon d)\cup(D_{3}\setminus B(0,\epsilon d))=:B_{1}\cup B_{2}.$

We rewrite the $I_{3}(x)$ as

$\displaystyle
I_{3}(x)=\int_{B_{2}}\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy+\int_{B_{1}}$
$\displaystyle\frac{\varrho_{+}(\xi,x,y)}{|y|^{N+sp}}\,dy$ (3.35)
$\displaystyle+\int_{B_{1}}\frac{\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy$

On $B_{2}$ the estimation of $I_{3}(x)$ is similar to $I_{2_{+}}(x)$ hence

$\displaystyle\int_{B_{2}}\frac{\varrho_{+}(\xi,x,y)+\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy\leq-
C_{4}d^{\beta(p-1)-sp}$ (3.36)

We only estimate
$\displaystyle\int_{B_{1}}\frac{\varrho_{+}(\xi,x,y)}{|y|^{N+sp}}\,dy$ , as
the estimation of
$\displaystyle\int_{B_{1}}\frac{\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy$ follows
using the similar arguments.

Since $\xi$ is a $C^{2}$ function, using Taylor’s expansion, we have

$\xi(x+y)=\xi(x)+\nabla\xi(x)\cdot y+y^{\top}\cdot D^{2}\xi(\alpha)\cdot
y\quad\text{for some}\quad\alpha=x+\theta y,\;\theta\in(0,1).$

We denote

$l(x+y)=\xi(x)+\nabla\xi(x)\cdot y,$

so that we have

$\xi(x+y)-l(x+y)=y^{\top}\cdot D^{2}\xi(\alpha)\cdot y.$

Similarly, we have

$\displaystyle\xi(x)=\xi(x+y)+\nabla\xi(x+y)\cdot(-y)+(-y)^{\top}\cdot$
$\displaystyle D^{2}\xi(\gamma)\cdot(-y)$ for some
$\displaystyle\gamma=x+y-\theta y,\theta\in(0,1).$

We denote

$l(x)=\xi(x+y)+\nabla\xi(x+y)\cdot(-y),$

so we have

$\xi(x)-l(x)=(-y)^{\top}\cdot D^{2}\xi(\gamma)\cdot(-y)$

and

$l(x+y)-l(x)=\nabla\xi(x)\cdot y.$

Let $g(t):=|t|^{p-2}t$ and using [24, Lemma 3.1 and 3.4], we have

$\begin{split}&\left|\int_{B(0,\epsilon
d)}\frac{(\xi(x+y)-\xi(x))^{p-1}}{|y|^{N+sp}}\,dy\right|\\\
&\leq\int_{B(0,\epsilon
d)}\frac{|g(\xi(x+y)-\xi(x))-g(l(x+y)-l(x))|}{|y|^{N+sp}}\,dy\\\ &\leq
C_{5}\int_{B(0,\epsilon
d)}\frac{(|l(x+y)-l(x)|+|\xi(x)-l(x)|)^{p-2}|\xi(x)-l(x)|}{|y|^{N+sp}}\,dy\\\
&\leq C_{5}\int_{B(0,\epsilon d)}\frac{(|\nabla\xi(x)\cdot
y|+|(-y)^{\top}\cdot D^{2}\xi(\gamma)\cdot(-y)|)^{p-2}|(-y)^{\top}\cdot
D^{2}\xi(\gamma)\cdot(-y)|}{|y|^{N+sp}}\,dy.\end{split}$ (3.37)

Since $x\in\Omega_{\delta}$, $\xi(x)=d^{\beta}(x)$. Then

$\begin{split}\nabla\xi(x)&=\beta d^{\beta-1}\nabla d(x)\mbox{ and
}\frac{\partial^{2}\xi(x)}{\partial x_{i}\partial
x_{j}}=\beta(\beta-1)d^{\beta-2}A_{ij}(x)\text{ for }1\leq i,j\leq
N,\end{split}$

where

$A_{ij}(x)=\frac{\partial d(x)}{\partial x_{i}}\frac{\partial d(x)}{\partial
x_{j}}+\frac{d(x)}{\beta-1}\frac{\partial^{2}d(x)}{\partial x_{i}\partial
x_{j}}.$

Then for $\gamma=x+y-\theta y,\thinspace\theta\in(0,1)$ and
$x\in\Omega_{\delta},y\in B(0,\epsilon d),$ we denote

$M_{i,j}:=\sup_{\gamma}\beta(\beta-1)A_{ij}(\gamma),$

Thus we have $|D^{2}\xi(\gamma)|\leq Md^{\beta-2}$ for some $M>0$. Hence from
(3.37) and with a change of variable $y=d(x)z$ we have,

$\begin{split}&\left|\int_{B(0,\epsilon
d)}\frac{(\xi(x+y)-\xi(x))^{p-1}}{|y|^{N+sp}}\,dy\right|\\\ &\leq
C_{6}d^{\beta(p-1)-sp}\int_{B(0,\epsilon)}\frac{(|\beta\nabla d(x)\cdot
z|+M|z|^{2})^{p-2}M|z|^{2}}{|z|^{N+sp}}\,dz\\\ &\leq
C_{6}d^{\beta(p-1)-sp}\int_{0}^{\epsilon}\int_{S^{N}}\left(|\beta\nabla
d(x)\cdot\omega|r+Mr^{2}\right)^{p-2}Mr^{1-sp}\,d\omega\,dr\\\ &\leq
C_{6}d^{\beta(p-1)-sp}\int_{0}^{\epsilon}\int_{S^{N}}\left(\frac{|\beta\nabla
d(x)\cdot\omega|}{|\beta\nabla d(x)|}+\frac{Mr}{|\beta\nabla
d(x)|}\right)^{p-2}Mr^{p-1-sp}|\beta\nabla d(x)|^{p-2}\,d\omega\,dr\\\ &\leq
C_{6}Md^{\beta(p-1)-sp}\int_{0}^{\epsilon}\left(1+\frac{Mr}{|\beta\nabla
d(x)|}\right)^{p-2}r^{p-1-sp}|\beta\nabla d(x)|^{p-2}\,dr\end{split}$

The last inequality here is obtained by applying [24, Lemma 3.5]. We can now
directly use the estimates exactly as in [24, Lemma 3.6] and for the case
$p\geq 2$ we get

$\displaystyle\bigg{|}\int_{B(0,\epsilon d)}$
$\displaystyle\frac{(\xi(x+y)-\xi(x))^{p-1}}{|y|^{N+sp}}\,dy\bigg{|}$ (3.38)
$\displaystyle\hskip 62.59596pt\leq C_{5}Md^{\beta(p-1)-sp}\left(|\beta\nabla
d(x)|^{p-2}\epsilon^{p-sp}+M^{p-2}\epsilon^{p-2+p(1-s)}\right)$
$\displaystyle\hskip 244.69386pt\leq C_{6}d^{\beta(p-1)-sp}\epsilon^{p-sp}$

and for $\frac{2}{2-s}<p<2,$

$\left|\int_{B(0,\epsilon
d)}\frac{(\xi(x+y)-\xi(x))^{p-1}}{|y|^{N+sp}}\,dy\right|\leq
C_{6}d^{\beta(p-1)-sp}\epsilon^{p-2+p(1-s)}.$

In both the cases choosing $\epsilon$ small enough we have

$\displaystyle\int_{B_{1}}\frac{\varrho_{+}(\xi,x,y)}{|y|^{N+sp}}\,dy+\int_{B_{1}}\frac{\varrho_{-}(\xi,x,y)}{|y|^{N+sp}}\,dy\leq\frac{C_{4}}{2}d^{\beta(p-1)-sp}$
(3.39)

Using (3.35) (3.36) and (3.39) we conclude that

$\displaystyle I_{3}(x)\leq-C_{7}d^{\beta(p-1)-sp}$ (3.40)

This concludes the proof of the Lemma.

∎

We define for $x_{0}\in\partial\Omega,\thickspace\tau>0$

$\Omega^{\tau}_{x_{0}}:=\left\\{x\in\mathbb{R}^{N}:x_{0}+\tau
x\in\Omega\right\\}.$

Also for the function $\xi$ defined in (3.25), we set
${\xi^{\tau}_{x_{0}}}(x):=\xi(x_{0}+\tau x)$ and define

$d_{\tau}(x):=dist(x,\partial\Omega^{\tau}_{x_{0}})=\tau^{-1}dist(x_{0}+\tau
x,\partial\Omega).$ (3.41)

######  Lemma .

Let $s\in(0,1)$ and $\frac{2}{2-s}<p<\infty$. Also let $\beta=sp-\theta$ in
(3.25) for $\theta\in\left(sp-\beta_{0},sp\right)$ for some
$\beta_{0}\in(0,\frac{sp}{p-1})$. Then there exist $c_{0},\delta>0$ such that

$(-\Delta)_{p}^{s}\xi^{\tau}_{x_{0}}\geq
c_{0}d^{sp^{2}-2sp-\theta(p-1)}_{\tau}\quad\text{ in
}\thickspace(\Omega^{\tau}_{x_{0}})_{\delta},\thickspace 0<\tau<1.$ (3.42)

Moreover, if $u\in L^{\infty}(\mathbb{R}^{N})$ satisfies the following in
viscosity sense for $c_{1}>0$

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&\leq
c_{1}d^{sp^{2}-2sp-\theta(p-1)}_{\tau}\quad\text{ in
}\Omega^{\tau}_{x_{0}},\\\ u&=0\quad\hskip 73.97733pt\text{ on
}(\Omega^{\tau}_{x_{0}})^{c}.\end{array}\right.$ (3.45)

then

$u(x)\leq
c_{2}\left(c_{1}+\|u\|_{L^{\infty}(\Omega^{\tau}_{x_{0}})}\right)d^{sp-\theta}_{\tau},\thickspace
x\in(\Omega^{\tau}_{x_{0}})_{\delta}$ (3.46)

for some $c_{2}>0$ is only depending on $s,\delta,\theta,p$ and $c_{0}$.

###### Proof.

Observe that for $x\in(\Omega^{\tau}_{x_{0}})_{\delta}$ we have $x_{0}+\tau
x\in(\Omega)_{\tau\delta}\subset(\Omega)_{\delta}$ as $\tau<1$. The rest of
the argument for establishing (3.42) follows using similar calculations as in
the proof of the Lemma 3.1 by taking $\beta=sp-\theta$ and translating
$\xi(x)$ to $\xi^{\tau}_{x_{0}}(x)$ and $(\Omega)_{\delta}$ to
$(\Omega^{\tau}_{x_{0}})_{\delta}$.
Next, to obtain (3.46), we define $v=R\xi^{\tau}_{x_{0}}$ where

$\displaystyle
R=\left(\frac{c_{1}}{c_{0}}\right)^{1/(p-1)}+(\tau\delta)^{\theta-
sp}\|u\|_{L^{\infty}(\Omega^{\tau}_{x_{0}})}.$

and observe that

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}v&\geq(-\Delta)_{p}^{s}u\quad\text{
in }(\Omega^{\tau}_{x_{0}})_{\delta},\\\ u=v&=0\quad\hskip 34.14322pt\text{ on
}(\Omega^{\tau}_{x_{0}})^{c}\\\ v&\geq u\hskip 45.52458pt\text{ in
}\Omega^{\tau}_{x_{0}}\setminus(\Omega^{\tau}_{x_{0}})_{\delta}.\end{array}\right.$
(3.50)

Using the comparison principle we can get

$u(x)\leq
c_{2}\left((c_{1})^{1/(p-1)}+\|u\|_{L^{\infty}(\Omega^{\tau}_{x_{0}})}\right)d^{sp-\theta}_{\tau}\quad\text{
in }\thickspace(\Omega^{\tau}_{x_{0}})_{\delta}$

where $c_{2}=\left((\tau\delta)^{\theta-sp}+(c_{0})^{-1/(p-1)}\right)$. ∎

##  4\. An apriori uniform boundedness estimate for subcritical problem

In this section we give the uniform $L^{\infty}$ estimates for the solutions
of the semipositone subcritical problem. We recall the local Hölder estimates
for viscosity solutions from [26, Theorem 1].

######  Theorem 4.1.

For $1<p<\infty$, assume $f\in C(B_{2}(0))\cap L^{\infty}(B_{2}(0))$ and let
$u\in L^{\infty}(\mathbb{R}^{N})$ be viscosity solution of

$(-\Delta)_{p}^{s}u=f\quad\text{ in }B_{2}(0).$

Then $u$ is Hölder continuous in $B_{1}(0)$ and in particular there exist
$\alpha\in(0,1)$ and $c$ depending on $s,p$ such that

$\|u\|_{C^{\alpha}(B_{1}(0))}\leq
c\left(\|u\|_{L^{\infty}(\mathbb{R}^{N})}+\|f\|^{\frac{1}{p-1}}_{L^{\infty}(B_{2}(0))}\right).$
(4.51)

######  Theorem 4.2.

Assume that, $0<s<1,$ and $\frac{2}{2-s}<p<\infty,\thinspace
p-1<q<p_{s}^{*}-1$, and $g\in C(\overline{\Omega}\times\mathbb{R})$ satisfies
that $|g(x,z)|\leq c|z|^{r},\thinspace x\in\Omega,\thinspace z\in\mathbb{R}$
where $c>0$ and $0<r<q$. Let $u$ be a positive viscosity solution to the
problem

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u(x)&=u^{q}+g(x,u)\hskip
14.22636pt\text{ in }\Omega,\\\ u(x)&=0\hskip 68.8556pt\text{ on
}\Omega^{c}.\end{array}\right.$ (4.54)

Then there exists a constant $C>0$ such that

$\|u\|_{L^{\infty}(\Omega)}\leq C.$

###### Proof.

We begin our proof by assuming the existence of a sequence of positive
solutions $\\{u_{k}\\}$ of problem (4.54) such that
$M_{k}=\|u_{k}\|_{L^{\infty}(\Omega)}\rightarrow\infty$. For $x_{k}\in\Omega$
such that $u_{k}(x_{k})=M_{k}$, we define the functions

$v_{k}(y)=\frac{u_{k}(x_{k}+\mu_{k}y)}{M_{k}},\thickspace y\in\Omega^{k},$

where

$\displaystyle\Omega^{k}:=\left\\{y\in\mathbb{R}^{N}:x_{k}+\mu_{k}y\in\Omega\right\\},$
$\displaystyle\mu_{k}=M^{\frac{-(q-(p-1))}{sp}}_{k}\in\mathbb{R}\longrightarrow
0,$

and functions $v_{k}$ satisfies $0<v_{k}\leq 1$ and $v_{k}(0)=1$. Next we have

$(-\Delta)_{p}^{s}v_{k}(y)=v^{q}_{k}+h_{k}\quad\text{in}\quad\Omega^{k}.$
(4.55)

where $|h_{k}|\leq CM^{r-q}_{k}$ and $h_{k}\in C(\Omega^{k})$. Next passing to
subsequences, there are two cases, either
$d(x_{k})\mu^{-1}_{k}\rightarrow\infty$ or
$d(x_{k})\mu^{-1}_{k}\rightarrow\tilde{d}$ for some non-negative constant
$\tilde{d}$. The first case implies $\Omega_{k}\rightarrow\mathbb{R}^{N}$. The
right hand side in (4.55) is uniformly bounded, now as in [2] using estimates
in (4.51) with an application of Arzela-Ascoli’s theorem and a diagonal
argument we get $v_{k}\rightarrow v$ locally uniformly in $\mathbb{R}^{N}$ up
to a subsequence. Thanks to [7, Corolary 4.7], the limiting function $v$ is
indeed a nontrivial positive viscosity solution to the problem
$(-\Delta)^{s}_{p}v=v^{p}$ in $\mathbb{R}^{N}$. Then by Theorem 4.1, we get
$v\in C^{\alpha}(\mathbb{R}^{N})$ for some $\alpha\in(0,1)$ which contradicts
the nonexistence hypothesis $(\mathcal{NA}).$
Next we shall consider the case $d(x_{k})\mu^{-1}_{k}\rightarrow\tilde{d}\geq
0$ which implies $x_{k}\rightarrow x_{0}$ for some $x_{0}\in\partial\Omega.$
Without loss of generality, we assume outward normal at $x_{0},$ i.e
$\nu(x_{0})=-e^{N}$. Define

$w_{k}(y)=\frac{u_{k}(\zeta_{k}+\mu_{k}y)}{M_{k}}\quad y\in D^{k},$

where $\zeta_{k}\in\partial\Omega$ is the projection of $x_{k}$ on
$\partial\Omega$ and

$D^{k}:=\\{y\in\mathbb{R}^{N}:\zeta_{k}+\mu_{k}y\in\Omega\\}.$

Observe that

$0\in\partial D^{k},$ (4.56)

and

$D^{k}\rightarrow\mathbb{R}^{N}_{+}=\\{y\in\mathbb{R}^{N}:y_{N}>0\\}\thickspace\text{as}\medspace
k\rightarrow+\infty.$

It is easy to show that $w_{k}$ satisfies (4.55) in $D^{k}$ with a different
function $h_{k}$, but with the same bounds. Setting

$y_{k}:=\frac{x_{k}-\zeta_{k}}{\mu_{k}},$

so that $|y_{k}|=d(x_{k})\mu^{-1}_{k}$, we observe that $w_{k}(y_{k})=1$.
Since $|y_{k}|\rightarrow\tilde{d},$ by passing to further subsequence
$y_{k}\rightarrow y_{0}$; $|y_{0}|=\tilde{d}\geq 0$. Next we claim the
following:
Claim: $y_{0}$ is in the interior of half-space $\mathbb{R}^{N}_{+}$. For this
it is sufficient to show that

$\tilde{d}=\lim_{k\rightarrow\infty}d(x_{k})\mu^{-1}_{k}>0.$ (4.57)

Now our estimate, namely Lemma 3.1, plays a crucial role in establishing this
claim. We observe that by (4.55), and since $r<q$, we get

$(-\Delta)_{p}^{s}w_{k}\leq C\leq
C_{1}d^{sp^{2}-2sp-\theta(p-1)}_{k}\thickspace\text{in}\medspace D^{k},$

where $d_{k}(y)=dist(y,\partial D^{k})$ and for a fixed
$\theta\in\left(sp-\beta_{0},sp\right)$ as given in Lemma 3.1. If
$d_{k}(y)<\delta$, by (3.46) we have $w_{k}(y)\leq C_{0}d_{k}(y)^{sp-\theta}$
for some $C_{0}>0.$ Now from (4.56) and using the definition of $d_{k}(y_{k})$
clearly $|y_{k}|\geq d_{k}(y_{k}).$ Combining all these we get the following
if $d_{k}(y_{k})<\delta$

$1=w_{k}(y_{k})\leq C_{0}d_{k}(y_{k})^{sp-\theta}\leq
C_{0}|y_{k}|^{sp-\theta}.$

This asserts that $|y_{k}|$ is bounded below by a positive constant, thus the
claim follows.
Next, similar to [2, Page no 208] we can use the estimates in (4.51) with an
application of Arzela-Ascoli’s theorem and a diagonal argument to conclude
$w_{k}\rightarrow w$ uniformly on a compact set of $\mathbb{R}^{N}_{+}$, with
$0\leq w\leq 1.$ Since $y_{k}\rightarrow y_{0},$ for some $y_{0}$ is in the
interior of $\mathbb{R}^{N}_{+},$ the uniform convergence of $w_{k}$ in the
compact subsets of $\mathbb{R}^{N}_{+}$ implies that $w(y_{0})=1.$ and
$w(y)\leq Cy^{sp-\theta}_{N}$ for $y_{N}<\delta.$ Thus $w\in
C(\mathbb{R}^{N}_{+})$ is non-negative, bounded solution of

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}w&=w^{q}\quad\text{
in }\mathbb{R}^{N}_{+},\\\ w&=0\hskip 19.91684pt\text{ in
}\mathbb{R}^{N}\setminus\mathbb{R}^{N}_{+}.\end{array}\right.$

Again using Theorem 4.1 we get $w\in C^{\alpha}(\mathbb{R}^{N})$ for some
$\alpha\in(0,1)$. Since $w(y_{0})=1$, the strong maximum principle implies
$w>0$. So we conclude our theorem by contradiction to the nonexistence
assumption $(\mathcal{NA})$. ∎

######  Remark .

Results discussed in section 3 and section 4 do not use the convexity of the
domain $\Omega.$

##  5\. Nonlocal superlinear semipositone problem with subcritical growth

In this section we look for a positive solution to following the Dirichlet
boundary value problem

$\displaystyle(P^{\mu})\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=\mu(u^{r}-1)\text{
in }\Omega,\\\ u&>0\hskip 39.83368pt\text{ in }\Omega,\\\ u&=0\hskip
39.83368pt\text{ on }\Omega^{c},\end{array}\right.$ (5.62)

where $p-1<r<p^{*}_{s}-1$. Using the substitution $w=\gamma u$ where
$\gamma^{r+1-p}=\mu$, we see that (5.62) is equivalent to the nonlocal problem

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}w&=w^{r}-\gamma^{r}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\Omega,\\\ w&>0\hskip 39.83368pt\text{ in
}\Omega,\\\ w&=0\hskip 39.83368pt\text{ on }\Omega^{c}.\end{array}\right.$
(5.66)

We use this observation to study the equivalent problem (5.66).

######  Definition .

We define the operator $K:C(\overline{\Omega})\rightarrow
C(\overline{\Omega})\cap W_{0}^{s,p}(\Omega)$ as $K(f)=u$ where $u$ is the
unique weak solution of $(-\Delta)_{p}^{s}\,u=f$ in $\Omega$ and $u=0$ on
$\Omega^{c}.$

The weak solution $u$ can be obtained as the minimizer of the associated
functional in $W^{s,p}_{0}(\Omega)$. Using [21, Theorem 1.1], $u\in
C^{\alpha}(\overline{\Omega})$ for some $\alpha\in(0,s]$ and thus the map $K$
is well defined. From Theorem [3, Theorem 1.3], we also infer that the weak
solution $u$ is in fact a viscosity solution. Now, if we set

$F(\mu,u)=\mu(|u|^{r}-1)$ (5.67)

then finding a weak solution to the nonlinear problem

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=F(\mu,u)\leavevmode\nobreak\
\text{ in }\Omega,\\\ u&>0\hskip 34.14322pt\text{ in }\Omega,\\\ u&=0\hskip
34.14322pt\text{ on }\Omega^{c}.\end{array}\right.$ (5.71)

is equivalent to finding a fixed point for the map $K(F(\mu,u)).$ By the
regularity results for subcritical problem (see [20]), the weak solution of
(5.71) is infact a viscosity solution. Using the rescaling argument once
again, we see that $u=K(F(\mu,u))$ iff $w=K(\tilde{F}(\gamma,w))$ where
$\tilde{F}(\gamma,w)=|w|^{r}-\gamma^{r}$ i.e.

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}w&=|w|^{r}-\gamma^{r}\text{
in }\Omega,\\\ w&=0\hskip 34.14322pt\text{ on }\Omega^{c}.\end{array}\right.$
(5.74)

We shall denote

$S(\gamma,w)=w-K(\tilde{F}(\gamma,w))\mbox{ for }0\leq\gamma<\infty.$ (5.75)

For $\gamma=0,$ the map $S(0,w)$ is denoted as $S_{0}(w).$ The solutions of
$S_{0}(w)=0$ are nothing but the solutions of

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}w&=|w|^{r}\text{ in
}\Omega,\\\ w&=0\hskip 11.38092pt\text{ on }\Omega^{c}.\end{array}\right.$
(5.78)

Next we state [13, Proposition 2.1 and Remark 2.1] here which will be used in
the proof of Lemma 5.

######  Proposition .

Let $C$ be a cone in a Banach space $X$ and $\Phi:C\rightarrow C$ be a compact
map such that $\Phi(0)=0.$ Assume there exists $0<R_{1}<R_{2}$ such that

  * (i)

$x\neq t\Phi(x)$ for all $t\in[0,1]$ and $\|x\|_{X}=R_{1}$.

  * (ii)

There exists a map $T:\overline{B}_{R_{2}}\times[0,\infty)\rightarrow C$ such
that $T(x,0)=\Phi(x)$ for all $\|x\|_{X}=R_{2},T(x,t)\neq x$ for
$\|x\|_{X}=R_{2}$ and $0\leq t<\infty$ and $T(x,t)=x$ has no solution for
$x\in\overline{B}_{R_{2}},\,t\geq t_{0}.$

If we denote $U=\\{x\in C:R_{1}<\|x\|_{X}<R_{2}\\}$ and $B_{\rho}=\\{x\in
C:\|x\|_{X}<\rho\\}$, then
$deg(I-\Phi,B_{R_{2}},0)=0,deg(I-\Phi,B_{R_{1}},0)=1$ and
$deg(I-\Phi,U,0)=-1.$

######  Lemma .

There exists $0<R_{1}<R_{2}$ such that $S_{0}(w)\neq 0$ for all $w$ belonging
to the set $\\{w:\|w\|_{\infty}=R_{1}\;\text{or}\;R_{2}\\}$ and
$deg(S_{0},B_{R_{2}}\setminus\overline{B}_{R_{1}},0)=-1.$

###### Proof.

Let us consider $X=C(\overline{\Omega}),$ and $C=\\{u\in X:u(x)\geq 0\\}$. We
define a map $\Phi:C\rightarrow C$ by setting
$\Phi(\cdot):=K(\tilde{F}(0,\cdot))$. It is straightforward to show that
$\Phi$ is a compact map. Suppose that $t\Phi(u)=u$ for some $t\in[0,1]$ then,

$\displaystyle\left\\{\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=t^{p-1}|u|^{r}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ in }\Omega,\\\ u&=0\hskip 39.83368pt\text{ on
}\Omega^{c}.\end{array}\right.$ (5.81)

Using the variational characterization of the principal eigenvalue and the
$L^{\infty}$ regularity of the weak solution of (5.81)

$\lambda_{1}^{p}\int_{\Omega}|u|^{p}\leq\|u\|_{W_{0}^{s,p}(\Omega)}=t^{p-1}\int_{\Omega}|u|^{r+1}\leq\|u\|_{L^{\infty}(\Omega)}^{r+1-p}\int_{\Omega}|u|^{p}.$

which imples, $\|u\|_{L^{\infty}(\Omega)}\geq c(s,p,r).$ Now by choosing
$R_{1}$ small enough we have condition $(i)$ listed in Proposition 5.
Next we define $T(u,t)=K(\tilde{F}(0,(|u|+t))).$ Then $T(u,0)=\Phi(u)$ and we
will verify two more conditions of the map $T$, viz.,

  * (a)

$T(u,t)\neq u$ for all $\|u\|_{L^{\infty}(\Omega)}=R_{2}$ and $0\leq
t<\infty.$

  * (b)

$T(u,t)=u$ has no solution for $u\in\overline{B}_{R_{2}}$ and $t\geq t_{0}.$

To verify (b), we infact claim that $T(u,t)=u$ has no solution if $t\geq
t_{0}.$ Suppose that for any arbitrary $t$ there exists a solution $u_{t}\in
C(\overline{\Omega})$ of $T(u_{t},t)=u_{t}.$ Taking
$\frac{\varphi_{1}^{p}}{u_{t}^{p-1}}$ as the test function and using [5,
Proposition 4.2] we have

$\displaystyle\int_{\Omega}$
$\displaystyle\frac{(u_{t}+t)^{r}\varphi_{1}^{p}}{u_{t}^{p-1}}$
$\displaystyle=\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{t}(x)-u_{t}(y)|^{p-2}(u_{t}(x)-u_{t}(y))}{|x-y|^{N+sp}}\left(\frac{\varphi_{1}(x)^{p}}{u_{t}(x)^{p-1}}-\frac{\varphi_{1}(y)^{p}}{u_{t}(y)^{p-1}}\right)$
$\displaystyle\hskip
125.19194pt\leq\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\varphi_{1}(x)-\varphi_{1}(y)|^{p}}{|x-y|^{N+sp}}=\lambda_{1}\int_{\Omega}|\varphi_{1}|^{p}.$

where $\varphi_{1}$ is the first eigenfunction of $(-\Delta)_{p}^{s}.$ Using
the strict convexity of the domain and the monotonicity of the solutions
$u_{t}$ as given in [12, Theorem 1.1] we appeal to [13, eqn. 8] and obtain for
any $x\in\Omega_{\epsilon}$

$\displaystyle\gamma(\inf_{\Omega\setminus\Omega_{\frac{\epsilon}{2}}}\varphi_{1}^{p})u_{t}(x)^{r-p+1}$
$\displaystyle\leq\int_{I_{x}}u_{t}^{r-p+1}(\xi)\varphi_{1}^{p}(\xi)d\xi\leq\int_{\Omega}u_{t}^{r-p+1}(\xi)\varphi_{1}^{p}(\xi)dx$
$\displaystyle\hskip
85.35826pt\leq\int_{\Omega}\frac{(u_{t}+t)^{r}\varphi_{1}^{p}}{u_{t}^{p-1}}\;\leq\;\lambda_{1}\int_{\Omega}|\varphi_{1}|^{p}.$

where $I_{x}$ is a measurable set as defined in [13]. Thus we get that
$\|u_{t}\|_{L^{\infty}(\Omega_{\epsilon})}\leq C_{\epsilon}$ for all $t.$ Now
for $t$ large enough, using (5) we have the following estimate

$t\int_{\Omega_{\epsilon}}\frac{\varphi_{1}^{p}}{C_{\epsilon}^{p-1}}\leq
t\int_{\Omega_{\epsilon}}\frac{\varphi_{1}^{p}}{u_{t}^{p-1}}\leq
t\int_{\Omega}\frac{\varphi_{1}^{p}}{u_{t}^{p-1}}\leq\lambda_{1}\int_{\Omega}|\varphi_{1}|^{p}.$
(5.82)

Thus, from (5.82), we infer that $T(u,t)=u$ has no solution for $t\geq t_{0}.$
This proves (b).
We will next show that if $u$ solves $T(u,t)=u$ for $t\in[0,\infty),$ then
$\|u\|_{{\infty}}\leq M$ (independent of $t$) and this verifies condition
$(a)$. We proceed as in [18]. On the contrary let us assume that there exists
$t_{k}\in[0,\infty)$ such that for the corresponding solutions $u_{k}$,
$\|u_{k}\|_{L^{\infty}(\Omega)}\rightarrow\infty$. Denote
$M_{k}=\|u_{k}\|_{L^{\infty}(\Omega)}\rightarrow\infty$ and let
$x_{k}\in\Omega$ be points with $M_{k}=u_{k}(x_{k})$. First we claim that, up
to a subsequence,

$\frac{t_{k}}{\|u_{k}\|_{L^{\infty}(\Omega)}}\rightarrow 0\text{ as
}k\rightarrow\infty.$

Indeed, without loss of generality, we assume that $t_{k}>0$ for all $k$ and
$t_{k}\rightarrow\infty$. Define $w_{k}:=\frac{u_{k}}{t_{k}}$ and
$\lambda_{k}:=t_{k}^{r-p+1}$. Then, it is easy to check that
$(w_{k},\lambda_{k})$ satisfies weakly

$\displaystyle(-\Delta)_{p}^{s}w_{k}=\lambda_{k}(w_{k}+1)^{r}$

Then from the comparison principle, we have $w_{k}\geq\bar{w}_{k}$ where

$\displaystyle(-\Delta)_{p}^{s}\bar{w}_{k}=\lambda_{k}\mbox{ in
}\Omega\;\;\;\mbox{ and }\;\;\;\;\bar{w}_{k}=0\mbox{ in }\;\Omega^{c}$ (5.83)

Suppose $\sup_{k}\|\bar{w}_{k}\|_{L^{\infty}(\Omega)}$ is bounded by $C$, then
using the weak formulation of (5.83), we get

$\|\bar{w}_{k}\|_{W_{0}^{s,p}(\Omega)}^{p}\leq C\lambda_{k}|\Omega|.$ (5.84)

Let $\phi\geq 0$ be a nontrivial function in $C_{c}^{\infty}(\Omega),$ then
using Hölder’s inequality we get

$\displaystyle\displaystyle 0<\int_{\Omega}\phi$
$\displaystyle\leq\frac{1}{\lambda_{k}}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\bar{w}_{k}(x)-\bar{w}_{k}(y)|^{p-2}(\bar{w}_{k}(x)-\bar{w}_{k}(y))(\phi(x)-\phi(y))}{|x-y|^{N+sp}}$
$\displaystyle\displaystyle\leq$
$\displaystyle\,{\lambda_{k}^{-1}}\|\bar{w}_{k}\|_{W^{s,p}_{0}(\Omega)}^{p-1}\|\phi\|_{W^{s,p}_{0}(\Omega)}\;\leq{C_{1}\lambda_{k}^{-1/p}}\rightarrow
0,$ (5.85)

which is a contradiction. Thus $\sup_{k}\|\bar{w}_{k}\|_{L^{\infty}(\Omega)}$
must be unbounded. Therefore we have

$\displaystyle\sup_{k}\|w_{k}\|_{L^{\infty}(\Omega)}\geq\sup_{k}\|\bar{w}_{k}\|_{L^{\infty}(\Omega)}=\infty.$

This verifies the claim since
$\|w_{k}\|_{\infty}=\frac{\|u_{k}\|_{\infty}}{t_{k}}.$ Now we introduce the
Gidas-Spruck translated function

$v_{k}(y)=\frac{u_{k}(x_{k}+\mu_{k}y)}{M_{k}},\quad y\in\Omega^{k}$

where

$\Omega^{k}=\left\\{y\in\mathbb{R}^{N}:x_{k}+\mu_{k}y\in\Omega\right\\}\text{
and }\mu_{k}=M^{-\frac{r-(p-1)}{sp}}_{k}.$

A straightforward calculation yields

$(-\Delta)^{s}_{p}v_{k}(x)=\left(v_{k}(x)+\frac{t_{k}}{M_{k}}\right)^{r},$
(5.86)

Since $v_{k}$’s are also viscosity solutions of (5.86) and as
$\thickspace\frac{t_{k}}{M_{k}}\rightarrow 0,$ using the arguments as in the
proof of Theorem 4.2, we can pass through the limit to get a contradiction to
the nonexistence results for the sub-critical problem (1.9). Hence, if
$T(u,t)=u$ then $\|u\|_{L^{\infty}(\Omega)}$ is bounded independent of $t.$
Finally, we use Proposition 5 with the proper choice of $R_{1}$ and $R_{2}$
and conclude that $deg(S_{0},B_{R_{2}}\setminus\overline{B}_{R_{1}},0)=-1.$ ∎

Next we prove the main result of our paper for which we proceed as in [11]. We
determine the degree of $S(\gamma,.)$ by connecting $S(\gamma,.)$ and $S(0,.)$
using the homotopy invariance of degree with respect to $\gamma$. Since
$deg(S_{0},B_{R_{2}}\setminus\overline{B}_{R_{1}},0)\neq 0,$ in particular,
this will imply that $S(\gamma,w)$ has a solution $w$ satisfying
$R_{1}<\|w\|_{L^{\infty}(\Omega)}<R_{2}$. Finally, we will show that the
solution hence obtained for (5.74) is infact positive for $\gamma$ small
enough.

######  Theorem 5.1.

For $p\geq 2$, the problem (5.74) admits a positive solution for
$\gamma\in[0,\gamma_{0}].$

###### Proof.

We prove the theorem in two steps:
STEP I:  There exists a $\gamma_{0}>0$ such that
$deg(S(\gamma,\cdot),B_{R_{2}}\setminus\overline{B}_{R_{1}},0)=-1$ for all
$\gamma\in[0,\gamma_{0}]$.

If $S(\gamma,w)\neq 0$ for $\|w\|_{\infty}\in\\{R_{1},R_{2}\\},$ then using
Lemma 5 and the homotopy invariance of degree we know that
$deg(S(\gamma,\cdot),B_{R_{2}}\setminus\overline{B}_{R_{1}},0)=-1.$ Suppose
that $S(\gamma_{n},w_{n})=0$ for some sequence $\gamma_{n}\rightarrow 0$ and
$\|w_{n}\|\in\\{R_{1},R_{2}\\}.$ Since
$K\tilde{F}(\gamma,\cdot):C(\overline{\Omega})\rightarrow
C(\overline{\Omega})$ is compact, we can find a function $w_{0}\in
C(\overline{\Omega})$ with $\|w_{0}\|\in\\{R_{1},R_{2}\\}$ and $S(0,w_{0})=0$.
This contradicts the previous lemma 5 and hence Step I is proved.
Clearly, Step I implies that the set of solutions of $S(\gamma,w)=0$ is
nonempty for $\gamma$ small enough. If $w$ is a positive function solving the
equation $S(\gamma,w)=0,$ then $w$ solves the PDE (5.74). With this
observation, the proof of Theorem 5.1 is complete if we prove Step II given
below.
STEP II:  For $\gamma$ small enough, $S(\gamma,w)=0$ for some $w$ in
$B_{R_{2}}\setminus\overline{B}_{R_{1}}$ implies $w>0.$
Let $S(\gamma_{n},w_{n})=0$ for $\gamma_{n}\rightarrow 0$ and $w_{n}\in
B_{R_{2}}\setminus\overline{B}_{R_{1}}$. Since
$\|w_{n}\|_{C(\overline{\Omega})}$ is bounded, using [21, Theorem 1.1 ] and
[22, Theorem 1.1 ], we have

$\|w_{n}\|_{C^{\alpha}(\overline{\Omega})}\leq C\mbox{ and
}\left\|\frac{w_{n}}{d^{s}(x)}\right\|_{C^{\alpha}(\overline{\Omega})}\leq C.$

Thus, by Ascoli Arzela theorem, up to a subsequence

$w_{n}\rightarrow w_{0}\mbox{ in }C^{0}(\overline{\Omega})\;\;\mbox{ and
}\;\;\frac{w_{n}}{d^{s}(x)}\rightarrow\frac{w_{0}}{d^{s}(x)}\mbox{ in
}C^{0}(\overline{\Omega}).$

We also know that $w_{0}$ is a non-zero solution of $S(w_{0},0)=0.$ By
regularity results $w_{0}\in C_{d^{s}}^{0}(\overline{\Omega})$ and Hopf type
lemma [14], $\inf_{x\in\Omega}\frac{w_{0}(x)}{d^{s}(x)}>0.$ Finally, the
positivity of $w_{n}$ follows by using the above uniform convergence and the
fact that $\displaystyle\frac{w_{0}(x)}{d^{s}(x)}>0$ in $\overline{\Omega}.$
Hence we conclude that (5.74) admits a positive solution for
$\gamma\in[0,\gamma_{0}].$ ∎

Proof of the Theorem 1.1: As per the discussion presented at the beginning of
section 5, we can infer that the existence of positive solutions for
$(P^{\mu})$ is equivalent to the existence of positive solutions for (5.74).
Since theorem 5.1 has been established, it follows that problem $(P^{\mu})$
has a positive solution for all $\mu\in(0,\mu_{0})$.

######  Remark .

We remark that with slight modification, the above theorem can be proved for
more general Dirichlet problems like $(-\Delta)_{p}^{s}\,u=\lambda f(u)$ in
$\Omega$ and $u=0$ in $\mathbb{R}^{N}\setminus\Omega$ where
$f:[0,\infty)\rightarrow\mathbb{R}$ is such that $f(0)<0,$ $f(s)\geq 0$ for
$s>>1$ and an additional growth assumption
$\lim_{s\rightarrow\infty}\frac{f(s)}{s^{q-1}}=b$ for some $b>0$ and
$q\in(p-1,p_{s}^{*}-1).$ The details are exactly as discussed in [11].

######  Remark .

Notably, in the local case of the $p$-Laplacian when $p\geq 2$, Theorem 1.1
represents a unique contribution as it employs degree theory, which has not
been utilized previously to establish similar results. Prior research by [11]
could only examine the case where $p\in(1,2]$ due to the absence of a uniform
$L^{\infty}$ bound. However, by employing the ideas discussed previously for
the local scenario, we can now expand the results of [11] to encompass the
case of p Laplacian when $p>2.$

##  6\.  Semipositone Fractional p-Laplace Problem with Critical Growth

Semipositone problems with critical exponents have been extensively studied in
the literature. It is well-established that, by using the Pohozaev identity
for the p-Laplacian, one can prove the non-existence of positive solutions for
the equation $-\Delta_{p}u=\mu(u^{p^{*}-1}-1)$ subject to zero Dirichlet
boundary conditions in any star-shaped domain $\Omega$. For the fractional
Laplace equation, where the Pohozaev identity is available (as shown in [33]),
similar non-existence results can be proven for the semipositone critical
exponent problem. However, for the fractional p Laplacian, a Pohozaev identity
in a bounded domain has not been established, which makes it difficult to
confirm this fact completely. Nonetheless, recent results in [1] have made
progress towards understanding the Pohozaev identity for the fractional p
Laplacian in $\mathbb{R}^{N}$.

Despite this uncertainty, we can still draw motivation from the work of [30]
and consider a perturbed multiparameter problem in the spirit of the Brezis-
Nirenberg problem as described in $(\ref{nscpt1})$. In [30], for the local
case, a positive solution to this perturbed problem was shown to exist. In
this section, we extend this study to a similar problem involving the
fractional p Laplacian operator.

The main objective of this section is to establish the existence of a positive
solution to the $p$-superlinear semipositone problem with critical growth
involving the fractional $p$-Laplace operator, as presented in equation
($P_{\lambda}^{\mu}$) for the conditions $p\geq 2$, $N\geq sp^{2}$, and
$\lambda\in(0,\lambda_{1})$. Therefore, from now on in this section, we will
assume that these conditions hold. To prove Theorem 1.2, we follow a similar
approach to that of [30] by utilizing variational tools as the background
framework for the mountain pass lemma. However, a significant challenge we
face in this study is demonstrating the positivity of the solution. This issue
is resolved through the implementation of Hopf’s Lemma, which is discussed in
[20], specifically when $p\geq 2$. We now consider the modified problem for
$\lambda,\mu>0$

$\displaystyle\left.\begin{array}[]{rl}(-\Delta)_{p}^{s}u&=\lambda
u^{p-1}_{+}+u_{+}^{p^{*}_{s}-1}-\mu f(u)\text{ in }\Omega,\\\ u&=0\hskip
109.54327pt\text{ on }\Omega^{c},\end{array}\right\\}$ (6.89)

where $u_{+}(x)=\max\\{u(x),0\\}$ and

$f(t)=\begin{cases}1&\quad t\geq 0,\\\ 1-|t|^{p-1}&\quad-1<t<0,\\\ 0&\quad
t\leq-1.\end{cases}$

We are interested in finding the critical points of the $C^{1}$-functional

$\begin{split}E_{\mu}(u)=&\frac{1}{p}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy+\int_{\Omega}\left(-\frac{\lambda
u^{p}_{+}}{p}-\frac{u^{p^{*}_{s}}_{+}}{p^{*}_{s}}\right)\,dx\\\
&+\mu\left[\int_{\\{-1<u<0\\}}\left(u-\frac{u|u|^{p-1}}{p}\right)\,dx-\left(1-\frac{1}{p}\right)|\\{u\leq-1\\}|\right]\\\
&\hskip 227.62204pt+\mu\int_{\\{u\geq 0\\}}u\,dx\end{split}$

where $|.|$ denotes the Lebesgue measure in $\mathbb{R}^{N}$. For all $v\in
D_{0}^{s,p}(\Omega)$, we also have

$\displaystyle\langle DE_{\mu}(u){,}v\rangle=$
$\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\,dx\,dy$
$\displaystyle\hskip 51.21504pt+\mu\left[\int_{\\{u\geq
0\\}}v\,dx+\int_{\\{-1<u<0\\}}\left(1-|u|^{p-1}\right)v\,dx\right]\quad$
$\displaystyle\hskip 165.02606pt+\int_{\Omega}\left(\lambda
u_{+}^{p-1}v-u_{+}^{p^{*}_{s}-1}v\right)\,dx.$

We now define $S_{s,p}$, the best constant in Sobolev inequality by

$S_{s,p}=\inf_{u\in
D^{s,p}_{0}(\Omega)\setminus{\\{0\\}}}\dfrac{\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy}{\left(\displaystyle\int_{\Omega}|u(x)|^{p^{*}_{s}}\,dx\right)^{p/p^{*}_{s}}}$
(6.90)

We encourage the readers to go through [28] for more results regarding the
minimization problem (6.90). We also recall the definition of the fractional
gradient here.

######  Definition .

[4] $(s,p)$ gradient of a function $v\in D_{0}^{s,p}(\mathbb{R}^{N})$ is
defined as

$|D^{s}v(x)|^{p}=\int_{\mathbb{R}^{N}}\frac{|v(x+h)-v(x)|^{p}}{|h|^{N+sp}}\,dh.$

We note that $(s,p)$ gradient is well defined a.e in $\mathbb{R^{N}}$ and
$|D^{s}v|\in L^{p}(\mathbb{R^{N}})$. Next we recall the concentration
compactness theorem [4].

######  Theorem 6.1.

[4, Theorem 1.1] Let $(u_{n})\subset D^{s,p}_{0}(\mathbb{R^{N}})$ be a weakly
convergent subsequence with weak limit $u$. Then there exist two bounded
measures $\kappa$ and $\nu$, an atmost enumerable set of indices $I$, and
positive real numbers $\kappa_{i},\nu_{i}$, points
$x_{i}\in\overline{\Omega},i\in I$, such that the following convergence hold
weakly∗ in the sense of measures.

$\displaystyle|D^{s}u_{n}|^{p}dx$
$\displaystyle\rightharpoonup\kappa\geq|D^{s}u|^{p}dx+\sum_{i\in
I}\kappa_{i}\delta_{x_{i}}$ $\displaystyle|u_{n}|^{p^{*}_{s}}dx$
$\displaystyle\rightharpoonup\nu=|u|^{p^{*}_{s}}dx+\sum_{i\in
I}\nu_{i}\delta_{x_{i}}$ $\displaystyle S^{1/p}_{s,p}\nu^{1/p^{*}_{s}}_{i}$
$\displaystyle\leq\kappa^{1/p}_{i}\quad\text{for all}\quad i\in I$

where $S_{s,p}$ as in (6.90).

We start this section by finding the level of PS-condition.

######  Lemma .

For any fixed $\lambda,\mu>0$, $E_{\mu}$ satisfies the $(PS)_{c}$ condition
for all

$c<\frac{s}{N}S^{N/sp}_{s,p}-\left(1-\frac{1}{p}\right)\mu|\Omega|.$ (6.91)

###### Proof.

Let $(u_{n})$ be a $(PS)_{c}$ sequence. We can show that sequence $(u_{n})$ is
bounded in $D_{0}^{s,p}(\Omega)$ following similar steps of [30, Lemma 2.1].
Since $(u_{n})$ is bounded so is $(u_{n+})$, a renamed subsequence which
converges to some $u_{+}\geq 0$ weakly in $D_{0}^{s,p}(\Omega)$, strongly in
$L^{q}(\Omega)$ for all $q\in[1,p^{*}_{s})$, $a.e.$ in $\Omega$ and

$|D^{s}u_{n+}|^{p}dx\rightharpoonup\kappa,\quad
u_{n+}^{p^{*}_{s}}dx\rightharpoonup\nu$ (6.92)

The convergence holds weakly* in the sense of measure, where $\kappa$ and
$\nu$ are bounded measures. Using concentration compactness theorem 6.1 there
exists a countable index set $I$ and positive real numbers
$\kappa_{i},\nu_{i}$, points $x_{i}\in\overline{\Omega},\thinspace i\in I$
such that

$\kappa\geq|D^{s}u_{+}|^{p}dx+\sum_{i\in
I}\kappa_{i}\delta_{x_{i}},\quad\nu=u_{+}^{p^{*}_{s}}dx+\sum_{i\in
I}\nu_{i}\delta_{x_{i}}$ (6.93)

and $S^{1/p}_{s,p}\nu^{1/p^{*}_{s}}_{i}\leq\kappa^{1/p}_{i}$ for all $i\in I$.
Our aim is to prove $I=\emptyset$. Suppose by contradiction we fix a
concentration point $x_{i}$. We define a smooth function
$\varphi:\mathbb{R^{N}}\rightarrow[0,1]$ such that $\varphi(x)=1$ for $|x|\leq
1$ and $\varphi(x)=0$ for $|x|\geq 2$ and for $i\in I$ and $\rho>0$ set

$\varphi_{i,\rho}(x)=\varphi\left(\frac{x-x_{i}}{\rho}\right),\thickspace
x\in\mathbb{R}^{N}.$

Clearly $\varphi_{i,\rho}:\mathbb{R}^{N}\rightarrow[0,1]$ is a smooth
function. We note that the sequence $(\varphi_{i,\rho}u_{n+})$ is bounded in
$D^{s,p}_{0}(\Omega)$. Taking $v=\varphi_{i,\rho}u_{n+}$ in $\langle
DE_{\mu}(u_{n}){,}v\rangle$, we get

$\begin{split}&\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\varphi_{i,\rho}u_{n+}(x)-\varphi_{i,\rho}u_{n+}(y))}{|x-y|^{N+sp}}\,dx\,dy\\\
&=\int_{\Omega}\left(\lambda
u_{n+}^{p-1}\varphi_{i,\rho}u_{n+}+u_{n+}^{p^{*}_{s}-1}\varphi_{i,\rho}u_{n+}\right)\,dx-\mu\int_{\Omega}\varphi_{i,\rho}u_{n+}\,dx+o_{n}(1)\|u_{n+}\|.\end{split}$
(6.94)

The term on the left-hand side of (6.94) can be estimated as

$\begin{split}&\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\varphi_{i,\rho}u_{n+}(x)-\varphi_{i,\rho}u_{n+}(y))}{|x-y|^{N+sp}}\,dx\,dy\\\
&\geq\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+sp}}u_{n+}(y)\,dx\,dy\\\
&\hskip
170.71652pt+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{n+}(x)-u_{n+}(y)|^{p}}{|x-y|^{N+sp}}\varphi_{i,\rho}(x)\,dx\,dy.\end{split}$
(6.95)

For the second term on right hand side of (6.95), for a fixed $x$, taking the
transformation $y-x=h$ and using the definition (6) and (6.92), we have

$\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{n+}(x)-u_{n+}(y)|^{p}}{|x-y|^{N+sp}}\varphi_{i,\rho}(x)\,dx\,dy\rightarrow\int_{\mathbb{R}^{N}}\varphi_{i,\rho}\,d\kappa.$
(6.96)

For the first term in right hand side of (6.95), using the definition of
$(s,p)$ gradient, we have

$\begin{split}&\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+sp}}u_{n+}(y)\,dx\,dy\\\
&\hskip 199.16928pt\leq
C\left(\int_{\mathbb{R}^{N}}|u_{n+}|^{p}|D^{s}\varphi_{i,\rho}|^{p\,dx}\right)^{\frac{1}{p}}.\end{split}$
(6.97)

Now we consider the first term on right hand side of (6.94) i.e.,

$\int_{\Omega}\left(\lambda
u_{n+}^{p}\varphi_{i,\rho}+u_{n+}^{p^{*}_{s}}\varphi_{i,\rho}-\mu\varphi_{i,\rho}u_{n+}\right)\,dx.$

Using (6.92) we have

$\int_{\Omega}\varphi_{i,\rho}u_{n+}^{p^{*}_{s}}\,dx\rightarrow\int_{\Omega}\varphi_{i,\rho}\,d\nu.$
(6.98)

Now we observe that

$\begin{split}\bigg{|}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}&\dfrac{|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))(\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+sp}}u_{n+}(y)\,dx\,dy\\\
&\hskip 165.02606pt+\int_{\Omega}\left(-\lambda
u_{n+}^{p}\varphi_{i,\rho}+\mu\varphi_{i,\rho}u_{n+}\right)\,dx\bigg{|}\end{split}$
$\leq
C\left[\left(\int_{\mathbb{R}^{N}}|u_{n+}|^{p}|D^{s}\varphi_{i,\rho}|^{p}\,dx\right)^{\frac{1}{p}}+\int_{\Omega\cap
B_{2\rho}(x_{i})}u_{n+}^{p}\,dx+\mu\int_{\Omega\cap
B_{2\rho}(x_{i})}u_{n+}\,dx\right].$

We also have

$\int_{\Omega\cap B_{2\rho}(x_{i})}u_{n+}^{p}\,dx\rightarrow\int_{\Omega\cap
B_{2\rho}(x_{i})}u_{+}^{p}\,dx.$ (6.99)

Next passing to the limit in (6.94) and using (6.95)-(6.99), we get

$\begin{split}&\int_{\mathbb{R}^{N}}\varphi_{i,\rho}\,d\kappa-\int_{\Omega}\varphi_{i,\rho}\,d\nu\\\
&\leq
C\left[\left(\int_{\mathbb{R}^{N}}|u_{n+}|^{p}|D^{s}\varphi_{i,\rho}|^{p}\,dx\right)^{\frac{1}{p}}+\int_{\Omega\cap
B_{2\rho}(x_{i})}u_{+}^{p}\,dx+\mu\int_{\Omega\cap
B_{2\rho}(x_{i})}u_{+}\,dx\right].\end{split}$

Now letting $\rho\rightarrow 0$, the right hand side of the above inequality
goes to $0$. This implies $\kappa_{i}\leq\nu_{i}$, which together with
$\nu_{i}>0$ and $S^{1/p}_{s,p}\nu^{1/p^{*}_{s}}_{i}\leq\kappa^{1/p}_{i}$ gives
$\nu_{i}\geq S^{N/sp}_{s,p}$. On the other hand, similar to [30, Lemma 2.1]
and using (6.92) and (6.93) we get

$\nu_{i}\leq\frac{N}{s}\left[\left(1-\frac{1}{p}\right)\mu|\Omega|+c\right]<S^{N/sp}_{s,p}$

a contradiction. Hence $I=\emptyset$ and

$\int_{\Omega}u_{n+}^{p^{*}_{s}}\,dx\rightarrow\int_{\Omega}u_{+}^{p^{*}_{s}}\,dx.$
(6.100)

Next passing to further subsequence, $u_{n}$ converges weakly to $u$ i.e.,
$u_{n}\rightharpoonup u$ in $D^{s,p}_{0}(\Omega)$, strongly in $L^{r}(\Omega)$
for $r\in[1,p^{*}_{s})$ and a.e. in $\Omega$.
We note that $u$ weakly solves the problem (6.89) and using the Brezis-Lieb
type lemma and (6.100) we can show that $u_{n}\rightarrow u$ in
$D^{s,p}_{0}(\Omega)$. This completes the proof of Lemma 6. ∎

Next, we state some results regarding the minimization problem (6.90) , which
is employed to establish mountain pass results, and introduce
$u_{\epsilon,\delta}$ in the same way as depicted in [28, Equation (2.16)].
For the sake of completeness, we shall discuss the definition here. Let $U$ be
a normalised radially symmetric nonnegative decreasing minimizer of $S_{s,p}$
( see [28, propsition 2.1]) which satisfies

$(-\Delta)^{s}_{p}U=U^{p^{*}_{s}-1},\thickspace\text{and}\thickspace\|U\|^{p}=\|U\|_{L_{p^{*}_{s}}(\mathbb{R}^{N})}^{p^{*}_{s}}=S^{N/sp}_{s,p}.$
(6.101)

For any $\epsilon>0$,

$U_{\epsilon}(x)=\frac{1}{\epsilon^{(N-sp)/p}}U\left(\frac{|x|}{\epsilon}\right)$
(6.102)

denotes the associated family of minimisers for $S_{s,p}$ and also satisfies
(6.101). Due to the absence of explicit formula for $U$, we have the following
asymptotic estimates.

######  Lemma .

There exists $c_{1},c_{2}>0$ and $\theta>1$ such that for all $r\geq 1$,

$\dfrac{c_{1}}{r^{(N-sp)/(p-1)}}\leq U(r)\leq\dfrac{c_{2}}{r^{(N-sp)/(p-1)}}$

and

$\frac{U(\theta
r)}{U(r)}\leq\frac{c_{2}}{c_{1}}\dfrac{1}{\theta^{(N-sp)/(p-1)}}.$

We have some auxiliary estimates from [28]. For $\epsilon,\delta>0$, and
$\theta$ as in Lemma 6, set

$m_{\epsilon,\delta}=\dfrac{U_{\epsilon}(\delta)}{U_{\epsilon}(\delta)-U_{\epsilon}(\theta\delta)}$

and

$g_{\epsilon,\delta}(t)=\begin{cases}0&\quad\text{if}\thickspace 0\leq t\leq
U_{\epsilon}(\theta\delta)\\\
m^{p}_{\epsilon,\delta}(t-U_{\epsilon}(\theta\delta))&\quad\text{if}\thickspace
U_{\epsilon}(\theta\delta)\leq t\leq U_{\epsilon}(\delta)\\\
t+U_{\epsilon}(\delta)(m^{p-1}_{\epsilon,\delta}-1)&\quad\text{if}\thickspace
t\geq U_{\epsilon}(\delta),\end{cases}$

and let

$G_{\epsilon,\delta}(t)=\int_{0}^{t}g^{\prime}_{\epsilon,\delta}(\tau)^{\frac{1}{p}}\,d\tau=\begin{cases}0&\quad\text{if}\thickspace
0\leq t\leq U_{\epsilon}(\theta\delta)\\\
m_{\epsilon,\delta}(t-U_{\epsilon}(\theta\delta))&\quad\text{if}\thickspace
U_{\epsilon}(\theta\delta)\leq t\leq U_{\epsilon}(\delta)\\\
t&\quad\text{if}\thickspace t\geq U_{\epsilon}(\delta).\end{cases}$ (6.103)

The functions $g_{\epsilon,\delta}$ and $G_{\epsilon,\delta}$ are
nondecreasing and absolutely continuous. Consider the radially symmetric
nonincreasing function

$u_{\epsilon,\delta}(r)=G_{\epsilon,\delta}(U_{\epsilon}(r)),$

which satisfies

$u_{\epsilon,\delta}(r)=\begin{cases}U_{\epsilon}(r)&\quad\text{if}\thickspace
r\leq\delta\\\ 0&\quad\text{if}\thickspace r\geq\theta\delta.\end{cases}$

Now similar to [30, Lemma 3.1] and thanks to $N\geq sp^{2}$, we can show that
for all sufficiently small $\mu>0,$ $E_{\mu}$ has a uniformly positive
mountain pass level below the threshold for compactness given in Lemma 6 .

######  Lemma .

There exist positive $\mu_{0},\rho,c_{0}>0,R>\rho$ and
$\beta<\frac{s}{N}S^{N/sp}_{s,p}$ such that the following hold for all
$\mu\in(0,\mu_{0})$ and $\lambda\in(0,\lambda_{1})$:

  * ${(i)}$

$E_{\mu}(0)=0$ and $E_{\mu}(u)\geq c_{0}$ for all $u$ such that $\|u\|=\rho$,

  * ${(ii)}$

$E_{\mu}(tu_{\epsilon,\delta})\leq 0$ for all $t\geq R$ and
$\epsilon\leq\delta/2$ and $\delta\in(0,1]$,

  * ${(iii)}$

denoting by $\Gamma=\left\\{\gamma\in
C([0,1],D^{s,p}_{0}(\Omega)):\gamma(0)=0,\gamma(1)=Ru_{\epsilon,\delta}\right\\}$
the class of paths joining the origin to $Ru_{\epsilon,\delta}$,

$c_{0}\leq c_{\mu}:=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq
1}E_{\mu}(\gamma(t))\leq\beta-\left(1-\frac{1}{p}\right)\mu|\Omega|$ (6.104)

for all sufficiently small $\epsilon>0$,

  * ${(iv)}$

$E_{\mu}$ has a critical point $u_{\mu}$ at the level $c_{\mu}$.

From the above lemma we conclude that there exists $u_{\mu}$ which is a weak
solution of (6.89). Now we shall prove some more properties of $u_{\mu}.$

######  Lemma .

There exists $\mu_{*}\in(0,\mu_{0}]$ such that the following hold for all
$\mu\in(0,\mu_{*})$:

  * ${(i)}$

$u_{\mu}$ is uniformly bounded in $D^{s,p}_{0}(\Omega)$,

  * ${(ii)}$

$\int_{E}|u_{\mu}|^{p^{*}_{s}}\,dx\rightarrow
0\thickspace\text{as}\thickspace|E|\rightarrow 0,\thickspace\text{uniformly
in}\thickspace\mu$,

  * ${(iii)}$

$u_{\mu}$ is uniformly bounded in $C_{d}^{0,\alpha}(\overline{\Omega})$ for
some $\alpha\in(0,s]$.

###### Proof.

As mentioned in the proof of Lemma 6 the sequence $\\{u_{\mu}\\}$ is uniformly
bounded in $D_{0}^{s,p}(\Omega)$ for $\mu\in(0,\mu_{0})$ which proves $(i).$
Next we shall prove $(ii).$ On contrary let us suppose that for some sequence
${\mu_{j}}\rightarrow 0$ and $|E_{j}|\rightarrow 0$ we have
$\liminf_{j\rightarrow\infty}\int_{E_{j}}|u_{\mu_{j}}|^{p_{s}^{*}}>0.$ Since
$(u_{\mu_{j}})$ is bounded, a renamed subsequence $(u_{\mu_{j+}})$ converges
to some $u_{+}\geq 0$ weakly in $D_{0}^{s,p}(\Omega)$. Following the argument
as in Lemma 6 we get

$\int_{\Omega}u_{\mu_{j+}}^{p^{*}_{s}}\,dx\rightarrow\int_{\Omega}u_{+}^{p^{*}_{s}}\,dx.$
(6.105)

This implies, $(u_{\mu_{j}})$ then converges to $u$ in $D_{0}^{s,p}(\Omega)$,
and so also in $L^{p^{*}_{s}}(\Omega)$. Then

$\|u_{\mu_{j}}\|_{L^{p^{*}_{s}}({E_{j}})}\leq\|u_{\mu_{j}}-u\|_{L^{p^{*}_{s}}({\Omega})}+\|u\|_{L^{p^{*}_{s}}({E_{j}})}\rightarrow
0,$

which is a contradiction and hence (ii).

Now Theorem A.1 is applicable and $\|u_{\mu_{j}}\|_{\infty}$ is uniformly
bounded. Since $p\geq 2,$ [22, Theorem 1.1] is applicable and $u_{\mu}$ is
uniformly bounded in $C_{d}^{0,\alpha}(\overline{\Omega}).$ ∎

Proof of the Theorem 1.2: We claim that $u_{\mu}$ is positive in $\Omega$ and
hence a solution of (1.17). It is sufficient to show that for every sequence
$\mu_{j}\rightarrow 0$, a subsequence $u_{\mu_{j}}$ is positive in $\Omega$.
By Lemma 6, we have

$\displaystyle E_{\mu_{j}}(u_{\mu_{j}})=c_{\mu_{j}}\geq c_{0}$

and

$\displaystyle\langle
DE_{\mu_{j}}(u_{\mu_{j}}){,}v\rangle=0\quad\forall\thinspace v\in
D_{0}^{s,p}(\Omega).$

From Lemma 6, we have upto a subsequence $u_{\mu_{j}}\rightharpoonup u$ in
$D^{s,p}_{0}(\Omega).$ Now with the help of the ideas of [9, Theorem 2.15] and
Brezis-Lieb type lemma, we can conclude for $\mu_{j}\rightarrow 0$

$\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u_{\mu_{j}}(x)-u_{\mu_{j}}(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\rightarrow\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy.$

We also have $u_{\mu_{j}}\rightarrow u$ in $C^{\alpha}(\overline{\Omega})$ for
$\mu_{j}\rightarrow 0$. So we have

$\frac{1}{p}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy+\int_{\Omega}\left(-\frac{\lambda
u^{p}_{+}}{p}-\frac{u^{p^{*}_{s}}_{+}}{p^{*}_{s}}\right)\,dx\geq c_{0}$
(6.106)

and $u$ weakly satisfy (6.89) with $\mu=0$. From mountain pass Lemma 6 we have
$c_{0}>0.$ Since $u$ satisfy (6.106), this implies that $u$ can not be
identically zero. For each $x\in\Omega,\lambda
u^{p-1}_{+}(x)+u_{+}^{p^{*}_{s}-1}(x)\geq 0$ and using strong minimum
principle and Hopf’s lemma [14], we have

$u(x)\geq cd^{s}(x)>0\thickspace\text{in}\thickspace\Omega.$

Now upto a subsequence $u_{\mu_{j}}\rightarrow u$ in
$C_{d}^{0,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,s).$ This implies
that $u_{\mu_{j}}>0$ in $\Omega$ for sufficiently large $j$. So we conclude
that for small $\mu,\thinspace u_{\mu}>0$ is solution of the problem (1.17)
with $DE_{\mu}(u_{\mu})=0$ and $E_{\mu}(u_{\mu})=c_{\mu}$ where $c_{\mu}$ is
as given in Lemma 6 viz.,

$c_{\mu}:=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}E_{\mu}(\gamma(t))$

where $\Gamma=\left\\{\gamma\in
C([0,1],D^{s,p}_{0}(\Omega)):\gamma(0)=0,\gamma(1)=Ru_{\epsilon,\delta}\right\\}$.
Now we can show that $u_{\mu}$ is ground state solution by following the ideas
of [30, Theorem 1.1]. This completes the proof of the Theorem 1.2. ∎.

## Appendix A

We first state some of the well known auxiliary results.
Auxiliary results

######  Lemma .

Let $1<p<\infty$ and $\beta\geq 1.$ For every $a,b,t\geq 0,$ it holds that

$|a-b|^{p-2}(a-b)(a_{t}^{\beta}-b_{t}^{\beta})\geq\frac{{\beta}p^{p}}{({\beta}+p-1)^{p}}\left|a_{t}^{\frac{{\beta}+p-1}{p}}-b_{t}^{\frac{{\beta}+p-1}{p}}\right|^{p},$

where $a_{t}=\min\\{a,t\\}$ and $b_{t}=\min\\{b,t\\}.$

######  Lemma .

Let $1<p<\infty$ and $\phi:\mathbb{R}\to\mathbb{R}$ be a differentiable convex
function. Then

$\displaystyle|a-b|^{p-2}(a-b)\left[c\leavevmode\nobreak\
|\phi^{\prime}(a)|^{p-2}\phi^{\prime}(a)-t\leavevmode\nobreak\
|\phi^{\prime}(b)|^{p-2}\phi^{\prime}(b)\right]$ $\displaystyle\hskip
170.71652pt\geq|\phi(a)-\phi(b)|^{p-2}(\phi(a)-\phi(b))(c-t),$

for every $a,b\in\mathbb{R}$ and every $c,t\geq 0.$

Now, we state the regularity result and prove it.

######  Theorem A.1.

Let $\Omega\subset\mathbb{R}^{N},N\geq 2$ be a bounded domain and $u\in
D^{s,p}_{0}(\Omega)$ weakly solve $(-\Delta)_{p}^{s}u=f_{\mu}(x,u)$ in
$\Omega$ for a parameter $\mu>0$ and $f$ satisfying the following growth
condition

$|f_{\mu}(x,t)|\leq C_{1}\left(1+|t|^{p^{*}_{s}-1}\right),$ (1.107)

where $C_{1}>0$ is constant. If $u$ is uniformly bounded in
$D^{s,p}_{0}(\Omega)$ and $\int_{E}|u|^{p^{*}_{s}}\,dx\rightarrow
0\thickspace\text{as}\thickspace|E|\rightarrow 0,\thickspace\text{uniformly
in}\thickspace\mu$ then there exist two positive constants $C_{*}$ and $C^{*}$
depending upon $s,p,N,\Omega$ such that

$\displaystyle\|u\|_{L^{\infty}(\Omega)}\leq\left(C_{*}\right)^{\displaystyle\frac{1}{p_{s}^{*}-p}}\left(C^{*}\right)^{\displaystyle\frac{p-1}{\sqrt{p}(\sqrt{p_{s}^{*}}-\sqrt{p})}}\|u\|_{L^{p_{s}^{*}}(\Omega)}.$

that is, $\|u\|_{L^{\infty}(\Omega)}$are uniformly bounded.

###### Proof.

For every $0<\epsilon<<1,$ we define the smooth convex Lipschitz function

$h_{\epsilon}(t)=(\epsilon^{2}+t^{2})^{\frac{1}{2}}$

and take $\psi|h_{\epsilon}^{\prime}(u)|^{p-2}h_{\epsilon}^{\prime}(u)$ as the
test function in weak formulation, where $\psi\in
C_{c}^{\infty}(\Omega),\psi>0.$ By Lemma A, we obtain

$\displaystyle\int_{{\mathbb{R}^{N}}}\int_{\mathbb{R}^{N}}\frac{|h_{\epsilon}(u(x))-h_{\epsilon}(u(y))|^{p-2}(h_{\epsilon}(u(x))-h_{\epsilon}(u(y)))(\psi(x)-\psi(y))}{|x-y|^{N+sp}}dxdy$
$\displaystyle\hskip 162.18062pt\leq\int_{\Omega}|f(x,u)|\leavevmode\nobreak\
|h_{\epsilon}^{\prime}(u(x))|^{p-1}\psi(x)\;dx.$ (1.108)

Since $h_{\epsilon}(t)$ converges to $h(t)=|t|$ as $\epsilon\to 0^{+}$ and
$|h_{\epsilon}^{\prime}(t)|\leq 1$, We note that, using Young’s inequality and
Lipschitz continuity of $h_{\epsilon},$

$\displaystyle\frac{|h_{\epsilon}(u(x))-h_{\epsilon}(u(y))|^{p-2}(h_{\epsilon}(u(x))-h_{\epsilon}(u(y)))(\psi(x)-\psi(y))}{|x-y|^{N+sp}}$
$\displaystyle\hskip
113.81102pt\geq-\frac{|h_{\epsilon}(u(x))-h_{\epsilon}(u(y))|^{p-1}|\psi(x)-\psi(y)|}{|x-y|^{N+sp}}$
$\displaystyle\hskip
113.81102pt\geq-\frac{(p-1)}{p}\frac{|h_{\epsilon}(u(x))-h_{\epsilon}(u(y))|^{p}}{|x-y|^{N+sp}}-\frac{1}{p}\frac{|\psi(x)-\psi(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle\hskip
153.6447pt\geq-\frac{(p-1)}{p}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}-\frac{1}{p}\frac{|\psi(x)-\psi(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

By using generalized Fatou’s lemma in (A), we get

$\displaystyle\int_{{\mathbb{R}^{N}}}\int_{\mathbb{R}^{N}}\frac{\Big{|}|u(x)|-|u(y)|\Big{|}^{p-2}\Big{(}|u(x)|-|u(y)|\Big{)}\Big{(}\psi(x)-\psi(y)\Big{)}}{|x-y|^{N+sp}}dxdy$
$\displaystyle\hskip 210.55022pt\leq\int_{\Omega}\;|f(x,u)|\psi(x)\;dx.$
(1.109)

Since $C_{c}^{\infty}(\Omega)$ is dense in $D_{0}^{s,p}(\Omega),$ (A) holds
true, for $0\leq\psi\in D_{0}^{s,p}(\Omega).$ Next, for $l\in\mathbb{N}$, we
define $u_{l}=\min\\{l,|u(x)|\\}.$ Clearly $u_{l}\in D_{0}^{s,p}(\Omega).$ For
$k\geq 1,$ let us set $\beta:=kp-p+1.$ So, $\beta>1.$ Choosing
$\psi=u_{l}^{\beta}$ in (A) and using Lemma A, we obtain

$\displaystyle\frac{\beta
p^{p}}{(\beta+p-1)^{p}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}$
$\displaystyle\frac{\left|(u_{l}(x))^{\frac{\beta+p-1}{p}}-(u_{l}(y))^{\frac{\beta+p-1}{p}}\right|^{p}}{|x-y|^{N+sp}}dxdy$
$\displaystyle\hskip
96.73918pt\leq\int_{\Omega}\;|f(x,u)|(u_{l}(x))^{\beta}\;dx.$ (1.110)

By observing that

$\frac{1}{\beta}\left({\frac{\beta+p-1}{p}}\right)^{p}\leq\left({\frac{\beta+p-1}{p}}\right)^{p-1},\text{\leavevmode\nobreak\
\leavevmode\nobreak\ for large\leavevmode\nobreak\ }\beta$

and using the the relation $k=\frac{\beta+p-1}{p}$ along with continuous
embedding $D_{0}^{s,p}(\Omega)\hookrightarrow L^{p_{s}^{*}}(\Omega)$, from
(A), we get

$\displaystyle\left\|u_{l}^{k}\right\|^{p}_{L^{p_{s}^{*}}(\Omega)}\leq\frac{(k)^{p-1}}{S_{s,p}^{p}}\int_{\Omega}\;|f(x,u)|(u_{l}(x))^{\beta}\;dx,$
(1.111)

where the best Sobolev constant $S_{s,p}$. Now we will estimate the right-hand
side in (1.111). Using the fact $u_{l}\leq|u|,$ we deduce

$\displaystyle\int_{\Omega}\;|f(x,u)|(u_{l}(x))^{\beta}\leavevmode\nobreak\
dx\leq
C_{1}\left(\int_{\Omega}|u_{l}|^{\beta}\,dx+\int_{\Omega}|u|^{p_{s}^{*}-1}|u_{l}^{\beta}|\,dx\right)$
$\displaystyle=C_{1}\Bigg{[}\int_{\Omega\cap\\{|u|<\Lambda\\}}|u_{l}|^{\beta}dx+\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u_{l}|^{\beta}dx+\int_{\Omega\cap\\{|u|<\Lambda\\}}|u|^{p_{s}^{*}-1}|u_{l}^{\beta}|\,dx$
$\displaystyle\hskip
256.0748pt+\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}-1}|u_{l}^{\beta}|\,dx\Bigg{]}$
$\displaystyle\leq
C_{1}\Bigg{[}\int_{\Omega\cap\\{|u|<\Lambda\\}}|u|^{\beta}\,dx+\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}+\beta-1}\,dx+\int_{\Omega\cap\\{|u|<\Lambda\\}}|u|^{p_{s}^{*}+\beta-1}\,dx$
$\displaystyle\hskip
261.76526pt+\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}+\beta-1}\,dx\Bigg{]}$
$\displaystyle\leq
C_{1}\Bigg{[}\Lambda^{1-p}\int_{\Omega\cap\\{|u|<\Lambda\\}}|u|^{p+\beta-1}\,dx+\Lambda^{p_{s}^{*}-p}\int_{\Omega\cap\\{|u|<\Lambda\\}}|u|^{p+\beta-1}\,dx$
$\displaystyle\hskip
221.93158pt+2\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}-p}\;|u|^{p+\beta-1}\,dx\Bigg{]}$
$\displaystyle\hskip 14.22636pt\leq
C_{1}\Big{[}(\Lambda^{1-p}+\Lambda^{p_{s}^{*}-p})\;\|u\|_{L^{kp}(\Omega)}^{kp}+2\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}-p}\;|u|^{p+\beta-1}\,dx\Big{]},$
(1.112)

where $\Lambda>1$ will be chosen later, $C_{1}>0$ is a constant. By plugging
(A) into (1.111) and applying Fatou’s lemma, we get

$\displaystyle\|u\|_{L^{kp_{s}^{*}}(\Omega)}^{kp}$ $\displaystyle\leq
C_{1}\frac{k^{p-1}}{(S_{s,p})^{p}}\Bigg{[}(\Lambda^{1-p}+\Lambda^{p_{s}^{*}-p})\;\|u\|_{L^{kp}(\Omega)}^{kp}$
$\displaystyle\hskip
119.50148pt+2\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}-p}\;|u|^{p+\beta-1}\,dx\Bigg{]}.$
(1.113)

Now we estimate the second term on the right-hand side in (A). For this, using
Hölder inequality for constant exponents, we obtain

$\displaystyle\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}-p}\;|u|^{kp}\,dx\leq$
$\displaystyle
C_{2}\left(\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}}dx\right)^{\frac{p_{s}^{*}-p}{p_{s}^{*}}}\left(\int_{\Omega}|u|^{kp_{s}^{*}}dx\right)^{\frac{p}{p_{s}^{*}}}$
$\displaystyle\hskip
85.35826pt=C(\Lambda,u)\|u\|_{L^{kp_{s}^{*}}(\Omega)}^{kp},$ (1.114)

where $C_{2}>0$ is a constant and
$C(\Lambda,u)=C_{2}\left(\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}}dx\right)^{\frac{p_{s}^{*}-p}{p_{s}^{*}}}.$
Using uniform boundedness and uniform integrability of $u$ , $C(\Lambda,u)$
only depends on $u$ for $\Lambda$ large so we write this as $C(\Lambda)$ and
get

$\displaystyle\int_{\Omega\cap\\{|u|\geq\Lambda\\}}|u|^{p_{s}^{*}-p}\;|u|^{kp}\,dx\leq
C(\Lambda)\|u\|_{L^{kp_{s}^{*}}(\Omega)}^{kp}$ (1.115)

Combining (A) and (1.115), we have

$\displaystyle\|u\|_{L^{kp_{s}^{*}}(\Omega)}^{kp}\leq
C_{1}\frac{k^{p-1}}{(S_{s,p})^{p}}\left[(\Lambda^{1-p}+\Lambda^{p_{s}^{*}-p})\|u\|_{L^{kp}(\Omega)}^{kp}+2C(\Lambda)\|u\|_{L^{kp_{s}^{*}}(\Omega)}^{kp}\right].$
(1.116)

Now by applying Lebesgue dominated convergence theorem in (1.115), we can
choose $\Lambda>1$ large enough such that $\displaystyle
C(\Lambda)<\frac{(S_{s,p})^{p}}{4\tilde{C}(k)^{p-1}}$ and hence, from (1.116),
it follows that

$\displaystyle\|u\|_{L^{kp_{s}^{*}}(\Omega)}\leq\left(C_{*}^{\frac{1}{k}}\right)^{\frac{1}{p}}(k^{\frac{1}{k}})^{\frac{p-1}{p}}\|u\|_{L^{kp}(\Omega)},$
(1.117)

where $\displaystyle C_{*}=\frac{2C_{1}\leavevmode\nobreak\
(\Lambda^{1-p}+\Lambda^{p_{\mu,s}^{*}-p})}{(S_{s})^{p}}>1.$ Next, we start
bootstrap argument on (1.117).
Choose $k=k_{1}:=\frac{p_{s}^{*}}{p}>1$ as the first iteration. Thus, (1.117)
yields that

$\displaystyle\|u\|_{L^{k{{}_{1}}p_{s}^{*}}(\Omega)}\leq\left(C_{*}^{\frac{1}{k{{}_{1}}}}\right)^{\frac{1}{p}}(k_{1}^{\frac{1}{k_{1}}})^{\frac{p-1}{p}}\|u\|_{L^{p_{s}^{*}}(\Omega)}.$
(1.118)

Again by taking $k=k_{2}:=k_{1}\frac{p_{s}^{*}}{p}$ as the second iteration in
(1.117) and then inserting (1.118) in (1.117), we get

$\displaystyle\|u\|_{L^{k_{2}p_{s}^{*}}(\Omega)}$
$\displaystyle\leq\left(C_{*}^{\frac{1}{k_{2}}}\right)^{\frac{1}{p}}\left[(k_{2})^{\frac{1}{k_{2}}}\right]^{\frac{p-1}{p}}\|u\|_{L^{k_{1}p_{s}^{*}}(\Omega)}$
$\displaystyle\leq\left(C_{*}^{\frac{1}{k_{1}}+\frac{1}{k_{2}}}\right)^{\frac{1}{p}}\left[(k_{1})^{\frac{1}{k_{1}}}.(k_{2})^{\frac{1}{k_{2}}}\right]^{\frac{p-1}{p}}\|u\|_{L^{p_{s}^{*}}(\Omega)}.$
(1.119)

In this fashion, taking $k=k_{n}:=k_{n-1}\frac{p_{s}^{*}}{p}$ as the $n^{th}$
iteration and iterating for $n$ times, we obtain

$\displaystyle\|u\|_{L^{k_{n}p_{s}^{*}}(\Omega)}$
$\displaystyle\leq\left(C_{*}^{\frac{1}{k_{n}}}\right)^{\frac{1}{p}}\left[(k_{n})^{\frac{1}{k_{n}}}\right]^{\frac{p-1}{p}}\|u\|_{L^{k_{n-1}p_{s}^{*}}(\Omega)}$
$\displaystyle\leq\left(C_{*}^{\displaystyle\frac{1}{k_{1}}+\frac{1}{k_{2}}\cdots+\frac{1}{k_{n}}}\right)^{\frac{1}{p}}\left[(k_{1})^{\displaystyle\frac{1}{k_{1}}}.(k_{2})^{\displaystyle\frac{1}{k_{2}}}\cdots(k_{n})^{\displaystyle\frac{1}{k_{n}}}\right]^{\frac{p-1}{p}}\|u\|_{L^{p_{s}^{*}}(\Omega)}$
$\displaystyle=\left(C_{*}^{\displaystyle\sum_{j=1}^{n}{\frac{1}{k_{j}}}}\right)^{\frac{1}{p}}\left(\prod_{j=1}^{n}\left(k_{j}^{\sqrt{1/{k_{j}}}}\right)^{\sqrt{1/{k_{j}}}}\right)^{\frac{p-1}{p}}\|u\|_{L^{p_{s}^{*}}(\Omega)},$
(1.120)

where $k_{j}=\left(\frac{p_{s}^{*}}{p}\right)^{j}.$ Since
$\frac{p_{s}^{*}}{p}>1,$ we have $k_{j}^{\displaystyle\sqrt{1/{k_{j}}}}>1,$
for all $j\in\mathbb{N}$ and

$\lim_{j\to+\infty}k_{j}^{\displaystyle\sqrt{1/{k_{j}}}}=1.$

Hence, it follows that there exists a constant $C^{*}>1,$ independent of
$j,n\in\mathbb{N}$ such that $k_{j}^{\displaystyle\sqrt{1/{k_{j}}}}<C^{*}$ and
thus, (A) gives

$\displaystyle\|u\|_{L^{k_{n}p_{s}^{*}}(\Omega)}$
$\displaystyle\leq\left(C_{*}^{\displaystyle\sum_{j=1}^{n}{\frac{1}{k_{j}}}}\right)^{\frac{1}{p}}\left({C^{*}}^{\displaystyle\sum_{j=1}^{n}\sqrt{1/{k_{j}}}}\right)^{\frac{p-1}{p}}\|u\|_{L^{p_{s}^{*}}(\Omega)}.$
(1.121)

Observe that

$\sum_{j=1}^{\infty}{\frac{1}{k_{j}}}=\sum_{j=1}^{n}\left({\frac{p}{p_{s}^{*}}}\right)^{j}=\frac{p/p_{s}^{*}}{1-p/p_{s}^{*}}=\frac{p}{p_{s}^{*}-p}$

and

$\sum_{j=1}^{\infty}{\frac{1}{\sqrt{k_{j}}}}=\sum_{j=1}^{n}{\left(\sqrt{\frac{p}{p_{s}^{*}}}\right)^{j}}=\frac{\sqrt{p}}{\sqrt{p_{s}^{*}}-\sqrt{p}},$

from (1.121), we get that

$\displaystyle\|u\|_{L^{\nu_{n}}(\Omega)}\leq\left(C_{*}\right)^{\displaystyle\frac{1}{p_{s}^{*}-p}}\left(C^{*}\right)^{\displaystyle\frac{p-1}{\sqrt{p}(\sqrt{p_{s}^{*}}-\sqrt{p})}}\|u\|_{L^{p_{s}^{*}}(\Omega)},$
(1.122)

where $\nu_{n}:=k_{n}p_{s}^{*}.$ Note that, $\nu_{n}\to+\infty$ as
$n\to+\infty.$ Therefore, we claim that

$\displaystyle\|u\|_{L^{\infty}(\Omega)}=\|u\|_{L^{\infty}(\Omega)}\leq\left(C_{*}\right)^{\displaystyle\frac{1}{p_{s}^{*}-p}}\left(C^{*}\right)^{\displaystyle\frac{p-1}{\sqrt{p}(\sqrt{p_{s}^{*}}-\sqrt{p})}}\|u\|_{L^{p_{s}^{*}}(\Omega)}.$
(1.123)

Indeed, if not, let us assume
$\|u\|_{L^{\infty}(\Omega)}>C_{3}\|u\|_{L^{p_{s}^{*}}(\Omega)},$ where

$C_{3}=\left(C_{*}\right)^{\displaystyle\frac{1}{p_{s}^{*}-p}}\left(C^{*}\right)^{\displaystyle\frac{p-1}{\sqrt{p}(\sqrt{p_{s}^{*}}-\sqrt{p})}}.$

Then there exists $C_{4}>0$ and a subset $\mathcal{S}$ of $\Omega$ with
$\mathcal{|S|}>0$ such that

$u(x)>C_{3}\|u\|_{L^{p_{s}^{*}}(\Omega)}+C_{4},\text{\leavevmode\nobreak\
\leavevmode\nobreak\ for }x\in\mathcal{S}.$

The above implies

$\displaystyle\displaystyle\liminf_{\nu_{n}\to+\infty}\left(\int_{\Omega}|u(x)|^{\nu_{n}}dx\right)^{\frac{1}{\nu_{n}}}$
$\displaystyle\geq\displaystyle\liminf_{\nu_{n}\to+\infty}\left(\int_{\mathcal{S}}|u(x)|^{\nu_{n}}dx\right)^{\frac{1}{\nu_{n}}}$
$\displaystyle\geq\displaystyle\liminf_{\nu_{n}\to+\infty}\left(C_{3}\|u\|_{L^{p_{s}^{*}}(\Omega)}+C_{4}\right)\left(\mathcal{|S|}\right)^{\frac{1}{\nu_{n}}}$
$\displaystyle=\displaystyle\liminf_{\nu_{n}\to+\infty}\left(C_{3}\|u\|_{L^{p_{s}^{*}}(\Omega)}+C_{4}\right),$

a contradiction to (1.122). Therefore, (1.123) holds and hence, $u\in
L^{\infty}(\Omega).$ ∎

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