# Existence and uniqueness of solutions in fractional Orlicz-Sobolev spaces
for a nonlocal singular elliptic problem

Abdelaaziz Sbai, Youssef El hadfi∗ and Mounim El ouardy
Laboratory LIPIM
National School of Applied Sciences Khouribga
Sultan Moulay Slimane University, Morocco
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Corresponding author.

###### Abstract

In this article, we investigate the existence and uniqueness of a positive
solution for a class of singular nonlinear elliptic problem with boundary
condition. Our result holds in fractional Orlicz-Sobolev spaces.

##  1  Introduction

We are interested here with the existence and uniqueness of solution to a non-
homogeneous problem in fractional Orlicz-Sobolev spaces

$\left(P_{a}\right)\left\\{\begin{array}[]{ll}(-\Delta)_{a}^{s}u+a(|u|)u=\frac{g(x)}{u^{\gamma(x)}}&\text{
in }\Omega\\\ u>0&\text{ in }\Omega\\\ u=0&\text{ on
}\mathbb{R}^{N}\backslash\Omega.\end{array}\right.$ (1.1)

where $\Omega$ is an open bounded subset in $\mathbb{R}^{N}$ with Lipschitz
boundary $\partial\Omega,0<s<1$, $\gamma:\bar{\Omega}\rightarrow(0,1)$ is a
continuous function , $g$ is continuous functions and $(-\Delta)_{a}^{s}$ is
the fractional $a$ -Laplacian operator defined as

$(-\Delta)_{a}^{s}u(x)=2\lim_{\varepsilon\backslash
0}\int_{\mathbb{R}^{N}\backslash
B_{\varepsilon}(x)}a\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{u(x)-u(y)}{|x-y|^{s}}\frac{dy}{|x-y|^{N+s}},$

where $a:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ which will be specified
later. Recently, a great attention has been focused on the study of nonlocal
fractional operators of elliptic type (see, as an example [10, 12]). This sort
of operators arises in various types applications; for instance, continuum
mechanics, phase-change phenomena, population dynamics, minimal surfaces and
anomalous diffusion, since they’re the standard outcome of stochastically
stabilization of Levy processes, see [6, 16, 27] and the references cited
therein. Moreover, such equations and therefore the associated fractional
operators allow us to develop a generalization of quantum mechanics and also
to explain the motion of a chain or an array of particles that are connected
by elastic springs as well as unusual diffusion processes in turbulent fluid
motions and material transports in fractured media, for more details see [6,
16] and the references therein. Indeed, the nonlocal fractional operators have
been extensively studied by many authors in many various cases: bounded
domains and unbounded domains, different behavior of the nonlinearity, and so
on. Before giving our main results, allow us to briefly recall the literature
concerning related singular problems. The following research problem has been
investigated in several previous works:

$\left\\{\begin{array}[]{ll}(-\Delta)^{s}u=f(x,u)&\text{ in }\Omega,\\\ \quad
u=0&\text{ on }\mathbb{R}^{N}\backslash\Omega,\end{array}\right.$ (1.2)

where $f$ is a Carathéodory function. The questions of existence, multiplicity
and regularity of solutions to problem (1.2) have been extensively studied in
[1, 4, 25, 28, 30, 31] and references therein. Concerning the existence and
multiplicity of solutions to doubly nonlocal problems, a lot of works have
been done. For a detailed state of art, one can refer [18, 20, 21, 32, 37] and
references therein. On the other hand, Barrios et al. [13] started the work on
nonlocal equations with singular nonlinearity. Precisely, [13] deal with the
existence of solutions to the following problem

$\left\\{\begin{array}[]{ll}(-\Delta)^{s}u=\lambda\left(\frac{a(x)}{u^{r}}+f(x,u)\right)&\text{
in }\Omega,\\\ \quad u=0&\text{ on
}\mathbb{R}^{N}\backslash\Omega,\end{array}\right.$ (1.3)

where $\Omega$ is a bounded domain with smooth boundary,
$N>2s,0<s<1,r,\lambda>0,f(x,u)\sim$ $u^{p},1<p<2_{s}^{\ast}-1.$ We can also
refer to [8], the authors established the existence and uniqueness of a
positive ground state solution in Musielak-Orlicz-Sobolev spaces:

$\left\\{\begin{aligned} -\mathcal{A}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)\operatorname{div}(a(x,|\nabla u|)\nabla
u)&=g(x)u^{-\gamma(x)}\text{ in }\Omega,\\\ u&>0\text{ in }\Omega\\\
u&=0\text{ on }\partial\Omega,\end{aligned}\right.$

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N\geq 3)$ with smooth
boundary $\partial\Omega,\gamma:\bar{\Omega}\rightarrow(0,1)$ is a continuous
function, $\mathcal{A}$ and $g$ are continuous functions. Motivated by the
above papers, the aim of this paper is to extend this class of problem in
fractional Orlicz-Sobolev spaces and to study the existence and uniqueness of
a weak solution for problem $(P_{a})$. To our knowledge, this is the first
contribution to studying of nonlocal problems in this class of fractional
Orlicz-Sobolev spaces.

The rest of the paper is organized as follows. In the second section we make a
brief introduction for each of the following spaces: Orlicz-Sobolev spaces and
fractional Orlicz-Sobolev. Within the third section, applying several
important properties of fractional a-Laplacian operator, we obtain the
existence and uniqueness of weak solutions of problem $(P_{a}).$

##  2 Some preliminaries results

###  2.1 Orlicz-Sobolev Spaces.

First, we briefly recall the definitions and some elementary properties of the
Orlicz-Sobolev spaces. We refer the reader to [2, 3, 34], for additional
reference and for some of the proofs of the results in this subsection.
Let $\Omega$ be an open subset of $\mathbb{R}^{N},N\geq 1.$ We assume that
$a:\mathbb{R}\longrightarrow\mathbb{R}$ in $\left(P_{a}\right)$ is such that :
$\varphi:\mathbb{R}\longrightarrow\mathbb{R}$ defined by:

$\varphi(t)=\left\\{\begin{array}[]{ccc}a(|t|)t&\text{ for }&t\neq 0,\\\
0&\text{ for }&t=0\end{array}\right.$

is increasing homeomorphism from $\mathbb{R}$ onto itself. Let

$\Phi(t)=\int_{0}^{t}\varphi(\tau)d\tau.$

Then $\varPhi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be an N-function, that
is, $\Phi$ is continuous, convex, with $\Phi(t)>0$ for
$t>0,\frac{\Phi(t)}{t}\rightarrow 0$ as $t\rightarrow 0$ and
$\frac{\varPhi(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty.$ Equivalently,
$\Phi$ admits the representation;

$\Phi(t)=\int_{0}^{t}\varphi(s)ds,$

where $\varphi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is non-decreasing,
right continuous, with $\varphi(0)=0,\varphi(t)>0$ $\forall t>0$ and
$\varphi(t)\rightarrow\infty$ as $t\rightarrow\infty.$ The conjugate
N-function of $\varPhi$ is defined by

$\bar{\Phi}(t)=\int_{0}^{t}\bar{\varphi}(s)ds,$

where $\bar{\varphi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is given by
$\bar{\varphi}(t)=\sup\\{s:\varphi(s)\leq t\\}.$ Evidently we have

$st\leq\Phi(t)+\bar{\Phi}(s),$ (2.1)

which is known Young’s inequality. Equality holds in (2.1) if and only if
either $t=\bar{\varphi}(s)$ or $s=\varphi(t).$
We say that the N-function $\Phi$ satisfies the global $\Delta_{2}-$condition
if, for some $k>0$

$\Phi(2t)\leq k\Phi(t),\forall t\geq 0.$

When this inequality holds only for $t\geq t_{0}>0,\Phi$ is said to satisfy
the $\Delta_{2}-$condition near infinity. We call the pair $(\Phi,\Omega)$ is
$\Delta-$regular if either :

  * (a)

$\Phi$ satisfies a global $\Delta_{2}-$condition, or

  * (b)

$\Phi$ satisfies a $\Delta_{2}-$condition near infinity and $\Omega$ has
finite volume.

Throughout this paper, we assume that $\Phi:$

$1<\varphi^{-}:=\inf_{s\geq
0}\frac{s\varphi(s)}{\Phi(s)}\leq\varphi^{+}:=\sup_{s\geq
0}\frac{s\varphi(s)}{\Phi(s)}<+\infty,$ (2.2)

which assures that $\Phi$ satisfies the global $\Delta_{2}-$condition. We
assume that :

$\text{ the function }t\mapsto\varPhi(\sqrt{t})\text{ is convex for all }t\geq
0.$ (2.3)

The above relation assures that $L_{\Phi}(\Omega)$ is an uniformly convex
space (see [29]). Let $\Omega$ be an open subset of $\mathbb{R}^{N}.$ The
Orlicz class $K_{\varPhi}(\Omega)$ (resp. the Orlicz space
$L_{\varPhi}(\Omega)$ ) is defined as the set of ( equivalence classes of )
real-valued measurable functions $u$ on $\Omega$ such that

$\int_{\Omega}\varPhi(|u(x)|)dx<\infty\quad\left(\text{ resp.
}\int_{\Omega}\varPhi(\lambda|u(x)|)dx<\infty\text{ for some
}\lambda>0\right).$

$L_{\varPhi}(\Omega)$ is a Banach space under the Lexumburg norm

$\|u\|_{\varPhi}=\inf\left\\{\lambda>0:\int_{\Omega}\varPhi\left(\frac{|u(x)|}{\lambda}\right)dx\leq
1\right\\}$

and $K_{\varPhi}(\Omega)$ is a convex subset of $L_{\varPhi}(\Omega).$ The
closure in $L_{\varPhi}(\Omega)$ of the set of bounded measurable functions on
$\Omega$ with compact support in $\bar{\Omega}$ is denoted by
$E_{\varPhi}(\Omega)$ The equality $E_{\varPhi}(\Omega)=L_{\varPhi}(\Omega)$
holds if and only if $(\varPhi,\Omega)$ is $\Delta$-regular. Using Young’s
inequality, we have the following Hölder type inequality,

$\left|\int_{\Omega}uvdx\right|\leq
2\|u\|_{\varPhi}\|v\|_{\bar{\varPhi}}\quad\forall u\in
L_{\varPhi}(\Omega)\text{ and }v\in L_{\bar{\varPhi}}(\Omega).$

######  Proposition 2.1.

(see: [29]) Assume condition (2.2) is satisfied. Then the following relations
holds true

$\begin{array}[]{ll}\|u\|_{\Phi}^{\varphi^{-}}\leq\int_{\Omega}\Phi(|u|)dx\leq\|u\|_{\Phi}^{\varphi^{+}},\forall
u\in L_{\Phi}(\Omega)\text{ with }\|u\|_{\Phi}>1\\\
\|u\|_{\Phi}^{\varphi^{+}}\leq\int_{\Omega}\Phi(|u|)dx\leq\|u\|_{\Phi}^{\varphi^{-}},\forall
u\in L_{\Phi}(\Omega)\text{ with }\|u\|_{\Phi}<1\end{array}$

######  Lemma 2.2.

( c f .[15] ) Assume that $\Phi\in\Delta_{2}.$ Then we have,

$\bar{\Phi}(\varphi(t))\leq c\Phi(t)\text{ for all }t\geq 0\text{ where }c>0.$

To simplify the notation, we put

$h(x,y):=\frac{u(x)-u(y)}{|x-y|^{s}},\quad
d\mu=\frac{dxdy}{|x-y|^{N}}\quad\forall(x,y)\in\Omega\times\Omega,\quad\int_{\Omega^{2}}:=\int_{\Omega}\int_{\Omega}.$

###  2.2 Fractional Orlicz-Sobolev space.

This subsection is devoted to the definition of the fractional Orlicz-Sobolev
spaces, and we recall some result of continuous and compact embedding of
fractional Orlicz-Sobolev spaces. We refer the reader to [29, 33] for further
reference and for some of the proofs of these results.

$W^{s}L_{\Phi}(\Omega)=\left\\{u\in
L_{\Phi}(\Omega):\int_{\Omega^{2}}\Phi\left(\frac{\lambda|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dxdy}{|x-y|^{N}}<\infty\right\\}$

for some $\lambda>0.$ This space is equipped with the norm,

$\|u\|_{s,\Phi}=\|u\|_{\Phi}+[u]_{s,\Phi},$

where $[\cdot]_{s,\Phi}$ is the Gagliardo semi-norm, defined by

$[u]_{s,\Phi}=\inf\left\\{\lambda>0:\int_{\Omega^{2}}\Phi\left(\frac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\frac{dxdy}{|x-y|^{N}}\leq
1\right\\}.$

By [15], $W^{s}L_{\Phi}(\Omega)$ is Banach space, also separable (resp.
reflexive) space if and only if $\Phi\in\Delta_{2}$ (resp. $\Phi\in\Delta_{2}$
and $\bar{\Phi}\in\Delta_{2}$ ). Furthermore if $\Phi\in\Delta_{2}$ and
$\Phi(\sqrt{t})$ is convex, then the space $W^{s}L_{\Phi}(\Omega)$ is
uniformly convex.

######  Proposition 2.3.

(cf. [11]) Assume condition (2.2) is satisfied. Then the following relations
holds true,

$\begin{array}[]{l}{[u]_{s,\Phi}^{\varphi^{-}}\leq\phi(u)\leq[u]_{s,\Phi}^{\varphi^{+}},\forall
u\in W^{s}L_{\Phi}(\Omega)\text{ with }[u]_{s,\Phi}>1}\\\
{[u]_{s,\Phi}^{\varphi^{+}}\leq\phi(u)\leq[u]_{s,\Phi}^{\varphi^{-}},\forall
u\in W^{s}L_{\Phi}(\Omega)\text{ with }[u]_{s,\Phi}<1}\end{array}$

where
$\phi(u)=\int_{\Omega^{2}}\Phi\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dxdy}{|x-y|^{N}}$.

######  Proposition 2.4.

(cf. [11]) The following relations holds true

$\Psi(u)\geq\|u\|_{s,\Phi}^{\varphi^{-}}\,\,\forall u\in
W^{s}L_{\Phi}(\Omega),\|u\|_{s,\Phi}>1$ (2.4)
$\Psi(u)\geq\|u\|_{s,\Phi}^{\varphi^{+}}\,\,\forall u\in
W^{s}L_{\Phi}(\Omega),\|u\|_{s,\Phi}<1.$

Where
$\Psi(u):=\int_{\Omega^{2}}\Phi\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dxdy}{|x-y|^{N}}+\int_{\Omega}\Phi(|u(x)|)dx.$

######  Theorem 2.5.

( $cf.$ [11]) Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$. Then,

$C_{0}^{2}(\Omega)\subset W^{s}L_{\Phi}(\Omega).$

######  Remark 2.6.

A trivial consequence of Theorem 2.5 $,C_{0}^{\infty}(\Omega)\subset
W^{s}L_{\Phi}(\Omega)$ and $W^{s}L_{\Phi}(\Omega)$ is non-empty.

######  Theorem 2.7.

( $cf.$ [11]) Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N},N\geq
1$ with $C^{0,1}$ regularity and bounded boundary, let $0<s^{\prime}<s<1.$
Assume condition (2.2) is satisfied. We define

$\varphi_{s^{\prime}}^{\ast}=\left\\{\begin{array}[]{cc}\frac{N\varphi^{-}}{N-s^{\prime}\varphi^{-}}&\text{if
}N>s^{\prime}\varphi^{-}\\\ \infty&\text{ if }N\leq
s^{\prime}\varphi^{-}\end{array}\right.$

$\bullet$

If $s^{\prime}\varphi^{-}<N,$ then $W^{s}L_{\Phi}(\Omega)\hookrightarrow
L^{q}(\Omega),$ for all $q\in\left[1,\varphi_{s^{\prime}}^{\ast}\right]$ and
the embedding $W^{s}L_{\Phi}(\Omega)\hookrightarrow L^{q}(\Omega),$ is compact
for all $q\in\left[1,\varphi_{s^{\prime}}^{\ast}\right)$

$\bullet$

If $s^{\prime}\varphi^{-}=N,$ then $W^{s}L_{\Phi}(\Omega)\hookrightarrow
L^{q}(\Omega),$ for all $q\in[1,\infty]$ and the embedding
$W^{s}L_{\Phi}(\Omega)\hookrightarrow L^{q}(\Omega),$ is compact for all
$q\in[1,\infty)$

$\bullet$

If $s^{\prime}\varphi^{-}>N,$ then the embedding
$W^{s}L_{\Phi}(\Omega)\hookrightarrow L^{\infty}(\Omega),$ is compact.

######  Lemma 2.8.

( $cf.$ [11]) Assume condition (2.2)is satisfied. Then the following relations
holds true

$\begin{array}[]{c}\Phi(\sigma t)\geq\sigma^{\varphi^{-}}\Phi(t),\forall
t>0,\sigma>1\\\ \Phi(\tau t)\geq\tau^{\varphi^{+}}\Phi(t),\forall
t>0,\tau\in(0,1).\end{array}$

######  Remark 2.9.

First, we can apply Theorem 2.7 to obtain that, $W^{s}L_{\phi}(\Omega)$ is
continuously and compactly embedded in the classical Lebesgue space
$L^{r^{+}}(\Omega)$ for all $1\leq r^{+}<\varphi_{s^{\prime}}^{\ast}$. That
fact combined with continuous embedding of $L^{r^{+}}(\Omega)$ in
$L^{r(x)}(\Omega),$ with $r\in C(\bar{\Omega})$ and $1<r^{-}\leq r(x)\leq
r^{+}<+\infty,$ with $r^{+}=\sup_{x\in\bar{\Omega}}r(x)$ and
$r^{-}=\inf_{x\in\bar{\Omega}}r(x),$ as a result, $W^{s}L_{\varPhi}(\Omega)$
is compactly embedded in $L^{r(x)}(\Omega).$

######  Lemma 2.10.

(cf. [15]) The functional
$I_{1}:W^{s}L_{\Phi}(\Omega)\longrightarrow\mathbb{R}$ is of class $C^{1}$ and

$\left\langle
I_{1}^{\prime}(u),v\right\rangle=\int_{\Omega^{2}}a\left(\left|h_{x,y}(u)\right|\right)h_{x,y}(u)h_{x,y}(v)d\mu+\int_{\Omega}a(|u|)uvdx$

for all $u,v\in W^{s}L_{\Phi}(\Omega).$

The following Proposition generalizes the definition of convexity, and
therefore, we give a proof for the convenience.

######  Proposition 2.11.

Let $X$ be a vector space and let $I:X\rightarrow\mathbb{R}.$ Then $I$ is
convex if and only if

$I((1-\lambda)u+\lambda v)<(1-\lambda)\theta+\lambda\beta,\quad 0<\lambda<1$
(2.5)

whenever $I(u)<\theta$ and $I(v)<\beta,$ for all $u,v\in X$ and
$\theta,\beta\in\mathbb{R}.$

###### Proof.

Assume that functional $I:X\rightarrow\mathbb{R}$ is convex. Moreover, since
$I$ is a real-valued functional, there are real numbers
$\theta,\beta\in\mathbb{R}$ such that $I(u)<\theta$ and $I(v)<\beta.$ Then

$I((1-\lambda)u+\lambda v)<(1-\lambda)I(u)+\lambda
I(v)<(1-\lambda)\theta+\lambda\beta,\quad 0<\lambda<1$

On the other hand, assume that (2.5) holds. Since $I(u)<\theta$ and
$I(v)<\beta,$ we can write, for all $\varepsilon>0$

$\begin{array}[]{l}I(u)<I(u)+\varepsilon:=\theta\\\
I(v)<I(v)+\varepsilon:=\beta.\end{array}$

Therefore,

$I((1-\lambda)u+\lambda v)<(1-\lambda)I(u)+\lambda I(v)+\varepsilon,\quad
0<\lambda<1$ (2.6)

If we consider that (2.6) holds for any $\varepsilon>0,$ we conclude

$I((1-\lambda)u+\lambda v)\leq(1-\lambda)I(u)+\lambda I(v).$

∎

##  3 The main results and their proof

Now, we give the definition of weak solutions for problem $\left(P_{a}\right)$

######  Definition 3.1.

A function $u$ is called a weak solution to problem $\left(P_{a}\right)$ if
$u\in W^{s}L_{\Phi}(\Omega)$ such that $u>0$ in $\Omega$ and

$\int_{\Omega^{2}}a\left(\left|h_{x,y}(u)\right|\right)h_{x,y}(u)h_{x,y}(v)d\mu+\int_{\Omega}a(|u|)uvdx=\int_{\Omega}\frac{g(x)}{u^{\gamma(x)}}vdx,$
(3.1)

for all $v\in W^{s}L_{\Phi}(\Omega).$

The main result of present paper are the following.

######  Theorem 3.2.

Suppose that the following assumption hold: $(\mathrm{G}0):\,\,\,g(x)\in
C^{1}(\bar{\Omega})$ is a nontrivial nonnegative function. Then problem
$(P_{a})$ has a positive ground state solution in $W^{s}L_{\Phi}(\Omega)$ with
a negative energy level.

Let us consider the functional $J:W^{s}L_{\Phi}(\Omega)\rightarrow\mathbb{R}$
associated to problem $(P_{a})$ as follows

$J(u)=\int_{\Omega^{2}}\Phi\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)d\mu+\int_{\Omega}\Phi(|u(x)|)dx-\int_{\Omega}\frac{g(x)|u|^{1-\gamma(x)}}{1-\gamma(x)}dx.$

Now, we give an auxiliary result

######  Lemma 3.3.

Assume the hypotheses of Theorem are fulfilled. Then the functional $J$ is
coercive.

###### Proof.

For each $u\in W^{s}L_{\Phi}(\Omega)$ with $\|u\|>1,$ relations (2.4) and the
condition $\left(G_{0}\right)$ imply

$\displaystyle J(u)$
$\displaystyle=\int_{\Omega^{2}}\Phi\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)d\mu+\int_{\Omega}\Phi(|u|)dx-\int_{\Omega}\frac{g(x)|u|^{1-\gamma(x)}}{1-\gamma(x)}dx$
(3.2)
$\displaystyle\geq\|u\|_{s,\Phi}^{\varphi^{-}}-\frac{|g|_{\infty}}{1-\gamma^{+}}\|\left.\left.u\right|\right|_{L^{\frac{r(x)}{1-\gamma(x)}}(\Omega)}^{1-\gamma(x)}|\mathbf{1}|_{L^{\frac{r(x)}{r(x)+\gamma(x)-1}}(\Omega)}$
$\displaystyle\geq\|u\|_{s,\Phi}^{\varphi^{-}}-c\|u\|_{s,\Phi}^{1-\gamma^{-}},$

since $\varphi^{-}>1-\gamma^{-}$ the above inequality implies that
$J(u)\longrightarrow\infty$ as $\|u\|\rightarrow\infty,$ that is, $J$ is
coercive. ∎

######  Lemma 3.4.

Assume the hypotheses of Theorem 3.2 are fulfilled. Then there exists $\phi\in
W^{s}L_{\Phi}(\Omega)$ such that $\phi>0$ and $J(t\phi)<0$ for $t>0$ small
enough.

###### Proof.

Let $\Omega_{0}\subset\subset\Omega,$ for $x_{0}\in\Omega_{0},0<R<1$ satisfy
$B_{2R}\left(x_{0}\right)\subset\Omega_{0},$ where $B_{2R}\left(x_{0}\right)$
is the ball of radius $2R$ with center at the point $x_{0}$ in
$\mathbb{R}^{N}.$ Let $\phi\in
C_{0}^{\infty}\left(B_{2R}\left(x_{0}\right)\right)$ satisfies $0\leq\phi\leq
1$ and $\phi\equiv 1$ in $B_{2R}\left(x_{0}\right).$ Theorem 2.5 implies that
$\|\phi\|<\infty.$ Then for $0<t<1,$ we have

$\displaystyle J(t\phi)$
$\displaystyle=\int_{\Omega^{2}}\Phi\left(\frac{|t\phi(x)-t\phi(y)|}{|x-y|^{s}}\right)d\mu$
$\displaystyle+\int_{\Omega}\Phi(|t\theta|)dx-\int_{\Omega}\frac{g(x)|t\phi|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\|t\phi\|_{s,\Phi}^{\varphi^{-}}-\frac{t^{1-\gamma^{+}}}{1-\gamma^{+}}\int_{\Omega_{0}}g(x)|\phi|^{1-\gamma(x)}dx$
$\displaystyle\leq
t^{\varphi^{-}}\|\phi\|_{s,\Phi}^{\varphi^{-}}-\frac{t^{1-\gamma^{+}}}{1-\gamma^{+}}\int_{\Omega}g(x)|\phi|^{1-\gamma(x)}dx,$

since $\varphi^{-}>1-\gamma^{+}$ and
$\int_{\Omega_{0}}g(x)|\phi|^{1-\gamma(x)}dx>0$ we have
$J\left(t_{0}\phi\right)<0$ for $t_{0}\in(0,t)$ sufficiently small. ∎

######  Lemma 3.5.

The functional $J$ attains the global minimizer in $W^{s}L_{\varPhi}(\Omega),$
that is, there exists a function $u_{\ast}\in W^{s}L_{\varPhi}(\Omega)$ such
that

$m=J\left(u_{\ast}\right)=\inf_{u\in W^{s}L_{\varPhi}(\Omega)}J(u)<0.$

###### Proof.

By (G0), Hölder inequality, Proposition 2.4 and the continuous embedding
$W^{s}L_{\varPhi}(\Omega)\hookrightarrow L^{r(x)}(\Omega),$ it follows

$\displaystyle|J(u)|$ $\displaystyle\leq
I_{1}(u)+\frac{|g|_{\infty}}{1-\gamma^{+}}\int_{\Omega}|u|^{1-\gamma(x)}dx$
$\displaystyle\leq
2\|u\|_{s,\Phi}^{\varphi^{+}}+\frac{|g|_{\infty}}{1-\gamma^{+}}|||u|^{1-\gamma(x)}||_{L^{\frac{r(x)}{1-\gamma(x)}}(\Omega)}|\mathbf{1}|_{L^{\frac{1}{r(x)+\gamma(x)-1}}(\Omega)}$
$\displaystyle\leq
2\|u\|_{s,\Phi}^{\varphi^{+}}+\frac{c|g|_{\infty}}{1-\gamma^{+}}\|u\|_{s,\Phi}^{1-\gamma^{-}}<+\infty,$

which shows that $J$ is well-defined on $W^{s}L_{\varPhi}(\Omega).$
Considering the fact that the functional $I_{1}(u)$ is of class
$C^{1}\left(W^{s}L_{\varPhi}(\Omega),\mathbb{R}\right)$ (see [15]). Further,
by the well-known inequality

$\left|a^{p}-b^{p}\right|\leq|a-b|^{p},$

for any real numbers $a,b\geq 0$ and $0<p<1$, we obtain

$\displaystyle|J(u)-J(v)|$
$\displaystyle\leq|I_{1}(u)-I_{1}(v)|+\frac{|g|_{\infty}}{1-\gamma^{+}}\int_{\Omega}\|\left.u\right|^{1-\gamma(x)}-|v|^{1-\gamma(x)}\mid
dx$
$\displaystyle\leq|I_{1}(u)-I_{1}(v)|+\frac{|g|_{\infty}}{1-\gamma^{+}}\int_{\Omega}|u-v|^{1-\gamma(x)}dx$
$\displaystyle\leq|I_{1}(u)-I_{1}(v)|+\frac{|g|_{\infty}}{1-\gamma^{+}}||u-\left.\left.v\right|\right|_{L^{\frac{r(x)}{1-\gamma(x)}}(\Omega)}^{1-\gamma(x)}|\mathbf{1}|_{L^{\frac{r(x)}{r(x)+\gamma(x)-1}}(\Omega)}$
$\displaystyle\leq|I_{1}(u)-I_{1}(v)|+\frac{c|g|_{\infty}}{1-\gamma^{+}}\|u-v\|_{s,\Phi}^{1-\gamma^{-}},$

for any $u,v\in W^{s}L_{\varPhi}(\Omega).$ Therefore $J$ is continuous on
$W^{s}L_{\varPhi}(\Omega).$
Now, we shall show that $J$ is convex on $W^{s}L_{\varPhi}(\Omega).$ To this
end, considering the assumptions for $g$ and $\gamma,$ we have

$\displaystyle J(u)$
$\displaystyle=I_{1}(u)-\int_{\Omega}\frac{g(x)|u|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\max\left\\{\|u\|_{s,\Phi}^{\varphi_{-}},\|u\|_{s,\Phi}^{\varphi^{+}}\right\\}:=\theta$

and

$\displaystyle J(v)$
$\displaystyle=I_{1}(v)-\int_{\Omega}\frac{g(x)|v|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\max\left\\{\|v\|_{s,\Phi}^{\varphi_{-}},\|v\|_{s,\Phi}^{\varphi^{+}}\right\\}:=\beta,$

for all $u,v\in W^{s}L_{\varPhi}(\Omega).$ On the other hand, since $\Phi$ is
convex, so

$I_{1}(u)=\int_{\Omega}\Phi(x,|\nabla u|)dx+\int_{\Omega}\Phi(u)dx,$

( see[22])is also convex. Therefore, for $0<\lambda<1,$ we have

$(I_{1}((1-\lambda)u+\lambda v))\leq(1-\lambda)I_{1}(u)+\lambda I_{1}(u).$

Therefore, considering the all pieces of information obtained above along, it
follows

$\displaystyle J((1-\lambda)u+\lambda v))$
$\displaystyle=I_{1}((1-\lambda)u+\lambda
v)-\int_{\Omega}\frac{g(x)|(1-\lambda)u+\lambda
v|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq(1-\lambda)\max\left\\{\|u\|_{s,\Phi}^{\varphi_{-}},\|u\|_{s,\Phi}^{\varphi^{+}}\right\\}+\lambda\max\left\\{\|v\|_{s,\Phi}^{\varphi_{-}},\|v\|_{s,\Phi}^{\varphi^{+}}\right\\}$
$\displaystyle\leq(1-\lambda)\theta+\lambda\beta.$

Hence, by Proposition 2.11, $J$ is convex on $W^{s}L_{\varPhi}(\Omega).$ As
the functional $J$ is continuous, coercive and convex, it has a global minimum
belonging to $W^{s}L_{\varPhi}(\Omega),$ which in turn becomes a solution to
problem $(P_{a}).$ Let us denote

$m=\inf_{u\in W^{s}L_{\varPhi}(\Omega)}J(u),$

which is well-defined due to (3.2). Now, applying the Lemma 3.4, we obtain
that $J(t\phi)<0.$ If we set $t\phi=u$ with $\|u\|_{\Phi}<1,$ we obtain that

$m=\inf_{u\in W^{s}L_{\varPhi}(\Omega)}J(u)<0.$

On the other hand, if we take into account the definition of $m,$ there exists
a minimizing sequence $\left(u_{n}\right)$ of $W^{s}L_{\varPhi}(\Omega)$ such
that

$m=\lim_{n\rightarrow\infty}J\left(u_{n}\right)<0.$ (3.3)

Moreover, since $J\left(u_{n}\right)=J\left(\left|u_{n}\right|\right)$ we may
assume that $u_{n}\geq 0.$ Due to the coercivity of $J,$ $u_{n}$ must be
bounded in $W^{s}L_{\varPhi}(\Omega)$ other wise we would have that
$J\left(u_{n}\right)\rightarrow+\infty$ as
$\left\|u_{n}\right\|_{\Phi}\rightarrow\infty$ which contradicts $\eqref{34}.$
Since $W^{s}L_{\varPhi}(\Omega)$ is reflexive there exists a subsequence, not
relabelled, and $u_{\ast}\in W^{s}L_{\varPhi}(\Omega)$ such that

$u_{n}\rightharpoonup u_{\ast}\text{ in }W^{s}L_{\varPhi}(\Omega),$

since $J$ is continuous and convex on $W^{s}L_{\varPhi}(\Omega),$ it is weakly
lower semi-continuous on $W^{s}L_{\varPhi}(\Omega).$ Therefore,

$m\leq
J\left(u_{\ast}\right)=\int_{\Omega^{2}}\Phi\left(\frac{|u_{\ast}(x)-u_{\ast}(y)|}{|x-y|^{s}}\right)d\mu+\int_{\Omega}\Phi(|u_{\ast}|)dx-\int_{\Omega}\frac{g(x)\left|u_{\ast}\right|^{1-\gamma(x)}}{1-\gamma(x)}dx$
(3.4) $\leq\liminf_{n\rightarrow\infty}J\left(u_{n}\right)=m,$ (3.5)

which means

$m=J\left(u_{\ast}\right)=\inf_{W^{s}L_{\varPhi}(\Omega)}J(u)<0.$ (3.6)

∎

###### of Theorem 3.2..

Since $m=J\left(u_{\ast}\right)<0=J(0),$ it must be $u_{\ast}\geq 0,$
$u_{\ast}\neq 0.$ For $\phi\in W^{s}L_{\varPhi}(\Omega),\phi\geq 0$ and $t>0,$
we obtain

$\displaystyle 0\leq$ $\displaystyle\liminf_{t\rightarrow
0}\frac{J\left(u_{\ast}+t\phi\right)-J\left(u_{\ast}\right)}{t}$
$\displaystyle\leq\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})h_{x,y}(\phi)d\mu+\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx$ $\displaystyle-\limsup_{t\rightarrow
0}\int_{\Omega}g(x)\frac{\left(u_{\ast}+t\phi\right)^{1-\gamma(x)}-u_{\ast}^{1-\gamma(x)}}{1-\gamma(x)}dx.$

Or

$\limsup_{t\rightarrow
0}\int_{\Omega}g(x)\frac{\left(u_{\ast}+t\phi\right)^{1-\gamma(x)}-u_{\ast}^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\leq\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})h_{x,y}(\phi)d\mu+\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx.$ (3.7)

By the mean value theorem, there exists $\theta\in(0,1)$ such that

$\int_{\Omega}g(x)\frac{\left(u_{\ast}+t\phi\right)^{1-\gamma(x)}-u_{\ast}^{1-\gamma(x)}}{1-\gamma(x)}dx=\int_{\Omega}g(x)\left(u_{\ast}+t\theta\phi\right)^{-\gamma(x)}\phi
dx.$ (3.8)

Thus, we deduce that

$\left(u_{\ast}+t\theta\phi\right)^{-\gamma(x)}\phi\geq 0,\quad\forall
x\in\Omega$

and

$\left(u_{\ast}+t\theta\phi\right)^{-\gamma(x)}\phi\rightarrow
u_{\ast}^{-\gamma(x)}\phi,\text{ as }t\rightarrow 0,\text{ a.e. }x\in\Omega,$

using Fatou’s Lemma in (3.8), we find that,

$\displaystyle\lim_{t\rightarrow
0}\sup\int_{\Omega}g(x)\frac{\left(u_{\ast}+t\phi\right)^{1-\gamma(x)}-u_{\ast}^{1-\gamma(x)}}{1-\gamma(x)}dx$
(3.9) $\displaystyle\geq\liminf_{t\rightarrow
0}\int_{\Omega}g(x)\frac{\left(u_{\ast}+t\phi\right)^{1-\gamma(x)}-u_{\ast}^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle=\liminf_{t\rightarrow
0}\int_{\Omega}g(x)\left(u_{\ast}+t\theta\phi\right)^{-\gamma(x)}\phi
dx\geq\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx\geq 0.$

Taking into account relations (3.7) and (3.9), we deduce that

$\displaystyle\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})h_{x,y}(\phi)d\mu+\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx$ (3.10) $\displaystyle-\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx\geq
0,\quad\forall\phi\in W^{s}L_{\varPhi}(\Omega),\,\,\phi\geq 0.$

As above, we have that function $u_{\ast}\in W^{s}L_{\varPhi}(\Omega)$
satisfies

$(-\Delta)_{a}^{s}u_{\ast}+a(|u_{\ast}|)u_{\ast}\geq 0\,\,\,\text{ in
}\,\,\Omega,$ (3.11)

in the weak sense. since $u_{\ast}\geq 0$ and $u_{\ast}\neq 0,$ by the strong
maximum principle for weak solutions(see [7]), we must have
$u_{\ast}(x)>0,\quad\forall x\in\Omega.$ Next, we show that $u_{\ast}\in
W^{s}L_{\varPhi}(\Omega)$ satisfies (3.1). The proof below has been adapted
from one given in [26]. For given $\delta>0,$ define
$\Lambda:[-\delta,\delta]\rightarrow(-\infty,\infty)$ by
$\Lambda(t)=J\left(u_{\ast}+tu_{\ast}\right).$ Then $\Lambda$ achieves its
minimum at $t=0.$ Thus,

$\displaystyle\left.\frac{d}{dt}\Lambda(t)\right|_{t=0}=\left.\frac{d}{dt}J\left(u_{\ast}+tu_{\ast}\right)\right|_{t=0}=0.\text{
Or }$
$\displaystyle\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})h_{x,y}(u_{\ast})d\mu+\int_{\Omega}a(|u_{\ast}|)u_{\ast}^{2}dx-\int_{\Omega}g(x)u_{\ast}^{1-\gamma(x)}dx=0.$
(3.12)

Indeed, let $\phi\in W^{s}L_{\varPhi}(\Omega),$ and define $\varTheta\in
W^{s}L_{\varPhi}(\Omega)$ such that
$\varTheta:=\left(u_{\ast}+\varepsilon\phi\right)^{+}$
$=\max\left\\{0,u_{\ast}+\varepsilon\phi\right\\},\varepsilon>0.\text{
Evidently, }\varTheta\geq 0.$ If we replace $\varTheta$ both in (LABEL:311)
and (3), we have

$\displaystyle 0\leq\int_{\left\\{u_{\ast}+\varepsilon\phi\geq
0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi\geq
0\right\\}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}\left(u_{\ast}+\varepsilon\phi\right)d\mu$
$\displaystyle+\int_{\left\\{u_{\ast}+\varepsilon\phi\geq
0\right\\}}a(|u_{\ast}|)u_{\ast}\left(u_{\ast}+\varepsilon\phi\right)dx-\int_{\left\\{u_{\ast}+\varepsilon\phi\geq
0\right\\}}g(x)u_{\ast}^{-\gamma(x)}\left(u_{\ast}+\varepsilon\phi\right)dx$
$\displaystyle=\left(\int_{\Omega^{2}}-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\right)a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}\left(u_{\ast}+\varepsilon\phi\right)d\mu$
$\displaystyle+\left(\int_{\Omega}-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\right)a(|u_{\ast}|)u_{\ast}\left(u_{\ast}+\varepsilon\phi\right)dx$
$\displaystyle-\left(\int_{\Omega}-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\right)g(x)u_{\ast}^{-\gamma(x)}\left(u_{\ast}+\varepsilon\phi\right)dx$
$\displaystyle=\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(u_{\ast})d\mu+\int_{\Omega}a(|u_{\ast}|)u_{\ast}^{2}dx-\int_{\Omega}g(x)u_{\ast}^{1-\gamma(x)}dx$
$\displaystyle+\varepsilon\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu+\varepsilon\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx-\varepsilon\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx$
$\displaystyle-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}\left(u_{\ast}+\varepsilon\phi\right)d\mu$
$\displaystyle-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a(|u_{\ast}|)u_{\ast}\left(u_{\ast}+\varepsilon\phi\right)dx+\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}g(x)u_{\ast}^{-\gamma(x)}\left(u_{\ast}+\varepsilon\phi\right)dx$
$\displaystyle=\varepsilon\left(\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu\right)$
$\displaystyle+\varepsilon\left(\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx-\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx\right)$
$\displaystyle-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}\left(u_{\ast}+\varepsilon\phi\right)d\mu$
$\displaystyle-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a(|u_{\ast}|)u_{\ast}\left(u_{\ast}+\varepsilon\phi\right)dx+\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}g(x)u_{\ast}^{-\gamma(x)}\left(u_{\ast}+\varepsilon\phi\right)dx$
$\displaystyle\leq\ell_{0}(\varepsilon)-\ell_{1}(\varepsilon)$

where

$\displaystyle\ell_{0}(\varepsilon)=\varepsilon\left(\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu\right)$
$\displaystyle+\varepsilon\left(\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx-\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx\right)$
$\displaystyle\ell_{1}(\varepsilon)=\varepsilon\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu$
$\displaystyle-\varepsilon\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a(|u_{\ast}|)u_{\ast}\phi
dx.$

Considering that $u_{\ast}>0$ and Lebesgue measure of the domain of
integration $\left\\{u_{\ast}+\varepsilon\phi<0\right\\}$ tends to zero as
$\varepsilon\rightarrow 0,$ we have

$\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu\rightarrow 0,\text{ as }\varepsilon\rightarrow 0.$

and

$\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a(|u_{\ast}|)u_{\ast}\phi
dx\rightarrow 0,\text{ as }\varepsilon\rightarrow 0.$

On the other hand, considering that $a(\cdot)\in(0,\infty)$, we can drop the
term

$-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(u_{\ast})d\mu-\int_{\left\\{u_{\ast}+\varepsilon\phi<0\right\\}}a(|u_{\ast}|)u_{\ast}^{2}dx,$

in $\ell_{0}(\varepsilon)$ since it is negative. Thus, dividing
$\ell_{1}(\varepsilon)$ by $\varepsilon$ and letting $\varepsilon\rightarrow
0,$ we find

$\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu$ $+\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx-\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx\geq 0.$ (3.13)

Considering that $\phi\in W^{s}L_{\varPhi}(\Omega)$ is arbitrary, (3.13) holds
for $-\phi$ as well. As a conclusion, we obtain

$\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})\cdot
h_{x,y}(\phi)d\mu$ $+\int_{\Omega}a(|u_{\ast}|)u_{\ast}\phi
dx-\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\phi dx=0$ (3.14)

that is to say, $u_{\ast}\in W^{s}L_{\varPhi}(\Omega)$ is a weak solution to
problem $(P_{a})$. Additionally, since $J$ is coercive and bounded below on
$W^{s}L_{\varPhi}(\Omega),u_{\ast}$ is a positive ground state solution to
problem $(P_{a}),$ i.e, a solution with minimum action among all nontrivial
solutions. Additionally, since $J\left(u_{\ast}\right)<0$ this solution has a
negative energy level. ∎

######  Theorem 3.6.

Assume the hypotheses of Theorem 3.2 hold. Additionally, assume the following
conditions hold:
(a1) There exists a real number $\underline{a}>0$ such that
$a(x,t)\geq\underline{a}>0$ holds for any $t\in\mathbb{R}.$ Then $u_{\ast}\in
W^{s}L_{\varPhi}(\Omega)$ is the unique solution to problem $(P_{a})$.

###### Proof.

Let us assume $v_{\ast}$ is an another solution to problem $(P_{a}).$ Then,
from (3.1), we have

$\displaystyle\int_{\Omega^{2}}a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})h_{x,y}\left(u_{\ast}-v_{\ast}\right)d\mu+\int_{\Omega}a(|u_{\ast}|)u_{\ast}\left(u_{\ast}-v_{\ast}\right)dx$
$\displaystyle-\int_{\Omega}g(x)u_{\ast}^{-\gamma(x)}\left(u_{\ast}-v_{\ast}\right)dx-\int_{\Omega^{2}}a\left(\left|h_{x,y}(v_{\ast})\right|\right)h_{x,y}(v_{\ast})h_{x,y}\left(u_{\ast}-v_{\ast}\right)d\mu$
$\displaystyle-\int_{\Omega}a(|v_{\ast}|)v_{\ast}\left(u_{\ast}-v_{\ast}\right)dx+\int_{\Omega}g(x)v_{\ast}^{-\gamma(x)}\left(u_{\ast}-v_{\ast}\right)dx=0.$

Or

$\int_{\Omega^{2}}\left(a\left(\left|h_{x,y}(u_{\ast})\right|\right)h_{x,y}(u_{\ast})-a\left(\left|h_{x,y}(v_{\ast})\right|\right)h_{x,y}(v_{\ast})\right)\cdot
h_{x,y}\left(u_{\ast}-v_{\ast}\right)d\mu$
$+\int_{\Omega}\left(a(|u_{\ast}|)u_{\ast}-a(|v_{\ast}|)v_{\ast}\right)\left(u_{\ast}-v_{\ast}\right)dx$
(3.15)
$=\int_{\Omega}g(x)\left(u_{\ast}^{-\gamma(x)}-v_{\ast}^{-\gamma(x)}\right)\left(u_{\ast}-v_{\ast}\right)dx.$
(3.16)

For $\alpha\in(0,1)$ and $x,y\geq 0,$ we have the elementary inequality

$\left(x^{-\alpha}-y^{-\alpha}\right)(x-y)\leq 0.$ (3.17)

On the other hand, by Lemma 2.4 given in [29], we have the following
inequality: for any $k,l>0,$ there exists a positive constant
$C(\delta),\delta=\min\left\\{1,a_{0},k,l\right\\},$ such that

$(ka(|\xi|)\xi-la(|\eta|)\eta)\cdot(\xi-\eta)\geq
C(\delta)\Phi(|\xi-\eta|)\quad\forall\xi,\eta\in\mathbb{R}^{N},$ (3.18)

holds, provided that $(\mathrm{a}1)$ hold. Therefore, if we apply (3.18) and
(3.17) to the lines (3.15)and (3.16) respectively, we deduce that

$0\leq
C(\delta)\int_{\Omega^{2}}\Phi\left(h_{x,y}(u_{\ast}-v_{\ast})\right)d\mu+\int_{\Omega}\Phi(|u_{\ast}-v_{\ast}|)dx\leq
0.$

Or

$\Psi\left(u_{\ast}-v_{\ast}\right)=0.$

As consequence of Proposition 2.3, we deduce that

$0\leq\min\left\\{\left\|u_{\ast}-v_{\ast}\right\|^{\varphi^{-}},\left\|u_{\ast}-v_{\ast}\right\|^{\varphi^{+}}\right\\}\leq\Psi\left(u_{\ast}-v_{\ast}\right),$

it follows that

$\left\|u_{\ast}-v_{\ast}\right\|=0.$

Therefore, we obtain $u_{\ast}=v_{\ast}$ in $W^{s}L_{\varPhi}(\Omega),$ that
is, $u_{\ast}$ is the unique solution to problem $(P_{a})$. ∎

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