# Embedding and extension results in Fractional Musielak-Sobolev spaces

E. Azroul1, A. Benkirane2 M. SHIMI3 and M. Srati4 E. Azroul, A. Benkirane, M
Shimi and M. Srati
Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz,
Laboratory of Mathematical Analysis and Applications, Fez, Morocco.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we are concerned with some qualitative properties of the new
fractional Musielak-Sobolev spaces $W^{s}L_{\varPhi_{x,y}}$ such that the
generalized Poincaré type inequality and some continuous and compact embedding
theorems of these spaces. Moreover, we prove that any function in
$W^{s}L_{\varPhi_{x,y}}(\Omega)$ may be extended to a function in
$W^{s}L_{\varPhi_{x,y}}(\mathbb{R}^{N})$, with $\Omega\subset\mathbb{R}^{N}$
is a bounded domain of class $C^{0,1}$. In addition, we establish a result
relates to the complemented subspace in
$W^{s}{L_{\varPhi_{x,y}}}\left(\mathbb{R}^{N}\right)$. As an application,
using the mountain pass theorem and some variational methods, we investigate
the existence of a nontrivial weak solution for a class of nonlocal fractional
type problems with Dirichlet boundary data.

###### Key words and phrases:

Fractional Musielak-Sobolev spaces, Nonlocal problems, Direct variational
method.

###### 2010 Mathematics Subject Classification:

46E35, 35R11, 35J20, 47G20.

##  1\. Introduction and preliminaries results

In the last decades, the fractional Sobolev spaces have been a classical topic
in functional and harmonic analysis all along, see for instance [34, 47] which
treat the topic in detail. On the other hand, fractional spaces, and the
corresponding nonlocal problems, are now experiencing impressive applications
in different subjects, both for the pure mathematical research and in view of
concrete applications, such as, phase transitions [3], crystal dislocation
[20], conservation laws [21], mathematical finance [27], anomalous diffusion
[38], stratified materials [43], etc.

As well known, when $s\in(0,1)$, $p\in[1,\infty)$ and $\Omega$ is an open
subset of $\mathbb{R}^{N}$, with $C^{0,1}$-regularity, any function in
fractional Sobolev space $W^{s,p}(\Omega)$ may be extended to a function in
$W^{s,p}(\mathbb{R}^{N})$ (see [30]). Extension results are quite important in
applications and for improve certain embeddings theorems, in the integer
classical case [26, 28, 32, 45, 46], as well as in the fractional case [29,
30, 31, 48]. All these previous results are held under certain crucial
regularity assumptions on the domain $\Omega$.

The problem of characterizing the class of sets that are extension domains for
$W^{s}L_{\varPhi_{x,y}}$ is open, where $W^{s}L_{\varPhi_{x,y}}$ is the new
fractional Musielak-Sobolev space introduced recently in [5]. In general, an
arbitrary open set $\Omega$ is not an extension domain for
$W^{s}L_{\varPhi_{x,y}}$. When $s$ is an integer, we cite [32] for a complete
characterization in the special case when $s=1$, $\varPhi_{x,y}(t)=t^{2}$ and
$N=2$, and we refer the interested reader to the nice monograph of Leoni [35]
in which this problem is very well discussed (see, in particular, Chapter 11
and Chapter 12).

The study of variational problems where the modular function satisfies
nonpolynomial growth conditions instead of having the usual $p$-structure
arouses much interest in the development of applications to electrorheological
fluids as an important class of non-Newtonian fluids (sometimes referred to as
smart fluids). The electro-rheological fluids are characterized by their
ability to drastically change the mechanical properties under the influence of
an external electromagnetic field. A mathematical model of electro-rheological
fluids was proposed by Rajagopal and Ruzicka (we refer the reader to [41, 42]
for more details). Another important application is related to image
processing [44] where this kind of diffusion operator is used to underline the
borders of the distorted image and to eliminate the noise.

Recently, great attention has been devoted to the study of a new class of
fractional Sobolev spaces and related nonlocal problems, in particular, in the
fractional Orlicz-Sobolev spaces $W^{s}L_{\varPhi}(\Omega)$ (see [6, 7, 13,
14, 17, 19, 24, 25]) and in the fractional Sobolev spaces with variable
exponents $W^{s,p(x,y)}(\Omega)$ (see [8, 9, 10, 11, 12, 15, 16, 33]), in
which the authors establish some basic properties of these modular spaces and
the associated nonlocal operators, they also obtained certain existence
results for nonlocal problems involving this type of integro-differential
operators. Furthermore, in that context, the authors in [5] introduced a new
functional framework which can be seen as a natural generalization of the
above mentioned functional spaces.

Our main purpose in this paper is to continue the study of the aforementioned
new class of fractional Sobolev spaces and the related nonlocal operators, by
introducing the new fractional Musielak-Sobolev spaces
$W^{s}L_{\varPhi_{x,y}}(\Omega)$. In addition, we establish certain basic
properties of those spaces and we prove some continuous and compact embedding
theorems of these spaces into Musielak spaces. Moreover, we show that any
function in fractional Musielak-Sobolev spaces
$W^{s}L_{\varPhi_{x,y}}(\Omega)$ may be extended to a function in
$W^{s}L_{\varPhi_{x,y}}(\mathbb{R}^{N})$, with $\Omega\subset\mathbb{R}^{N}$
is a bounded domain of class $C^{0,1}$. As an application, we are concerned
with the existence of weak solutions of the following nonlocal problem

$(\mathcal{P}_{a})\hskip
14.22636pt\left\\{\begin{array}[]{clclc}(-\Delta)^{s_{1}}_{a_{(x,.)}}u+(-\Delta)^{s_{2}}_{a_{(x,.)}}u&=&f(x,u)&\text{
in }&\Omega,\\\ \\\ \hskip 28.45274pt\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u&=&0&\text{ in
}&\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $\Omega$ is an open bounded subset in $\mathbb{R}^{N}$, $N\geqslant 1$,
with Lipschitz boundary $\partial\Omega$, $0<s_{2}\leqslant s_{1}<1$,
$f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a Carathéodory function
satisfies some suitable conditions. Moreover, for any $i=1,2$,
$(-\Delta)^{s_{i}}_{a_{(x,.)}}$ is the nonlocal integro-differential operator
of elliptic type defined as follows

$\displaystyle(-\Delta)^{s_{i}}_{a_{(x,.)}}u(x)=2\lim\limits_{\varepsilon\searrow
0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}a_{(x,y)}\left(\dfrac{|u(x)-u(y)|}{|x-y|^{s_{i}}}\right)\dfrac{u(x)-u(y)}{|x-y|^{s_{i}}}\dfrac{dy}{|x-y|^{N+s_{i}}},$

for all $x\in\mathbb{R}^{N}$, where $(x,y,t)\mapsto
a_{(x,y)}(t):=a(x,y,t):\overline{\Omega}\times\overline{\Omega}\times\mathbb{R}\longrightarrow\mathbb{R}$
is such that :
$\varphi(.,.,.):\overline{\Omega}\times\overline{\Omega}\times\mathbb{R}\longrightarrow\mathbb{R}$
defined by

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

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

$\varPhi_{x,y}(t):=\varPhi(x,y,t)=\int_{0}^{t}\varphi_{x,y}(\tau)d\tau\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all
}(x,y)\in\overline{\Omega}\times\overline{\Omega},\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and all }t\geqslant 0.$

Then, $\varPhi_{x,y}$ is a Musielak function (see [39]), that is

  * •

$\varPhi(x,y,.)$ is a $\varPhi$-function for every
$(x,y)\in\overline{\Omega}\times\overline{\Omega}$, i.e., is continuous,
nondecreasing function with $\varPhi(x,y,0)=0$, $\varPhi(x,y,t)>0$ for $t>0$
and $\varPhi(x,y,t)\rightarrow\infty$ as $t\rightarrow\infty$.

  * •

For every $t\geqslant 0$,
$\varPhi(.,.,t):\overline{\Omega}\times\overline{\Omega}\longrightarrow\mathbb{R}$
is a measurable function.

We introduce the function $\widehat{a}_{x}(t):=\widehat{a}(x,t)=a_{(x,x)}(t)$
for all $(x,t)\in\overline{\Omega}\times\mathbb{R}$. Then, we define the
function
$\widehat{\varphi}(.,.):\overline{\Omega}\times\mathbb{R}\longrightarrow\mathbb{R}$
by

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

is increasing homeomorphism from $\mathbb{R}$ onto itself. If we set

$\widehat{\varPhi}_{x}(t):=\widehat{\varPhi}(x,t)=\int_{0}^{t}\widehat{\varphi}_{x}(\tau)d\tau\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all}\leavevmode\nobreak\ \leavevmode\nobreak\
t\geqslant 0.$ (1.1)

Then, $\widehat{\varPhi}_{x}$ is also a Musielak function.

######  Remark 1.1.

  * •

For all $(x,y)\in\overline{\Omega}\times\overline{\Omega}$, $\varPhi_{x,y}$
and $\widehat{\varPhi}_{x}$ are two convex and increasing functions from
$\mathbb{R}^{+}$ to $\mathbb{R}^{+}$.

  * •

Note that $(-\Delta)^{s}_{a_{(x,.)}}$ is a nonlocal integro-differential
operator of elliptic type which can be seen as a generalization of the
fractional $p(x,.)$-Laplacian operator $(-\Delta_{p(x,.)})^{s}$ (when
$a_{(x,y)}(t)=|t|^{p(x,y)-2}$, see for instance [8, 33]), and of the
fractional $p$-Laplacian operator $(-\Delta)_{p}^{s}$ in the constant exponent
case ( when $p(x,y)=p=\text{constant}$, i.e. $a_{(x,y)}(t)=|t|^{p-2}$).

  * •

The operator $(-\Delta)^{s}_{a_{(x,.)}}$ reduces to the fractional
$a$-Laplacian if $a_{(x,y)}(t)=a(t)$, i.e. the function $a$ is independent of
variables $x,y$ (see for example [6, 24]). On the other hand, we remark that
is the fractional version of the well-known $a$-Laplacian operator
$div\big{(}a(x,|\nabla u(x)|)\nabla u(x)\big{)}$ which is associated with the
Musielak-Sobolev spaces (see [37]).

  * •

In contrast to the classical $p(x)$-Laplacian and $a$-Laplacian, which are
local operators, the integro-differential operator $(-\Delta)^{s}_{a_{(x,.)}}$
is a paradigm of the vast family of nonlocal nonlinear operators, and this has
immediate consequences in the formulation of basic questions such as the
Dirichlet problem. For this, the Dirichlet datum is given in
$\mathbb{R}^{N}\setminus\Omega$ (which is different from the classical case of
the $p(x)$-Laplacian and $a$-Laplacian) and not simply on $\partial\Omega$.
Moreover, the value of $(-\Delta)^{s}_{a_{(x,.)}}u(x)$ at any point
$x\in\Omega$ depends not only on the values of $u$ on $\Omega$, but actually
on the entire space $\mathbb{R}^{N}$, which implies that the first equation in
$(\mathcal{P}_{a})$ is no longer a pointwise equation, it is no longer a
pointwise identity. Hence, it is often called nonlocal problem. This causes
some mathematical difficulties which make the study of such a problem
particularly interesting.

This paper is organized as follows, In rest of this Section, we recall some
preliminaries on the new fractional Musielak-Sobolev spaces which will be used
frequently. In Section 2, we establish some qualitative properties of these
new spaces as the generalized Poincaré type inequality and some continuous and
compact embedding theorems. Moreover, we prove that any function in fractional
Musielak-Sobolev spaces $W^{s}L_{\varPhi_{x,y}}(\Omega)$ may be extended to a
function in $W^{s}L_{\varPhi_{x,y}}(\mathbb{R}^{N})$, with
$\Omega\subset\mathbb{R}^{N}$ is a bounded domain of class $C^{0,1}$.
Furthermore, we establish a result relates to the complemented subspace in
$W^{s}{L_{\varPhi_{x,y}}}\left(\mathbb{R}^{N}\right)$. In Section 3, as an
application, we obtain the existence of a nontrivial weak solution for problem
$(\mathcal{P}_{a})$, by using a version of mountain pass theorem.

Next, For the function $\widehat{\varPhi}_{x}$ given in (1.1), we introduce
the Musielak class as follows

$K_{\widehat{\varPhi}_{x}}(\Omega)=\left\\{u:\Omega\longrightarrow\mathbb{R}\text{
mesurable }:\int_{\Omega}\widehat{\varPhi}_{x}(|u(x)|)dx<\infty\right\\},$

and the Musielak space

$L_{\widehat{\varPhi}_{x}}(\Omega)=\left\\{u:\Omega\longrightarrow\mathbb{R}\text{
mesurable }:\int_{\Omega}\widehat{\varPhi}_{x}(\lambda|u(x)|)dx<\infty\text{
for some }\lambda>0\right\\}.$

The space $L_{\widehat{\varPhi}_{x}}(\Omega)$ is a Banach space endowed with
the Luxemburg norm

$||u||_{\widehat{\varPhi}_{x}}=\inf\left\\{\lambda>0\text{ :
}\int_{\Omega}\widehat{\varPhi}_{x}\left(\dfrac{|u(x)|}{\lambda}\right)dx\leqslant
1\right\\}.$

The conjugate function of $\varPhi_{x,y}$ is defined by
$\overline{\varPhi}_{x,y}(t)=\int_{0}^{t}\overline{\varphi}_{x,y}(\tau)d\tau\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all
}(x,y)\in\overline{\Omega}\times\overline{\Omega}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and all }t\geqslant 0$, where
$\overline{\varphi}_{x,y}:\mathbb{R}\longrightarrow\mathbb{R}$ is given by
$\overline{\varphi}_{x,y}(t):=\overline{\varphi}(x,y,t)=\sup\left\\{s\text{ :
}\varphi(x,y,s)\leqslant t\right\\}.$ Furthermore, we have the following
Hölder type inequality

$\left|\int_{\Omega}uvdx\right|\leqslant
2||u||_{\widehat{\varPhi}_{x}}||v||_{\overline{\widehat{\varPhi}}_{x}}\hskip
14.22636pt\text{ for all }u\in L_{\widehat{\varPhi}_{x}}(\Omega)\text{ and
}v\in L_{\overline{\widehat{\varPhi}}_{x}}(\Omega).$ (1.2)

Throughout this paper, we assume that

$1<\varphi^{-}:=\inf_{t\geqslant
0}\dfrac{t\varphi_{x,y}(t)}{\varPhi_{x,y}(t)}\leqslant\varphi^{+}:=\sup_{t\geqslant
0}\dfrac{t\varphi_{x,y}(t)}{\varPhi_{x,y}(t)}<+\infty\text{ for all
}(x,y)\in\overline{\Omega}\times\overline{\Omega}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\varPhi_{1})$

This relation implies that

$1<\widehat{\varphi}^{-}:=\inf_{t\geqslant
0}\dfrac{t\widehat{\varphi}_{x}(t)}{\widehat{\varPhi}_{x}(t)}\leqslant\widehat{\varphi}^{+}:=\sup_{t\geqslant
0}\dfrac{t\widehat{\varphi}_{x}(t)}{\widehat{\varPhi}_{x}(t)}<+\infty,\text{
for all }x\in\overline{\Omega}.$ (1.3)

It follows that $\varPhi_{x,y}$ and $\widehat{\varPhi}_{x}$ satisfy the global
$\Delta_{2}$-condition (see [37]), written $\varPhi_{x,y}\in\Delta_{2}$ and
$\widehat{\varPhi}_{x}\in\Delta_{2}$, that is,

$\varPhi_{x,y}(2t)\leqslant K_{1}\varPhi_{x,y}(t)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all
}(x,y)\in\overline{\Omega}\times\overline{\Omega},\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and all }t\geqslant 0,$ (1.4)

and

$\widehat{\varPhi}_{x}(2t)\leqslant
K_{2}\widehat{\varPhi}_{x}(t)\leavevmode\nobreak\ \leavevmode\nobreak\ \text{
for any }x\in\overline{\Omega},\leavevmode\nobreak\ \leavevmode\nobreak\
\text{ and all }t\geqslant 0,$ (1.5)

where $K_{1}$ and $K_{2}$ are two positive constants. The inequality
$(\ref{rr1})$ implies that
$L_{\widehat{\varPhi}_{x}}(\Omega)=K_{\widehat{\varPhi}_{x}}(\Omega)$ (see
[39]).

An important role in manipulating the Musielak spaces is played by the modular
of the space $L_{\widehat{\varPhi}_{x}}(\Omega)$. It is worth noticing that
the relation between the norm and the modular shows an equivalence between the
topology defined by the norm and that defined by the modular.

######  Proposition 1.1.

([37, Proposition 2.1]). Assume that the condition $(\varPhi_{1})$ is
satisfied. Then, for all $u\in L_{\widehat{\varPhi}_{x}}(\Omega)$, the
following relations hold true

  1. (i)

$||u||_{\widehat{\varPhi}_{x}}>1$ $\Rightarrow$
$||u||_{\widehat{\varPhi}_{x}}^{\widehat{\varphi}^{-}}\leqslant\int_{\Omega}\widehat{\varPhi}_{x}(|u|)dx\leqslant||u||_{\widehat{\varPhi}_{x}}^{\widehat{\varphi}^{+}},$

  2. (ii)

$||u||_{\widehat{\varPhi}_{x}}<1$ $\Rightarrow$
$||u||_{\widehat{\varPhi}_{x}}^{\widehat{\varphi}^{+}}\leqslant\int_{\Omega}\widehat{\varPhi}_{x}(|u|)dx\leqslant||u||_{\widehat{\varPhi}_{x}}^{\widehat{\varphi}^{-}}.$

######  Definition 1.1.

Let $A_{x}(t)$,
$B_{x}(t):\mathbb{R}^{+}\times\Omega\longrightarrow\mathbb{R}^{+}$ be two
Musielak functions. $A_{x}$ is stronger $($resp essentially stronger$)$ than
$B_{x}$, $A_{x}\succ B_{x}$ (resp $A_{x}\succ\succ B_{x}$) in symbols, if for
almost every $x\in\overline{\Omega}$

$B(x,t)\leqslant A(x,at),\text{ , }t\geqslant t_{0}\geqslant 0,$

for some $($resp for each$)$ $a>0$ and $t_{0}$ (depending on $a$).

######  Remark 1.2.

$($[1, Section 8.5]$)$. $A_{x}\succ\succ B_{x}$ is equivalent to the condition

$\lim_{t\rightarrow\infty}\left(\sup\limits_{x\in\overline{\Omega}}\dfrac{B(x,\lambda
t)}{A(x,t)}\right)=0,$

for all $\lambda>0$.

######  Theorem 1.1.

$($[18]$)$. Let $\Omega$ be an open subset of $\mathbb{R}^{N}$ which has a
finite volume. Let $A_{x}(t)$, $B_{x}(t):\leavevmode\nobreak\
\mathbb{R}^{+}\times\Omega\longrightarrow\mathbb{R}^{+}$ be two Musielak
functions such that $B_{x}\prec\prec A_{x}$.Then any bounded subset $S$ of
$L_{A_{x}}(\Omega)$ which is precompact in $L^{1}(\Omega)$, is also precompact
in $L_{B_{x}}(\Omega)$.

Now, due to the nonlocality of the operator $(-\Delta)^{s}_{a_{(x,.)}}$, we
define the new fractional Musielak-Sobolev space as introduce in [5] as
follows

$W^{s}{L_{\varPhi_{x,y}}}(\Omega)=\Bigg{\\{}u\in
L_{\widehat{\varPhi}_{x}}(\Omega):\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{\lambda|u(x)-u(y)|}{|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}<\infty\text{
for some }\lambda>0\Bigg{\\}}.$

This space can be equipped with the norm

$||u||_{s,\varPhi_{x,y}}=||u||_{\widehat{\varPhi}_{x}}+[u]_{s,\varPhi_{x,y}},$
(1.6)

where $[.]_{s,\varPhi_{x,y}}$ is the Gagliardo seminorm defined by

$[u]_{s,\varPhi_{x,y}}=\inf\Bigg{\\{}\lambda>0:\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}\leqslant
1\Bigg{\\}}.$

######  Remark 1.3.

Fractinal Orlicz-Sobolev spaces (see e.g. [6, 7, 24]) and fractional Sobolev
spaces with variable exponent (see e.g. [8, 16, 33]) are two distinct
extensions of fractional Sobolev spaces $W^{s,p}(\Omega)$ (see [22, 30]), and
they are two special kinds of the new fractional Musielak-Sobolev spaces (see
e.g. [5]). For the reader convenience, we recall the definition of the
aforementioned spaces.

a$)-$ For the case: $\varPhi_{x,y}(t)=\varPhi(t)$, i.e. $\varPhi$ is
independent of variables $x,y$, we say that $L_{\varPhi}$ and
$W^{s}L_{\varPhi}$ are Orlicz spaces and fractional Orlicz-Sobolev spaces
respectively such that

$W^{s}{L_{\varPhi}}(\Omega)=\Bigg{\\{}u\in
L_{\varPhi}(\Omega):\int_{\Omega}\int_{\Omega}\varPhi\left(\dfrac{\lambda|u(x)-u(y)|}{|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}<\infty\textnormal{
for some }\lambda>0\Bigg{\\}}.$

b$)-$ For the case: $\varPhi_{x,y}(t)=|t|^{p(x,y)}$ for all
$(x,y)\in\overline{\Omega}\times\overline{\Omega}$, where
$p:\overline{\Omega}\times\overline{\Omega}\longrightarrow(1,+\infty)$ is a
continuous bounded function such that

$1<p^{-}=\underset{(x,y)\in\overline{\Omega}\times\overline{\Omega}}{\min}p(x,y)\leqslant
p(x,y)\leqslant
p^{+}=\underset{(x,y)\in\overline{\Omega}\times\overline{\Omega}}{\max}p(x,y)<+\infty,$
(1.7)

and

$p\leavevmode\nobreak\ \text{is symmetric, that is, }\leavevmode\nobreak\
\leavevmode\nobreak\ p(x,y)=p(y,x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all
}(x,y)\in\overline{\Omega}\times\overline{\Omega}.$ (1.8)

If denoted by $\bar{p}(x)=p(x,x)$ for all $x\in\overline{\Omega}.$ Then, we
replace $L_{\varPhi_{x}}$ by $L^{\overline{p}(x)}$, and
$W^{s}L_{\varPhi_{x,y}}$ by $W^{s,p(x,y)}$ and we refer them as variable
exponent Lebesgue spaces, and fractional Sobolev spaces with variable exponent
respectively, defined by

$L^{\overline{p}(x)}(\Omega)=\bigg{\\{}u:\Omega\longrightarrow\mathbb{R}\leavevmode\nobreak\
\leavevmode\nobreak\
\text{measurable}:\int_{\Omega}|u(x)|^{\overline{p}(x)}dx<+\infty\bigg{\\}},$

and

$W=W^{s,p(x,y)}(\Omega)$ $\hskip 17.07182pt=\bigg{\\{}u\in
L^{\bar{p}(x)}(\Omega):\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{\lambda^{p(x,y)}|x-y|^{sp(x,y)+N}}\leavevmode\nobreak\
dxdy<+\infty,\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for
some}\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda>0\bigg{\\}}.$

with the norm

$\|u\|_{W}=\|u\|_{L^{\bar{p}(x)}(\Omega)}+[u]_{W},$

where $[.]_{W}$ is a Gagliardo seminorm with variable exponent given by

$[u]_{W}=[u]_{s,p(x,y)}=\inf\bigg{\\{}\lambda>0:\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{\lambda^{p(x,y)}|x-y|^{N+sp(x,y)}}\leavevmode\nobreak\
dxdy\leqslant 1\bigg{\\}}.$

When $p(x,y)=p=\text{constant}\in(1,+\infty)$, the space $W$ reduce to the
fractional Sobolev space $W^{s,p}(\Omega)$. For a comprehensive introduction
to the study of these spaces and the related nonlocal problems, we refer the
reader to [22, 30].

######  Theorem 1.2.

$($[5]$)$. Let $\Omega$ be an open subset of $\mathbb{R}^{N}$, and let
$s\in(0,1)$. The space $W^{s}L_{\varPhi_{x,y}}(\Omega)$ is a Banach space with
respect to the norm $(\ref{r2})$, and a separable $($resp. reflexive$)$ space
if and only if $\varPhi_{x,y}\in\Delta_{2}$ $($resp.
$\varPhi_{x,y}\in\Delta_{2}$ and $\overline{\varPhi}_{x,y}\in\Delta_{2}$$)$.
Furthermore, if $\varPhi_{x,y}\in\Delta_{2}$ and $\varPhi_{x,y}(\sqrt{t})$ is
convex, then the space $W^{s}L_{\varPhi_{x,y}}(\Omega)$ is an uniformly convex
space.

######  Definition 1.2.

$($[5]$)$. We say that $\varPhi_{x,y}$ satisfies the fractional boundedness
condition, written $\varPhi_{x,y}\in\mathcal{B}_{f}$, if

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sup\limits_{(x,y)\in\overline{\Omega}\times\overline{\Omega}}\varPhi_{x,y}(1)<\infty.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\varPhi_{2})$

######  Theorem 1.3.

$($[5]$)$. Let $\Omega$ be an open subset of $\mathbb{R}^{N}$, and $0<s<1$.
Assume that $\varPhi_{x,y}\in\mathcal{B}_{f}$. Then,

$C^{2}_{0}(\Omega)\subset W^{s}L_{\varPhi_{x,y}}(\Omega).$

######  Lemma 1.1.

$($[5]$)$. Assume that $(\varPhi_{1})$ is satisfied. Then

$\overline{\varPhi}_{x,y}(\varphi_{x,y}(t))\leqslant\varphi^{+}\varPhi_{x,y}(t)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all
}(x,y)\in\overline{\Omega}\times\overline{\Omega}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and all }t\geqslant 0.$

######  Lemma 1.2.

$($[5]$)$. Assume that $(\varPhi_{1})$ is satisfied. Then the following
inequalities hold true:

$\varPhi_{x,y}(\sigma
t)\geqslant\sigma^{\varphi^{-}}\varPhi_{x,y}(t)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }t>0\text{ and any }\sigma>1,$ (1.9)
$\varPhi_{x,y}(\sigma
t)\geqslant\sigma^{\varphi^{+}}\varPhi_{x,y}(t)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }t>0\text{ and any }\sigma\in(0,1),$
(1.10) $\varPhi_{x,y}(\sigma
t)\leqslant\sigma^{\varphi^{+}}\varPhi_{x,y}(t)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }t>0\text{ and any }\sigma>1,$ (1.11)
$\varPhi_{x,y}(t)\leqslant\sigma^{\varphi^{-}}\varPhi_{x,y}\left(\dfrac{t}{\sigma}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }t>0\text{ and any }\sigma\in(0,1).$
(1.12)

For any $u\in W^{s}L_{\varPhi_{x,y}}(\Omega)$, we define the modular function
on $W^{s}L_{\varPhi_{x,y}}(\Omega)$ as follows

$\varPsi(u)=\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}+\int_{\Omega}\widehat{\varPhi}_{x}\left(|u(x)|\right)dx.$

######  Proposition 1.2.

$($[5]$)$. Assume that $(\varPhi_{1})$ is satisfied. Then, for any $u\in
W^{s}L_{\varPhi_{x,y}}(\Omega)$, the following relations hold true:

$||u||_{s,\varPhi_{x,y}}>1\Longrightarrow||u||_{s,\varPhi_{x,y}}^{\varphi^{-}}\leqslant\varPsi(u)\leqslant||u||_{s,\varPhi_{x,y}}^{\varphi^{+}},$
(1.13)
$||u||_{s,\varPhi_{x,y}}<1\Longrightarrow||u||_{s,\varPhi_{x,y}}^{\varphi^{+}}\leqslant\varPsi(u)\leqslant||u||_{s,\varPhi_{x,y}}^{\varphi^{-}}.$
(1.14)

######  Proposition 1.3.

$($[5]$)$ Suppose that $(\varPhi_{1})$ is satisfied. Then, for any $u\in
W^{s}L_{\varPhi_{x,y}}(\Omega)$, the following assertions hold true:

$[u]_{s,\varPhi_{x,y}}>1\Longrightarrow[u]^{\varphi^{-}}_{s,\varPhi_{x,y}}\leqslant\phi(u)\leqslant[u]^{\varphi^{+}}_{s,\varPhi_{x,y}},$
(1.15)
$[u]_{s,\varPhi_{x,y}}<1\Longrightarrow[u]^{\varphi^{+}}_{s,\varPhi_{x,y}}\leqslant\phi(u)\leqslant[u]^{\varphi^{-}}_{s,\varPhi_{x,y}},$
(1.16)

where
$\phi(u)=\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$.

We conclude this section by recalling a version of the mountain pass theorem
as given in [4] which is the main tool to prove our existence result.

######  Theorem 1.4.

Let $X$ be a real Banach space and $J\in C^{1}(X,\mathbb{R})$ with $J(0)=0$.
Suppose that the following conditions hold:

$(G_{1})$ There exists $\rho>0\text{ and }r>0\text{ such that }J(u)\geqslant
r\text{ for }||u||=\rho$.

$(G_{2})$ There exists $e\in X\text{ with }||e||>\rho\text{ such that
}J(e)\leqslant 0$.
Let

$c:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))\text{ with
}\Gamma=\left\\{\gamma\in C([0,1],X);\gamma(0)=0,\gamma(1)=e\right\\}.$

Then there exists a sequence $\left\\{u_{n}\right\\}$ in $X$ such that

$J(u_{n})\rightarrow c\text{ \hskip 28.45274pt and \hskip
28.45274pt}J^{\prime}(u_{n})\rightarrow 0.$

##  2\. Embeddings and extension results

In this section, we will establish some embeddings and extension results of
the new fractional Musielak Sobolev spaces.

###  2.1. Some embedding results

This subsection is devoted to the embedding results of the new fractional
Musielak-Sobolev spaces $W^{s}L_{\varPhi_{x,y}}(\Omega)$. To this end, we
follow the same approach used to obtain the embedding results in the
fractional Orlicz-Sobolev space $W^{s}L_{\varPhi}(\Omega)$ established in [6].

First, it is worth noticing that, as in the classical case with $s_{1},s_{2}$
being an integers, the space $W^{s_{1}}L_{\varPhi_{x,y}}(\Omega)$ is
continuously embedded in $W^{s_{2}}L_{\varPhi_{x,y}}(\Omega)$ when
$s_{2}\leqslant s_{1}$, as next result points out.

######  Proposition 2.1.

Let $\Omega$ be an open subset of $\mathbb{R}^{N}$ and let $0<s_{2}\leqslant
s_{1}<1$. Assume that $(\varPhi_{1})$-$(\varPhi_{2})$ holds true. Then,
$W^{s_{1}}{L_{\varPhi_{x,y}}}(\Omega)$ continuously embedded in
$W^{s_{2}}{L_{\varPhi_{x,y}}}(\Omega)$.

Proof. Let $u\in W^{s_{1}}{L_{\varPhi_{x,y}}}(\Omega)$ and $\lambda>0$, If we
define

$\Omega_{1}=\left\\{(x,y)\subset\Omega\times\Omega\leavevmode\nobreak\
\leavevmode\nobreak\ \ \leavevmode\nobreak\ \leavevmode\nobreak\
\dfrac{|D^{s_{1}}u|}{\lambda}\leqslant 1\right\\}\text{ and
}\Omega_{2}=\Omega\times\Omega\setminus\Omega_{1},$

where $D^{s_{i}}u=\dfrac{u(x)-u(y)}{|x-y|^{s_{i}}},i=1,2,$ and we set
$d\mu=\dfrac{dxdy}{|x-y|^{N}}$. Then

$\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|D^{s_{2}}u|}{\lambda}\right)d\mu$
$\displaystyle=\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\dfrac{1}{|x-y|^{s_{2}-s_{1}}}\right)d\mu$
$\displaystyle\leqslant\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\right)\dfrac{dxdy}{|x-y|^{N+p(s_{2}-s_{1})}}$
$\displaystyle=\left(\int\int_{\Omega_{1}}+\int\int_{\Omega_{2}}\right)\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\right)\dfrac{dxdy}{|x-y|^{N+p(s_{2}-s_{1})}}$
$\displaystyle=I_{\Omega_{1}}+I_{\Omega_{2}}$

where $p=1$ if $|x-y|\geqslant 1$ or $p=\varphi^{+}$ if $|x-y|<1$ for all
$(x,y)\in\Omega\times\Omega$. Notice that

$I_{\Omega_{1}}=\int\int_{\Omega_{1}}\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\right)\dfrac{dxdy}{|x-y|^{N+p(s_{2}-s_{1})}}\leqslant\sup\limits_{(x,y)\in\overline{\Omega}\times\overline{\Omega}}\varPhi_{x,y}(1)\int\int_{\Omega_{1}}\dfrac{dxdy}{|x-y|^{N+p(s_{2}-s_{1})}}:=c_{1}.$

Since $N>N+p(s_{2}-s_{1})$, it follows that the above integral is finite. On
the other hand, we have

$\displaystyle I_{\Omega_{2}}$
$\displaystyle\leqslant\int\int_{\Omega_{2}}\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\right)\dfrac{dxdy}{|x-y|^{N+p(s_{2}-s_{1})}}$
$\displaystyle\leqslant
d^{p(s_{1}-s_{2})}\int\int_{\Omega_{2}}\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\leqslant
d^{p(s_{1}-s_{2})}\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|D^{s_{1}}u|}{\lambda}\right)d\mu<\infty,$

where $d=\sup\limits_{(x,y)\in\overline{\Omega}\times\overline{\Omega}}|x-y|$.
Hence,

$\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|D^{s_{2}}u|}{[u]_{s_{1},\varPhi_{x,y}}}\right)d\mu$
$\displaystyle\leqslant c_{1}+d^{p(s_{1}-s_{2})}.$

This fact implies that

$[u]_{s_{2},\varPhi_{x,y}}\leqslant\left(c_{1}+d^{p(s_{1}-s_{2})}\right)[u]_{s_{1},\varPhi_{x,y}}.$
(2.1)

Therefore,

$||u||_{s_{2},\varPhi_{x,y}}\leqslant c||u||_{s_{1},\varPhi_{x,y}},$

where $c=(1+c_{1}+d^{p(s_{1}-s_{2})})$.

Given $s\in(0,1)$ and let $\widehat{\varPhi}_{x}$ as defined in (1.1). We
denote by $\widehat{\varPhi}_{x}^{-1}$ the inverse function of
$\widehat{\varPhi}_{x}$ which satisfies the following conditions:

$\int_{0}^{1}\dfrac{\widehat{\varPhi}_{x}^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau<\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }x\in\overline{\Omega},$ (2.2)

$\int_{1}^{\infty}\dfrac{\widehat{\varPhi}_{x}^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau=\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }x\in\overline{\Omega}.$ (2.3)

Note that, if $\varphi_{x,y}(t)=|t|^{p(x,y)-1}$, then (2.2) holds precisely
when $sp(x,y)<N$ for all $(x,y)\in\overline{\Omega}\times\overline{\Omega}$.
If (2.3) is satisfied, we define the inverse Musielak conjugate function of
$\widehat{\varPhi}_{x}$ as follows

$(\widehat{\varPhi}^{*}_{x,s})^{-1}(t)=\int_{0}^{t}\dfrac{\widehat{\varPhi}_{x}^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau.$
(2.4)

######  Theorem 2.1.

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ with
$C^{0,1}$-regularity and bounded boundary. If $(\ref{15})$ and $(\ref{16n})$
hold, then

$W^{s}{L_{\varPhi_{x,y}}}(\Omega)\hookrightarrow
L_{\widehat{\varPhi}^{*}_{x,s}}(\Omega).$ (2.5)

######  Theorem 2.2.

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ and
$C^{0,1}$-regularity with bounded boundary. If $(\ref{15})$ and $(\ref{16n})$
hold, then the embedding

$W^{s}{L_{\varPhi_{x,y}}}(\Omega)\hookrightarrow L_{B_{x}}(\Omega),$ (2.6)

is compact for all $B_{x}\prec\prec\widehat{\varPhi}^{*}_{x,s}$.

The proof will be carried out in a several lemmas. The first one establishes
an estimate for the Musielak conjugate function $\widehat{\varPhi}^{*}_{x,s}$
defined in (2.4).

######  Lemma 2.1.

Let $s\in(0,1)$, we assume that $(\ref{15})$\- $(\ref{16n})$ holds true and
let $\widehat{\varPhi}^{*}_{x,s}$ be defined by $(\ref{17})$. Then for all
$s^{\prime}\in(0,s)$, the following conclusions may be drawn.

  1. (1)

$[\widehat{\varPhi}^{*}_{x,s}(t)]^{\frac{N-s^{\prime}}{N}}$ is a Musielak
function, in particular, $\widehat{\varPhi}^{*}_{x,s}$ is a Musielak function.

  2. (2)

For every $\varepsilon>0$, there exists a constant $K_{\varepsilon}>0$ such
that for every $t$,

$[\widehat{\varPhi}^{*}_{x,s}(t)]^{\frac{N-s^{\prime}}{N}}\leqslant\dfrac{1}{2\varepsilon}\widehat{\varPhi}^{*}_{x,s}(t)+\dfrac{K_{\varepsilon}}{\varepsilon}t.$
(2.7)

Proof. The proof is similar to [6, Lemma 3].

######  Lemma 2.2.

Let $\Omega$ be an open subset of $\mathbb{R}^{N}$, and $0<s<1$. Let $f$
satisfies a Lipschitz-condition on $\mathbb{R}$ and $f(0)=0$, then,

  1. (i)

For every $u\in W^{s,1}_{loc}(\Omega)$, if $g(x)=f(|u(x)|)$, then $g\in
W^{s,1}_{loc}(\Omega)$.

  2. (ii)

For every $u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$, if $g(x)=f(|u(x)|)$, then
$g\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$.

In particular, for every $u\in W^{s,1}(\Omega)$, if $g(x)=f(|u(x)|)$, then
$g\in W^{s,1}(\Omega)$.

Proof To prove Lemma 2.2, we follow the same approach as in [6, Lemma 4].

######  Lemma 2.3.

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ and let
$0<s^{\prime}<s<1$. Assume that the condition $(\varPhi_{1})$-$(\varPhi_{2})$
are satisfied, then the space $W^{s}L_{\varPhi_{x,y}}(\Omega)$ is continuously
embedded in $W^{s^{\prime},q}(\Omega)$ for all $q\in[1,\varphi^{-}]$.

Proof By $(\ref{A2})$ there exist $c>0$ such that

$|t|^{\widehat{\varphi}^{-}}\leqslant
c\widehat{\varPhi}_{x}(t)\leavevmode\nobreak\ \leavevmode\nobreak\
t>1\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ for all
}x\in\overline{\Omega}.$ (2.8)

Indeed, form $(\ref{A2})$ we have

$\widehat{\varphi}^{-}\leqslant\dfrac{t\widehat{\varphi}_{x}(t)}{\widehat{\varPhi}_{x}(t)}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t>1,$

so,

$\widehat{\varphi}^{-}\left[\ln(t)\right]^{\prime}\leqslant\left[\ln(\widehat{\varPhi}_{x}(t))\right]^{\prime}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t>1,$

this implies

$|t|^{\varphi^{-}}\leqslant\dfrac{\widehat{\varPhi}_{x}(t)}{\widehat{\varPhi}_{x}(1)}\leqslant
c\widehat{\varPhi}_{x}(t),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
t>1$

where $c=\dfrac{1}{\inf\limits_{x\in\Omega}\widehat{\varPhi}_{x}(1)}$, note
that by definition of $\widehat{\varPhi}_{x}$ and $(\varPhi_{2})$,
$0<c<\infty$.

Then, for $u\in W^{s}L_{\varPhi_{x,y}}(\Omega)$, we have

$\displaystyle\int_{\Omega}|u(x)|^{\widehat{\varphi}^{-}}dx$
$\displaystyle\leqslant\int_{\Omega\cap\left\\{|u|\leqslant
1\right\\}}|u(x)|^{\widehat{\varphi}^{-}}dx+\int_{\Omega\cap\left\\{|u|>1\right\\}}|u(x)|^{\widehat{\varphi}^{-}}dx$
$\displaystyle\leqslant|\Omega|+c\int_{\Omega}\widehat{\varPhi}_{x}(|u(x)|)dx.$

Hence,

$||u||_{\widehat{\varphi}^{-}}\leqslant c_{1}||u||_{\widehat{\varPhi}_{x}},$
(2.9)

where $c_{1}=|\Omega|+c$. On the other hand, similar to Proposition 2.1 we
have

$[u]_{s^{\prime},\varphi^{-}}\leqslant c_{2}[u]_{s,\varPhi_{x,y}}.$ (2.10)

Then, combining $(\ref{lab1})$ with $(\ref{lab2})$, we obtain

$||u||_{s^{\prime},\varphi^{-}}\leqslant c[u]_{s,\varPhi_{x,y}}.$

This completes the proof.

Proof of Theorem 2.1. Let $0<s^{\prime}<s<1$,
$\sigma(t)=[\widehat{\varPhi}^{*}_{x,s}(t)]^{\frac{N-s^{\prime}}{N}}$ and
$u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$, we suppose for the moment that $u$ is
bounded on $\Omega$ and not equal to zero in
$L_{\widehat{\varPhi}_{x}}(\Omega)$, then
$\displaystyle\int_{\Omega}\widehat{\varPhi}^{*}_{x,s}\left(\dfrac{|u(x)|}{\lambda}\right)dx$
decreases continuously from infinity to zero as $\lambda$ increases from zero
to infinity and according, assumes the value unity for some positive value $k$
of $\lambda$, thus

$\int_{\Omega}\widehat{\varPhi}^{*}_{x,s}\left(\dfrac{|u(x)|}{k}\right)dx=1\text{
, }k=||u||_{\widehat{\varPhi}^{*}_{x,s}}.$ (2.11)

Let $f(x)=\sigma\left(\dfrac{u(x)}{k}\right)$. Using Lemma 2.2 $u\in
W^{s^{\prime},1}(\Omega)$, and $\sigma$ is Lipschitz function (see Lemma 2.1),
so from Lemma 2.2 we have $f\in W^{s^{\prime},1}(\Omega)$, and since
$N>s^{\prime}$, then by [29, Theroem 4.58], one has,

$W^{s^{\prime},1}(\Omega)\hookrightarrow L^{\frac{N}{N-s^{\prime}}}(\Omega).$

Hence

$||f||_{L^{\frac{N}{N-s^{\prime}}}}\leqslant
k_{1}\left(||f||_{L^{1}}+[f]_{s^{\prime},1}\right),$

and by (2.11), we get

$1=\left(\int_{\Omega}\widehat{\varPhi}^{*}_{x,s}\left(\dfrac{|u(x)|}{k}\right)dx\right)^{\frac{N-s^{\prime}}{N}}=||f||_{L^{\frac{N}{N-s^{\prime}}}}.$

This implies that

$\displaystyle 1$ $\displaystyle\leqslant
k_{1}\left(||f||_{L^{1}}+[f]_{s^{\prime},1}\right)$ (2.12)
$\displaystyle=k_{1}\left(\int_{\Omega}\sigma\left(\dfrac{u(x)}{k}\right)dx+\int_{\Omega}\int_{\Omega}\dfrac{|f(x)-f(y)|}{|x-y|^{N+s^{\prime}}}dxdy\right)$
$\displaystyle=k_{1}\left(\int_{\Omega}\sigma\left(\dfrac{u(x)}{k}\right)dx+\int_{\Omega}\int_{\Omega}\dfrac{\left|\sigma\left(\dfrac{u(x)}{k}\right)-\sigma\left(\dfrac{u(y)}{k}\right)\right|}{|x-y|^{N+s^{\prime}}}dxdy\right)$
$\displaystyle=k_{1}I_{1}+k_{1}I_{2}.$

From (2.7) and since $L_{\widehat{\varPhi}_{x}}(\Omega)\hookrightarrow
L^{1}(\Omega)$, then for $\varepsilon=k_{1}$, we have

$k_{1}I_{1}\leqslant\dfrac{1}{2}\int_{\Omega}\widehat{\varPhi}^{*}_{x,s}\left(\dfrac{|u(x)|}{k}\right)dx+\dfrac{k_{\varepsilon}}{k}\int_{\Omega}|u(x)|dx\leqslant\dfrac{1}{2}+\dfrac{k^{\prime}_{\varepsilon}}{k}||u||_{\widehat{\varPhi}_{x}},$
(2.13)

On the other hand, as $\sigma$ is a Lipschitz function, then there exists
$C>0$ such that

$k_{1}I_{2}\leqslant\dfrac{C}{k}\int_{\Omega}\int_{\Omega}\dfrac{|u(x)-u(y)|}{|x-y|^{N+s^{\prime}}}dxdy.$

Next, by Lemma 2.2, we have

$\int_{\Omega}\int_{\Omega}\dfrac{|u(x)-u(y)|}{|x-y|^{N+s^{\prime}}}dxdy\leqslant
C^{\prime}[u]_{s,\varPhi_{x,y}},$ (2.14)

thus,

$k_{1}I_{2}\leqslant\dfrac{C}{k}C^{\prime}[u]_{s,\varPhi_{x,y}}.$ (2.15)

We pose $k_{3}=Ck_{1}C^{\prime}$. Combining (2.13)-(2.15), we obtain

$1\leqslant\dfrac{1}{2}+\dfrac{k^{\prime}_{\varepsilon}}{k}||u||_{\widehat{\varPhi}_{x}}+\dfrac{k_{3}}{k}[u]_{s,\varPhi_{x,y}},$

this implies that,

$\dfrac{k}{2}\leqslant
k^{\prime}_{\varepsilon}||u||_{\widehat{\varPhi}_{x}}+k_{3}[u]_{s,\varPhi_{x,y}}.$

Hence, we obtain,

$k=||u||_{\widehat{\varPhi}^{*}_{x,s}}\leqslant k_{4}||u||_{s,\varPhi_{x,y}},$

where $k_{4}=\max\left\\{2k^{\prime}_{\varepsilon},2k_{3}\right\\}$.
Now, for $u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ arbitrary, we define

$u_{n}(x)=\left\\{\begin{array}[]{lcl}u(x)\hskip 28.45274pt\text{ if
}|u(x)|\leqslant n,\\\ n\text{ sgn }u(x)\text{ if
}|u(x)|>n.\end{array}\right.$

Then, $\left\\{u_{n}\right\\}$ is bounded and by Lemma 2.2, $u_{n}\in
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$. Moreover

$||u_{n}||_{\widehat{\varPhi}^{*}_{x,s}}\leqslant
k_{4}||u_{n}||_{s,\varPhi_{x,y}}\leqslant k_{4}||u||_{s,\varPhi_{x,y}}.$

Let
$\lim\limits_{n\rightarrow\infty}||u_{n}||_{\widehat{\varPhi}^{*}_{x,s}}=k$,
then $k\leqslant k_{4}||u||_{s,\varPhi_{x,y}}$. Using the Fatou’s Lemma, we
get,

$\int_{\Omega}\widehat{\varPhi}^{*}_{x,s}\left(\dfrac{|u(x)|}{k}\right)dx\leqslant\lim\limits_{n\rightarrow\infty}\int_{\Omega}\widehat{\varPhi}_{x,s}\left(\dfrac{|u_{n}(x)|}{||u_{n}||_{\widehat{\varPhi}^{*}_{x,s}}}\right)dx<1,$

Consequently, $u\in L_{\widehat{\varPhi}^{*}_{x,s}}(\Omega)$ and
$||u||_{\widehat{\varPhi}^{*}_{x,s}}\leqslant k\leqslant
k_{4}||u||_{s,\varPhi_{x,y}}.$

Proof of Theorem 2.2. From Lemma 2.3, we get

$W^{s}{L_{\varPhi_{x,y}}}(\Omega)\hookrightarrow
W^{s^{\prime},1}(\Omega)\hookrightarrow L^{1}(\Omega).$

By [29, Theroem 4.58], the latter embedding being compact. Since a bounded
subset $S$ of $W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ is also a bounded subset of
$L_{\widehat{\varPhi}^{*}_{x,s}}(\Omega)$ and precompact in $L^{1}(\Omega)$.
Hence, by Theorem 1.1 it is precompact in $L_{B_{x}}(\Omega)$.

Now, combining Lemma 2.3 and [29, Theroem 4.58], we obtain the following
results.

######  Corollary 2.1.

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ with
$C^{0,1}$-regularity and bounded boundary. Let $0<s^{\prime}<s<1$ and let
$\varPhi_{x,y}$ be a Musielak function satisfies the condition
$(\varPhi_{1})$, we define

$\varphi_{s^{\prime}}^{*}=\hskip
2.84544pt\left\\{\begin{array}[]{clclc}\frac{N\varphi^{-}}{N-s^{\prime}\varphi^{-}}\hskip
14.22636pt\text{ if }N>s^{\prime}\varphi^{-}\\\ \\\ \infty\hskip
14.22636pt\text{ if }N\leqslant s^{\prime}\varphi^{-}.\end{array}\right.$

  * $\bullet$

If $s^{\prime}\varphi^{-}<N$, then
$W^{s}{L_{\varPhi_{x,y}}}(\Omega)\hookrightarrow L^{q}(\Omega),$ for all
$q\in[1,\varphi^{*}_{s^{\prime}}]$ and the embedding
$W^{s}{L_{\varPhi_{x,y}}}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact for
all $q\in[1,\varphi_{s^{\prime}}^{*})$.

  * $\bullet$

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

  * $\bullet$

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

Next, we introduce a closed linear subspace of
$W^{s}L_{\varPhi_{x,y}}(\Omega)$ as follows

$W^{s}_{0}L_{\varPhi_{x,y}}(\Omega)=\left\\{u\in
W^{s}L_{\varPhi_{x,y}}(\mathbb{R}^{N})\text{ : }u=0\text{ a.e in
}\mathbb{R}^{N}\setminus\Omega\right\\}.$

Then, we have the following generalized Poincaré type inequality.

######  Theorem 2.3.

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ with
$C^{0,1}$-regularity and bounded boundary, let $s\in(0,1)$. Then there exists
a positive constant $\gamma$ such that

$||u||_{\widehat{\varPhi}_{x}}\leqslant\gamma[u]_{s,\varPhi_{x,y}}\text{ for
all }u\in W^{s}_{0}L_{\varPhi_{x,y}}(\Omega).$

Proof. Let
$\widehat{\sigma}(t)=[\widehat{\varPhi}_{x}(t)]^{\frac{N-s^{\prime}}{N}}$
where $s<s^{\prime}$ and $u\in W_{0}^{s}{L_{\varPhi_{x,y}}}(\Omega)$, then
similarly to Lemma 2.2, for
$f(x)=\widehat{\sigma}\left(\dfrac{u(x)}{k}\right)$ with
$k=||u||_{\widehat{\varPhi}_{x}}$, we have $f\in
W_{0}^{s^{\prime},1}(\Omega)$, and since $N>s^{\prime}$, then by [29, Theroem
4.58], one has,

$W_{0}^{s^{\prime},1}(\Omega)\hookrightarrow
L^{\frac{N}{N-s^{\prime}}}(\Omega).$

It follows that

$||f||_{L^{\frac{N}{N-s^{\prime}}}}\leqslant k_{5}[f]_{s^{\prime},1},$

and

$1=\left(\int_{\Omega}\widehat{\varPhi}_{x}\left(\dfrac{|u(x)|}{k}\right)dx\right)^{\frac{N-s^{\prime}}{N}}=||f||_{L^{\frac{N}{N-s^{\prime}}}}.$

this implies that,

$\begin{aligned} 1&\leqslant k_{5}[f]_{s^{\prime},1}\\\
&=k_{5}\int_{\Omega}\int_{\Omega}\dfrac{|f(x)-f(y)|}{|x-y|^{N+s^{\prime}}}dxdy\\\
&=k_{5}\int_{\Omega}\int_{\Omega}\dfrac{|\sigma(\dfrac{u(x)}{k})-\sigma(\dfrac{u(y)}{k})|}{|x-y|^{N+s^{\prime}}}dxdy\\\
&\leqslant\dfrac{k_{5}c}{k}\int_{\Omega}\int_{\Omega}\dfrac{|u(x)-u(y)|}{|x-y|^{N+s^{\prime}}}dxdy.\\\
&\leqslant\dfrac{k_{5}c}{k}[u]_{s,\varPhi_{x,y}}.\end{aligned}.$ (2.16)

Thus

$||u||_{\widehat{\varPhi}_{x}}\leqslant\gamma[u]_{s,\varPhi_{x,y}}$

where $\gamma=k_{5}c$.

######  Remark 2.1.

  * $a)-$

As a trivial consequence of Theorem 2.3, for a bounded open subset $\Omega$ of
$\mathbb{R}^{N},$ there exists a positive constant $\lambda_{1}$ such that,

$\int_{\Omega}\widehat{\varPhi}_{x}\left(u(x)\right)dx\leqslant\lambda_{1}\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}},$
(2.17)

for all $u\in W_{0}^{s}{L_{\varPhi_{x,y}}}(\Omega)$.

  * $b)-$

From Theorem 2.3, we deduce that $[.]_{s,\varPhi_{x,y}}$ is a norm on
$W^{s}_{0}L_{\varPhi_{x,y}}(\Omega)$ which is equivalent to the norm
$||u||_{s,\varPhi_{x,y}}$.

###  2.2. Extending a $W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ function to the whole
of $\mathbb{R}^{N}$

As usual, for any $k\in\mathbb{N}$ and $\alpha\in(0,1]$, we say that $\Omega$
is of class $C^{k,\alpha}$ if there exists $M>0$ such that for any
$x\in\partial\Omega$ there exists a ball $B=B_{r}(x)$, $r>0$, and an
isomorphism $T:Q\rightarrow B$ such that

$T\in C^{k,\alpha}(\overline{Q}),\hskip 5.69046ptT^{-1}\in
C^{k,\alpha}(\overline{B}),\hskip 5.69046ptT(Q_{+})=B\cap\Omega,\hskip
5.69046ptT(Q_{0})=B\cap\partial\Omega$ (2.18) $\text{and
}||T||_{C^{k,\alpha}(\overline{Q})}+||T^{-1}||_{C^{k,\alpha}(\overline{B})}\leqslant
M,$

where

$Q:=\left\\{x=(x^{\prime},x_{N})\in\mathbb{R}^{N-1}\times\mathbb{R}:|x^{\prime}|<1\text{
and }|x_{N}|<1\right\\},$
$Q_{+}:=\left\\{x=(x^{\prime},x_{N})\in\mathbb{R}^{N-1}\times\mathbb{R}:|x^{\prime}|<1\text{
and }0<x_{N}<1\right\\},$ $\text{ and }Q_{0}:=\left\\{x\in
Q:x_{N}=0\right\\}.$

Given an open bounded domain $\Omega\in\mathbb{R}^{N}$, For any $s\in(0,1)$
and any Musielak function $\varPhi_{x,y}$, we say that an open set $\Omega$ is
an extension domains for $W^{s}L_{\varPhi_{x,y}}(\Omega)$ if there exists a
positive constant $C=C(N,s,\Omega)$ such that for every function $u\in
W^{s}L_{\varPhi_{x,y}}(\Omega)$ there exists $\widetilde{u}\in
W^{s}L_{\varPhi_{x,y}}(\mathbb{R}^{N})$ with $\widetilde{u}(x)=u(x)$ for all
$x$ in $\Omega$.

Hence, our aim in this subsection is to show that any open bounded set
$\Omega$ of class $C^{0,1}$ with bounded boundary is an extension domain for
$W^{s}{L_{\varPhi_{x,y}}}$. To this end, we follow the same approach as in [2,
30].

In this subsection we assume that $\varPhi_{x,y}\in\mathcal{B}_{f}$ and

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\lim\limits_{t\rightarrow\infty}\dfrac{|t|^{\varphi^{+}}}{\widehat{\varPhi}_{x,s}^{*}(kt)}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \forall k>0.\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (\varPhi_{3})$

We start with some preliminary lemmas, in which we will construct the
extension to the whole of $\mathbb{R}^{N}$ of a function $u$ defined on
$\Omega$ in two separated cases: when the function $u$ is identically zero in
a neighborhood of the boundary $\partial\Omega$ and when $\Omega$ coincides
with the half-space $\mathbb{R}^{N}_{+}$.

######  Lemma 2.4.

Assume that $(\varPhi_{1})$-$(\varPhi_{3})$ holds. Then for all $u\in
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$

$\int_{\Omega}\varPhi_{x,y}(|u(x)|)dx<\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \forall y\in\overline{\Omega}.$

Proof. First, seminary to $(\ref{si})$, we have

$\varPhi_{x,y}(t)\leqslant|t|^{\varphi^{+}}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t>1.$

Then,

$\displaystyle\int_{\Omega}\varPhi_{x,y}(|u(x)|)dx$
$\displaystyle\leqslant\int_{\Omega}\varPhi_{x,y}(1)dx+\int_{\Omega}|u(x)|^{\varphi^{+}}$
$\displaystyle\leqslant\sup\limits_{(x,y)\in\overline{\Omega}\times\overline{\Omega}}\varPhi_{x,y}(1)|\Omega|+\|u\|^{\varphi^{+}}_{\varphi^{+}}$

On the other hand, by $(\varPhi_{3})$ we can used Theorem 2.2 and we have

$\|u\|_{\varphi^{+}}\leqslant c\|u\|_{s,\varPhi_{x,y}}.$

So,

$\displaystyle\int_{\Omega}\varPhi_{x,y}(|u(x)|)dx$
$\displaystyle\leqslant\sup\limits_{(x,y)\in\overline{\Omega}\times\overline{\Omega}}\varPhi_{x,y}(1)|\Omega|+\|u\|_{\varphi^{+}}^{\varphi^{+}}$
$\displaystyle\leqslant\sup\limits_{(x,y)\in\overline{\Omega}\times\overline{\Omega}}\varPhi_{x,y}(1)|\Omega|+c\|u\|^{\varphi^{+}}_{s,\varPhi_{x,y}}$
$\displaystyle<\infty.$

######  Lemma 2.5.

Let $\Omega$ be an open bounded subset of $\mathbb{R}^{N}$, $s\in(0,1)$, and
$u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$. If there exists a compact subset
$K\subset\Omega$ such that $u\equiv 0$ in $\Omega\setminus K$, then the
extension function $\widetilde{u}$ defined as

$\widetilde{u}(x)=\left\\{\begin{array}[]{clclc}u(x)&if&x\in\Omega,\\\ \\\
0&if&x\in\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

belongs to $W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$.

Proof. Let $u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$. So, clearly
$\widetilde{u}\in L_{\widehat{\varPhi}_{x}}(\mathbb{R}^{N})$. On the other
hand, since $\widetilde{u}$ in $\mathbb{R}^{N}\setminus\Omega$, then for some
$\lambda>0$, we have

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\varPhi_{x,y}\left(\dfrac{|\widetilde{u}(x)-\widetilde{u}(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle=\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\hskip
5.69046pt+2\int_{\mathbb{R}^{N}\setminus\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}},$

where the first term in the right hand-side is finite. Moreover, for any
$y\in\mathbb{R}^{N}\setminus K$ we have

$\displaystyle\int_{\mathbb{R}^{N}\setminus\Omega}\int_{\Omega}$
$\displaystyle\varPhi_{x,y}\left(\dfrac{|u(x)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\leqslant\int_{\mathbb{R}^{N}\setminus\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)|}{\lambda}\right)\dfrac{dxdy}{|x-y|^{sp+N}}$
$\displaystyle\leqslant\int_{\mathbb{R}^{N}\setminus\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{\mathds{1}_{K}|u(x)|}{\lambda}\right)\dfrac{dxdy}{dis(y,\partial
K)^{sp+N}}$
$\displaystyle\leqslant\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)|}{\lambda}\right)dx\int_{\mathbb{R}^{N}\setminus\Omega}\dfrac{dy}{dis(y,\partial
K)^{sp+N}}.$

where $p=1$ if $|x-y|\geqslant 1$ or $p=\varphi^{+}$ if $|x-y|<1$ for some
$(x,y)\in\Omega\times\Omega$. Since $dis(\partial\Omega,\partial K)>0$ and
$N+sp>N$. Then, by Lemma 2.4

$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\varPhi_{x,y}\left(\dfrac{|\widetilde{u}(x)-\widetilde{u}(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}<\infty.$

######  Lemma 2.6.

Let $s\in(0,1)$ and let $\Omega$ be an open subset of $\mathbb{R}^{N}$,
symmetric with respect to the coordinate $x_{N}$, and consider the sets
$\Omega_{+}=\left\\{x\in\Omega:x_{N}>0\right\\}$ and
$\Omega_{-}=\left\\{x\in\Omega:x_{N}\leqslant 0\right\\}$. Let $u$ be a
function in $W^{s}{L_{\varPhi_{x,y}}}(\Omega_{+})$, we define

$\widetilde{u}(x)=\left\\{\begin{array}[]{clclc}u(x^{\prime},x_{N})&if&x_{N}\geqslant
0,\\\ \\\ u(x^{\prime},-x_{N})&if&x_{N}<0.\end{array}\right.$

Then $\widetilde{u}$ belongs to $W^{s}{L_{\varPhi_{x,y}}}(\Omega).$

Proof. We set $\widehat{x}=(x^{\prime},-x_{N})$, then by splitting the
integrals, for some $\lambda>0$, we get

$\displaystyle\int_{\Omega}\widehat{\varPhi}_{x}\left(\dfrac{|\widetilde{u}(x)|}{\lambda}\right)dx$
$\displaystyle=\int_{\Omega_{+}}\widehat{\varPhi}_{x}\left(\dfrac{|u(x^{\prime},x_{N})|}{\lambda}\right)dx+\int_{\Omega_{-}}\widehat{\varPhi}_{x}\left(\dfrac{|u(x^{\prime},-x_{N})|}{\lambda}\right)dx$
$\displaystyle=\int_{\Omega_{+}}\widehat{\varPhi}_{x}\left(\dfrac{|u(x)|}{\lambda}\right)dx+\int_{\Omega_{+}}\widehat{\varPhi}_{x}\left(\dfrac{|u(\widehat{x}^{\prime},\widehat{x}_{N})|}{\lambda}\right)dx$
$\displaystyle=2\int_{\Omega_{+}}\widehat{\varPhi}_{x}\left(\dfrac{|u(x)|}{\lambda}\right)dx$
$\displaystyle<\infty.$

On the other hand, for all $\lambda>0$, we have

$\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|\widetilde{u}(x)-\widetilde{u}(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}=\int_{\mathbb{R}^{N}\setminus\Omega_{+}}\int_{\mathbb{R}^{N}\setminus\Omega_{+}}\varPhi_{x,y}\left(\dfrac{|u(x^{\prime},-x_{N})-u(y^{\prime},-y_{N})|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle+2\int_{\Omega_{+}}\int_{\mathbb{R}^{N}\setminus\Omega_{+}}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y^{\prime},-y_{N})|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}+\int_{\Omega_{+}}\int_{\Omega_{+}}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}.$

By changing variable $\widehat{x}=(x^{\prime},-x_{N})$ and
$\widehat{y}=(y^{\prime},-y_{N})$ we get

$\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|\widetilde{u}(x)-\widetilde{u}(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle=4\int_{\Omega_{+}}\int_{\Omega_{+}}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle<\infty.$

This concludes the proof.

######  Lemma 2.7.

Let $\Omega$ be an open subset of $\mathbb{R}^{N}$ and $s\in(0,1)$. Let $u\in
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ and $\psi\in C^{0,1}(\Omega)$,
$0\leqslant\psi\leqslant 1$. Then $\psi u\in
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$.

Proof. Let $u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$. Then, it is clear that
$\psi u\in L_{\widehat{\varPhi}_{x}}(\Omega)$. In addition, since
$|\psi|\leqslant 1$, it follows that $||\psi
u||_{L_{\widehat{\varPhi}_{x}}(\Omega)}\leqslant||u||_{L_{\widehat{\varPhi}_{x}}(\Omega)}$
. On the other hand, adding and subtracting the factor $\psi(x)u(y)$, we have

$\displaystyle\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|\psi(x)u(x)-\psi(y)u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\leqslant\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{2|\psi(x)u(x)-\psi(x)u(y)|}{2\lambda|x-y|^{s}}+\dfrac{2|\psi(x)u(y)-\psi(y)u(y)|}{2\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\leqslant\dfrac{2^{\varphi^{+}}}{2}\left(\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|\psi(x)u(x)-\psi(x)u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}+\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|\psi(x)u(y)-\psi(y)u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}\right)$
$\displaystyle\leqslant
2^{\varphi^{+}-1}\left(\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}+\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(y)(\psi(x)-\psi(y))|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}\right).$

Since $\psi\in C^{0,1}(\Omega)$, we obtain

$\displaystyle\int_{\Omega}\int_{\Omega}$
$\displaystyle\varPhi_{x,y}\left(\dfrac{|u(y)(\psi(x)-\psi(y))|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle=\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|\geqslant
1\right\\}}\varPhi_{x,y}\left(\dfrac{|u(y)(\psi(x)-\psi(y))|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\hskip
5.69046pt+\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|\leqslant
1\right\\}}\varPhi_{x,y}\left(\dfrac{u(y)(\psi(x)-\psi(y))}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\leqslant\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|\geqslant
1\right\\}}\varPhi_{x,y}\left(\dfrac{|u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\hskip
5.69046pt+\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|\leqslant
1\right\\}}\varPhi_{x,y}\left(\dfrac{L|u(y)||x-y|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}$
$\displaystyle\leqslant
C\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(y)|}{\lambda}\right)dy,$

where $L$ denotes the Lipschitz constant of $\psi$ and

$C=\displaystyle\int_{\Omega\cap\left\\{|z|\geqslant
1\right\\}}\dfrac{1}{|z|^{s+N}}dz+\int_{\Omega\cap\left\\{|z|\leqslant
1\right\\}}\dfrac{L^{\alpha}}{|z|^{(s-1)+N}}dz$

with $\alpha=1$ if $L<1$ and $\alpha=\varphi^{+}$ if $L>1$. Note that the two
above integrals are finite. In fact, the kernel $|z|^{-(s+N)}$ is summable
when $|z|\geqslant 1$ since $N+s>N$. Moreover, as $(s-1)+N<N$, then the kernel
$|z|^{-((s-1)+N)}$ is summable when $|z|\leqslant 1$. Therefore $\psi u\in
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$.

Now, we are in position to state and prove the main theorem of this
subsection.

######  Theorem 2.4.

Let $\Omega$ be an open subset of $\mathbb{R}^{N}$ with $C^{0,1}$-regularity
and bounded boundary. Then, $W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ is continuously
embedded in $W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$, namely for any $u\in
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ there exists $\widetilde{u}\in
W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$.

Proof. Since $\partial\Omega$ is compact, we can find a finite number of balls
$B_{j}$ such that $\partial\Omega\subset\cup_{j=1}^{k}B_{j}$ and so we can
write
$\mathbb{R}^{N}=\cup_{j=1}^{k}B_{j}\cup(\mathbb{R}^{N}\setminus\partial\Omega)$.

If we consider this covering, there exists a partition of unity related to it,
that is, there exist $k+1$ smooth functions $\psi_{0},\psi_{1},...,\psi_{k}$
such that $supp(\psi_{0})\subset\mathbb{R}^{N}$, $supp(\psi_{j})\subset B_{j}$
for all $j\in\left\\{0,...,k\right\\}$ and $\sum\limits_{j=0}^{k}\psi_{j}=1$.
Clearly,

$u=\sum\limits_{j=0}^{k}\psi_{j}u.$

From Lemma 2.7, we know that $\psi_{0}u\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$.
Furthermore, since $\psi_{0}u=0$ in a neighborhood of $\partial\Omega$, we can
extend it to the whole of $\mathbb{R}^{N}$ by setting

$\left\\{\begin{array}[]{clclc}\widetilde{\psi_{0}u}(x)=\psi_{0}u&if&x\in\Omega,\\\
\\\ 0&if&x\in\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

and $\widetilde{\psi_{0}u}\in W^{s}{L_{\varPhi_{x,y}}}(\Omega)$. For all
$j\in\left\\{1,...,k\right\\}$, let us consider $u_{|B_{j}\cap\Omega}$ and set

$v_{j}(y):=u(T_{j}(y))\text{ for any }y\in Q_{+},$

where $T_{j}:Q\rightarrow B_{j}$ is the isomorphism of class $C^{0,1}(\Omega)$
defined in $(\ref{11.})$. Note that such a $T_{j}$ exists because $\Omega$ is
an open subset of class $C^{0,1}$.

Next, we claim that $v_{j}\in W^{s}{L_{\varPhi_{x,y}}}(Q_{+})$. Indeed, using
the standard changing variable formula by setting $x=T_{j}(\widehat{x})$, we
obtain

$\displaystyle\int_{Q_{+}}\int_{Q_{+}}$
$\displaystyle\varPhi_{x,y}\left(\dfrac{|v_{j}(\widehat{x})-v_{j}(\widehat{y})|}{\lambda|\widehat{x}-\widehat{y}|^{s}}\right)\dfrac{d\widehat{x}d\widehat{y}}{|\widehat{x}-\widehat{y}|^{N}}$
(2.19)
$\displaystyle=\int_{Q_{+}}\int_{Q_{+}}\varPhi_{x,y}\left(\dfrac{|u(T_{j}(\widehat{x}))-u(T_{j}(\widehat{y}))|}{\lambda|\widehat{x}-\widehat{y}|^{s}}\right)\dfrac{d\widehat{x}d\widehat{y}}{|\widehat{x}-\widehat{y}|^{N}}$
$\displaystyle=\int_{B_{j}\cap\Omega}\int_{B_{j}\cap\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|T^{-1}_{j}(x)-T^{-1}_{j}(y)|^{s}}\right)\det(T_{j}^{-1})\dfrac{dxdy}{|T^{-1}_{j}(x)-T^{-1}_{j}(y)|^{N}}$
$\displaystyle\leqslant\bar{C}\int_{B_{j}\cap\Omega}\int_{B_{j}\cap\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{\lambda|x-y|^{s}}\right)\dfrac{dxdy}{|x-y|^{N}}.$

Hence $v_{j}\in W^{s}{L_{\varPhi_{x,y}}}(Q_{+})$. Moreover, using Lemma 2.6,
we can extend $v_{j}$ to all $Q$, then the extension $\widetilde{v_{j}}\in
W^{s}{L_{\varPhi_{x,y}}}(Q)$.

We set

$w_{j}(x):=\widetilde{v_{j}}(T_{j}^{-1}(x))\text{ for any }x\in B_{j}.$

By arguing as above, since $T_{j}$ is bi-Lipschitz, it follows that $w_{j}\in
W^{s}{L_{\varPhi_{x,y}}}(B_{j})$. Note that $w_{j}\equiv u$ (and consequently
$\psi_{j}w_{j}\equiv\psi_{j}u$) on $B_{j}\cap\Omega$. By definition
$\psi_{j}w_{j}$ has compact support in $B_{j}$ and therefore, as done for
$\psi_{0}u$, we can consider the extension $\widetilde{\psi_{j}w_{j}}$ to all
$\mathbb{R}^{N}$ in such a way that $\widetilde{\psi_{j}w_{j}}\in
W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$.

Finally, we set

$\widetilde{u}=\widetilde{\psi_{0}u}+\sum_{j=1}^{k}\widetilde{\psi_{j}\omega_{j}}$

which is the extension of $u$ defined on all $\mathbb{R}^{N}$. By
construction, it is clear that $\widetilde{u}|_{\Omega}=u$.

###  2.3. Complemented subspaces in $W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$

Let $(M,\|.\|_{M})$ be a Banach space of measurable functions on
$\mathbb{R}^{N}$ and $A$ is a measurable subset of $\mathbb{R}^{N}$ with
positive Lebesgue measure. We define the trace space $M_{|A}$ as follows

$M_{|A}:=\left\\{u:A\rightarrow\mathbb{R}\text{ such that }\exists U\in
M\text{ with }U_{|A}=u\text{ a.e. }\right\\}.$

This space is endowed with the norm

$\|u\|_{M_{|A}}=\inf\left\\{\|U\|_{M}:U\in M,U_{|A}=u\text{ a.e.}\right\\}.$

We denote by $\mathcal{T}U=U|_{A}$ the trace operator on $A$. In particular,
the aforementioned construction can be applies to the new fractional Musielak-
Sobolev space $W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$.
Note that the definitions of $L_{\widehat{\varPhi}_{x}}(\mathbb{R}^{N})$ and
$W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$ is analogous to
$L_{\widehat{\varPhi}_{x}}(\Omega)$ and $W^{s}{L_{\varPhi_{x,y}}}(\Omega)$,
one just replace occurrence of $\Omega$ by $\mathbb{R}^{N}$.

Next, let us consider the trace operator

$\mathcal{T}:W^{s}{L_{\varPhi_{x,y}}}\left(\mathbb{R}^{N}\right)\longrightarrow
W^{s}{L_{\varPhi_{x,y}}}(\Omega)$ $\hskip
85.35826pt\tilde{u}\longrightarrow\mathcal{T}\tilde{u}=\tilde{u}|_{\Omega}=u.$

######  Remark 2.2.

On other words, an open domain $\Omega\subset\mathbb{R}^{N}$ is a
$W^{s}{L_{\varPhi_{x,y}}}$-extension domain if there exists a continuous
linear extension operator

$\mathcal{E}:W^{s}{L_{\varPhi_{x,y}}}(\Omega)\longrightarrow
W^{s}{L_{\varPhi_{x,y}}}\left(\mathbb{R}^{N}\right)$ $\hskip
36.98866ptu\longrightarrow\mathcal{E}u=\tilde{u}$

######  Definition 2.1.

A closed subspace $Y$ of a Banach space $X$ is complemented if there is
another closed subspace $Z$ of $X$ such that $X=Y\oplus Z$. That is, $Y\cap
Z={0}$ and every element $x\in X$ can be written as $x=y+z$, with $y\in Y$ and
$z\in Z$.

In this subsection, our result relates to the complemented subspace in
$W^{s}{L_{\varPhi_{x,y}}}\left(\mathbb{R}^{N}\right)$. More precisely, we have
the following theorem,

######  Theorem 2.5.

Suppose that $\Omega\subset\mathbb{R}^{N}$ is an open bounded set of class
$C^{0,1}$. Then

$W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})=\ker\mathcal{T}\oplus\mathcal{E}(W^{s}{L_{\varPhi_{x,y}}}(\Omega)).$

Proof. From the Theorem 2.4 and Remark 2.2, we know that there exists a linear
operator $\mathcal{E}:W^{s}{L_{\varPhi_{x,y}}}(\Omega)\longrightarrow
W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$ such that

  1. (1)

$\mathcal{E}u_{|\Omega}=u$,

  2. (2)

$\|\mathcal{E}u\|_{W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})}\leqslant
C||u||_{W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})}.$

On the one hand, $\mathcal{E}(W^{s}{L_{\varPhi_{x,y}}}(\Omega))$ is a closed
subspace of $W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$. On the other hand, for
every element $u$ in $W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})$ can be written
as $u=u-\mathcal{E}(\mathcal{T}(u))+\mathcal{E}(\mathcal{T}(u))$, as
$u-\mathcal{E}(\mathcal{T}(u))\in\ker\mathcal{T}$ and
$\varepsilon(\mathcal{T}(u))\in\mathcal{E}(W^{s}{L_{\varPhi_{x,y}}}(\Omega))$.
Moreover,
$\ker\mathcal{T}\cap\mathcal{E}(W^{s}{L_{\varPhi_{x,y}}}(\Omega))={0}$, we
deduce that

$W^{s}{L_{\varPhi_{x,y}}}(\mathbb{R}^{N})=\ker\mathcal{T}\oplus\mathcal{E}(W^{s}{L_{\varPhi_{x,y}}}(\Omega)).$

##  3\. Application to a nonlocal problem

In this section, we investigate the existence of weak solution for problem
$(\mathcal{P}_{a})$ in the new fractional Musielak-Sobolev space.

We start by considering the function
$\alpha_{x}(t):=\alpha(x,t):\overline{\Omega}\times\mathbb{R}\longrightarrow\mathbb{R}$
such that $g(.,.):\overline{\Omega}\times\mathbb{R}\longrightarrow\mathbb{R}$
defined by

$g_{x}(t):=g(x,t)=\left\\{\begin{array}[]{clclc}\alpha(x,|t|)t&\text{ for
}&t\neq 0,\\\ \\\ 0&\text{ for }&t=0,\end{array}\right.$

which is an increasing homeomorphism from $\mathbb{R}$ onto itself. We assume
that $f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a Carathéodory
function satisfies the following condition

$(f_{1}):\hskip 108.12054pt|f(x,t)|\leqslant c_{0}(1+g_{x}(|t|)),\hskip
108.12054pt$

where $c_{0}$ is a nonnegative constant. Furthermore, if we set

$G_{x}(t):=G(x,t)=\int_{0}^{t}g_{x}(\tau)d\tau,\hskip 42.67912pt\text{
}\overline{G}_{x}(t)=\int^{t}_{0}\overline{g}_{x}(\tau)d\tau\hskip 14.22636pt$
(3.1)

with $\overline{g_{x}}(t)=\sup\left\\{s:g_{x}(s)\leqslant t\right\\}$, then we
obtain complementary Musielak function which define their corresponding
Musielak spaces $L_{G_{x}}$ and $L_{\overline{G}_{x}}$. Moreover, we suppose
that

$(f_{2}):\hskip 48.36958pt1<g^{-}=\inf_{t\geqslant
0}\dfrac{tg_{x}(t)}{G_{x}(t)}\leqslant g^{+}=\sup_{t\geqslant
0}\dfrac{tg_{x}(t)}{G_{x}(t)}<+\infty.\hskip 65.44142pt\text{.}$

$(f_{3}):\hskip
65.44142pt\lim_{t\rightarrow\infty}\dfrac{G_{x}(kt)}{\widehat{\varPhi}_{x,s_{2}}(x,t)}=0\text{
for all }k>0.\hskip 119.50148pt$

$(f_{4})$ : There exist $\theta>\varphi^{+}$ and $r>0$ such that $\text{for
all }|t|\geqslant r\text{ and a.e. }x\in\Omega$

$tf(x,t)\geqslant\theta F(x,t)\geqslant 0,$

where $F(x,t):=\displaystyle\int_{0}^{t}f(x,\tau)d\tau$.

$(f_{5}):\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \hskip
79.6678pt\limsup_{t\rightarrow
0}\dfrac{F(x,t)}{\varPhi_{x,y}(t)}<\dfrac{1}{\lambda_{1}}\text{ uniformly for
a.e. }x\in\Omega,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \hskip 19.91684pt$

where $\lambda_{1}$ is as in $(\ref{lambda1})$.

######  Remark 3.1.

In Problem $(\mathcal{P}_{a})$, since $0<s_{2}\leqslant s_{1}<1$, then by
Proposition 2.1, $W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ is continuously
embedded in $W^{s_{2}}_{0}L_{\varPhi_{x,y}}(\Omega)$. Thus a solution for a
problem of type $(\mathcal{P}_{a})$ will be sought in
$W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$.

To simplify the notations, for any $i=1,2$, we set

$D^{s_{i}}u:=\dfrac{u(x)-u(y)}{|x-y|^{s_{i}}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Psi_{s_{i}}(u):=\int_{\Omega}\int_{\Omega}\varPhi_{x,y}\left(\dfrac{|u(x)-u(y)|}{|x-y|^{s_{i}}}\right)\dfrac{dxdy}{|x-y|^{N}},$
$d\mu=\dfrac{dxdy}{|x-y|^{N}},\leavevmode\nobreak\ \leavevmode\nobreak\ \text{
and }\leavevmode\nobreak\ \leavevmode\nobreak\
||u||_{s_{i}}:=||u||_{W^{s_{i}}_{0}L_{\varPhi_{x,y}}(\Omega)}.$

Next, we give the definition of weak solutions for problem
$(\mathcal{P}_{a})$.

######  Definition 3.1.

$u\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ is called a weak solution of
problem $(\mathcal{P}_{a})$ if,

$\int_{\Omega}\int_{\Omega}a(|D^{s_{1}}u|)D^{s_{1}}uD^{s_{1}}vd\mu+\int_{\Omega}\int_{\Omega}a(|D^{s_{2}}u|)D^{s_{2}}uD^{s_{2}}vd\mu=\int_{\Omega}f(x,u)vdx,$
(3.2)

for all $v\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$.

Now, we are ready to state our existence result.

######  Theorem 3.1.

Suppose that $(f_{1})$-$(f_{5})$ hold true. Then problem $(\mathcal{P}_{a})$
has a nontrivial weak solution $u\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$
which is a critical point of mountain pass type for the energy functional
$J:W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)\longrightarrow\mathbb{R}$ defined by

$J(u)=\Psi_{s_{1}}(u)+\Psi_{s_{2}}(u)-\int_{\Omega}F(x,u)dx.$ (3.3)

Let us denote by
$\Psi,I:W^{s_{1}}_{0}{L_{\varPhi}}(\Omega)\longrightarrow\mathbb{R}$ the
functionals

$\Psi(u)=\Psi_{s_{1}}(u)+\Psi_{s_{2}}(u)\text{ }\text{ and
}I(u)=\int_{\Omega}F(x,u)dx.$

######  Remark 3.2.

We note that the functional
$J:W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)\longrightarrow\mathbb{R}$ in
$(\ref{14.})$ is well defined. Indeed, if $u\in
W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$, then by Proposition 2.1, we have
$u\in W^{s_{2}}_{0}L_{\varPhi_{x,y}}(\Omega)$, thus, $\Psi(u)<\infty$.
Moreover, by condition (f3), we have that $u\in L_{G_{x}}(\Omega)$ and thus
$u\in L^{1}(\Omega)$. Hence, by the condition (f1),

$|F(x,u)|\leqslant\int_{0}^{u}|f(x,t)|dt=c_{0}(|u|+G_{x}(|u|)).$

It follows that

$I(u)=\int_{\Omega}|F(x,u)|dx<\infty.$

By a standard argument (see for instance [6, 7]) we have $J\in
C^{1}\left(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega),\mathbb{R}\right)$ and its
Gâteaux derivative is given by

$\displaystyle\left\langle J^{\prime}(u),v\right\rangle=$
$\displaystyle\int_{\Omega}\int_{\Omega}a(|D^{s_{1}}u|)D^{s_{1}}uD^{s_{1}}vd\mu+\int_{\Omega}\int_{\Omega}a(|D^{s_{2}}u|)D^{s_{2}}uD^{s_{2}}vd\mu$
$\displaystyle-\int_{\Omega}f(x,u)vdx,$

where $\langle.,.\rangle$ denotes the usual duality between
$\left(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega),\|.\|_{s_{1}}\right)$ and its
dual space
$\left(\left(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)\right)^{*},\|.\|_{s_{1},*}\right).$
Next, we show an important lemma, namely that if the functional $J$ defined in
$(\ref{14.})$ satisfies the conclusion of Theorem 1.4, then it has a critical
point.

######  Lemma 3.1.

Let $(f_{1})$-$(f_{3})$ hold true. Let $J$ be the functional defined in
$(\ref{14.})$, and let $\left\\{u_{n}\right\\}$ be a sequence in
$W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ such that

$\hskip 14.22636ptJ(u_{n})\longrightarrow c>0\text{ , }\hskip 14.22636pt\hskip
14.22636pt||J^{\prime}(u_{n})||_{s_{1},*}\longrightarrow 0.$ (3.4)

Then there exists $u\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ such that

$J(u)=c,\hskip 14.22636ptJ^{\prime}(u)=0.$

Proof. It follows from $(3.4)$ that there exists $C>0$ such that
$|J(u_{n})|\leqslant C$ and $|\left\langle
J^{\prime}(u_{n}),u_{n}\right\rangle|\leqslant C||u_{n}||.$ By assumptions
$(\varPhi_{1})$ and $(f_{1})$-$(f_{3})$, we have

$0<t\varphi_{x,y}(t)\leqslant\varphi^{+}\varPhi_{x,y}(t)\text{ for all }t>0,$
(3.5) $0<tg_{x}(t)\leqslant g^{+}G_{x}(t)\text{ for all }t>0,$ (3.6)

and

$\left|\int_{\Omega\cap\left\\{|u_{n}|\leqslant
r\right\\}}(F(x,u_{n})-\theta^{-1}f(x,u_{n})u_{n})dx\right|\leqslant
c_{0}\left[(1+\theta^{-1})r+(1+\theta^{-1}g^{+})G_{x}(r)\right]\leqslant C.$
(3.7)

Thus, by (3.5)-(3.7) and Proposition 1.3, we get

$\displaystyle C+C||u_{n}||$ $\displaystyle\geqslant
J(u_{n})-\dfrac{1}{\theta}\left\langle J^{\prime}(u_{n}),u_{n}\right\rangle$
(3.8)
$\displaystyle\geqslant\Psi_{s_{1}}(u_{n})+\Psi_{s_{2}}(u_{n})-\dfrac{1}{\theta}\int_{\Omega}\int_{\Omega}\varphi_{x,y}\left(D^{s_{1}}u_{n}\right)D^{s_{1}}u_{n}d\mu$
$\displaystyle-\dfrac{1}{\theta}\int_{\Omega}\int_{\Omega}\varphi_{x,y}\left(D^{s_{2}}u_{n}\right)D^{s_{2}}u_{n}d\mu$
$\displaystyle-\left|\int_{\Omega\cap\left\\{|u_{n}|\leqslant
r\right\\}}(F(x,u_{n})-\theta^{-1}f(x,u_{n})u_{n})dx\right|$
$\displaystyle\geqslant\Psi_{s_{1}}(u_{n})+\Psi_{s_{2}}(u_{n})-\dfrac{\varphi^{+}}{\theta}\Psi_{s_{1}}(u_{n})-\dfrac{\varphi^{+}}{\theta}\Psi_{s_{2}}(u_{n})-C$
$\displaystyle\geqslant\left(1-\dfrac{\varphi^{+}}{\theta}\right)\left(\Psi_{s_{1}}(u_{n})+\Psi_{s_{2}}(u_{n})\right)-C$
$\displaystyle\geqslant\left(1-\dfrac{\varphi^{+}}{\theta}\right)\Psi_{s_{1}}(u_{n})-C$
$\displaystyle\geqslant\left(1-\dfrac{\varphi^{+}}{\theta}\right)||u_{n}||^{\beta}-C$

where $C$ denote various positive constants and
$\beta=\left\\{\varphi^{+},\varphi^{-}\right\\}$. Hence,
$\left\\{u_{n}\right\\}$ is bounded in
$W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$. Since
$W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ is a reflexive space, we may assume
that ${u_{n}}$ converges weakly to $u$ in $W^{s_{1}}_{0}L_{\varPhi}(\Omega)$.
Further, since the embedding of $W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ into
$L_{G_{x}}(\Omega)$ is compact, we obtain that $u_{n}\longrightarrow u$ in
$L_{G_{x}}(\Omega)$.
Then, since $I\in C^{1}(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega),\mathbb{R})$,
using $(f_{1})$-$(f_{3})$ , we get
$\lim\limits_{n\rightarrow\infty}I(u_{n})=I(u)$ and
$\lim\limits_{n\rightarrow\infty}I^{\prime}(u_{n})=I^{\prime}(u)$ in
$(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega))^{*}$. Moreover, from (3.4), we have
$J^{\prime}(u_{n})\rightarrow 0$ in
$(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega))^{*}$. Hence,

$\Psi^{\prime}(u_{n})\longrightarrow I^{\prime}(u)\text{ in
}(W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega))^{*}.$ (3.9)

Now, since $\Psi$ is convex, then we have

$\Psi(u_{n})\leqslant\Psi(u)+\left\langle\Psi^{\prime}(u_{n}),u_{n}-u\right\rangle.$

Therefore, using $(\ref{37})$, we may deduce that

$\limsup_{n\rightarrow\infty}\Psi(u_{n})\leqslant\Psi(u).$

It further follows from the convexity of $\Psi$ that it is weakly lower
semicontinuous and hence

$\liminf_{n\rightarrow\infty}\Psi(u_{n})\geqslant\Psi(u),$

which implies that

$\lim_{n\rightarrow\infty}\Psi(u_{n})=\Psi(u).$

Thus

$\lim_{n\rightarrow\infty}J(u_{n})=J(u).$

Consequently, by the uniqueness of the limit, we deduce that,
$J^{\prime}(u)=0$. The convexity of $\Psi$ implies that $\Psi^{\prime}$ is
monotone and hence

$\left\langle\Psi^{\prime}(u_{n}),u_{n}-v\right\rangle\geqslant\left\langle\Psi^{\prime}(v),u_{n}-v\right\rangle\text{
for all }v\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega).$

By $(\ref{37})$, we have

$\left\langle I^{\prime}(u)-\Psi^{\prime}(v),u-v\right\rangle\geqslant 0\text{
for all }v\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega).$

Setting $v=u-th$ with $h\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$ and
$t\in\mathbb{R}^{+}$, then, we get

$\left\langle I^{\prime}(u)-\Psi^{\prime}(u-th),h\right\rangle\geqslant 0$

for all $h\in W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$. Letting $t\rightarrow
0$ and using the fact that $h$ is arbitrary in
$W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)$, we find that

$J^{\prime}(u)=\Psi^{\prime}(u)-I^{\prime}(u)=0.$

It follows that $u$ is a critical point of $J$.

###  3.1. On the geometry of the functional $J$

In this subsection, we will show that under the conditions which we have
imposed on the functions $a_{x,y}$ and $f$, the geometric conditions $(G_{1})$
and $(G_{2})$ of Theorem 1.4 will hold.

######  Lemma 3.2.

Under the assumptions of Theorem 3.1, the first geometric condition $(G_{1})$
of the mountain pass Theorem 1.4 hold for the functional $J$ defined in
$(\ref{14.})$.

Proof. For all $u\in
W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)\setminus\left\\{0\right\\}$, the
functional $I$ is satisfied:

$\displaystyle J(u)$ $\displaystyle=\Psi(u)-\int_{\Omega}F(x,u)dx$ (3.10)
$\displaystyle=\Psi(u)\left[1-\dfrac{\displaystyle\int_{\Omega}F(x,u)dx}{\Psi(u)}\right].$

Using the condition $(f_{5})$, we have that there exists $\varepsilon\in(0,1)$
and $t_{0}>0$ such that

$F(x,t)\leqslant\frac{1-\varepsilon}{\lambda_{1}}\varPhi_{x,y}(t)\text{ for
all }|t|\leqslant t_{0}\text{ and all }x\in\overline{\Omega}.$

We pose $\Omega_{0}:=\left\\{x\in\Omega:|u(x)|\geqslant t_{0}\right\\}$, then
we have

$\int_{\Omega}F(x,u(x))dx\leqslant\frac{1-\varepsilon}{\lambda_{1}}\int_{\Omega\setminus\Omega_{0}}\varPhi_{x,y}(u(x))dx+\int_{\Omega_{0}}F(x,u(x))dx.$
(3.11)

By $(\ref{11})$, we have

$\dfrac{(1-\varepsilon)\displaystyle\int_{\Omega\setminus\Omega_{0}}\varPhi(|u(x)|)dx}{\lambda_{1}\Psi(u)}\leqslant\dfrac{(1-\varepsilon)\displaystyle\int_{\Omega\setminus\Omega_{0}}\varPhi(|u(x)|)dx}{\lambda_{1}\varPsi_{s_{1}}(u)}\leqslant
1-\varepsilon.$ (3.12)

Next, from $(f_{1})$, we have

$F(x,t)\leqslant c_{0}(|t|+G_{x}(|t|)),\text{ for all }|t|\geqslant
t_{0}\text{ and for a.e. }x\in\Omega.$

By proposition 1.1, it follows that

$\displaystyle\int_{\Omega_{0}}F(x,u)dx$ $\displaystyle\leqslant
c_{0}\left(||u||_{L^{1}(\Omega)}+\int_{\Omega}G_{x}(|u|)dx\right)$
$\displaystyle\leqslant
c_{0}\left(||u||_{L^{1}(\Omega)}+||u||^{g^{-}}_{G_{x}}+||u||^{g^{+}}_{G_{x}}\right).$

Since $W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)\hookrightarrow
L_{G_{x}}(\Omega)$ and $W^{s_{1}}_{0}L_{\varPhi_{x,y}}(\Omega)\hookrightarrow
L^{1}(\Omega)$, we obtain

$\int_{\Omega_{0}}F(x,u(x))dx\leqslant
c_{0}c_{1}(||u||_{s_{1}}+||u||_{s_{1}}^{g^{-}}+||u||_{s_{1}}^{g^{+}}).$ (3.13)

Then, for $||u||_{s_{1}}\leqslant 1$, we find

$\int_{\Omega_{0}}F(x,u(x))dx\leqslant 3c_{0}c_{1}||u||_{s_{1}},$ (3.14)

where $c_{1}$ denote various positive constants. By Proposition 1.3, we have

$\dfrac{\displaystyle\int_{\Omega_{0}}F(x,u)dx}{\Psi_{s_{1}}(u)}\leqslant
3c_{0}c_{1}||u||_{s_{1}}^{1-\varphi^{+}}.$ (3.15)

Now, using $(\ref{16})$, $(\ref{21.})$, $(\ref{22..})$ and $(\ref{22.})$, we
obtain that

$\displaystyle J(u)$
$\displaystyle\geqslant\Psi(u)\left(\varepsilon-3c_{0}c_{1}||u||_{s_{1}}^{1-\varphi^{+}}\right)$
$\displaystyle\geqslant\dfrac{\varepsilon}{2}\Psi(u)$

whenever

$\rho\leqslant\min\left\\{1,\left(\dfrac{\varepsilon}{6c_{0}c_{1}}\right)^{\frac{1}{1-\varphi^{+}}}\right\\}.$
(3.16)

Finally, by Proposition 1.3, we get

$||u||\longrightarrow 0\Longleftrightarrow\Psi(u)\longrightarrow 0.$

Hence for $\rho>0$ as given in $(\ref{r23})$, there exists a
$\alpha=\alpha(\rho)>0$ such that for all $u$ with $||u||_{s_{1}}=\rho$, we
have

$\Psi(u)\geqslant\alpha.$

We therefore obtain

$J(u)\geqslant\alpha\dfrac{\varepsilon}{2}.$

Thus, if we set $r=\alpha\dfrac{\varepsilon}{2}$, we obtain that $(G_{1})$ is
satisfied.

######  Lemma 3.3.

Under the assumptions of Theorem 3.1, the second geometric condition $(G_{2})$
of the mountain pass Theorem 1.4 hold for the functional $J$ defined in
$(\ref{14.})$.

Proof. First, by $(f_{4})$, it follows that

$F(x,\xi)\geqslant
r^{-\theta}\min\left\\{F(x,r),F(x,-r)\right\\}|\xi|^{\theta}$ (3.17)

for all $|\xi|>r$ and a.e. $x\in\Omega$. Thus by $(\ref{r1})$ and
$F(x,\xi)\leqslant\max_{|\xi|\leqslant r}F(x,\xi)$ for all $|\xi|\leqslant r$,
we obtain

$F(x,\xi)\geqslant
r^{-\theta}\min\left\\{F(x,r),F(x,-r)\right\\}|\xi|^{\theta}-\max_{|\xi|\leqslant
r}F(x,\xi)-\min\left\\{F(x,r),F(x,-r)\right\\}$ (3.18)

for any $\xi\in\mathbb{R}$ and a.e. $x\in\Omega$.
From Theorem 1.3, we can fix $u_{0}\in C^{\infty}_{0}(\Omega)$ such that
$||u_{0}||_{s_{1}}=1$ and let $t\geqslant 1$. By $(\ref{r12})$, we have

$\displaystyle J(tu_{0})$
$\displaystyle=\Psi_{s_{1}}(tu_{0})+\Psi_{s_{2}}(tu_{0})-\int_{\Omega}F(x,tu_{0})dx$
$\displaystyle\leqslant||tu_{0}||_{s_{1}}^{\varphi^{+}}+||tu_{0}||_{s_{2}}^{\varphi^{+}}+||tu_{0}||_{s_{2}}^{\varphi^{-}}-\int_{\Omega}F(x,tu_{0})dx$
$\displaystyle\leqslant||tu_{0}||_{s_{1}}^{\varphi^{+}}+c^{\varphi^{+}}||tu_{0}||_{s_{1}}^{\varphi^{+}}+c^{\varphi^{-}}||tu_{0}||_{s_{1}}^{\varphi^{-}}-\int_{\Omega}F(x,tu_{0})dx$
$\displaystyle\leqslant(1+c^{\varphi^{+}})t^{\varphi^{+}}+c^{\varphi^{+}}t^{\varphi^{-}}-r^{-\theta}|t|^{\theta}\int_{\Omega}\min\left\\{F(x,r),F(x,-r)\right\\}|u_{0}(x)|^{\theta}dx$
$\displaystyle\hskip 28.45274pt+\int_{\Omega}\max_{|\xi|\leqslant
r}F(x,\xi)+\min\left\\{F(x,r),F(x,-r)\right\\}dx.$

From assumptions $(f_{1})$ and $(f_{5})$, we get $0<F(x,\xi)\leqslant
c_{0}(|r|+G_{x}(|r|))$ for $|\xi|\leqslant r$ a.e. $x\in\Omega$. Thus,
$0<\min\left\\{F(x,r),F(x,-r)\right\\}<c_{0}(|r|+G_{x}(|r|))$, a.e.
$x\in\Omega$. Since $\theta>\varphi^{+}\geqslant\varphi^{-}$ by assumption
$(f_{4})$, passing to the limit as $t\rightarrow\infty$, we obtain that
$J(tu_{0})\rightarrow-\infty$. Thus, the assertion $(G_{2})$ follows by taking
$e=Tu_{0}$ with $T$ sufficiently large.

###  3.2. Proof of Theorem 3.1.

It follows from Lemma 3.2 and Lemma 3.3 that the hypotheses of Theorem 1.4 are
satisfied. So Lemma 3.1 implies the existence of a nontrivial critical point
of the functional $J$ which is a weak solutions to our problem
$(\mathcal{P}_{a})$.

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