# Existence results for singular elliptic problem involving a fractional
$p$-Laplacian.

Hanaâ ACHOURa and Sabri BENSIDb

###### Abstract

In this article, the problems to be studied are the following

$\left\\{\begin{array}[]{ll}(-\Delta)_{p}^{s}u\pm\dfrac{|u|^{p-2}u}{|x|^{sp}}=\lambda
f(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (P±)

where $\Omega$ is a bounded regular domain in $\mathbb{R}^{N}(N\geq 2)$
containing the origin, $p>1$, $s\in(0,1)$, $(N>ps)$, $\lambda>0$,
$f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a Carathéodory function
satisfying a suitable growth condition and $(-\Delta)_{p}^{s}$ is the
fractional p-Laplacian defined as

$(-\Delta)_{p}^{s}u(x)\coloneqq\displaystyle 2\lim_{\varepsilon\rightarrow
0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dy,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{N},$

where $B_{\varepsilon}(x)$ is the open $\varepsilon$-ball of centre $x$ and
radius $\varepsilon$. Using the critical point theory combining to the
fractional Hardy inequality, we show that the problem $(P_{+})$ admits at
least two distinct nontrivial weak solutions. For the problem $(P_{-}),$ we
use the concentration-compactness principle for fractional Sobolev spaces to
give a weak lower semicontinuity result and prove that problem $(P_{-})$
admits at least one non-trivial weak solution.

aDynamical Systems and Applications Laboratory

Department of Mathematics, Faculty of Sciences

University of Tlemcen, B.P. 119

Tlemcen 13000, Algeria

Mail<EMAIL_ADDRESS>
bDynamical Systems and Applications Laboratory

Department of Mathematics, Faculty of Sciences

University of Tlemcen, B.P. 119

Tlemcen 13000, Algeria

Mail<EMAIL_ADDRESS>
AMS Classification: 34R35, 35J25, 35B38.

Keywords: Singular problem, fractional $p$-Laplacian, critical point,
variational method, fractional Hardy inequality.

##  1 Introduction

Singular elliptic problems arise in several context like chemical
heterogeneous catalysts, in the theory of pseudoplastic fluid, non-Newtonian
fluid and in the study of relativistic matter in magnetic fluid. for more
details, see [22, 10, 11].

On the other hand, recently, a great deal of works is devoted to the
fractional nonlocal problem arising in many fields such as anomalous diffusion
in plasma, flames propagation, geophysical fluid dynamics and American options
in finances. See [8, 12] and the references therein.

In this work, we continue the study of semilinear singular elliptic problem
with the nonlocal operator theory. more precisely, we are interested by the
fractional $p$-Laplacian operator which up to normalization functions may be
defined as

$(-\Delta)_{p}^{s}u(x)\coloneqq\displaystyle 2\lim_{\varepsilon\rightarrow
0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dy,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{N},$

where $B_{\varepsilon}(x)$ is the open $\varepsilon$-ball of centre $x$ and
radius $\varepsilon$. We refer to [9, 17] for more details.

In the last years, many works are devoted to the quasi-linear problem

$\left\\{\begin{array}[]{ll}(-\Delta)_{p}^{s}u=f(x,u)&\quad\mbox{in }\
\Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$ (1.1)

For $p=2$, problem (1.1) reduces to the fractional Laplacian problem. we refer
to the series of papers of Servadei and Valdinoci [24, 25, 26].

When the nonlinearity is discontinuous, the second author study problem (1.1)
and prove the existence and multiplicity results for the following problem

$\left\\{\begin{array}[]{ll}(-\Delta)^{s}u=f(u)H(u-\mu)&\quad\mbox{in }\
\Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $H$ is the Heaviside function, $f$ is a given function and $\mu>0$.

For $p\neq 2$ and when $f$ is regular nonlinearity, there is a large number of
papers treated problem (1.1) using different techniques. we refer the reader
to the papers [3, 2, 4, 9, 17, 21, 24, 28] and to the references therein.

When $f$ is discontinuous nonlinearity with respect to $u$, problem (1.1) was
treated by the authors in [1]. More precisely, $f$ is given by

$f(x,u)=m(x)\displaystyle\sum_{i=1}^{n}\leavevmode\nobreak\ H(u-\mu_{i}),$

for some $\mu_{i}>0$ verifying the condition

$\mu_{1}<\mu_{2}<\cdots<\mu_{n},\leavevmode\nobreak\ \text{ for }n\geq 1$

and $m\in L^{\infty}(\Omega)$ changes sign.

The authors prove the existence and multiplicity result via the nonsmooth
critical point theory.

In this paper, we are interested to study the existence of weak solutions for
problem (1.1) with singular absorption term of the following types

$\left\\{\begin{array}[]{ll}(-\Delta)_{p}^{s}u+\dfrac{|u|^{p-2}u}{|x|^{sp}}=\lambda
f(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (1.2)

and

$\left\\{\begin{array}[]{ll}(-\Delta)_{p}^{s}u-\dfrac{|u|^{p-2}u}{|x|^{sp}}=\lambda
f(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (1.3)

where $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in
$\mathbb{R}^{N}$, $p>1$, $s\in(0,1)$, $(N>ps)$ containing the origin and with
smooth boundary, $f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a
Carathéodory function satisfying the following subcritical growth condition

  1. (H1)

$\left|f(x,t)\right|\leq\alpha+\beta|t|^{q-1}$,
$\forall(x,t)\in\Omega\times\mathbb{R}$

for some non-negative constant $\alpha,\beta$ and $q\in]p,p^{*}[$, where the
fractional critical exponent be defined as

$p^{*}\coloneqq\left\\{\begin{array}[]{ll}\frac{Np}{N-sp}&\quad\mbox{if }\
N>ps\\\\[2.84544pt] \infty&\quad\mbox{if }\ N\leq ps.\end{array}\right.$

In the local setting $(s=1)$, M. Khodabakhshi and al. [18] studied the
existence of solutions to the problem (1.2) which was motivated by the work of
Ferrara and Bisci [13]. They studied the existence of at least one nontrivial
solution of the following elliptic problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=\mu\dfrac{|u|^{p-2}u}{|x|^{p}}+\lambda
f(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\partial\Omega,\end{array}\right.$ (1.4)

where $-\Delta_{p}u\coloneqq\text{div }(|\nabla u|^{p-2}\nabla u)$ denotes the
$p$-Laplacian, $\lambda,\mu>0$.

Note also the work of Khodabakhshi and Hadjian in [19] when the authors prove
the existences of three weak solutions of the following problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u+\dfrac{|u|^{p-2}u}{|x|^{p}}=\lambda
f(x,u)+\mu g(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\partial\Omega,\end{array}\right.$ (1.5)

where $f$ and $g$ are Carathéodory functions.

Hence, in this paper, we consider the nonlocal counter part of problem (1.5)
when $\mu=0$ and the counter part of problem (1.4). Using a variational
structure of our problems and based on a version of critical point theorem
contained in [5, 23] (See respectively Theorem 2.1 and Theorem 2.2), we prove
the existence of two weak solutions of problem (1.2) and we show that the
problem (1.3) admits at least one nontrivial solution .

The paper is organized as follows. In Section 2, we recall some basic
definitions. Section 3 is denoted to state and prove our main results and
finally some useful comments related to problems (1.2) and (1.3) are given.

##  2 Preliminaries.

In this section, we provide the variational setting for the problem (1.2),
jointly with some preliminary results for the fractional $p$-Laplacian. Also,
we recall certain definitions and essential results on the critical point
theory.

###  2.1 Variational Formulation of the Problem.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain containing the origin
and with smooth boundary $\partial\Omega$, for $p\in(1,\infty)$ and
$s\in(0,1)$, we define the fractional Sobolev space

$W^{s,p}(\mathbb{R}^{N})\coloneqq\left\\{u\in
L^{p}(\mathbb{R}^{N})\leavevmode\nobreak\ :\leavevmode\nobreak\
\dfrac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in
L^{p}(\mathbb{R}^{2N})\right\\},$

endowed with the norm

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

where the following term is the Gagliardo semi-norm of u defined as

$\left[u\right]_{W^{s,p}(\mathbb{R}^{N})}\coloneqq\left(\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{\frac{1}{p}},$

We shall work in the following closed linear subspace

$W_{0}\coloneqq\left\\{u\in W^{s,p}(\mathbb{R}^{N})\leavevmode\nobreak\
:\leavevmode\nobreak\ u=0\leavevmode\nobreak\ \mbox{ a.e. in
}\leavevmode\nobreak\ \mathbb{R}^{N}\setminus\Omega\right\\},$

which can be equivalently renormed by

$\|u\|_{W_{0}}\coloneqq\left[u\right]_{W^{s,p}(\mathbb{R}^{N})}.$

Let $1<ps<N$, then there exist a positive constant $c_{H}$ such that we
present the fractional Hardy inequality, which says that

$\int_{\mathbb{R}^{N}}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx\leq
c_{H}\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall u\in W_{0}.$ (2.6)

For more details, see [15].
Now, let $\mathbb{C}_{c}^{\infty}(\mathbb{R}^{N})$ the space of functions on
$\mathbb{R}$ that are infinitely differentiable and have compact support
contained in $\mathbb{R}^{N}$. If we denote by
$\mathbb{D}^{s,p}(\mathbb{R}^{N})$ the closure of
$\mathbb{C}_{c}^{\infty}(\mathbb{R}^{N})$ with respect to the Gagliardo semi-
norm $\left[u\right]_{W^{s,p}(\mathbb{R}^{N})},$ then for $sp<N,$ we define
the Sobolev constant $S_{p}$ by

$S_{p}=\displaystyle\inf_{v\in\mathbb{D}^{s,p}(\mathbb{R}^{N})}\frac{\left[u\right]_{W^{s,p}}^{p}}{\|v\|_{L^{p_{s}^{*}}}^{p}},$
(2.7)

where $p_{s}^{*}=\frac{Np}{N-sp}$ is the critical Sobolev exponent.

Also, It is well-known that $(W_{0},\|\cdot\|_{W_{0}})$ is a uniformly convex
(i.e. reflexive) Banach space, continuously embedded into $L^{p}(\Omega)$ for
all $p\in[1,p^{*}]$ and compactly injected in $L^{p}(\Omega)$ for all
$p\in[1,p^{*})$, there exist a $c_{p}>0$ which is the best constant of the
embedding, such that

$||u||_{L^{p}(\Omega)}\leq c_{p}||u||_{W_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall u\in
W_{0}.$ (2.8)

Next, let us denote by $(W^{*}_{0},\|\cdot\|_{W^{*}_{0}})$ the dual space of
$(W_{0},\|\cdot\|_{W_{0}})$ and we define the nonlinear operators
$A_{p,m}:W_{0}\rightarrow W^{*}_{0}$ as

$\langle
A_{p,m}(u),v\rangle=\displaystyle\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\frac{|u(x)|^{p-2}}{|x|^{sp}}u(x)v(x)\leavevmode\nobreak\
dx,$

where $\langle\cdot,\cdot\rangle$ denotes the inner product on $W_{0}$ and
$m=1$ or $m=-1.$

######  Lemma 2.1.

For $u,v\in W_{0}$ and there exist a constant $k\geq 1$, then the nonlinear
operators $A_{p,m}$ are well defined and verify the following:

$\langle A_{p,m}(u),v\rangle\leq k\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}$

and

$\|A_{p,m}(u)\|_{W^{*}_{0}}\leq k\|u\|_{W_{0}}^{p-1}.$

Proof. For all $u,v\in W_{0}$, we have

$\displaystyle\langle A_{p,m}(u),v\rangle$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p-2}}{|x|^{sp}}u(x)v(x)\leavevmode\nobreak\
dx$
$\displaystyle\leq\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p-2}|u(x)-u(y)||v(x)-v(y)|}{|x-y|^{(N+sp)\left(\frac{p-1+1}{p}\right)}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p-2}}{|x|^{(sp)\left(\frac{p-1+1}{p}\right)}}|u(x)||v(x)|\leavevmode\nobreak\
dx$
$\displaystyle\leq\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p-1}}{|x-y|^{(N+sp)\left(\frac{p-1}{p}\right)}}\dfrac{|v(x)-v(y)|}{|x-y|^{\left(\frac{N+sp}{p}\right)}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p-1}}{|x|^{(sp)\left(\frac{p-1}{p}\right)}}\dfrac{|v(x)|}{|x|^{\frac{ps}{p}}}\leavevmode\nobreak\
dx.$

Then, by Holder inequality we get

$\displaystyle\langle A_{p,m}(u),v\rangle$
$\displaystyle\leq\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{\frac{p-1}{p}}\left(\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{\frac{1}{p}}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+m\left(\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)^{\frac{p-1}{p}}\left(\int_{\Omega}\dfrac{|v(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)^{\frac{1}{p}}.$

For any $\beta\in(0,1)$ and $a,b,c,d>0$, we use the following inequality

$a^{\beta}c^{1-\beta}+m\leavevmode\nobreak\
b^{\beta}d^{1-\beta}\leq(a+m\leavevmode\nobreak\
b)^{\beta}(c+m\leavevmode\nobreak\ d)^{1-\beta},$

by setting $\beta=\frac{p-1}{p}$ and

$\begin{array}[]{l}a=\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\\\\[11.38092pt]
b=\displaystyle\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\\\\[11.38092pt]
c=\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\\\\[11.38092pt]
d=\displaystyle\int_{\Omega}\dfrac{|v(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx,\end{array}$

we can deduce that

$\displaystyle\langle A_{p,m}(u),v\rangle$
$\displaystyle\leq\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)^{\frac{p-1}{p}}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\times\left(\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|v(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)^{\frac{1}{p}}.$

Then, according to the fractional Hardy inequality (2.6) we get

$\displaystyle\langle A_{p,1}(u),v\rangle$
$\displaystyle\leq\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+c_{H}\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{\frac{p-1}{p}}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\times\left(\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+c_{H}\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy\right)^{\frac{1}{p}}$
$\displaystyle\leq(c_{H}+1)\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}$
$\displaystyle\leq k\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}$ $\displaystyle<+\infty.$

Where $k=c_{H}+1$ and $k\geq 1$, we have

$\displaystyle\langle A_{p,-1}(u),v\rangle$
$\displaystyle\leq\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)^{\frac{p-1}{p}}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\times\left(\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\int_{\Omega}\dfrac{|v(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)^{\frac{1}{p}}$ $\displaystyle\leq\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}$
$\displaystyle<+\infty.$

Moreover, respectively we have

$\|A_{p,1}(u)\|_{W^{*}_{0}}\leq k\|u\|_{W_{0}}^{p-1}$

and

$\|A_{p,-1}(u)\|_{W^{*}_{0}}\leq\|u\|_{W_{0}}^{p-1}.$

$\square$

######  Lemma 2.2.

For all $u,v\in W_{0}$ and there exist a constant $\mu\leq 1$, the operators
$A_{p,m}$ satisfy the following inequalities :

$\langle A_{p,1}(u)-A_{p,1}(v),u-v\rangle\geq
k\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right)$

and

$\langle
A_{p,-1}(u)-A_{p,-1}(v),u-v\rangle\geq\mu\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right).$

Proof. By direct computation, we have
$\displaystyle\langle A_{p,m}(u)-A_{p,m}(v),u-v\rangle$ $\displaystyle=\langle
A_{p,m}(u),u-v\rangle-\langle A_{p,m}(v),u-v\rangle$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))((u-v)(x)-(u-v)(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+m\int_{\Omega}\dfrac{|u(x)|^{p-2}}{|x|^{sp}}u(x)(u-v)(x)\leavevmode\nobreak\
dx$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|^{p-2}(v(x)-v(y))((u-v)(x)-(u-v)(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+m\int_{\Omega}\dfrac{|v(x)|^{p-2}}{|x|^{sp}}v(x)(u-v)(x)\leavevmode\nobreak\
dx$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx+\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|v|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx$
$\displaystyle-\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p-2}}{|x|^{sp}}u(x)v(x)\leavevmode\nobreak\
dx$
$\displaystyle-\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|^{p-2}(v(x)-v(y))(u(x)-u(y))}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|v(x)|^{p-2}}{|x|^{sp}}v(x)u(x)\leavevmode\nobreak\
dx$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx+\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|v|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ -\langle A_{p,m}(u),v\rangle-\langle
A_{p,m}(v),u\rangle.$
Furthermore, by Lemma 2.1, there exist $k\geq 1$ such that

$\displaystyle\langle A_{p,1}(u)-A_{p,1}(v),u-v\rangle$
$\displaystyle\geq\|u\|_{W_{0}}^{p}+\|v\|_{W_{0}}^{p}-\langle
A_{p,1}(u),v\rangle-\langle A_{p,1}(v),u\rangle$
$\displaystyle\geq\|u\|_{W_{0}}^{p}+\|v\|_{W_{0}}^{p}-k\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}-k\|v\|_{W_{0}}^{p-1}\|u\|_{W_{0}}$
$\displaystyle=-k\left(-\frac{1}{k}\|u\|_{W_{0}}^{p}-\frac{1}{k}\|v\|_{W_{0}}^{p}+\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}+\|v\|_{W_{0}}^{p-1}\|u\|_{W_{0}}\right)$
$\displaystyle\geq-k\left(-\|u\|_{W_{0}}^{p}-\|v\|_{W_{0}}^{p}+\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}+\|v\|_{W_{0}}^{p-1}\|u\|_{W_{0}}\right)$
$\displaystyle\geq
k\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right).$

Likewise, according to the fractional Hardy inequality (2.6) and Lemma 2.1 we
get

$\displaystyle\langle A_{p,-1}(u)-A_{p,-1}(v),u-v\rangle$
$\displaystyle\geq(1-c_{H})\|u\|_{W_{0}}^{p}+(1-c_{H})\|v\|_{W_{0}}^{p}-\langle
A_{p,-1}(u),v\rangle-\langle A_{p,-1}(v),u\rangle$
$\displaystyle\geq\mu\|u\|_{W_{0}}^{p}+\mu\|v\|_{W_{0}}^{p}-\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}-\|v\|_{W_{0}}^{p-1}\|u\|_{W_{0}}$
$\displaystyle=-\mu\left(-\|u\|_{W_{0}}^{p}-\|v\|_{W_{0}}^{p}+\dfrac{1}{\mu}\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}+\dfrac{1}{\mu}\|v\|_{W_{0}}^{p-1}\|u\|_{W_{0}}\right)$
$\displaystyle\geq-\mu\left(-\|u\|_{W_{0}}^{p}-\|v\|_{W_{0}}^{p}+\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}}+\|v\|_{W_{0}}^{p-1}\|u\|_{W_{0}}\right)$
$\displaystyle\geq\mu\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right),$

where $\mu=1-c_{H}$ and $\mu\leq 1$. $\square$

######  Proposition 2.1.

The nonlinear operators $A_{p,m}$ have the following properties :

  1. 1)

$A_{p,m}:W_{0}\rightarrow W^{*}_{0}$ are a continuous, bounded and strictly
monotone operators, i.e: if $\langle A_{p,m}(u)-A_{p,m}(v),u-v\rangle>0$,
$\forall u\neq v$.

  2. 2)

$A_{p,m}$ are a mappings of type $(S)$, i.e: if $u_{n}\rightarrow u$ weakly in
$W_{0}$ and $\displaystyle\limsup_{n\rightarrow\infty}\langle
A_{p,m}(u_{n})-A_{p,m}(u),u_{n}-u\rangle\leq 0$, then $u_{n}\rightarrow u$
strongly in $W_{0}$.

  3. 3)

$A_{p,m}:W_{0}\rightarrow W^{*}_{0}$ are homomorphisms.

Proof.

1) By Lemma 2.1, there exist a constant $k\geq 1$ such that

$\left|\langle A_{p,m}(u),v\rangle\right|\leq
k\|u\|_{W_{0}}^{p-1}\|v\|_{W_{0}},\leavevmode\nobreak\ \forall u,v\in W_{0}$

From this inequality, it is obvious that $A_{p,m}$ are continuous and bounded.

Now, using the well-known Simon inequality (see [27, formula (2.2)]): for all
$\xi,\eta\in\mathbb{R}^{N}$, for $p>1$ there exists a positive constant
$C_{p}$ such that

$C_{p}\langle|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta\rangle\geq\left\\{\begin{array}[]{ll}\left|\xi-\eta\right|^{p}&\quad\mbox{if
}\ p\geq 2,\\\
\left|\xi-\eta\right|^{2}\left(|\xi|^{p}+|\eta|^{p}\right)^{(p-2)/p}&\quad\mbox{if
}\ 1<p<2.\end{array}\right.$ (2.9)

Then, using Lemma 2.2 and applying inequality (2.9) for any $u,v\in W_{0}$,
with $u\neq v$, we have if $p\geq 2$

$C_{p}\langle A_{p,1}(u)-A_{p,1}(v),u-v\rangle\geq
k\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right)\geq
0$

equally,

$C_{p}\langle
A_{p,-1}(u)-A_{p,-1}(v),u-v\rangle\geq\mu\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right)\geq
0.$

and if $1<p<2$, we have

$C_{p}^{p/2}\left[\langle
A_{p,1}(u)-A_{p,1}(v),u-v\rangle\right]^{p/2}\left(\|u\|_{W_{0}}^{p}-\|v\|_{W_{0}}^{p}\right)^{(2-p)/2}\geq
k\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right),$

likewise,

$C_{p}^{p/2}\left[\langle
A_{p,-1}(u)-A_{p,-1}(v),u-v\rangle\right]^{p/2}\left(\|u\|_{W_{0}}^{p}-\|v\|_{W_{0}}^{p}\right)^{(2-p)/2}\geq\mu\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right),$

thus,

$C\left[\langle A_{p,1}(u)-A_{p,-1}(v),u-v\rangle\right]^{p/2}\geq
k\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right)\geq
0$

and

$C\left[\langle
A_{p,-1}(u)-A_{p,-1}(v),u-v\rangle\right]^{p/2}\geq\mu\left(\|u\|_{W_{0}}^{p-1}-\|v\|_{W_{0}}^{p-1}\right)\left(\|u\|_{W_{0}}-\|v\|_{W_{0}}\right)\geq
0,$

where $C>0$ is a constant, which lead us to conclude that $A_{p,m}$ are
strictly monotone.

2) Since $W_{0}$ is a reflexive Banach space, it is isometrically isomorphic
to a locally uniformly convex space. So as it was already proved, weak
convergence and norm convergence imply strong convergence. Therefore we only
need to show that $\|u_{n}\|_{W_{0}}\rightarrow\|u\|_{W_{0}}$.

further, we have that if $u_{n}\rightarrow u$ weakly in $W_{0}$ and

$\displaystyle\limsup_{n\rightarrow\infty}\langle
A_{p,m}(u_{n})-A_{p,m}(u),u_{n}-u\rangle\leq 0.$

Then

$\displaystyle\lim_{n\rightarrow+\infty}\langle
A_{p,m}(u_{n})-A_{p,m}(u),u_{n}-u\rangle=\lim_{n\rightarrow+\infty}\langle
A_{p,m}(u_{n}),u_{n}-u\rangle-\langle A_{p,m}(u),u_{n}-u\rangle=0.$

By Lemma 2.2 and 1), we have

$\langle A_{p,1}(u_{n})-A_{p,1}(u),u_{n}-u\rangle\geq
k\left(\|u_{n}\|_{W_{0}}^{p-1}-\|u\|_{W_{0}}^{p-1}\right)\left(\|u_{n}\|_{W_{0}}-\|u\|_{W_{0}}\right)\geq
0$

and

$\langle
A_{p,-1}(u_{n})-A_{p,-1}(u),u_{n}-u\rangle\geq\mu\left(\|u_{n}\|_{W_{0}}^{p-1}-\|u\|_{W_{0}}^{p-1}\right)\left(\|u_{n}\|_{W_{0}}-\|u\|_{W_{0}}\right)\geq
0.$

Which imply that $\|u_{n}\|_{W_{0}}\rightarrow\|u\|_{W_{0}}$ as
$n\rightarrow+\infty$ and lead us to conclude that $u_{n}\rightarrow u$
strongly in $W_{0}$ as $n\rightarrow+\infty$.

3) By 1), we know that $A_{p,m}$ are strictly monotone, which implicate that
$A_{p,m}$ are injective. Also, according to Lemma 2.1, we have

$\displaystyle\lim_{\|u\|_{W_{0}}\rightarrow+\infty}\dfrac{\langle
A_{p,m}(u),u\rangle}{\|u\|_{W_{0}}}$
$\displaystyle=\lim_{\|u\|_{W_{0}}\rightarrow+\infty}\dfrac{\|u\|_{W_{0}}^{p}+m\left\|\frac{u}{|x|^{s}}\right\|_{L^{p}(\Omega)}^{p}}{\|u\|_{W_{0}}}$
$\displaystyle=\lim_{\|u\|_{W_{0}}\rightarrow+\infty}\|u\|_{W_{0}}^{p-1}+m\left\|\frac{u}{|x|^{s}}\right\|_{L^{p}(\Omega)}^{p}\|u\|_{W_{0}}^{-1}$
$\displaystyle=\lim_{\|u\|_{W_{0}}\rightarrow+\infty}\|u\|_{W_{0}}^{p-1}$
$\displaystyle=+\infty.$

Thanks to $1<p<\frac{N}{s}$, hence $A_{p,m}$ are Coercive on $W_{0}$. Since
$A_{p,m}$ are continuous and bounded by 1). Then, by the Minty-Browder Theorem
(see [29, Theorem 26.A]), we conclude that $A_{p,m}$ are a surjections.
Thus, $A_{p,m}$ have inverse mappings $A_{p,m}^{-1}:W_{0}^{*}\rightarrow
W_{0}$. Therefore, the continuity of $A_{p,m}^{-1}$ is sufficient to ensure
$A_{p,m}$ to be homeomorphisms.
Assume, $g_{n},g\in W_{0}$ with $g_{n}\rightarrow g$ in $W_{0}$. Let
$u_{n}=A_{p,m}^{-1}(g_{n})$ and $u=A_{p,m}^{-1}(g)$. Then
$A_{p,m}(u_{n})=g_{n}$ and $A_{p,m}(u)=g$. Clearly, $\\{u_{n}\\}$ is bounded
in $W_{0}$. Thus there exist $u_{0}\in W_{0}$ and a subsequence of
$\\{u_{n}\\}$ still denoted by $\\{u_{n}\\}$ such that $u_{n}\rightharpoonup
u_{0}$ since $g_{n}\rightarrow g$, we have

$\lim_{n\rightarrow+\infty}\langle
A_{p,m}(u_{n})-A_{p,m}(u_{0}),u_{n}-u_{0}\rangle=\lim_{n\rightarrow+\infty}\langle
g_{n},u_{n}-u_{0}\rangle=0.$

In view of $A_{p,m}$ satisfying the $(S)$ condition by 2), we get
$u_{n}\rightarrow u_{0}$ in $W_{0}$. Moreover, $u=u_{0}$ a.e. in $\Omega$.
Hence, $u_{n}\rightarrow u$ in $W_{0}$, so that $A_{p,m}^{-1}$ are continuous.
$\square$

Initially, let us introduce the energy functionals
$E_{\lambda}^{p,m}:W_{0}\rightarrow\mathbb{R}$ associated with problem (1.2)
as

$E_{\lambda}^{p,m}(u)\coloneqq\Phi_{p,m}(u)-\lambda\Psi(u),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall u\in
W_{0},$

where

$\Phi_{p,m}(u)\coloneqq\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx\right)$

and

$\Psi(u)\coloneqq\int_{\Omega}F(x,u(x))\leavevmode\nobreak\ dx,$

where $F(x,t)=\int_{0}^{t}f(x,s)\leavevmode\nobreak\ ds$, for every
$(x,t)\in\Omega\times\mathbb{R}$.

Moreover, from the fractional Hardy’s inequality (2.6), we have

$\displaystyle\Phi_{p,1}(u)$
$\displaystyle=\dfrac{1}{p}\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+\dfrac{1}{p}\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx$
$\displaystyle\leq\dfrac{1}{p}\|u\|^{p}_{W_{0}}+\dfrac{c_{H}}{p}\|u\|^{p}_{W_{0}}$
$\displaystyle\leq\left(\dfrac{c_{H}+1}{p}\right)\|u\|^{p}_{W_{0}}$

and

$\displaystyle\Phi_{p,-1}(u)$
$\displaystyle=\dfrac{1}{p}\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\dfrac{1}{p}\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx$
$\displaystyle\geq\dfrac{1}{p}\|u\|^{p}_{W_{0}}-\dfrac{c_{H}}{p}\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx$ $\displaystyle\geq\left(\dfrac{1-c_{H}}{p}\right)\|u\|^{p}_{W_{0}}.$
(2.10)

Also,

$\Phi_{p,1}(u)=\dfrac{1}{p}\|u\|^{p}_{W_{0}}+\dfrac{1}{p}\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\geq\dfrac{1}{p}\|u\|^{p}_{W_{0}}$ (2.11)

and

$\displaystyle\Phi_{p,-1}(u)$
$\displaystyle=\dfrac{1}{p}\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\dfrac{1}{p}\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx$ $\displaystyle\leq\dfrac{1}{p}\|u\|^{p}_{W_{0}}.$

It follows that

$\dfrac{1}{p}\|u\|^{p}_{W_{0}}\leq\Phi_{p,1}(u)\leq\left(\dfrac{c_{H}+1}{p}\right)\|u\|^{p}_{W_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall u\in
W_{0}$

equally,

$\left(\dfrac{1-c_{H}}{p}\right)\|u\|^{p}_{W_{0}}\leq\Phi_{p,-1}(u)\leq\dfrac{1}{p}\|u\|^{p}_{W_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall u\in
W_{0}.$

So, $\Phi_{p,m}(u)$ are well defined and coercive in $W_{0}$.

Lastly, it’s evident that our energy functional $E_{\lambda}^{p,m}$ are well
defined and of class $C^{1}$. The derivative of $E_{\lambda}^{p,m}$ is given
by

$\langle\left(E_{\lambda}^{p,m}\right)^{{}^{\prime}}(u),v\rangle=\langle\Phi_{p,m}^{{}^{\prime}}(u),v\rangle-\lambda\langle\Psi^{{}^{\prime}}(u),v\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall u,v\in W_{0}.$

######  Definition 2.1.

Fixing the real parameter $\lambda$, a function
$u:\Omega\rightarrow\mathbb{R}$ is said to be a weak solution of the problem
(1.2) and (1.3), if $u\in W_{0}$ and

$\langle\Phi_{p,m}^{{}^{\prime}}(u),v\rangle=\langle
A_{p,m}(u),v\rangle=\lambda\langle\Psi^{{}^{\prime}}(u),v\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ v\in W_{0}$

where

$\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}(v(x)-v(y))\leavevmode\nobreak\
dxdy+m\int_{\Omega}\dfrac{|u(x)|^{p-2}}{|x|^{sp}}u(x)v(x)\leavevmode\nobreak\
dx$ $=\lambda\int_{\Omega}f(x,u(x))v(x)\leavevmode\nobreak\ dx,$

for every $v\in W_{0}$. Hence, the critical points of $E_{\lambda}^{p,m}$ are
exactly the weak solutions of problems (1.2) and (1.3).

######  Definition 2.2.

Let $X$ be a real Banach space. A Gâteaux differentiable function $E$
satisfies the Palais-Smale condition (in short (PS)-condition), if any
sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ such that

  1. (a)

$\left\\{E(u_{n})\right\\}$ is bounded, i.e. $E(u_{n})\leq
d\coloneqq\displaystyle\sup_{n\in\mathbb{N}}\left\\{E(u_{n})\right\\}$,

  2. (b)

$\|E^{{}^{\prime}}(u_{n})\|_{X^{*}}\longrightarrow 0$ as
$n\rightarrow+\infty$,

has a convergent subsequence.

Our main tools are the following critical point theorems.

######  Theorem 2.1.

[5, Theorem 3.2.]
Let $X$ be a real Banach space and let $\Phi,\Psi:X\rightarrow\mathbb{R}$ be
two continuously Gâteaux differentiable functionals such that $\Phi$ is
bounded from below and $\Phi(0)=\Psi(0)=0$. Fix $r>0$ such that

$\sup_{\\{\Phi(u)<r\\}}\Psi(u)<+\infty$

and assume that, for each

$\lambda\in\left]0,\frac{r}{\sup_{\\{\Phi(u)<r\\}}\Psi(u)}\right[,$

the functional $E_{\lambda}\coloneqq\Phi-\lambda\Psi$ satisfies the
(PS)-condition and it is unbounded from below. Then, for each
$\lambda\in]0,\frac{r}{\sup_{\\{\Phi(u)<r\\}}\Psi(u)}[$, the functional
$E_{\lambda}$ admits two distinct critical points.

######  Theorem 2.2.

[23, Theorem 2.5.]
Let $X$ be a real Banach space and let $\Phi,\Psi:X\rightarrow\mathbb{R}$ be
two continuously Gâteaux differentiable functionals such that $\Phi$ is
strongly continuous, sequentially weakly lower semicontinious and coercive.
Further, assume that $\Psi$ is sequentially weakly upper semicontinious. For
every $r>\inf_{X}\Phi$, put

$\varphi(r)\coloneqq\displaystyle\inf_{u\in\Phi^{-1}(]-\infty,r[)}\dfrac{\left(\sup_{\\{v\in\Phi^{-1}(]-\infty,r[)\\}}\Psi(v)\right)-\Psi(u)}{r-\Phi(u)}.$

Then, for every $r>\inf_{X}\Phi$ and every
$\lambda\in\left]0,1/\varphi(r)\right[$, the restriction
$E_{\lambda}\coloneqq\Phi-\lambda\Psi$ to
$\Phi^{-1}(]-\infty,r[)$ admits a global minimum, which is a critical point
(local minima) of $E_{\lambda}$ in $X$.

Now, we state the following Theorem which provides the Concentration-
Compactness principle in fractional Sobolev spaces. This result is useful for
the proof of the semicontinuity property of functionals.

######  Theorem 2.3.

Let $\\{u_{n}\\}_{n\in\mathbb{N}}\subset W_{0}$ be a weakly convergent
sequence with weak limit $u$. Then, there exist two bounded measures $\mu$ and
$\nu$, an at most enumerable set of indices $I$ of distinct points
$\\{x_{i}\\}_{i\in I}\subset\mathbb{R}^{N}$, and positive real numbers
$\\{\mu_{i}\\}_{i\in I},\leavevmode\nobreak\ \\{\nu_{i}\\}_{i\in I}$, such
that the following convergence hold weakly in the sense of measures,

$\displaystyle\int_{\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dydx\rightharpoonup\mu\geq\displaystyle\int_{\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dydx+\sum_{i\in I}\mu_{i}\delta_{x_{i}},$ (2.12)
$\left|u_{n}(x)\right|^{p}dx\rightharpoonup\nu=\left|u(x)\right|^{p}dx+\sum_{i\in
I}\nu_{i}\delta_{x_{i}},$ (2.13) $S_{p}\leavevmode\nobreak\
\nu_{i}\leq\mu_{i},\leavevmode\nobreak\ \forall i\in I,$ (2.14)

where $S_{p}$ is the Sobolev constant given by (2.7) and $\delta_{x_{i}}$
denotes the Dirac mass at $x_{i}$. Moreover, if we define

$\mu_{\infty}=\displaystyle\lim_{R\to\infty}\limsup_{n\to\infty}\int_{|x|\geq
R}\displaystyle\int_{\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dydx,$ (2.15)
$\nu_{\infty}=\displaystyle\lim_{R\to\infty}\limsup_{n\to\infty}\int_{|x|\geq
R}\left|u_{n}(x)\right|^{p}dx,$ (2.16)

then

$\displaystyle\limsup_{n\to\infty}\int_{|x|\geq
R}\displaystyle\int_{\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dydx=\mu(\mathbb{R}^{N})+\mu_{\infty},$ (2.17)
$\displaystyle\limsup_{n\to\infty}\int_{|x|\geq
R}\left|u_{n}(x)\right|^{p}dx=\nu(\mathbb{R}^{N})+\nu_{\infty},$ (2.18)
$S_{p}\leavevmode\nobreak\ \nu_{\infty}\leq\mu_{\infty},$ (2.19)

This result was recently proved in the following mentioned papers of J. F.
Bonder and al. in [6, Theorem 1.1.] and K. Ho & Y. H. Kim in [16, Theorem
4.1.].

##  3 Main results

In this section, we establish the main results of this paper. the first result
is the following

######  Theorem 3.1.

Let $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Carathéodory function
such that condition (H1) holds. Moreover, assume that

  1. (H2)

there exist $\theta>p$ and $M>0$ such that

$0<\theta F(x,t)\leq tf(x,t),$

for each $x\in\Omega$ and $|t|\geq M$.

Then, for each $\lambda\in]0,\overline{\lambda}[$, the following initial case
of problem (1.2)

$\left\\{\begin{array}[]{ll}(-\Delta)_{p}^{s}u+\dfrac{|u|^{p-2}u}{|x|^{sp}}=\lambda
f(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{on }\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (P+)

admits at least two distinct weak solutions, where

$\overline{\lambda}\coloneqq\dfrac{q}{qa_{1}c_{1}p^{\frac{1}{p}}+a_{2}c^{q}_{q}p^{\frac{q}{p}}}.$

To achieve the proof of this result, we may split into a sequence of
propositions the proof of Theorem 3.1

######  Proposition 3.1.

For every $\lambda>0$ and the assumptions (H1), (H2) hold. Then,
$E_{\lambda}^{p,1}=\Phi_{p,1}-\lambda\Psi$ satisfies the Palais-Smale
condition.

Proof. To prove that $E_{\lambda}^{p,1}$ satisfies the Palais-Smale condition
for every $\lambda>0$. Namely, we need to show that any sequence of Palais-
Smale is bounded in $W_{0}$ and admits a convergent subsequence. We proceed by
steps.

Step 1. The sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded in $W_{0}$.
For $n$ large enough, by (a) in Definition 2.2, we have

$\displaystyle E_{\lambda}^{p,1}(u_{n})$
$\displaystyle=\Phi_{p,1}(u_{n})-\lambda\Psi(u_{n})$
$\displaystyle=\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+\int_{\Omega}\dfrac{|u_{n}(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)-\lambda\int_{\Omega}F(x,u_{n}(x))\leavevmode\nobreak\ dx$
$\displaystyle\leq d.$

In the other hand, by (H2) we have

$\displaystyle E_{\lambda}^{p,1}(u_{n})$
$\displaystyle\geq\dfrac{1}{p}\|u_{n}\|^{p}_{W_{0}}-\dfrac{\lambda}{\theta}\int_{\Omega}f(x,u_{n}(x))u_{n}(x)\leavevmode\nobreak\
dx$
$\displaystyle\geq\frac{1}{p}\|u_{n}\|^{p}_{W_{0}}-\frac{\lambda}{\theta}\int_{\Omega}f(x,u_{n}(x))u_{n}(x)\leavevmode\nobreak\
dx+\frac{1}{\theta}\|u_{n}\|^{p}_{W_{0}}-\frac{1}{\theta}\|u_{n}\|^{p}_{W_{0}}$
$\displaystyle>\left(\frac{1}{p}-\frac{1}{\theta}\right)\|u_{n}\|^{p}_{W_{0}}+\frac{1}{\theta}\left(\|u_{n}\|^{p}_{W_{0}}-\lambda\int_{\Omega}f(x,u_{n}(x))u_{n}(x)\leavevmode\nobreak\
dx\right)$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{\theta}\right)\|u_{n}\|^{p}_{W_{0}}+\frac{1}{\theta}\langle\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n}),u_{n}\rangle.$

Due to (b) in Definition 2.2, we have $\theta>p>1$ and $\varepsilon\to 0$ such
that

$\|\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n})\|_{W_{0}^{*}}\leq\varepsilon$

and

$\displaystyle-\left|\frac{1}{\theta}\right|\langle\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n}),u_{n}\rangle$
$\displaystyle\leq\|\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n})\|_{W_{0}^{*}}\|u_{n}\|_{W_{0}}$
$\displaystyle\leq\varepsilon\|u_{n}\|_{W_{0}}.$

Thus,

$\left(\frac{1}{p}-\frac{1}{\theta}\right)\|u\|^{p}_{W_{0}}\leq
E_{\lambda}^{p,1}(u_{n})-\frac{1}{\theta}\langle\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n}),u_{n}\rangle\leq
d+\varepsilon\|u_{n}\|_{W_{0}}.$

It follows from this inequality that $\\{u_{n}\\}$ is bounded in $W_{0}$.

Step 2. the sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ possesses a convergent
subsequence.
By the Eberlian-Smulyan theorem (see [30, Theorem 21.D.]), passing to a
subsequence if necessary, we can assume that $u_{n}\rightharpoonup u$. Then,
because of the compactness of $\Psi^{{}^{\prime}}$ thanks to condition (H1)
and to the compact embedding $W_{0}\hookrightarrow L^{q}(\Omega)$, for every
$q\in[1,p^{*})$ we have

$\Psi^{{}^{\prime}}(u_{n})\longrightarrow\Psi^{{}^{\prime}}(u),$

Since

$\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n})=\Phi_{p,1}^{{}^{\prime}}(u_{n})-\lambda\Psi^{{}^{\prime}}(u_{n})\longrightarrow
0,$

then

$\Phi_{p,1}^{{}^{\prime}}(u_{n})=\left(E_{\lambda}^{p,1}\right)^{{}^{\prime}}(u_{n})+\lambda\Psi^{{}^{\prime}}(u_{n})\longrightarrow
0+\lambda\Psi^{{}^{\prime}}(u).$

Moreover, as $\Phi_{p,1}^{{}^{\prime}}$ is a homeomorphism according to Lemma
2.1, then $u_{n}\rightarrow u$ in $W_{0}$ and so $E_{\lambda}^{p,1}$ satisfies
the (PS)-condition. $\square$

######  Lemma 3.1.

Assume that $f$ satisfies (H2). Then, there exist a positive $c$ constant such
that

$F(x,t)\geq c|t|^{\theta},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
x\in\Omega,\leavevmode\nobreak\ \leavevmode\nobreak\ |t|>M.$ (3.20)

Proof. We start by setting $a(x)\coloneqq\displaystyle\min_{|\xi|=M}F(x,\xi)$
and

$\varphi_{t}(s)\coloneqq F(x,st),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall s>0.$ (3.21)

By the assumption (H2), we have for every $x\in\Omega$ and $|t|>M$ such that

$0<\theta\varphi_{t}(s)=\theta F(x,st)\leq
stf(x,st)=s\varphi_{t}^{{}^{\prime}}(s),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall s>\frac{M}{|t|}.$

Therefore,

$\theta\varphi_{t}(s)\leq s\varphi_{t}^{{}^{\prime}}(s).$

Also,

$\int^{1}_{\frac{M}{|t|}}\frac{\theta}{s}\leavevmode\nobreak\
ds\leq\int^{1}_{\frac{M}{|t|}}\frac{\varphi_{t}^{{}^{\prime}}(s)}{\varphi_{t}(s)}\leavevmode\nobreak\
ds.$

We see that,

$\int^{1}_{\frac{M}{|t|}}\left(\ln|s|^{\theta}\right)^{{}^{\prime}}\leavevmode\nobreak\
ds\leq\int^{1}_{\frac{M}{|t|}}\left(\ln\left|\varphi_{t}(s)\right|\right)^{{}^{\prime}}\leavevmode\nobreak\
ds,$

$\left[\ln|s|^{\theta}\right]^{1}_{\frac{M}{|t|}}\leq\left[\ln\left|\varphi_{t}(s)\right|\right]^{1}_{\frac{M}{|t|}},$

$-\ln\left[\frac{M}{|t|}\right]^{\theta}\leq\ln\left|\varphi_{t}(1)\right|-\ln\left|\varphi_{t}\left(\frac{M}{|t|}\right)\right|.$

Then,

$\ln\varphi_{t}\left(\frac{M}{|t|}\right)-\ln\frac{M^{\theta}}{|t|^{\theta}}\leq\ln\varphi_{t}(1).$

We have

$-\ln\frac{M^{\theta}}{|t|^{\theta}}=-\left[\ln
M^{\theta}-\ln|t|^{\theta}\right]=\ln|t|^{\theta}-\ln
M^{\theta}=\ln\frac{|t|^{\theta}}{M^{\theta}}.$

So,

$\ln\varphi_{t}\left(\frac{M}{|t|}\right)+\ln\frac{|t|^{\theta}}{M^{\theta}}\leq\ln\varphi_{t}(1),$

$e^{\ln\varphi_{t}\left(\frac{M}{|t|}\right)+\ln\frac{|t|^{\theta}}{M^{\theta}}}=e^{\ln\left[\varphi_{t}\left(\frac{M}{|t|}\right)\frac{|t|^{\theta}}{M^{\theta}}\right]}\leq
e^{\ln\varphi_{t}(1)}.$

Thus,

$\varphi_{t}\left(\frac{M}{|t|}\right)\frac{|t|^{\theta}}{M^{\theta}}\leq\varphi_{t}(1).$

Taking into account of (3.21), we obtain

$c|t|^{\theta}\leq a(x)\frac{|t|^{\theta}}{M^{\theta}}\leq
F(x,\frac{M}{|t|}t)\frac{|t|^{\theta}}{M^{\theta}}\leq F(x,t),$

where $c\geq 0$ is a constant. Thus, (3.20) is proved. $\square$

Now, we prove the following Proposition 3.2 using the previous proved Lemma
3.1.

######  Proposition 3.2.

For every $\lambda>0$ and the assumption (H2) hold. Then, $E_{\lambda}^{p,1}$
is unbounded from below.

Proof. We fix $u_{0}\in W_{0}\backslash\\{0\\}$ and for each $t>1$, we have

$\displaystyle E_{\lambda}^{p,1}(tu_{0})$
$\displaystyle=\Phi_{p,1}(tu_{0})-\lambda\Psi(tu_{0})$
$\displaystyle=\dfrac{t^{p}}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u_{0}(x)-u_{0}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+\int_{\Omega}\dfrac{|u_{0}(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)-\lambda\int_{\Omega}F(x,u_{0}(x))\leavevmode\nobreak\ dx.$

So, from Lemma 3.1 and Hardy’s inequality (2.6) one has

$\displaystyle E_{\lambda}^{p,1}(tu_{0})$
$\displaystyle\leq\dfrac{t^{p}(c_{H}+1)}{p}\|u_{0}\|^{p}_{W_{0}}-\lambda\int_{\Omega}c\left|tu_{0}(x)\right|^{\theta}\leavevmode\nobreak\
dx$ $\displaystyle\leq\dfrac{t^{p}(c_{H}+1)}{p}\|u_{0}\|^{p}_{W_{0}}-\lambda
ct^{\theta}\displaystyle\int_{\Omega}\left|u_{0}(x)\right|^{\theta}\leavevmode\nobreak\
dx$ $\displaystyle<0$

The assumption (H2) ensures that $\theta>p$, this condition guarantees that

$E_{\lambda}^{p,1}(tu_{0})\longrightarrow-\infty,\leavevmode\nobreak\ \text{
as }t\to+\infty,$

which lead us to deduce that $E_{\lambda}^{p,1}$ is unbounded from below.
$\square$

Proof of Theorem 3.1. Setting $\lambda\in]0,\overline{\lambda}[$, we seek to
apply Theorem 2.1 to problem (P+) in the case $r=1$ to the space $X\coloneqq
W_{0}$ and to the functionals

$\Phi_{p,1}(u)\coloneqq\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx\right)$

and

$\Psi(u)\coloneqq\int_{\Omega}F(x,u(x))\leavevmode\nobreak\ dx.$

The functional $\Phi_{p,1}$ is continuous and
$\Phi^{{}^{\prime}}_{p,1}:W_{0}\rightarrow W_{0}^{*}$ is a homeomorphism
according to Proposition 2.1. Moreover, thanks to condition (H1) and to the
compact embedding $W_{0}\hookrightarrow L^{q}(\Omega)$, for every
$q\in[1,p^{*})$. Then, $\Psi$ is continuous and has a compact derivative.

Via Proposition 3.1 and Proposition 3.2, we established that
$E_{\lambda}^{p,1}$ is unbounded from below and satisfies the Palais-Smale
condition. Therefore, we can apply our main tool Theorem 2.1, and it only
remains to prove that $\lambda\in\leavevmode\nobreak\
]0,\overline{\lambda}[\leavevmode\nobreak\ \subseteq\leavevmode\nobreak\
]0,\frac{1}{\sup_{\\{\Phi_{p,1}(u)<1\\}}\Psi(u)}[$ where
$\overline{\lambda}\coloneqq\frac{q}{qa_{1}c_{1}p^{\frac{1}{p}}+a_{2}c^{q}_{q}p^{\frac{q}{p}}}$.

From (2.11), we have

$\dfrac{1}{p}\|u\|^{p}_{W_{0}}\leq\Phi_{p,1}(u)<r,\text{ such that
}u\in\Phi_{p,1}^{-1}(]-\infty,r[).$

For $r=1$, we get

$\|u\|^{p}_{W_{0}}<p.$

Then, for each $u\in W_{0}$

$\|u\|_{W_{0}}<p^{\frac{1}{p}},\text{ such that,
}u\in\Phi_{p,1}^{-1}(]-\infty,1[).$ (3.22)

Moreover, according to (H1), we have

$\displaystyle|F(x,t)|$
$\displaystyle=\left|\int_{0}^{t}f(x,s)\leavevmode\nobreak\ ds\right|$
$\displaystyle\leq\alpha\left|\int_{0}^{t}\leavevmode\nobreak\
ds\right|+\beta\left|\int_{0}^{t}|s|^{q-1}\leavevmode\nobreak\ ds\right|$
$\displaystyle=\alpha|t|+\beta\dfrac{|t|^{q}}{q}.$

Then

$\displaystyle\Psi(u)$ $\displaystyle=\int_{\Omega}|F(x,u(x))|dx$
$\displaystyle\leq\int_{\Omega}\left[\alpha|u(x)|+\beta\dfrac{|u(x)|^{q}}{q}\right]\leavevmode\nobreak\
dx$ $\displaystyle\leq\int_{\Omega}\alpha|u(x)|\leavevmode\nobreak\
dx+\dfrac{\beta}{q}\int_{\Omega}|u(x)|^{q}\leavevmode\nobreak\ dx$
$\displaystyle=\alpha\|u\|_{L^{1}(\Omega)}+\dfrac{\beta}{q}\|u\|_{L^{q}(\Omega)}^{q}$

Using the compact embedding $W_{0}\hookrightarrow L^{q}(\Omega)$ for every
$q\in[1,p^{*})$ and for each $u\in\Phi_{p,1}^{-1}(]-\infty,1[)$, we have

$\Psi(u)\leq\alpha
c_{1}\|u\|_{W_{0}}+\beta\left(c_{q}\|u\|_{W_{0}}\right)^{q}.$

And by (3.22), we obtain

$\displaystyle\Psi(u)$ $\displaystyle<\alpha
c_{1}p^{\frac{1}{p}}+\dfrac{\beta}{q}c_{q}^{q}p^{\frac{q}{p}}$
$\displaystyle=\dfrac{q\alpha c_{1}p^{\frac{1}{p}}+\beta
c_{q}^{q}p^{\frac{q}{p}}}{q}.$

Since $\lambda\in]0,\overline{\lambda}[$, we get

$\sup_{\\{u\in\Phi_{p,1}^{-1}(]-\infty,1[)\\}}\Psi(u)<\dfrac{q\alpha
c_{1}p^{\frac{1}{p}}+\beta
c_{q}^{q}p^{\frac{q}{p}}}{q}\eqqcolon\dfrac{1}{\overline{\lambda}}<\dfrac{1}{\lambda},$

from, the latter one has

$0<\lambda<\overline{\lambda}\coloneqq\dfrac{q}{q\alpha
c_{1}p^{\frac{1}{p}}+\beta
c_{q}^{q}p^{\frac{q}{p}}}<\dfrac{1}{\sup_{\\{\Phi_{p,1}(u)<1\\}}\Psi(u)}.$

Then,

$\lambda\in]0,\overline{\lambda}[\subseteq\left]0,\dfrac{1}{\sup_{\\{\Phi_{p,1}(u)<1\\}}\Psi(u)}\right[.$

Now that all hypotheses of Theorem 2.1 are verified. We conclude that for each
$\lambda\in]0,\overline{\lambda}[$, the functional $E_{\lambda}^{p,1}$ admits
two distinct critical points that are weak solutions of problem (P+).
$\square$

######  Remark 3.1.

Remark that Theorem 3.1 confirm the existence of two positive weak solutions
for problem (P+), if the function $f$ is positive and $f(x,0)\neq 0$ in
$\Omega$.

The second result of this work is the following

######  Theorem 3.2.

Let $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a function with
$f(x,0)\neq 0$ in $\Omega$ satisfying condition (H1). Then, there exists a
positive number $\Lambda$ given by

$\Lambda\coloneqq q\sup_{\\{\rho>0\\}}\left\\{\dfrac{\rho^{p-1}}{q\alpha
c_{1}\left(\frac{p}{1-c_{H}}\right)^{\frac{1}{p}}+\beta
c_{q}^{q}\left(\frac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho^{q-1}}\right\\},$

such that, for every $\lambda\in]0,\Lambda[$, the following second case of
problem (1.2)

$\left\\{\begin{array}[]{ll}(-\Delta)_{p}^{s}u-\dfrac{|u|^{p-2}u}{|x|^{sp}}=\lambda
f(x,u)&\quad\mbox{in }\ \Omega\\\\[8.5359pt] u=0&\quad\mbox{in }\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ (P-)

admits at least one non-trivial weak solution $u_{\lambda}\in W_{0}$.
Moreover,

$\lim_{\lambda\rightarrow 0^{+}}\|u_{\lambda}\|_{W_{0}}=0$

and the function $g(\lambda)\coloneqq E_{\lambda}^{p,-1}(u_{\lambda})$ is
negative and strictly decreasing in $]0,\Lambda[$.

To achieve the proof of this result, we need to prove the following Lemmas:

######  Lemma 3.2.

Let $s\in(0,1)$ and $N>ps$. Then, the functional $\Phi_{p,-1}$ is coercive and
sequentially weakly lower semicontinious on $W_{0}$, i.e:

$\Phi_{p,-1}(u)\leq\liminf_{n\to+\infty}\Phi_{p,-1}(u_{n})\leavevmode\nobreak\
\text{ if }\leavevmode\nobreak\ u_{n}\longrightarrow u\leavevmode\nobreak\
\mbox{ weakly in }\leavevmode\nobreak\ W_{0}.$

Proof. Using the fractional Hardy inequality (2.6), we obtain that for any
$u\in W_{0}$,

$\displaystyle\Phi_{p,-1}(u)$
$\displaystyle=\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx\right)$
$\displaystyle\geq\left(\dfrac{1-c_{H}}{p}\right)\|u\|^{p}_{W_{0}}.$

We conclude that

$\Phi_{p,-1}(u)\longrightarrow+\infty,\text{ as
}\|u\|^{p}_{W_{0}}\longrightarrow+\infty,$

which means that $\Phi_{p,-1}(u)$ is coercive on $W_{0}$.

Now, by [14, Theorem 6.] we know that $\mathbb{C}_{c}^{\infty}(\Omega)$ is
dense subset of $W_{0}$. Hence, using density arguments, to prove that
$\Phi_{p,-1}$ is sequentially weakly lower semicontinious on $W_{0}$, it is
enough to show that the functional

$\Phi_{p,-1}\text{ is sequentially weakly lower semicontinious on
}\mathbb{C}_{c}^{\infty}(\Omega).$ (3.23)

So, let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be a sequence in
$\mathbb{C}_{c}^{\infty}(\Omega)$ such that

$\begin{array}[]{ll}u_{n}\longrightarrow u&\mbox{weakly in
}\leavevmode\nobreak\ W_{0},\mbox{ as }n\rightarrow+\infty,\end{array}$ (3.24)

Thus, according to Theorem 2.3 there exist two bounded measures $\mu$ and
$\nu$, an at most enumerable set of indices $I$ of distinct points
$\\{x_{i}\\}_{i\in I}\subset\overline{\Omega}$, and positive real numbers
$\\{\mu_{i}\\}_{i\in I},\leavevmode\nobreak\ \\{\nu_{i}\\}_{i\in I}$, such
that the following convergence hold weakly in the sense of measures,

$\displaystyle\int_{\mathbb{R}^{N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dydx\rightharpoonup\mu\geq\displaystyle\int_{\mathbb{R}^{N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dydx+\sum_{i\in I}\mu_{i}\delta_{x_{i}},$ (3.25)
$\left|u_{n}(x)\right|^{p}\rightharpoonup\nu=\left|u(x)\right|^{p}dx+\sum_{i\in
I}\nu_{i}\delta_{x_{i}},$ (3.26)

and finally,

$S_{p}\leavevmode\nobreak\ \nu_{i}\leq\mu_{i},\leavevmode\nobreak\ \forall
i\in I,$ (3.27)

where $S_{p}$ is the Sobolev constant given by (2.7) and $\delta_{x_{i}}$
denotes the Dirac mass at $x_{i}$.

From the continuity of the embedding $W_{0}\hookrightarrow L^{p}(\Omega)$, for
every $p\in[1,p^{*}]$, we hve that

$\begin{array}[]{ll}u_{n}\longrightarrow u&\mbox{ stronly in
}\leavevmode\nobreak\ L^{p}(\Omega),\mbox{ as
}n\rightarrow+\infty.\end{array}$

By (3.25), (3.26) and (3.27) we get

$\displaystyle\liminf_{n\to+\infty}\Phi_{p,-1}(u_{n})$
$\displaystyle=\liminf_{n\to+\infty}\left[\dfrac{1}{p}\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\dfrac{1}{p}\int_{\Omega}\dfrac{|u_{n}(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right]$
$\displaystyle\geq\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy+\sum_{i=1}^{k}\mu_{i}\right)-\dfrac{1}{p}\left(\int_{\Omega}\dfrac{|u_{n}(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx+\sum_{i=1}^{k}\nu_{i}\right)$
$\displaystyle=\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx\right)+\dfrac{1}{p}\sum_{i=1}^{k}\mu_{i}-\dfrac{1}{p}\sum_{i=1}^{k}\nu_{i}$
$\displaystyle\geq\Phi_{p,-1}(u)+\dfrac{1}{p}\sum_{i=1}^{k}\mu_{i}-\dfrac{1}{p}\sum_{i=1}^{k}\dfrac{\mu_{i}}{S_{p}}$
$\displaystyle=\Phi_{p,-1}(u)+\left(\dfrac{1-S_{p}^{-1}}{p}\right)\sum_{i=1}^{k}\mu_{i}$
$\displaystyle\geq\Phi_{p,-1}(u).$

This lead us to deduce statement stated in (3.23).

Now, let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be a sequence in $W_{0}$ satisfying
the same condition (3.24). Then, using density arguments, we have for any
$n\in\mathbb{N}$ there exists
$\\{u_{n}^{j}\\}_{j\in\mathbb{N}}\in\mathbb{C}_{c}^{\infty}(\Omega)$ such that

$\begin{array}[]{ll}u_{n}^{j}\longrightarrow u_{n}&\mbox{strongly in
}\leavevmode\nobreak\ W_{0},\mbox{ as }j\rightarrow+\infty.\end{array}$ (3.28)

From (3.24) and (3.28), we have that for any $\varphi\in W_{0}$

$\langle u_{n}^{j}-u,\varphi\rangle=\langle
u_{n}^{j}-u_{n},\varphi\rangle+\langle u_{n}-u,\varphi\rangle\rightarrow
0,\mbox{ as }n,j\rightarrow+\infty.$

Then,

$\begin{array}[]{ll}u_{n}^{j}\longrightarrow u&\mbox{weakly in
}\leavevmode\nobreak\ W_{0},\mbox{ as }n,j\rightarrow+\infty.\end{array}$
(3.29)

Since $\\{u_{n}^{j}\\}_{j\in\mathbb{N}}\in\mathbb{C}_{c}^{\infty}(\Omega)$ and
the statment (3.23) is satisfied, we deduce that

$\liminf_{n,j\to+\infty}\Phi_{p,-1}(u_{n}^{j})\geq\Phi_{p,-1}(u).$ (3.30)

Moreover, by (3.28) it is easy to see that for any $n\in\mathbb{N}$ we have

$\lim_{j\to+\infty}\Phi_{p,-1}(u_{n}^{j})=\Phi_{p,-1}(u_{n}),$

so that, passing to $\liminf$ we get

$\liminf_{n,j\to+\infty}\Phi_{p,-1}(u_{n}^{j})=\liminf_{n\to+\infty}\lim_{j\to+\infty}\Phi_{p,-1}(u_{n}^{j})=\liminf_{n\to+\infty}\Phi_{p,-1}(u_{n})$
(3.31)

By (3.30) and (3.31) we get that

$\liminf_{n\to+\infty}\Phi_{p,-1}(u_{n})\geq\Phi_{p,-1}(u)$

Therefore, $\Phi_{p,-1}$ is sequentially weakly lower semicontinious on
$W_{0}$. $\square$

######  Lemma 3.3.

Let (H1) be satisfied. Then, the functional $\Psi$ is sequentially weakly
continuous.

Proof. let $\\{u_{n}\\}_{n\in\mathbb{N}}$ is bounded in $W_{0}$ and $W_{0}$ is
a reflexive space. Then, up to a subsequence denoted by $u_{n}$, there exists
$u\in W_{0}$, such that

$\begin{array}[]{ll}u_{n}\longrightarrow u&\mbox{ weakly in
}\leavevmode\nobreak\ W_{0},\\\ u_{n}\longrightarrow u&\mbox{ stronly in
}\leavevmode\nobreak\ L^{p}(\Omega),\\\ u_{n}(x)\longrightarrow u(x)&\mbox{
a.e. in }\leavevmode\nobreak\ \Omega,\end{array}$

Now, by assumption (H1) we have that

$\displaystyle|F(x,t)|$
$\displaystyle=\left|\int_{0}^{t}f(x,s)\leavevmode\nobreak\ ds\right|$
$\displaystyle\leq\alpha\left|\int_{0}^{t}\leavevmode\nobreak\
ds\right|+\beta\left|\int_{0}^{t}|s|^{q-1}\leavevmode\nobreak\ ds\right|$
$\displaystyle=\alpha|t|+\beta\dfrac{|t|^{q}}{q}.$

further, according to the compact embedding $W_{0}\hookrightarrow
L^{q}(\Omega)$ for every $q\in[1,p^{*})$, we get

$|\Psi(u_{n})|\leq\int_{\Omega}|F(x,u_{n})|\leavevmode\nobreak\ dx\leq\alpha
c_{1}\|u_{n}\|_{W_{0}}+\beta\left(c_{q}\|u_{n}\|_{W_{0}}\right)^{q}<+\infty,$

Thus, we apply the Lebesgue dominated convergence Theorem, we obtain

$\displaystyle\lim_{n\rightarrow
0}\int_{\Omega}\Psi(u_{n})\leavevmode\nobreak\
dx=\int_{\Omega}\Psi(u)\leavevmode\nobreak\ dx.$

Then, the functional $\Psi$ is weakly semicontinuous. $\square$

Proof of Theorem 3.2. We set $\lambda\in]0,\Lambda[$. In order to apply
Theorem 2.2 to problem (P-) with the space $X=W_{0}$ and to the functionals

$\Phi_{p,-1}(u)\coloneqq\dfrac{1}{p}\left(\displaystyle\int_{\mathbb{R}^{2N}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\leavevmode\nobreak\
dxdy-\int_{\Omega}\dfrac{|u(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\ dx\right)$

and

$\Psi(u)\coloneqq\int_{\Omega}F(x,u(x))\leavevmode\nobreak\ dx.$

In view of Proposition 2.1 and Lemma 3.2, the functional $\Phi_{p,-1}$ is
continuous, coercive and sequentially weakly lower semicontinious, also its
$\displaystyle\inf_{u\in X}\Phi_{p,-1}(u)=0$. Moreover, the functional $\Psi$
is continuous, has a compact derivative and is sequentially weakly continuous
according to Lemma 3.3. We prove the theorem in the following steps.

Step 1. We start by proving that problem (P-) admits at least one non-trivial
weak solution $u_{\lambda}\in W_{0}$.
By (H1), we have

$\displaystyle|F(x,t)|$
$\displaystyle\leq\alpha|t|+\beta\dfrac{|t|^{q}}{q},\leavevmode\nobreak\
\forall(x,t)\in\Omega\times\mathbb{R}.$ (3.32)

Using the above inequality, we obtain

$\displaystyle\Psi(u)$ $\displaystyle=\int_{\Omega}|F(x,u(x))|dx$
$\displaystyle\leq\int_{\Omega}\left[\alpha|u(x)|+\beta\dfrac{|u(x)|^{q}}{q}\right]\leavevmode\nobreak\
dx$ $\displaystyle\leq\int_{\Omega}\alpha|u(x)|\leavevmode\nobreak\
dx+\dfrac{\beta}{q}\int_{\Omega}|u(x)|^{q}\leavevmode\nobreak\ dx$
$\displaystyle=\alpha\|u\|_{L^{1}(\Omega)}+\dfrac{\beta}{q}\|u\|_{L^{q}(\Omega)}^{q}.$

According to the compact embedding $W_{0}\hookrightarrow L^{q}(\Omega)$, for
every $q\in[1,p^{*})$, we have

$\displaystyle\Psi(u)$ $\displaystyle\leq\alpha
c_{1}\|u\|_{W_{0}}+\beta\left(c_{q}\|u\|_{W_{0}}\right)^{q}.$

On the other hand, we get from (2.1) that

$\|u\|_{W_{0}}<\left(\dfrac{pr}{1-c_{H}}\right)^{\frac{1}{p}},\leavevmode\nobreak\
\forall u\in W_{0},\leavevmode\nobreak\ \Phi_{p,-1}(u)<r.$ (3.33)

Now, from (3.33), one has

$\displaystyle\Psi(u)$ $\displaystyle<\alpha
c_{1}\left(\dfrac{pr}{1-c_{H}}\right)^{\frac{1}{p}}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{pr}{1-c_{H}}\right)^{\frac{q}{p}},$

for every $u\in W_{0}$ such that $\Phi_{p,-1}(u)<r$.

Then,

$\sup_{\\{u\in\Phi_{p,-1}^{-1}(]-\infty,r[)\\}}\Psi(u)<\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}r^{\frac{1}{p}}+\beta\dfrac{c_{q}^{q}}{p}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}r^{\frac{q}{q}}.$

Hence, for every $r\in]0,+\infty[$ we have

$\dfrac{\left(\sup_{\\{u\in\Phi_{p,-1}^{-1}(]-\infty,r[)\\}}\Psi(u)\right)}{r}\leq\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}r^{\frac{1-p}{p}}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}r^{\frac{q-p}{p}}.$

In particular, for $r=\rho^{p}$ we have

$\dfrac{\left(\sup_{\\{u\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)\\}}\Psi(u)\right)}{\rho^{p}}\leq\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}\rho^{1-p}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho^{q-p}.$
(3.34)

Now, setting the function $u_{0}\in W_{0}$ such that

$u_{0}\in\Phi_{p,-1}^{-1}(]-\infty,\gamma^{p}[).$ (3.35)

Also, we observe that

$\Phi_{p,-1}(u_{0})=\dfrac{1}{p}\|u_{0}\|^{p}_{W_{0}}+\dfrac{1}{p}\int_{\Omega}\dfrac{|u_{0}(x)|^{p}}{|x|^{sp}}\leavevmode\nobreak\
dx=0$ (3.36)

and

$\Psi(u_{0})=\int_{\Omega}F(x,u_{0}(x))\leavevmode\nobreak\ dx=0.$ (3.37)

Therefore, from (3.35), (3.36) and (3.37), we obtain that

$\displaystyle\varphi(\rho^{p})$
$\displaystyle\coloneqq\displaystyle\inf_{u\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)}\dfrac{\left(\sup_{\\{v\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)\\}}\Psi(v)\right)-\Psi(u)}{\rho^{p}-\Phi_{p,-1}(u)}$
$\displaystyle\leq\dfrac{\left(\sup_{\\{v\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)\\}}\Psi(v)\right)-\Psi(u_{0})}{\rho^{p}-\Phi_{p,-1}(u_{0})}$
$\displaystyle=\dfrac{\left(\sup_{\\{v\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)\\}}\Psi(v)\right)}{\rho^{p}}.$

Thus, by (3.34)we have

$\displaystyle\varphi(\rho^{p})$ $\displaystyle\leq\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}\rho^{1-p}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho^{q-p}.$

Furthermore, since $0<\lambda<\Lambda$

$\varphi(\rho^{p})\leq\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}\rho^{1-p}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho^{q-p}\eqqcolon\dfrac{1}{\Lambda(\rho)}<\dfrac{1}{\lambda}.$

So,

$0<\lambda<\Lambda(\rho)\coloneqq\dfrac{q\rho^{p-1}}{q\alpha
c_{1}\left(\frac{p}{1-c_{H}}\right)^{\frac{1}{p}}+\beta
c_{q}^{q}\left(\frac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho^{q-1}}\leq\dfrac{1}{\varphi(\rho^{p})}.$

Then,

$\lambda\in\left]0,\Lambda(\rho)\right[\subseteq\left]1,\dfrac{1}{\varphi(\rho^{p})}\right[$

In conclusion, according to Theorem 2.2, there exists a critical point
$u_{\lambda}\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)$ for $E_{\lambda}^{p,-1}$
in $W_{0}$ which is a global minimum of the restriction $E_{\lambda}^{p,-1}$
to $\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)$. Moreover, the function
$u_{\lambda}\neq 0$ since $f(x,0)\neq 0$ in $\Omega$.

Step 2. We show that $\lim_{\lambda\to 0^{+}}\|u_{\lambda}\|_{W_{0}}=0$ and
that the functional $g(\lambda)\coloneqq E_{\lambda}^{p,-1}(u_{\lambda})$ is
negative and strictly decreasing in $\left]0,\Lambda(\rho)\right[$.
As $\Phi_{p,-1}$ is coercive, then
$u_{\lambda}\in\Phi_{p,-1}^{-1}(]-\infty,\rho^{p}[)$ is bounded in $W_{0}$,
that is to say

$\|u_{\lambda}\|_{W_{0}}\leq K,\leavevmode\nobreak\ \text{for }K>0$

Thus, due to the compactness of the operator $\Psi^{{}^{\prime}}$, there
exists a constant $C>0$ such that

$\left|\langle\Psi^{{}^{\prime}}(u_{\lambda}),u_{\lambda}\rangle\right|\leq\|\Psi^{{}^{\prime}}(u_{\lambda})\|_{W_{0}^{*}}\|u_{\lambda}\|_{W_{0}}<CK^{2},\leavevmode\nobreak\
\forall\lambda\in\left]0,\Lambda(\rho)\right[.$ (3.38)

On the other hand, since $u_{\lambda}$ is a critical point of
$E_{\lambda}^{p,-1}$, then

$\langle\left(E_{\lambda}^{p,-1}\right)^{{}^{\prime}}(u_{\lambda}),u_{\lambda}\rangle=0,$

which implies that

$\langle\Phi_{p,-1}^{{}^{\prime}}(u_{\lambda})-\lambda\Psi^{{}^{\prime}}(u_{\lambda}),u_{\lambda}\rangle=0,\leavevmode\nobreak\
\forall\lambda\in\left]0,\Lambda(\rho)\right[.$

So,

$p\Phi_{p,-1}(u_{\lambda})=\langle\Phi_{p,-1}^{{}^{\prime}}(u_{\lambda}),u_{\lambda}\rangle=\lambda\langle\Psi^{{}^{\prime}}(u_{\lambda}),u_{\lambda}\rangle,\leavevmode\nobreak\
\forall\lambda\in\left]0,\Lambda(\rho)\right[.$ (3.39)

Therefore, by (3.38) and (3.39), we get

$\lim_{\lambda\rightarrow
0^{+}}p\Phi_{p,-1}(u_{\lambda})=\lim_{\lambda\rightarrow
0^{+}}\lambda\langle\Psi^{{}^{\prime}}(u_{\lambda}),u_{\lambda}\rangle=0,\leavevmode\nobreak\
\forall p<1.$ (3.40)

Moreover, by (2.1) one has

$\|u_{\lambda}\|^{p}_{W_{0}}\leq\dfrac{p\Phi_{p,-1}(u_{\lambda})}{1-c_{H}},\leavevmode\nobreak\
\forall\lambda\in\left]0,\Lambda(\rho)\right[.$ (3.41)

Then, we conclude by the conditions (3.40) and (3.41) that

$\lim_{\lambda\rightarrow 0^{+}}\|u_{\lambda}\|_{W_{0}}=0$

Furthermore, since the restriction $E_{\lambda}^{p,-1}$ to
$\Phi_{p,-1}^{-1}(]-\infty,\gamma^{p}[)$ admits a global minimum, which is a
local minimum of $E_{\lambda}^{p,-1}$ in $W_{0}$, the map $g(\lambda)\coloneqq
E_{\lambda}^{p,-1}(u_{\lambda})$ is negative in
$\left]0,\Lambda(\rho)\right[$, because $u_{\lambda}\neq 0$ and
$E_{\lambda}^{p,-1}(0)=0$.

Finally, we prove that the function $g(\lambda)\coloneqq
E_{\lambda}^{p,-1}(u_{\lambda})$ is negative and strictly decreasing in
$]0,\Lambda[$, by observing that

$E_{\lambda}^{p,-1}(u)=\lambda\left(\dfrac{\Phi_{p,-1}(u)}{\lambda}-\Psi(u)\right).$

Now, we assume $u_{\lambda_{1}},u_{\lambda_{2}}\in W_{0}$ are critical points
of $E_{\lambda}^{p,-1}$, for every
$\lambda_{1},\lambda_{2}\in\left]0,\Lambda(\rho)\right[$, with
$\lambda_{1}<\lambda_{2}$. Further, we set

$I_{\lambda_{i}}\coloneqq\inf_{u\in\Phi_{p,-1}^{-1}(]-\infty,\gamma^{p}[)}\left(\dfrac{\Phi_{p,-1}(u)}{\lambda_{i}}-\Psi(u)\right)=\dfrac{1}{\lambda_{i}}E_{\lambda_{i}}^{p,-1}(u_{\lambda_{i}}),\leavevmode\nobreak\
i=1,2.$

Obviously, as mentioned earlier $I_{\lambda_{i}}<0$ for $i=1,2$, and since
$\lambda_{1}<\lambda_{2}$, we have $I_{\lambda_{2}}\leq I_{\lambda_{1}}$.
Therefore,

$E_{\lambda_{2}}^{p,-1}(u_{\lambda_{2}})=\lambda_{2}I_{\lambda_{2}}\leq\lambda_{2}I_{\lambda_{1}}<\lambda_{1}I_{\lambda_{1}}=E_{\lambda_{1}}^{p,-1}(u_{\lambda_{1}}).$

We conclude that, as $\lambda\in\left]0,\Lambda\right[$ is arbitrary, the
above conclusions are still true in $\left]0,\Lambda\right[$. Hence the proof
is completed. $\square$

######  Remark 3.2.

To calculate the maximum of $\Lambda$, we need to look at its first derivative

$\Lambda^{{}^{\prime}}(\rho)=-\left[\dfrac{\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}(1-p)\rho^{-p}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}(q-p)\rho^{q-p-1}}{\left(\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}\rho^{1-p}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho^{q-p}\right)^{2}}\right].$

Further, we set $\Lambda^{{}^{\prime}}(\rho)$ equal to zero and obtain

$\alpha
c_{1}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{1}{p}}(1-p)\rho^{-p}+\beta\dfrac{c_{q}^{q}}{q}\left(\dfrac{p}{1-c_{H}}\right)^{\frac{q}{p}}(q-p)\rho^{q-p-1}=0.$

Then,

$\rho_{\max}\coloneqq\left(\dfrac{p}{1-c_{H}}\right)^{-\frac{1}{p}}\left[q\dfrac{\alpha
c_{1}}{\beta c_{q}^{q}}\left(\dfrac{1-p}{p-q}\right)\right]^{\frac{1}{q-1}}.$

Which lead us to conclude that $\Lambda$ is defined as follows

$\Lambda(\rho)=\left\\{\begin{array}[]{ll}+\infty&\quad\mbox{if }\ 1<q<p\\\
\frac{1-c_{H}}{\beta c_{q}^{q}}&\quad\mbox{if }\ q=p\\\
\frac{q\rho_{\max}^{p-1}}{q\alpha
c_{1}\left(\frac{p}{1-c_{H}}\right)^{\frac{1}{p}}+\beta
c_{q}^{q}\left(\frac{p}{1-c_{H}}\right)^{\frac{q}{p}}\rho_{\max}^{q-1}}&\quad\mbox{if
}\ q\in]p,p^{*}[.\end{array}\right.$

Notice from the latter that if $f$ satisfy the condition (H1) at infinity,
with $q\in]1,p[$. Then, from Theorem 3.2 we confirm that for each $\lambda>0$,
our problem (1.3) admits at least a non-trivial weak solution.

## References

  * [1] H. Achour and S. Bensid, On a fractional $p$-Laplacian problem with discontinuous nonlinearities, Mediterr. J. Math. 18, 241 (2021).
  * [2] V. Ambrosio, A multiplicity result for a fractional p-Laplacian problem without growth conditions, Riv. Math. Univ. Parma (N. S.) 9(2018), no. 1, 53–71.
  * [3] V. Ambrosio, Nontrivial solutions for a fractional p-Laplacian problem via Rabier Theorem, Complex Var. Elliptic Equ, 62 (2017), no. 6, 838–847.
  * [4] V. Ambrosio and T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst. 38(2018), no. 11, 5835–5881.
  * [5] G. Bonanno, Relations between the mountain pass theorem and local minima, Advances in Nonlinear Analysis. 1, No.3, 205–220 (2012).
  * [6] J. F. Bonder, N. Saintier and A. Silva, The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem, Nonlinear Differential Equations and Applications NoDEA. 25, No.6, 1–25 (2018).
  * [7] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society. 88, No.3, 486–490 (1983).
  * [8] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Mathematics. 224, No.5, 106, 2052–2093 (2010).
  * [9] L.A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia. 7, 37-52 (2012).
  * [10] J. I. Diaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Communications in Partial Differential Equations. 12, No.12, 1333–1344 (1987).
  * [11] J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I, Elliptic Equations, Research Notes in Math. No.1, 106 (1985).
  * [12] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des Sciences Mathematiques. 136, No.5, 521–573 (2012).
  * [13] M. Ferrara and , G. M. Bisci, Existence results for elliptic problems with Hardy potential, Bulletin des Sciences Mathématiques. 138, No.7, 846–859 (2014).
  * [14] A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 40, No.1, 235–253 (2015).
  * [15] R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, Journal of Functional Analysis. 255, No.12, 3407–3430 (2008).
  * [16] K. Ho and Y. H. Kim, The concentration-compactness principles for Ws, p and application, Advances in Nonlinear Analysis. 10, No.1, 816–848 (2021).
  * [17] A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory. Advances in Calculus of Variations. 9, No.2, 101-125 (2016).
  * [18] M. Khodabakhshi, A. M. Aminpour, G. A. Afrouzi and A. Hadjian, Existence of two weak solutions for some singular elliptic problems, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 110, No.2, 385–393 (2016).
  * [19] M. Khodabakhshi and A. Hadjian, Existence of three weak solutions for some singular elliptic problems, Complex Variables and Elliptic Equations, 63, No.1, 68–75 (2018).
  * [20] P. L. Lions, S. Liu, K. Perera and M. Squassina,The concentration compactness principle in the calculus of variations. The limit case. I, II. Rev. Mat. Iberoamericana. 1, 45–121 (1985).
  * [21] S. Mosconi, K. Perera, M. Squassina, and Y. Yang, The Brezis–Nirenberg problem for the fractional p-Laplacian, Calculus of Variations and Partial Differential Equations, 55, No.5, (2016), 105.
  * [22] A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM Journal on Applied Mathematics. 38, No.2, 275–281 (1980).
  * [23] B. Ricceri, A general variational principle and some of its applications. Journal of Computational and Applied Mathematics. 113, No.1-2, 401–410 (2000).
  * [24] R. Servadei, E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389, 887–898 (2012).
  * [25] R. Servadei, E. Valdinoci, The Brezis Nirenberg result for the fractional Laplacian, Trans. Am. Math. Soc, 367, 67–102 (2015).
  * [26] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete and Continuous Dynamical Systems, 33, No.5, 2105-2137 (2013).
  * [27] J. Simon, Régularité de la solution d’une équation non linéaire dans $\mathbb{R}^{N}$, Journées d’Analyse Non Linéaire, 205–227 (1978).
  * [28] M. Xiang, B. Zhang, and V. D. Rădulescu, Existence of solutions for perturbed fractional p-Laplacian equations. Journal of Differential Equations, 260, No.2, (2016), 1392–1413.
  * [29] E. Zeidler, Nonlinear Functional Analysis and Its Applications II / B: Nonlinear Monotone Operators, Springer. (1990).
  * [30] E. Zeidler and L. F. Boron, Nonlinear Functional Analysis and Its Applications II / A: Linear Monotone Operators, Springer. (1990).