# A singular elliptic problem involving fractional $p$-Laplacian and a
discontinuous critical nonlinearity

K. Saoudi†111Basic and Applied Scientific Research Center, Imam Abdulrahman
Bin Faisal University, Saudi Arabia<EMAIL_ADDRESS>Akasmika
Panda222Department of Mathematics, National Institute of Technology Rourkela
<EMAIL_ADDRESS>Debajyoti Choudhuri333Department of Mathematics, National
Institute of Technology Rourkela<EMAIL_ADDRESS>
###### Abstract

In this article, we prove the existence of solutions to a nonlinear nonlocal
elliptic problem with a singualrity and a discontinuous critical nonlinearity
which is given as follows.

$\displaystyle\begin{split}(-\Delta)_{p}^{s}u&=\mu
g(x,u)+\frac{\lambda}{u^{\gamma}}+H(u-\alpha)u^{p_{s}^{*}-1},~{}\text{in}~{}\Omega\\\
u&>0,~{}\text{in}~{}\Omega,\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\end{split}$ (0.1)

where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with Lipschitz
boundary, $s\in(0,1)$, $2<p<\frac{N}{s}$, $\gamma\in(0,1)$, $\lambda,\mu>0$,
$\alpha\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq
0$, $H(a)=1$ if $a>0$ and $p_{s}^{*}=\frac{Np}{N-sp}$ is the fractional
critical Sobolev exponent.
Under suitable assumptions on the function $g$, we prove the existence of
solution to the problem. Furthermore, we show that as $\alpha\rightarrow
0^{+}$, the sequence of solutions of $\eqref{main_prob}$ for each such
$\alpha$ converges to a solution of the problem for which $\alpha=0$.

Keywords: Fractional $p$-Laplacian, Heaviside function, Mountain pass theorem,
Critical exponent, Singularity.

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

##  1 Introduction

We will study the existence of solution to the following nonlinear, nonlocal
problem involving a singularity and a discontinuous critical nonlinearity.

$\displaystyle\begin{split}(-\Delta)_{p}^{s}u&=\mu
g(x,u)+\frac{\lambda}{u^{\gamma}}+H(u-\alpha)u^{p_{s}^{*}-1},~{}\text{in}~{}\Omega\\\
u&>0,~{}\text{in}~{}\Omega,\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\end{split}$ ($P_{\alpha}$)

We impose the hypotheses on $g$ which are as follows.

  * ($g_{1}$)

the function $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a
$N$-measurable function and $g(x,u)=0$ if $u\leq 0$.

  * ($g_{2}$)

there exist $K>0$ and $r\in(p,p_{s}^{*})$ such that $|g(x,u)|\leq
K(1+|u|^{r-1})$ for every $u\geq 0$.

  * ($g_{3}$)

there exists $b>p$ and $v>0$ such that for all $u\geq v$

$0<bG(x,u)=b\int_{0}^{u}g(x,\tau)d\tau\leq u\underline{g}(x,u).~{}\text{refer
Section}~{}\ref{priliminaries}~{}\text{for the definition of $\underline{g}$}$

  * ($g_{4}$)

there exists $\beta>0$ (that will be fixed later) such that $H(u-\beta)\leq
g(x,u)$ uniformly in $\Omega\times(0,\infty)$.

  * ($g_{5}$)

Let $\lambda_{1}$ be the first eigen value of $(-\Delta)_{p}^{s}$ defined in
$\eqref{first}$. Then $\lim\limits_{u\rightarrow
0}\frac{g(x,u)}{u^{p-1}}\leq\lambda_{1}$ uniformly in $\Omega$.

A prototype of $g$ satisfying the asuumptions $(g_{1})-(g_{5})$ is
$H(t-\beta)t^{r-1}/\beta^{r}$.
The problems of type $\eqref{main prob intro}$ having discontinuous
nonlinearities have many applications in free boundary problems of
mathematical physics. For instance, obstacle problem, Elenbaas equations, the
seepage surface problem etc. Refer [3, 11, 12, 13] for further details.
Elliptic problems involving critical and discontinuous nonlinearities can be
treated by different techniques. Amongst these methods, variational methods
for nondifferentiable functionals, dual variational principle, Palais
principle of symmetric criticality for locally Lipschitz functional, lower-
upper solution method, theory of multivalued mappings and global branching are
a few well known techniques. Badiale & Tarantello in [6] studied the following
class of problem using variational methods with lower-upper solution methods.

$\displaystyle\begin{split}(-\Delta)u&=\delta
H(u-\alpha)+u^{2_{s}^{*}-1},~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{on}~{}\partial\Omega.\end{split}$ (1.1)

Here $2^{*}=2N/(N-2)$, $\delta,\alpha>0$ and $H$ is the Heaviside function.
Later, the authors in [2] generalized the work of [6] in $\mathbb{R}^{N}$.
Badiale in [5] proved the existence result for the critical elliptic problem
given by

$\displaystyle\begin{split}(-\Delta)u&=g(u)+u^{2_{s}^{*}-1},~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{on}~{}\partial\Omega,\end{split}$ (1.2)

where $g$ can have discontinuities. The authors in [16] and [18] extended the
result of [5] for a Kirchhoff type problem involving critical Caffarelli-Kohn-
Nirenberg growth and for a Schrödinger-Kirchhoff equation, respectively.
Recently, Santos & Tavares in [17] considered the problem

$\displaystyle\begin{split}\mathcal{L}_{K}u&=g(x,u)+H(u-\alpha)u^{2_{s}^{*}-2}u,~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\\\ u&\geq
0,~{}\text{in}~{}\Omega,\end{split}$ (1.3)

where $2_{s}^{*}=2N/(N-2s)$, $\alpha>0$, $g$ is a discontinuous function and
$\mathcal{L}_{K}$ is a nonlocal operator

$\mathcal{L}_{K}u(x)=\iint_{\mathbb{R}^{2N}}(u(x+y)+u(x-y)-2u(x))K(y)dy.$

They used the nonsmooth version of Mountain pass theorem to investigate the
existence and the behavior of solution for problem $\eqref{prob 3}$. We also
cite [1, 3, 4, 7, 11, 13, 14, 32, 34] and the references therein for readers
to have a glimpse of the problems of the type as in $\eqref{prob
1}-\eqref{prob 3}$.
Inspired by the above works, specifically [5, 6, 11, 12, 17] we analyze our
problem $\eqref{main prob intro}$. The problem $\eqref{main prob intro}$ with
singularity, critical and discontinuous nonlinearities is a new and first work
in the literature, at least to our knowledge. But we find enormous works
dealing with the following class of problems involving singularity and
critical nonlinearity given by

$\displaystyle\begin{split}(-\Delta)_{p}^{s}u&=\frac{\lambda
g_{1}(x)}{u^{\gamma}}+\delta u^{q-1},~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\\\
u&>0,~{}\text{in}~{}\Omega,\end{split}$ (1.4)

where $\lambda,\delta>0$, $q\in(1,p_{s}^{*}]$, $g_{1}>0$ is bounded. Several
techniques like variational method, concentration compactness method, Nehari
manifold method etc. have been applied to study the problems of type
$\eqref{prob 4}$ for both local and nonlocal cases. Refer [15, 21, 22, 23, 24,
25, 27, 28, 30, 31] and the bibliography therein.
The main result of this article is the following.

######  Theorem 1.1.

Let $(g_{1})-(g_{5})$ hold. Then

  1. 1.

there exist $\bar{\alpha},\bar{\lambda},\bar{\mu}>0$ such that for every
$a\in(0,\bar{\alpha})$, every $\lambda\in(0,\bar{\lambda})$ and every
$\mu\in(0,\bar{\mu})$ the problem $\eqref{main prob intro}$ admits at least
one nontrivial weak solution $u_{\alpha}$. Furthermore, the lebesgue measure
of the set $\\{x\in\Omega:u_{\alpha}>\alpha\\}$ is positive

  2. 2.

for any sequence $\alpha_{n}\in(0,\bar{\alpha})$ with $\alpha_{n}\rightarrow
0^{+}$, we have, up to a subsequential level, $u_{\alpha_{n}}\rightarrow
u_{0}$ in $W_{0}^{s,p}(\Omega)$, where $u_{0}$ is a weak solution of the
problem $(P_{0})$, i.e.

$\displaystyle\begin{split}(-\Delta)_{p}^{s}u&=\mu
g(x,u)+\frac{\lambda}{u^{\gamma}}+u^{p_{s}^{*}-1},~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\\\
u&>0,~{}\text{in}~{}\Omega.\end{split}$ ($P_{0}$)

The proof of the main result, i.e Theorem 1.1, has been splitted into two
sections. Section 3 is devoted to the first part of Theorem 1.1, i.e. the
existence of a weak solution $u_{\alpha}$ to $\eqref{main prob intro}$. In
Section 4, we examine the nature of the sequence $(u_{\alpha})$ as
$\alpha\rightarrow 0^{+}$ and prove the second part of Theorem 1.1.

##  2 Nonsmooth critical point theory

Let us fix $0<s<1$, $2<p<\frac{N}{s}$, $\Omega$ be an open and bounded domain
of $\mathbb{R}^{N}$. We denote
$Q=(\mathbb{R}^{N}\times\mathbb{R}^{N})\setminus(\Omega^{c}\times\Omega^{c})$
where $\Omega^{c}=\mathbb{R}^{N}\setminus\Omega$. We define the fractional
Sobolev space by

$\displaystyle
W^{s,p}(\Omega)=\\{u:\mathbb{R}^{N}\rightarrow\mathbb{R}~{}\text{is
measurable}:u|_{\Omega}\in
L^{p}(\Omega),\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dydx<\infty\\}$

equipped with the norm

$\|u\|_{W^{s,p}(\Omega)}=\|u\|_{L^{p}(\Omega)}+\left(\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dydx\right)^{1/p}.$

We further define the space

$W_{0}^{s,p}(\Omega)=\\{u\in W^{s,p}(\Omega):u=0~{}\text{a.e.
in}~{}\mathbb{R}^{N}\setminus\Omega\\}$

and $(W_{0}^{s,p}(\Omega),\|\cdot\|_{W_{0}^{s,p}(\Omega)})$ is a reflexive
Banach space where the fractionl Sobolev norm is given by

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

Given below are a few well known embedding results for the space
$W_{0}^{s,p}(\Omega)$.

######  Theorem 2.1 ([33]).

The following results hold for the fractional Sobolev space
$W_{0}^{s,p}(\Omega)$.

  1. 1.

If $\Omega$ has a continuous boundary, then the embedding
$W_{0}^{s,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact for every
$q\in[1,p_{s}^{*})$.

  2. 2.

The embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{p_{s}^{*}}(\Omega)$ is
continuous.

We now define the best constant $S_{s,p}>0$ given by

$S_{s,p}=\underset{u\in
W_{0}^{s,p}(\Omega)\setminus\\{0\\}}{\inf}\frac{\int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dydx}{(\int_{\Omega}|u|^{p_{s}^{*}}dx)^{\frac{p}{p_{s}^{*}}}}$
(2.1)

and $S_{s,p}$ is well-defined due to Theorem 2.1.

######  Theorem 2.2 ([8], Thoerem 4.1 of [19]).

Let $s\in(0,1)$ and $p>1$. Then the eigenvalue problem

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

possesses a smallest eigenvalue $\lambda_{1}>0$ given by

$\lambda_{1}=\underset{u\in
W_{0}^{s,p}(\Omega):\|u\|_{L^{p}(\Omega)}=1}{\min}\|u\|_{W_{0}^{s,p}(\Omega)}^{p}.$
(2.2)

Let $V$ be a Banach space. Then a functional $J:V\rightarrow\mathbb{R}$ is
said to be locally Lipschitz continuous if for any $u\in V$ there exists an
open neighborhood $N:=N_{u}\subset V$ and some constant $C:=C_{N}>0$ such that

$|J(u_{1})-J(u_{2})|\leq C\|u_{1}=u_{2}\|_{V},~{}u_{1},u_{2}\in V.$

We define the directional derivative of $J$ at $u$ in the direction of $z\in
V$ by

$\tilde{J}(u;z)=\lim\limits_{h\rightarrow 0}\lim\limits_{\xi\downarrow
0}\frac{J(u+h+\xi z)-J(u+h)}{\xi}.$

Thus, $\tilde{J}(u;\cdot)$ is convex, continuous and its subdifferential at
$w\in V$ is the set

$\partial\tilde{J}(u;w)=\\{\nu\in
V^{*}:\tilde{J}(u;z)\geq\tilde{J}(u;w)+\langle\nu,z-w\rangle,~{}z\in V\\}.$

Here $\langle\cdot,\cdot\rangle$ is the duality pair between $V$ and $V^{*}$.
The generalized gradient of $J$ at $u$ is defined by

$\partial J(u)=\\{\nu\in V^{*}:\langle\nu,z\rangle\leq\tilde{J}(u;z),~{}z\in
V\\}$

and is convex and weak∗\- compact. Clearly, $\partial J(u)$ is nonempty and is
the subdifferential of $\partial\tilde{J}(u;0)$ as $\tilde{J}(u;0)=0$.
We say $\bar{u}$ to be a critical point of $J$ if $0\in\partial J(\bar{u})$
and $c\in\mathbb{R}$ to be a critical value of $J$ if $J(\bar{u})=c$ for some
critical point $\bar{u}\in V$. Moreover, if $J$ is a $C^{1}$ functonal then
$\partial J(u)=\\{J^{\prime}(u)\\}$. We now denote

$\Lambda_{J}(u)=\min\\{\|\nu\|_{V^{*}}:\nu\in\partial J(u)\\}.$

The following result is the Mountain Pass Theorem for locally Lipschitz non-
differentiable functional.

######  Theorem 2.3 ([26, 32]).

Let $V$ be a Banach space and $J$ be a locally Lipschitz functional with
$J(0)=0$. Assume that there exists $\rho_{1},\rho_{2}>0$ and $\sigma\in V$
such that

  1. 1.

$J(u)\geq\rho_{2}$, for every $u\in V$; $\|u\|_{V}=\rho_{1}$,

  2. 2.

$J(\sigma)<0$ and $\|\sigma\|_{V}>\rho_{1}$.

Let

$c=\underset{\zeta\in\Gamma}{\inf}\underset{t\in[0,1]}{\max}~{}J(\zeta(t)),~{}\Gamma=\\{\zeta\in
C([0,1];V):\zeta(0)=0,\zeta(1)=\sigma\\}.$

Then $c\geq\rho_{2}$ and there exists a Palais-Smale [(PS)c] sequence
$(u_{n})\subset V$ satisfying

$J(u_{n})\rightarrow c~{}\text{and}~{}\Lambda_{J}(u_{n})\rightarrow 0.$

Moreover, if $J$ satisfies the nonsmooth Palais-Smale [(PS)c] condition, i.e.
every $(PS)_{c}$ sequence has a convergent subsequence, then $c$ is a critical
value of $J$.

######  Proposition 2.4 ([26]).

Let $(u_{n})\subset V$ and $(\theta_{n})\subset V^{*}$ with
$\theta_{n}\in\partial J(u_{n})$. If $u_{n}\rightarrow u$ in $V$ and
$\theta_{n}\overset{*}{\rightharpoonup}\theta$ in $V^{*}$, then
$\theta\in\partial J(u)$.

A function $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is said to be a
$N$-measurable function if for every $u\in W_{0}^{s,p}(\Omega)$ the function
$g(\cdot,u(\cdot)):V\rightarrow\mathbb{R}$ is measurable (refer [11]). Let
$g(x,\cdot)\in L_{loc}^{1}(\Omega)$, then we denote

$\underline{g}(x,u)=\underset{\delta\downarrow
0}{\lim}~{}\underset{|v-u|<\delta}{essinf}~{}g(x,v),~{}\overline{g}(x,u)=\underset{\delta\downarrow
0}{\lim}~{}\underset{|v-u|<\delta}{esssup}~{}g(x,v).$

We now provide the notion of weak solution to $\eqref{main prob intro}$
(influenced by [17]). We say a function $u\in W_{0}^{s,p}(\Omega)$ to be a
weak solution of $\eqref{main prob intro}$ if $u>0$ a.e. in $\Omega$,
$u^{-\gamma}\in L^{1}_{loc}(\Omega)$ and there exist $\eta_{\alpha}\in
L^{\frac{r}{r-1}}(\Omega)$ and $\theta_{\alpha}\in
L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega)$ such that for every $\varphi\in
W_{0}^{s,p}(\Omega)$

$\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y)}{|x-y|^{N+sp}}(\varphi(x)-\varphi(y))dxdy-\mu\int_{\Omega}\eta_{\alpha}\varphi
dx-\int_{\Omega}\theta_{\alpha}\varphi
dx-\lambda\int_{\Omega}\frac{\varphi}{u^{\gamma}}dx=0.$ (2.3)

Here $\eta_{\alpha}(x)\in[\underline{g}(x,u(x)),\overline{g}(x,u(x))]$ and
$\theta_{\alpha}(x)\in[\underline{f}_{\alpha}(u(x)),\overline{f}_{\alpha}(u(x))]$
a.e. in $\Omega$ with $f_{\alpha}(t)=H(t-\alpha)t^{p_{s}^{*}-1}$.

##  3 Existence result for $\eqref{main prob intro}$ \- Proof of Theorem 1.1
(1)

The energy functional corresponding to the problem $\eqref{main prob intro}$
is $J_{\alpha}^{0}:W_{0}^{s,p}(\Omega)\rightarrow\mathbb{R}$ defined by

$J_{\alpha}^{0}(u)=\frac{1}{p}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy-\frac{\lambda}{1-\gamma}\int_{\Omega}(u^{+})^{1-\gamma}-\mu\int_{\Omega}G(x,u)-\int_{\Omega}F_{\alpha}(u),$

where $G(x,u)=\int_{0}^{u}g(x,\tau)d\tau$,
$F_{\alpha}(u)=\int_{0}^{u}f_{\alpha}(\tau)d\tau$ with
$f_{\alpha}(u)=H(u-\alpha)u^{p_{s}^{*}-1}$. Let us denote

$I_{\lambda}^{0}(u)=\frac{1}{p}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy-\frac{\lambda}{1-\gamma}\int_{\Omega}(u^{+})^{1-\gamma},~{}I_{\mu}(u)=\int_{\Omega}G(x,u),~{}I_{\alpha}(u)=\int_{\Omega}F_{\alpha}(u).$
(3.1)

We provide some properties of the functionals $I_{\mu}$ and $I_{\alpha}$ in
the following lemma and these results can be proved by following the arguments
of Lemma 3.1 of [16].

######  Lemma 3.1.

The functionals $I_{\mu}:L^{r}(\Omega)\rightarrow\mathbb{R}$,
$I_{\alpha}:L^{p_{s}^{*}}(\Omega)\rightarrow\mathbb{R}$ are locally Lipschitz
functionals and they satisfy the followings.

  1. 1.

$\partial I_{\mu}(u)\subset[\underline{g}(x,u(x)),\overline{g}(x,u(x))]$ a.e.
in $\Omega$.

  2. 2.

$\partial
I_{\alpha}(u)\subset[\underline{f}_{\alpha}(u(x)),\overline{f}_{\alpha}(u(x))]$
a.e. in $\Omega$.

  3. 3.

$\displaystyle\begin{split}[\underline{f}_{\alpha}(u),\overline{f}_{\alpha}(u)]=&\begin{cases}\\{0\\},&~{}\text{if}~{}u<a\\\
[0,u^{p_{s}^{*}-1}],&~{}\text{if}~{}u=a\\\
\\{u^{p_{s}^{*}-1}\\},&~{}\text{if}~{}u>a.\end{cases}\end{split}$ (3.2)

######  Remark 3.2.

The inclusion (1) of Lemma 3.1 imply that if $\eta\in\partial I_{\mu}(u)$ then
$\eta\in L^{\frac{r}{r-1}}(\Omega)$ and
$\eta(x)\subset[\underline{g}(x,u(x)),\overline{g}(x,u(x))]$ a.e. in $\Omega$.
The same argument follows for $(2)$.

The functional $I_{\lambda}^{0}$ is not of class $C^{1}$ due to the presesnce
of a singularity. To tackle this issue, we follow the truncation technique.
Let us consider the problem

$\displaystyle\begin{split}(-\Delta)_{p}^{s}u&=\frac{\lambda}{u^{\gamma}},~{}\text{in}~{}\Omega\\\
u&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\\\
u&>0,~{}\text{in}~{}\Omega.\end{split}$ (3.3)

According to Canino et al. in [10], we have the existence result for
$\eqref{subsolution}$ as given below.

######  Lemma 3.3.

Let $\gamma\in(0,1)$ and $\lambda>0$. Then $\eqref{subsolution}$ admits a
unique nontrivial weak solution $\underline{u}_{\lambda}$ in
$W_{0}^{s,p}(\Omega)$ such that for every $\omega\subset\subset\Omega$ we have
$\underset{\omega}{ess~{}inf}~{}\underline{u}_{\lambda}>0$.

We now define

$\displaystyle\begin{split}\psi(x,t)=&\begin{cases}t^{-\gamma},&~{}\text{if}~{}t>\underline{u}_{\lambda}\\\
\underline{u}_{\lambda}^{-\gamma},&~{}\text{if}~{}t\leq\underline{u}_{\lambda},\end{cases}\end{split}$
(3.4)

where $\underline{u}_{\lambda}$ is the unique solution to
$\eqref{subsolution}$. Further, define a function
$I_{\lambda}:W_{0}^{s,p}(\Omega)\rightarrow\mathbb{R}$ by

$I_{\lambda}(u)=\frac{1}{p}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy-\lambda\int_{\Omega}\Psi(x,u)dx$
(3.5)

where $\Psi(x,t)=\int_{0}^{t}\psi(x,\tau)d\tau$.
We now consider the following cutoff problem.

$\displaystyle\begin{split}(-\Delta)_{p}^{s}w&=\mu
g(x,w)+\lambda\psi(x,w)+H(w-\alpha)w^{p_{s}^{*}-1},~{}\text{in}~{}\Omega\\\
w&=0,~{}\text{in}~{}\mathbb{R}^{N}\setminus\Omega,\\\
w&>0,~{}\text{in}~{}\Omega.\end{split}$ (3.6)

A function $w\in W_{0}^{s,p}(\Omega)$ is said to be a weak solution of
$\eqref{main prob help}$ if $w>0$ a.e. in $\Omega$ and there exist
$\eta_{\alpha}\in L^{\frac{r}{r-1}}(\Omega)$ and $\theta_{\alpha}\in
L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega)$ such that for every $\varphi\in
W_{0}^{s,p}(\Omega)$

$\int_{\mathbb{R}^{2N}}\frac{|w(x)-w(y)|^{p-2}(w(x)-w(y)}{|x-y|^{N+sp}}(\varphi(x)-\varphi(y))dxdy-\mu\int_{\Omega}\eta_{\alpha}\varphi
dx-\int_{\Omega}\theta_{\alpha}\varphi dx-\lambda\int_{\Omega}\psi(x,w)\varphi
dx=0,$ (3.7)

where $\eta_{\alpha}(x)\in[\underline{g}(x,w(x)),\overline{g}(x,w(x))]$ and
$\theta_{\alpha}(x)\in[\underline{f}_{\alpha}(w(x)),\overline{f}_{\alpha}(w(x))]$
a.e. in $\Omega$.
The associated functional of $\eqref{main prob help}$ is
$J_{\alpha}:W_{0}^{s,p}(\Omega)\rightarrow\mathbb{R}$ defined by

$J_{\alpha}(w)=I_{\lambda}(w)-I_{\mu}(w)-I_{\alpha}(w).$ (3.8)

The functionals $I_{\lambda},I_{\mu}$ and $I_{\alpha}$ are given in $\eqref{I,
I alpha}$ and $\eqref{I lambda}$. By Lemma 3.1, the functional $J_{\alpha}$ is
a locally Lipschitz functional on $W_{0}^{s,p}(\Omega)$ and thus using
Proposition 1.3.12 and 1.3.13 of [20] we obtain

$\partial J_{\alpha}(w)\subset\\{I_{\lambda}^{\prime}(w)\\}-\partial
I_{\mu}(w)-\partial I_{\alpha}(w),~{}\text{for all}~{}w\in
W_{0}^{s,p}(\Omega).$ (3.9)

It is easy to check that if $w$ is a weak solution to $\eqref{main prob help}$
with $w\geq\underline{u}_{\lambda}$ a.e. in $\Omega$, then $w$ is also a weak
solution to $\eqref{main prob intro}$. Hence, with the help of a comparison
principle and non-smooth variational approach we prove our main result.

######  Lemma 3.4.

Any sequence $(w_{n})\subset W_{0}^{s,p}(\Omega)$ satisfying
$J_{\alpha}(w_{n})\rightarrow c$ and $\Lambda_{J_{\alpha}}(w_{n})\rightarrow
0$ is bounded in $W_{0}^{s,p}(\Omega)$.

###### Proof.

Suppose $(v_{n})\subset(W_{0}^{s,p}(\Omega))^{\prime}$ is a sequence with
$\|v_{n}\|_{(W_{0}^{s,p}(\Omega))^{\prime}}=\Lambda_{J_{\alpha}}(w_{n})$,
where $(W_{0}^{s,p}(\Omega))^{\prime}$ is the dual of $W_{0}^{s,p}(\Omega)$.
This implies $v_{n}\in\partial J_{\alpha}(w_{n})$. From
$\eqref{differential}$, there exist $\eta_{n}\in\partial I_{\mu}(w_{n})$ and
$\theta_{n}\in\partial I_{\alpha}(w_{n})$ that satisfy

$\displaystyle\langle v_{n},\varphi\rangle$
$\displaystyle=\int_{\mathbb{R}^{2N}}\frac{|w_{n}(x)-w_{n}(y)|^{p-2}(w_{n}(x)-w_{n}(y)}{|x-y|^{N+sp}}(\varphi(x)-\varphi(y))dxdy-\lambda\int_{\Omega}\psi(x,w_{n})\varphi
dx$
$\displaystyle~{}~{}~{}~{}-\mu\langle\eta_{n},\varphi\rangle-\langle\theta_{n},\varphi\rangle,~{}\forall\varphi\in
W_{0}^{s,p}(\Omega).$ (3.10)

By $\eqref{J alpha}$, $\eqref{representation}$ and Lemma 3.1, we get

$\displaystyle J_{\alpha}(w_{n})-\frac{1}{b}\langle v_{n},w_{n}\rangle=$
$\displaystyle\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}+\frac{\lambda}{b}\int_{\Omega}\psi(x,w_{n})w_{n}-\lambda\int_{\Omega}\Psi(x,w_{n})$
$\displaystyle~{}~{}~{}~{}+\mu\int_{\Omega}\left(\frac{1}{b}\eta_{n}w_{n}-G(x,w_{n})\right)+\int_{\Omega}\left(\frac{1}{b}\theta_{n}w_{n}-F_{\alpha}(w_{n})\right).$
(3.11)

According to $(g_{3})$,

$\int_{\Omega}\left(\frac{1}{b}\eta_{n}w_{n}-G(x,w_{n})\right)\geq\int_{\\{w_{n}\leq
v\\}}\left(\frac{1}{b}\eta_{n}w_{n}-G(x,w_{n})\right)$ (3.12)

and using $(g_{1})$, $(g_{2})$ we obtain the following uniform bound.

$\left|\int_{\\{w_{n}\leq
v\\}}\left(\frac{1}{b}\eta_{n}w_{n}-G(x,w_{n})\right)\right|\leq
K\left[\left(\frac{1}{b}+1\right)v+\left(\frac{1}{b}+\frac{1}{r}\right)v^{r}\right]|\Omega|=C_{1},~{}\forall
n\in\mathbb{N}.$ (3.13)

From Lemma 3.1 (3), we rewrite the last term of $\eqref{star}$ as follows.

$\int_{\Omega}\frac{1}{b}\theta_{n}w_{n}-F_{\alpha}(w_{n})=\left(\frac{1}{b}-\frac{1}{p_{s}^{*}}\right)\int_{\\{w_{n}>\alpha\\}}|w_{n}|^{p_{s}^{*}}+\int_{\\{w_{n}=\alpha\\}}\frac{1}{b}\theta_{n}w_{n}+\frac{1}{p_{s}^{*}}\int_{\\{w_{n}>\alpha\\}}\alpha^{p_{s}^{*}}.$
(3.14)

We have used the fact that
$F_{\alpha}(t)=\chi_{\\{t\geq\alpha\\}}\frac{1}{p_{s}^{*}}(|t|^{p_{s}^{*}}-\alpha^{p_{s}^{*}})$
for all $t\in\mathbb{R}$ to obatin $\eqref{star 3}$.
On combining $\eqref{star}-\eqref{star 3}$ we get

$\displaystyle J_{\alpha}(w_{n})-\frac{1}{b}\langle v_{n},w_{n}\rangle$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}\|w_{n}\|_{L^{1-\gamma}(\Omega)}^{1-\gamma}-\mu
C_{1}+\left(\frac{1}{b}-\frac{1}{p_{s}^{*}}\right)\int_{\\{w_{n}>\alpha\\}}|w_{n}|^{p_{s}^{*}}.$
(3.15)

Using Theorem 2.1 and $\eqref{best}$ in $\eqref{star 4}$ we establish the
following.

$\displaystyle J_{\alpha}(w_{n})-\frac{1}{b}\langle v_{n},w_{n}\rangle$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\|w_{n}\|_{W_{0}^{s,p}(\Omega)}^{1-\gamma}-\mu
C_{1}$
$\displaystyle~{}~{}~{}~{}+\left(\frac{1}{b}-\frac{1}{p_{s}^{*}}\right)\int_{\\{w_{n}>\alpha\\}}|w_{n}|^{p_{s}^{*}}.$
(3.16)

We already have $J_{\alpha}(w_{n})=c+o_{n}(1)$ and
$\|v_{n}\|_{(W_{0}^{s,p}(\Omega))^{\prime}}=o_{n}(1)$. Thus,

$\displaystyle J_{\alpha}(w_{n})-\frac{1}{b}\langle v_{n},w_{n}\rangle$
$\displaystyle\leq|J_{\alpha}(w_{n})|+\frac{1}{b}\|v_{n}\|_{(W_{0}^{s,p}(\Omega))^{\prime}}\|w_{n}\|_{W_{0}^{s,p}(\Omega)}$
$\displaystyle\leq c+1+\|w_{n}\|_{W_{0}^{s,p}(\Omega)}+o_{n}(1).$ (3.17)

Since $p<b<p_{s}^{*}$ and $\alpha>0$, from $\eqref{star 4-4}$ and $\eqref{star
5}$, we obtain

$c+1+\|w_{n}\|_{W_{0}^{s,p}(\Omega)}+o_{n}(1)\geq\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\|w_{n}\|_{W_{0}^{s,p}(\Omega)}^{1-\gamma}-\mu
C_{1}.$ (3.18)

With the consideration of the above inequality $\eqref{star 6}$, we conclude
that $(w_{n})$ is a bounded sequence in $W_{0}^{s,p}(\Omega)$. ∎

######  Proposition 3.5.

Let $(w_{n})\subset W_{0}^{s,p}(\Omega)$ be a non-smooth (PS)c sequence such
that $J_{\alpha}(w_{n})\rightarrow c$ and
$\Lambda_{J_{\alpha}}(w_{n})\rightarrow 0$ with

$c<\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-\mu
C_{1}=c^{*},$ (3.19)

where $S_{s,p}$ is defined in $\eqref{best}$ and $C_{1}$ is given in
$\eqref{star 2}$. Then $J_{\alpha}$ satisfies the non-smooth (PS)c condition,
i.e. $(w_{n})$ admits a strongly convergent subsequence. Further, $c^{*}>0$
for a sufficiently small $\mu$.

###### Proof.

According to Lemma 3.4, the (PS)c sequence $(w_{n})$ is bounded in
$W_{0}^{s,p}(\Omega)$. Let $v_{n},\eta_{n}$ and $\theta_{n}$ are same as used
in the proof of Lemma 3.4. Then by $(g_{2})$ and Lemma 3.1, we establish that
$(\eta_{n})$ and $(\theta_{n})$ are bounded in $L^{\frac{r}{r-1}}(\Omega)$ and
$L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega)$, respectively. Hence, up to a
subsequence,

$w_{n}\rightharpoonup
w~{}\text{in}~{}W_{0}^{s,p}(\Omega),~{}w_{n}(x)\rightarrow w(x)~{}\text{a.e.
in}~{}\Omega,\eta_{n}\overset{*}{\rightharpoonup}\eta~{}\text{in}~{}L^{\frac{r}{r-1}}(\Omega)~{}\text{and}~{}\theta_{n}\overset{*}{\rightharpoonup}\theta~{}\text{in}~{}L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega).$
(3.20) $\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}\rightarrow M.$ (3.21)

If $M=0$, then it implies $w_{n}\rightarrow w$ in $W_{0}^{s,p}(\Omega)$ as
$n\rightarrow\infty$ and hence the proof. Thus, we assume $M>0$. Noting that
$\eqref{star 7}$ indicates

$\int_{\Omega}|w_{n}|^{p_{s}^{*}-1}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}\varphi
dx\rightarrow\int_{\Omega}|w|^{p_{s}^{*}-1}\chi_{\\{w>\alpha\\}}\varphi
dx,~{}\forall~{}\varphi\in L^{p_{s}^{*}}(\Omega).$

Then, by Brézis-Lieb Lemma (see [9], Theorem 1), we have

$\|w_{n}\|_{W_{0}^{s,p}(\Omega)}^{p}=\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}+\|w\|_{W_{0}^{s,p}(\Omega)}^{p}+o_{n}(1),$
(3.22)
$\int_{\Omega}|w_{n}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}|^{p_{s}^{*}}=\int_{\Omega}|w_{n}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}-w\chi_{\\{w>\alpha\\}}|^{p_{s}^{*}}+\int_{\Omega}|w\chi_{\\{w_{n}>\alpha\\}}|^{p_{s}^{*}}.$
(3.23)

Since $w_{n}\rightarrow w$ strongly in $L^{q}(\Omega)$ for any $1\leq
q<p_{s}^{*}$ and
$\int_{\Omega}\psi(x,w_{n})<\int_{\Omega}\frac{1}{\underline{u}_{\lambda}^{\gamma}}$
(see the definition of $\psi$ given in $\eqref{defn}$), we obtain

$\left|\int_{\Omega}\psi(x,w_{n})(w_{n}-w)\right|\leq o_{n}(1).$ (3.24)

Using Lemma 3.1, $(g_{2})$ and $\eqref{starr 2}-\eqref{starr 4}$ we obtain

$\displaystyle o_{n}(1)$ $\displaystyle=\langle v_{n},w_{n}-w\rangle$
$\displaystyle=\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}-\int_{\mathbb{R}^{2N}}\frac{|w_{n}(x)-w_{n}(y)|^{p-2}(w_{n}(x)-w_{n}(y))}{|x-y|^{N+2s}}(w(x)-w(y))dxdy$
$\displaystyle~{}~{}~{}~{}-\int_{\Omega}\psi(x,w_{n})(w_{n}-w)-\int_{\Omega}\theta_{n}(w_{n}-w)+o_{n}(1)$
$\displaystyle=\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}-\int_{\Omega}|w_{n}|^{p_{s}^{*}-1}(w_{n}-w)\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}dx+o_{n}(1)$
$\displaystyle=\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}-\int_{\Omega}|w_{n}|^{p_{s}^{*}}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}dx+\int_{\Omega}|w|^{p_{s}^{*}}\chi_{\\{w>\alpha\\}}dx+o_{n}(1)$
$\displaystyle=\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}-\int_{\Omega}\left|w_{n}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}-w\chi_{\\{w>\alpha\\}}\right|^{p_{s}^{*}}dx+o_{n}(1).$

This implies

$\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}=\int_{\Omega}\left|w_{n}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}-w\chi_{\\{w>\alpha\\}}\right|^{p_{s}^{*}}dx+o_{n}(1).$
(3.25)

From $\eqref{best}$, $\eqref{starr 1}$, $\eqref{starr 2}$ and $\eqref{starr
5}$ we have

$\displaystyle S_{s,p}$
$\displaystyle\leq\frac{\|w_{n}-w\|_{W_{0}^{s,p}(\Omega)}^{p}}{\left(\int_{\Omega}\left|w_{n}\chi_{\\{w_{n}>\alpha-\frac{1}{n}\\}}-w\chi_{\\{w>\alpha\\}}\right|^{p_{s}^{*}}dx\right)^{p/p^{*}_{s}}}$
$\displaystyle\leq M^{\frac{sp}{N}}+o_{n}(1)$ (3.26)

and hence $M>S_{s,p}^{\frac{N}{sp}}+o_{n}(1)$. According to $\eqref{star 4}$
and the fact that $p<b<p_{s}^{*}$, we get

$\displaystyle J_{\alpha}(w_{n})-\frac{1}{b}\langle v_{n},w_{n}\rangle$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}\|w_{n}\|_{L^{1-\gamma}(\Omega)}^{1-\gamma}-\mu
C_{1},$ (3.27)

where $b$ is given in $(g_{3})$ and $C_{1}$ is obtained in $\eqref{star 2}$
which is independent of $\alpha$. Since $J_{\alpha}(w_{n})=c+o_{n}(1)$ and
$\|v_{n}\|_{(W_{0}^{s,p}(\Omega))^{\prime}}=o_{n}(1)$, using $\eqref{star
7}-\eqref{starr 2}$, $\eqref{starr 6}$, $\eqref{starr 7}$, Theorem 2.1,
$\eqref{best}$ and Young’s inequality we obtain

$\displaystyle c$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{b}\right)(M+\|w\|_{W_{0}^{s,p}(\Omega)}^{p})-\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\|w\|_{W_{0}^{s,p}(\Omega)}^{1-\gamma}-\mu
C_{1}+o_{n}(1)$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{b}\right)(M+\|w\|_{W_{0}^{s,p}(\Omega)}^{p})-\left(\frac{1}{p}-\frac{1}{b}\right)\|w\|_{W_{0}^{s,p}(\Omega)}^{p}$
$\displaystyle~{}~{}~{}~{}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-\mu
C_{1}+o_{n}(1)$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-\mu
C_{1}+o_{n}(1)$ $\displaystyle=c^{*}+o_{n}(1).$ (3.28)

The above inequality $\eqref{starr 8}$ is a contradiction to $\eqref{label}$.
Thus, $M=0$ and $w_{n}\rightarrow w$ strongly in $W_{0}^{s,p}(\Omega)$.
Moreover, $c^{*}>0$ for a sufficiently small $\mu>0$. ∎

In the next lemma, we prove the functional $J_{\alpha}$ satisfies all the
hypotheses of Theorem 2.3 with a suitable choice of $\beta$ given in
$(g_{4})$.

######  Lemma 3.6.

Let $(g_{1})-(g_{5})$ hold. Then there exist $\rho_{1}>0$,
$\bar{\mu}\in(0,1)$, $\bar{\lambda}=\bar{\lambda}(\rho_{1})>0$, $\rho_{2}>0$,
$m^{*}>0$ and $\sigma\in W_{0}^{s,p}(\Omega)$ such that for every $\alpha>0$,
every $\lambda\in(0,\bar{\lambda}]$ and every $\mu\in(0,\bar{\mu}]$ we have

  1. 1.

$\underset{m\in[0,m^{*}]}{\sup}J_{\alpha}(m\sigma)<\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-\mu
C_{1}$ where $S_{s,p}$ is defined in $\eqref{best}$ and $C_{1}$ is given in
$\eqref{star 2}$.

  2. 2.

$J_{\alpha}(w)\geq\rho_{2}$ for every $w\in W_{0}^{s,p}(\Omega)$,
$\|w\|_{W_{0}^{s,p}(\Omega)}=\rho_{1}$, where $\rho_{1}$ and $\rho_{2}$ are
independent of $\alpha$.

  3. 3.

$J_{\alpha}(m^{*}\sigma)<0$ for
$\|m^{*}\sigma\|_{W_{0}^{s,p}(\Omega)}>\rho_{1}$.

###### Proof.

Let us fix $\sigma\in W_{0}^{s,p}(\Omega)$ with $\sigma>0$ in $\Omega$ and
$\|\sigma\|_{W_{0}^{s,p}(\Omega)}=1$. By the hypothesis $(g_{4})$ we can write

$J_{\alpha}(m\sigma)=\frac{1}{p}m^{p}-m\int_{\\{m\sigma>\beta\\}}\sigma
dx+\beta\left|\\{m\sigma>\beta\\}\right|,~{}\forall~{}m\geq 0.$ (3.29)

It is easy to check that there exist $\bar{\mu}\in(0,1)$ and $\lambda^{*}>0$
such that for any $\lambda\in(0,\lambda^{*}]$ and $\mu\in(0,\bar{\mu}]$ we
have

$\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-{\mu}C_{1}>0.$

Hence, we choose $m^{*}>0$ such that

$\displaystyle\frac{1}{p}(m^{*})^{p}-m^{*}\int_{\\{m\sigma>\beta\\}}\sigma
dx<0$ (3.30)

and

$\displaystyle\frac{(m^{*})^{p}}{p}<\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-{\mu}C_{1}.$
(3.31)

Since $m^{*}$ does not depend on $\beta$, we get

$\int_{\Omega}\sigma dx=\int_{\\{m^{*}\sigma>\beta\\}}\sigma dx+o_{\beta}(1).$
(3.32)

Combining $\eqref{starrr 1}-\eqref{starrr 4}$ we choose $\beta>0$ very small
such that $J_{\alpha}(m^{*}\sigma)<0$ for every $\sigma>0$. This proves $(3)$
and also $(1)$.
According to $(g_{1})$, $(g_{5})$ and the fact that $J_{\alpha}(0)=0$, there
exist $\overline{K}$ and $c_{\overline{K}}>0$ (independent of $\alpha$) such
that $\overline{K}<\lambda_{1}$ and

$\left|g(x,t)\right|\leq\overline{K}|t|^{p-1}+c_{\overline{K}}|t|^{r-1},~{}\forall~{}t\in\mathbb{R}.$

Thus, from $\eqref{first}$ for every $w\in W_{0}^{s,p}(\Omega)$

$\int_{\Omega}G(x,w)dx\leq\frac{\overline{K}}{p\lambda_{1}}\iint_{\mathbb{R}^{2N}}\frac{|w(x)-w(y)|^{p}}{|x-y|^{N+sp}}dxdy+\frac{c_{\overline{K}}}{r}\int_{\Omega}|w|^{r}dx.$
(3.33)

Hence,

$J_{\alpha}(w)\geq\frac{1}{p}\left(1-\frac{\mu\overline{K}}{\lambda_{1}}\right)\|w\|^{p}_{W_{0}^{s,p}(\Omega)}-C_{1}c_{\overline{K}}\mu\|w\|^{r}_{W_{0}^{s,p}(\Omega)}-C_{2}\|w\|^{p_{s}^{*}}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\|w\|^{1-\gamma}_{W_{0}^{s,p}(\Omega)},$
(3.34)

for some constant $C_{1},C_{2}>0$ independent of $\alpha$. Since
$1-\gamma<1<p<r<p_{s}^{*}$, the function

$h(t)=\frac{1}{p}\left(1-\frac{\mu\overline{K}}{\lambda_{1}}\right)t^{p-1+\gamma}-C_{1}c_{\overline{K}}\mu
t^{r-1+\gamma}-C_{2}t^{p_{s}^{*}-1+\gamma},~{}t\in[0,1]$

admits a maximum at some $\rho_{1}\in(0,1]$ small enough, i.e.
$\underset{t\in[0,1]}{\max}h(t)=h(\rho_{1})>0$. Therefore, let

$\lambda^{**}=\frac{(1-\gamma)S_{s,p}^{\frac{1-\gamma}{p}}}{2|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}}h(\rho_{1}),$

then for every $w\in W_{0}^{s,p}(\Omega)$ with
$\|w\|_{W_{0}^{s,p}(\Omega)}=\rho_{1}\leq 1$ and for every
$\lambda\in(0,\lambda^{**}]$, we have
$J_{\alpha}(w)\geq\rho_{1}^{1-\gamma}h(\rho_{1})/2=\rho_{2}$. Let
$\bar{\lambda}=\min\\{\lambda^{*},\lambda^{**}\\}$. Then with
$\lambda\in(0,\bar{\lambda})$ and $\mu\in(0,\bar{\mu})$, we conclude the
proof. ∎

######  Proposition 3.7.

Let $(g_{1})-(g_{5})$ are satisfied. Then there exist
$\bar{\alpha},\bar{\lambda},\bar{\mu}>0$ such that for every
$a\in(0,\bar{\alpha})$, every $\lambda\in(0,\bar{\lambda})$ and every
$\mu\in(0,\bar{\mu})$ the problem $\eqref{main prob intro}$ admits at least
one nontrivial weak solution $u_{\alpha}$. Furthermore, the lebesgue measure
of the set $\\{x\in\Omega:u_{\alpha}>\alpha\\}$ is positive.

###### Proof.

Let

$c_{\alpha}=\underset{\zeta\in\Gamma}{\inf}~{}\underset{t\in[0,1]}{\max}~{}J_{\alpha}(\zeta(t))~{}\text{and}~{}\Gamma=\\{\zeta\in
C([0,1];W_{0}^{s,p}(\Omega)):\zeta(0)=0,\zeta(1)=m^{*}\sigma\\},$

where $m^{*}$, $\sigma$, $\rho_{1}$, $\rho_{2}$, $\bar{\lambda}$, $\bar{\mu}$
are obtained in Lemma 3.6. Since $J_{\alpha}$ satisfy the hypothese of Theorem
2.3 (refer Lemma 3.6), we guarantee the existence of a sequence
$(w_{n})\subset W_{0}^{s,p}(\Omega)$ that satisfy
$J_{\alpha}(w_{n})=c_{\alpha}+o_{n}(1)$ and
$\Lambda_{J_{\alpha}}(w_{n})=o_{n}(1)$. By $(1)$ and $(2)$ of Lemma 3.6, we
also have

$\rho_{2}\leq
c_{\alpha}<\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-{\mu}C_{1},~{}\forall~{}\alpha>0.$
(3.35)

From Proposition 3.5 there exists $w_{\alpha}\in W_{0}^{s,p}(\Omega)$ such
that, up to a subsequence, $w_{n}\rightarrow w_{\alpha}$ in
$W_{0}^{s,p}(\Omega)$ as $n\rightarrow\infty$. This implies
$J_{\alpha}(w_{\alpha})=c_{\alpha}$ and $0\in\partial J_{\alpha}(w_{\alpha})$.
Thus, by $\eqref{differential}$ and Lemma 3.1, there exist $\eta_{\alpha}\in
L^{\frac{r}{r-1}}(\Omega)$ and $\theta_{\alpha}\in
L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega)$ such that

$\int_{\mathbb{R}^{2N}}\frac{|w_{\alpha}(x)-w_{\alpha}(y)|^{p-2}(w_{\alpha}(x)-w_{\alpha}(y)}{|x-y|^{N+sp}}(\varphi(x)-\varphi(y))dxdy=\lambda\int_{\Omega}\psi(x,w_{\alpha})\varphi
dx+\mu\langle\eta_{\alpha},\varphi\rangle+\langle\theta_{\alpha},\varphi\rangle,$
(3.36)

for every $\varphi\in W_{0}^{s,p}(\Omega)$, where
$\eta_{\alpha}(x)\in[\underline{g}(x,w_{\alpha}(x)),\overline{g}(x,w_{\alpha}(x))]\geq
0$ and
$\theta_{\alpha}(x)\in[\underline{f}(w_{\alpha}(x)),\overline{g}(w_{\alpha}(x))]\geq
0$ a.e. in $\Omega$. According to the strong maximum principle (Lemma 2.3,
[29]) we have $w_{\alpha}>0$ a.e. in $\Omega$.
This proves that $w_{\alpha}$ is a weak solution to $\eqref{main prob help}$.
By the weak comparison principle for fractional $p$-Laplacian (Lemma 3.1,
[22]), we conclude that $\underline{u}_{\lambda}\leq w_{\alpha}$ a.e. in
$\Omega$. This implies $\psi(x,w_{\alpha})=w_{\alpha}^{-\gamma}$ a.e. in
$\Omega$ and $w_{\alpha}=u_{\alpha}$ is a weak solution to $\eqref{main prob
intro}$.
The next claim is to prove that the set
$\\{x\in\Omega:w_{\alpha}(x)>\alpha\\}$ has positive lebesgue measure in
$\mathbb{R}^{N}$. We prove this claim by method of contradiction. For this let
us assume that the set $\\{x\in\Omega:w_{\alpha}(x)>\alpha\\}$ is of zero
lebesgue measure in $\mathbb{R}^{N}$. Thus, $w_{\alpha}(x)\leq\alpha$ a.e. in
$\Omega$.
From Lemma 3.1, $\eqref{starrr 5}$ and $\eqref{starrr 8}$ we obtain

$\displaystyle\|w_{\alpha}\|_{W_{0}^{s,p}(\Omega)}^{p}$
$\displaystyle=\mu\int_{\Omega}\eta_{\alpha}w_{\alpha}+\int_{\\{w_{\alpha}=\alpha\\}}\theta_{\alpha}w_{\alpha}+\lambda\int_{\Omega}\psi(x,w_{\alpha})w_{\alpha}$
$\displaystyle\leq\mu\int_{\Omega}(\overline{K}\alpha^{p}+c_{\overline{K}}\alpha^{r})dx+\int_{\\{w_{\alpha}=\alpha\\}}\alpha^{p_{s}^{*}}+\lambda\int_{\Omega}\alpha^{1-\gamma}.$

Since $J_{\alpha}(w_{\alpha})=c_{\alpha}$ by Lemma 3.6 and $\eqref{starrr 7}$,
for $\alpha>0$ small enough, we establish

$p\rho_{2}\leq[\mu(\overline{K}+c_{\overline{K}})+1+\lambda]|\Omega|\alpha^{1-\gamma}.$

This contradicts the fact that $\rho_{2}$ is independent of $\alpha$. Thus,
there exists $\bar{\alpha}>0$ small such that for any
$\alpha\in(0,\bar{\alpha})$ the set $\\{x\in\Omega:w_{\alpha}(x)>\alpha\\}$
has positive lebesgue measure. ∎

##  4 Proof of Theorem 1.1 (2)

Let $u_{\alpha}$ be a nontrivial weak solution to $\eqref{main prob intro}$
given in Proposition 3.7. In this section, we prove the second part of Theorem
1.1, i.e. we examine the nature of $(u_{\alpha})$ as $\alpha\rightarrow
0^{+}$.
Consider the functional $J_{0}^{0}:W_{0}^{s,p}(\Omega)\rightarrow\mathbb{R}$
associated to $\eqref{main prob 0}$ defined by

$J_{0}^{0}(u)=\frac{1}{p}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy-\frac{\lambda}{1-\gamma}\int_{\Omega}u^{1-\gamma}-\mu\int_{\Omega}G(x,u)-\frac{1}{p_{s}^{*}}\int_{\Omega}(u^{+})^{p_{s}^{*}},~{}\forall~{}u\in
W_{0}^{s,p}(\Omega).$

Let us define

$c_{0}=\underset{\zeta\in\Gamma}{\inf}~{}\underset{t\in[0,1]}{\max}~{}J_{0}^{0}(\zeta(t))~{}\text{and}~{}\Gamma=\\{\zeta\in
C([0,1];W_{0}^{s,p}(\Omega)):\zeta(0)=0,\zeta(1)=m^{*}\sigma\\},$ (4.1)

where $m^{*},\sigma$ as obtained in Lemma 3.6.

######  Lemma 4.1.

$\lim\limits_{\alpha\rightarrow 0^{+}}c_{\alpha}=c_{0}\geq\rho_{2}$, where
$c_{\alpha}$, $c_{0}$ and $\rho_{2}$ are given in Proposition 3.7,
$\eqref{starrrr}$ and Lemma 3.6, respectively.

###### Proof.

Since
$F_{\alpha}(t)=\chi_{\\{t\geq\alpha\\}}\frac{1}{p_{s}^{*}}(|t|^{p_{s}^{*}}-\alpha^{p_{s}^{*}})$,
we obtain

$\displaystyle\left|\frac{1}{p_{s}^{*}}\int_{\Omega}(u^{+})^{p_{s}^{*}}-\int_{\Omega}F_{\alpha}(u)\right|$
$\displaystyle=\left|\frac{1}{p_{s}^{*}}\int_{\Omega}(u^{+})^{p_{s}^{*}}\chi_{\\{u\leq\alpha\\}}+\frac{1}{p_{s}^{*}}\int_{\Omega}\alpha^{p_{s}^{*}}\chi_{\\{u>\alpha\\}}\right|$
$\displaystyle\leq\frac{2\alpha^{p_{s}^{*}}|\Omega|}{p_{s}^{*}}.$ (4.2)

Clearly, $J_{0}^{0}(u)\leq J_{\alpha}^{0}(u)$, for all $u\in
W_{0}^{s,p}(\Omega)$. Thus, $c_{0}\leq c_{\alpha}$, for all $\alpha>0$.
According to $\eqref{starrrr 1}$ we establish

$J_{\alpha}^{0}(u)=J_{0}^{0}(u)+o_{\alpha}(1),~{}\forall~{}u\in
W_{0}^{s,p}(\Omega),$ (4.3)

where $o_{\alpha}(1)\rightarrow 0$ as $\alpha\rightarrow 0^{+}$ independently
of $u$. This gives,

$J_{\alpha}^{0}(\zeta(t))=J_{0}^{0}(\zeta(t))+o_{\alpha}(1),~{}\forall~{}\zeta\in\Gamma,~{}t\in[0,1]$
(4.4)

and hence $c_{\alpha}=c_{0}+o_{\alpha}(1)$.
With the consideration of $\eqref{starrrr 3}$ and Lemma 3.6 $(3)$, we conclude
that $\lim\limits_{\alpha\rightarrow 0^{+}}c_{\alpha}=c_{0}\geq\rho_{2}$. ∎

######  Proposition 4.2.

For any sequence $\alpha_{n}\in(0,\bar{\alpha})$ with $\alpha_{n}\rightarrow
0^{+}$, we have, up to a subsequence, $u_{\alpha_{n}}\rightarrow u_{0}$ in
$W_{0}^{s,p}(\Omega)$, where $u_{0}$ is a nontrivial weak solution to the
problem $\eqref{main prob 0}$.

###### Proof.

Let $u_{\alpha}$ be the weak solutionto $\eqref{main prob intro}$ given in
Proposition 3.7. Thus,
$J_{\alpha}^{0}(u_{\alpha})=J_{\alpha}(u_{\alpha})=c_{\alpha}$and

$\int_{\mathbb{R}^{2N}}\frac{|u_{\alpha}(x)-u_{\alpha}(y)|^{p-2}(u_{\alpha}(x)-u_{\alpha}(y)}{|x-y|^{N+sp}}(\varphi(x)-\varphi(y))dxdy-\mu\int_{\Omega}\eta_{\alpha}\varphi
dx-\int_{\Omega}\theta_{\alpha}\varphi
dx-\lambda\int_{\Omega}\frac{\varphi}{u_{\alpha}^{\gamma}}dx=0,$ (4.5)

where
$\eta_{\alpha}(x)\in[\underline{g}(x,u_{\alpha}(x)),\overline{g}(x,u_{\alpha}(x))]$
and
$\theta_{\alpha}(x)\in[\underline{f}_{\alpha}(u_{\alpha}(x)),\overline{f}_{\alpha}(u_{\alpha}(x))]$
a.e. in $\Omega$ with $f_{\alpha}(t)=H(t-\alpha)t^{p_{s}^{*}-1}$.
Consider the sequence $(w_{n})\subset W_{0}^{s,p}(\Omega)$ obtained in
Proposition 3.7 with $w_{n}\rightarrow u_{\alpha}$ in $W_{0}^{s,p}(\Omega)$.
From $\eqref{star 6}$ we have

$c_{\alpha}+1+\|w_{n}\|_{W_{0}^{s,p}(\Omega)}\geq\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{n}\|^{p}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\|w_{n}\|_{W_{0}^{s,p}(\Omega)}^{1-\gamma}-\mu
C_{1},~{}\forall~{}\alpha>0,$

where $C_{1}$ is independent of $\alpha$ (refer $\eqref{star 2}$). Thus,

$c_{\alpha}+1+\|w_{\alpha}\|_{W_{0}^{s,p}(\Omega)}\geq\left(\frac{1}{p}-\frac{1}{b}\right)\|w_{\alpha}\|^{p}_{W_{0}^{s,p}(\Omega)}-\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\|w_{\alpha}\|_{W_{0}^{s,p}(\Omega)}^{1-\gamma}-\mu
C_{1},~{}\forall~{}\alpha>0$

and the sequence $(u_{\alpha})$ is uniformly bounded in $W_{0}^{s,p}(\Omega)$.
By $(g_{2})$ and Lemma 3.1, we establish that $(\eta_{\alpha})$ and
$(\theta_{\alpha})$ are bounded in $L^{\frac{r}{r-1}}(\Omega)$ and
$L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega)$, respectively.
Consider a sequence $(\alpha_{n})\subset(0,\bar{\alpha})$ with
$\alpha_{n}\rightarrow 0^{+}$. Hence, up to a subsequence,

$\displaystyle u_{\alpha_{n}}\rightharpoonup
u_{0}~{}\text{in}~{}W_{0}^{s,p}(\Omega),$
$\displaystyle~{}u_{\alpha_{n}}(x)\rightarrow u_{0}(x)~{}\text{a.e.
in}~{}\Omega,~{}u_{\alpha_{n}}\rightarrow
u_{\alpha}~{}\text{in}~{}L^{q}(\Omega)~{}\text{for any}~{}1\leq q<p_{s}^{*},$
$\displaystyle\eta_{\alpha_{n}}\overset{*}{\rightharpoonup}\eta_{0}~{}\text{in}~{}L^{\frac{r}{r-1}}(\Omega)~{}$
$\displaystyle\text{and}~{}\theta_{\alpha_{n}}\overset{*}{\rightharpoonup}u_{0}^{p_{s}^{*}-1}~{}\text{in}~{}L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega).$
(4.6)

We already have $u_{\alpha_{n}}\geq\underline{u}_{\lambda}$ for all
$n\in\mathbb{N}$ (refer Proposition 3.7), where $\underline{u}_{\lambda}$ is a
weak solution to $\eqref{main prob help}$ given in Lemma 3.3. Thus, by
combining $\eqref{starrrr 4}$ and $\eqref{starrrr 5}$ we pass the limit
$\alpha_{n}\rightarrow 0^{+}$ to obtain

$\int_{\mathbb{R}^{2N}}\frac{|u_{0}(x)-u_{0}(y)|^{p-2}(u_{0}(x)-u_{0}(y)}{|x-y|^{N+sp}}(\varphi(x)-\varphi(y))dxdy-\mu\int_{\Omega}\eta_{0}\varphi
dx-\int_{\Omega}u_{0}^{p_{s}^{*}-1}\varphi
dx-\lambda\int_{\Omega}\frac{\varphi}{u_{0}^{\gamma}}dx=0,$

for every $\varphi\in W_{0}^{s,p}(\Omega)$. From Proposition 2.4,
$\eta_{0}\in\partial I_{\mu}(u_{0})$.
According to $\eqref{starrr 7}$ we have

$\rho_{2}\leq
J^{0}_{\alpha_{n}}(u_{\alpha_{n}})=c_{\alpha}<\left(\frac{1}{p}-\frac{1}{b}\right)S_{s,p}^{\frac{N}{sp}}-\left(\frac{1}{p}-\frac{1}{b}\right)^{-\frac{1-\gamma}{p-1+\gamma}}\left(\frac{\lambda}{1-\gamma}|\Omega|^{\frac{p_{s}^{*}-1+\gamma}{p_{s}^{*}}}S_{s,p}^{-\frac{1-\gamma}{p}}\right)^{\frac{p}{p-1+\gamma}}-{\mu}C_{1},~{}\forall~{}\alpha_{n}>0$
(4.7)

and from $\eqref{starrrr 2}$,

$c_{\alpha_{n}}=J^{0}_{\alpha_{n}}(u_{\alpha_{n}})=J^{0}_{0}(u_{\alpha_{n}})+o_{\alpha_{n}}(1).$
(4.8)

Considering $\eqref{starrrr 6}$, $\eqref{starrrr 7}$ and then following the
proof of Proposition 3.5, we get

$u_{\alpha_{n}}\rightarrow u_{0}~{}\text{ in
}W_{0}^{s,p}(\Omega)~{}\text{as}~{}\alpha_{n}\rightarrow 0^{+}.$ (4.9)

Combining $\eqref{starrrr 8}$, Lemma 4.1 and $\eqref{starrrr 2}$, we have
$J_{0}^{0}(u_{0})=c_{0}\geq\rho_{2}>0$. This proves that $u_{0}$ is a
nontrivial weak solution to $\eqref{main prob 0}$ and we conclude the proof. ∎

## Acknowledgement

The author Debajyoti Choudhuri thanks the grant received from Council of
Scientific and Industrial Research (CSIR), India for the research grant
(09/983(0013)/2017-EMR-I). The author Akasmika Panda thanks the financial
assistantship received from the Ministry of Human Resource Development
(M.H.R.D.), Govt. of India.

##  5 References

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