# Existence of solutions for a nonlocal Kirchhoff type problem in Fractional
Orlicz-Sobolev spaces

E. Azroul1, A. Benkirane2, M. Srati3 and M. Shimi4 E. Azroul, A. Benkirane, M.
Srati and M.Shimi
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>
###### Key words and phrases:

Fractional Orlicz-Sobolev spaces, Fractional $\displaystyle A$-Laplacian
operator, Kirchhoff type problem, Direct method in variational methods.

###### 2010 Mathematics Subject Classification:

46E30, 58E05, 35R11, 35J60.

Abstract In this paper, we investigate the existence of weak solution for a
Kirchhoff type problem driven by a nonlocal operator of elliptic type in a
fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary
conditions

$\displaystyle(D_{K,A})\hskip
14.22636pt\left\\{\begin{array}[]{clclc}M\left(\displaystyle\int_{\mathbb{R}^{2N}}A\left([u(x)-u(y)]K(x,y)\right)dxdy\right)\mathcal{L}^{K}_{A}u&=&f(x,u)&\text{
in }&\Omega,\\\ \\\ \hskip 199.16928ptu&=&0&\text{ in
}&\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

Where $\displaystyle\mathcal{L}^{K}_{A}$ is a nonlocal operator with singular
kernel $\displaystyle K$ and $\displaystyle A$ is an $\displaystyle
N$-function, $\displaystyle\Omega$ is an open bounded subset in
$\displaystyle\mathbb{R}^{N}$ with Lipschitz boundary
$\displaystyle\partial\Omega$.

##  1\. Introduction

The equation,

$\rho\dfrac{\partial^{2}u}{\partial
t^{2}}-\left(\dfrac{P_{0}}{h}+\dfrac{E}{2L}\int_{0}^{L}|\dfrac{\partial
u}{\partial x}|^{2}dx\right)\dfrac{\partial^{2}u}{\partial x^{2}}$ (1)

presented by Kirchhoff [25] in 1883 is an extension of the classical
d’Alembert’s wave equation by considering the changes in the length of the
string during vibrations. In $\displaystyle(\ref{1.})$, $\displaystyle L$ is
the length of string, $\displaystyle h$ is the area of the cross section,
$\displaystyle E$ is the Young modulus of the material, $\displaystyle\rho$ is
the mass density, and $\displaystyle P_{0}$ is the initial tension. The
Kirchhoff’s model takes into account the length changes of the string produced
by transverse vibrations. Some interesting results can be found, for example
in [13]. On the other hand, Kirchhoff-type boundary value problems model
several physical and biological systems where $\displaystyle u$ describes a
process which depend on the average of itself, as for example, the population
density. We refer the reader to [5, 22, 33] for some related works. In [17],
the authors showed the existence and multiplicity of solutions to a class of
$\displaystyle p(x$)-Kirchhoff type equations via variational methods. In
[25], the author obtained the existence of infinite solutions to the
$\displaystyle p$-Kirchhoff type quasilinear elliptic equations via the
fountain theorem. In [15], the authors investigated higher order
$\displaystyle p(x)$-Kirchhoff type equations via symmetric mountain pass
Theorem, even in the degenerate case. However, they did not consider the
existence of solutions for Kirchhoff type problems in the fractional setting.
In the very recent paper [21], the authors first provided a detailed
discussion about the physical meaning underlying the fractional Kirchhoff
models and their applications, see [21] for further details.

Problems of this type have been intensively studied in the last few years, due
to numerous and relevant applications in many fields of mathematics, such as
approximation theory, mathematical physics (electrorheological fluids),
calculus of variations, nonlinear potential theory, the theory of
quasiconformalmappings, differential geometry, geometric function theory,
probability theory and image processing (see for instance [12, 19, 23]).

The study of nonlinear elliptic equations involving quasilinear homogeneous
type operators is based on the theory of Sobolev spaces and fractional Sobolov
spaces $\displaystyle W^{s,p}(\Omega)$ in order to find weak solutions. In
certain equations, precisely in the case of nonhomogeneous differential
operators, when trying to relax some conditions on these operators (as growth
conditions), the problem can not be formulated with classical Lebesgue and
Sobolev spaces. Hence, the adequate functional spaces is the so-called Orlicz
spaces. These spaces consists of functions that have weak derivatives and
satisfy certain integrability conditions. Many properties of Orlicz-Sobolev
spaces and fractional Orlicz-Sobolev spaces come in [1, 6, 20, 24, 32]. For
this, many researchers have studied the existence of solutions for the
eigenvalue problems involving nonhomogeneous operators in the divergence form
through Orlicz-Sobolev spaces by using variational methods and critical point
theory, monotone operator methods, fixed point theory and degree theory (see
for instance [2, 3, 4, 7, 8, 9, 10, 11, 14, 28, 29]).

We consider the following problem,

$\displaystyle(D_{K,p})\hskip
14.22636pt\left\\{\begin{array}[]{clclc}M\left(\displaystyle\int_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}K(x-y)dxdy\right)\mathcal{L}^{p}_{K}u&=&f(x,u)&\text{
in }&\Omega\\\ \\\ \hskip 199.16928ptu&=&0&\text{ in
}&\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $\displaystyle N>sp$ with $\displaystyle s\in(0,1)$,
$\displaystyle\Omega$ is an open bounded subset in
$\displaystyle\mathbb{R}^{N}$ with Lipschitz boundary
$\displaystyle\partial\Omega$, $\displaystyle
M:[0,\infty)\longrightarrow(0,\infty)$ is a continuous function,
$\displaystyle f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a
Carathéodory function and $\displaystyle\mathcal{L}^{p}_{K}u$ is a non-local
operator defined as follows:

$\displaystyle\mathcal{L}^{p}_{K}u(x)$
$\displaystyle=2P.V\int_{\mathbb{R}^{N}}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x-y)dy,\hskip
5.69046ptx\in\mathbb{R}^{N},$ $\displaystyle=2\lim\limits_{\varepsilon\searrow
0}\int_{\mathbb{R}^{N}\smallsetminus
B_{\varepsilon}(x)}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x-y)dy,\hskip
5.69046ptx\in\mathbb{R}^{N},$

where $\displaystyle 1<p<\infty$ and $\displaystyle
K:\mathbb{R}^{N}\setminus\left\\{0\right\\}\longrightarrow(0,\infty)$ is a
measurable function with the following property :

$\displaystyle\left\\{\begin{array}[]{clclc}&K(x)=K(-x)&&\text{ for all
}&x\in\mathbb{R}^{N}\setminus\left\\{0\right\\},\\\ \\\ &\text{ there exist
}k_{0}>0\text{ such that }&K(x)\geqslant\dfrac{k_{0}}{|x|^{N+sp}}&\text{ for
all }&x\in\mathbb{R}^{N}\setminus\left\\{0\right\\},\\\ \\\ &\delta K\in
L^{1}(\mathbb{R}^{N})&&\text{ where
}&\delta(x)=\min\left\\{1,|x|^{p}\right\\}.\end{array}\right.$

In [36], the authors discuss the above-mentioned problem in two cases: when
$\displaystyle f$ satisfies sublinear growth condition, the existence of
nontrivial weak solutions is obtained by applying the direct method in
variational methods; when $\displaystyle f$ satisfies suplinear growth
condition, the existence of two nontrivial weak solutions is obtained by using
the mountain pass Theorem.

In this paper, our purpose is to generalize the previous works in the setting
of fractional Orlicz-Sobolev spaces. That is we are interested to study the
existence of weak solution for a Kirchhof type problem driven by a nonlocal
operator of elliptic type, with homogeneous Dirichlet boundary conditions as
follows:

$\displaystyle(D_{K,A})\hskip
14.22636pt\left\\{\begin{array}[]{clclc}M\left(\displaystyle\int_{\mathbb{R}^{2N}}A\left([u(x)-u(y)]K(x,y)\right)dxdy\right)\mathcal{L}^{K}_{A}u&=&f(x,u)&\text{
in }&\Omega\\\ \\\ \hskip 199.16928ptu&=&0&\text{ in
}&\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $\displaystyle N>1$, $\displaystyle\Omega$ is an open bounded subset in
$\displaystyle\mathbb{R}^{N}$ with Lipschitz boundary
$\displaystyle\partial\Omega$, $\displaystyle
M:[0,\infty)\longrightarrow(0,\infty)$ is a continuous function $\displaystyle
f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a Carathéodory function
satisfying the Ambrosetti-Rabinowitz type condition and
$\displaystyle\mathcal{L}^{K}_{A}u$ is a nonlocal integro-differential
operator of elliptic type defined as follows:

$\displaystyle\mathcal{L}^{K}_{A}u(x)$
$\displaystyle=2P.V\int_{\mathbb{R}^{N}}A^{\prime}\left((u(x)-u(y))K(x,y)\right)K(x,y)dy,\hskip
5.69046ptx\in\mathbb{R}^{N},$ $\displaystyle=2\lim\limits_{\varepsilon\searrow
0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}a\left((u(x)-u(y))K(x,y)\right)K(x,y)dy,\hskip
5.69046ptx\in\mathbb{R}^{N},$

with $\displaystyle A$ is an $\displaystyle N$-function and the singular
kernel $\displaystyle
K:\mathbb{R}^{N}\times\mathbb{R}^{N}\longrightarrow(0,\infty)$ is a measurable
function satisfies the following properties :

$\displaystyle(K_{0})\left\\{\begin{array}[]{clclc}&K(x,y)=K(y,x)&&\text{ for
all }&(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N},\\\ \\\ &\exists
k_{0}>0\text{ :
}K(x,y)\geqslant\dfrac{k_{0}}{|x-y|^{s}A^{-1}(|x-y|^{N})}&&\text{ for all
}&(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N},\\\ \\\ &\delta K\in
L_{M}(\mathbb{R}^{N}\times\mathbb{R}^{N})&&\text{ where
}&\delta(x,y)=\min\left\\{1,|x-y|\right\\}.\end{array}\right.$

A typical example for $\displaystyle K$ is given by singular kernel
$\displaystyle K(x,y)=\frac{1}{|x-y|^{s}A^{-1}(|x-y|^{N})}$. In this case
$\displaystyle\mathcal{L}^{K}_{A}=(-\Delta)^{s}_{A}$ which is introduced in
[6] and the problem $\displaystyle(D_{K,A})$ becomes

$\displaystyle(D_{s,A})\left\\{\begin{array}[]{clclc}M\left(\displaystyle\int_{\mathbb{R}^{2N}}A\left(\dfrac{u(x)-u(y)}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy\right)(-\Delta)^{s}_{A}u&=&f(x,u)&\text{
in }&\Omega\\\ \\\ \hskip 199.16928ptu&=&0&\text{ in
}&\mathbb{R}^{N}\smallsetminus\Omega,\end{array}\right.$

where $\displaystyle(-\Delta)^{s}_{A}$ is the fractional $\displaystyle
A$-Laplacian operator defined on smooth functions by :

$\displaystyle(-\Delta)^{s}_{A}u(x)$
$\displaystyle=2P.V\int_{\mathbb{R}^{N}}A^{\prime}\left(\dfrac{(u(x)-u(y))}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)\dfrac{dy}{|x-y|^{s}A^{-1}(|x-y|^{N})}$
$\displaystyle=2\lim\limits_{\varepsilon\searrow
0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}a\left(\dfrac{(u(x)-u(y))}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)\dfrac{dy}{|x-y|^{s}A^{-1}(|x-y|^{N})}$

Note that for $\displaystyle A(t)=\frac{|t|^{p}}{p}$, the problem
$\displaystyle(D_{K,A})$ coïncide with the problem $\displaystyle(D_{K,p})$.

This paper is organized as follows : in the second section, we recall some
properties of Orlicz-Sobolev and fractional Orlicz-Sobolev spaces into
Lebesgue spaces. In the third section, we will introduce some necessary
definitions and properties of space $\displaystyle W^{K}L_{A}$. Moreover, we
prove the continuous and compact embedding theorems of these spaces. Finally,
using the direct method in calculus variations, we obtain the existence of a
weak solution of problem $\displaystyle(D_{K,A})$.

##  2\. Some preliminaries results

To deal with this situation we introduce the fractional Orlicz-Sobolev space
to investigate problem $\displaystyle(D_{K,A})$. Let us recall the definitions
and some elementary properties of these spaces. For more details we refer the
reader to [1, 6, 18, 30, 24].

Let $\displaystyle\Omega$ be an open subset of $\displaystyle\mathbb{R}^{N}$.
Let $\displaystyle A$ : $\displaystyle\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$
be an N-function, that is, $\displaystyle A$ is continuous, convex, with
$\displaystyle A(t)>0$ for $\displaystyle t>0$,
$\displaystyle\frac{A(t)}{t}\rightarrow 0$ as $\displaystyle t\rightarrow 0$
and $\displaystyle\frac{A(t)}{t}\rightarrow\infty$ as $\displaystyle
t\rightarrow\infty$. Equivalently, $\displaystyle A$ admits the representation
: $\displaystyle A(t)=\displaystyle\int_{0}^{t}a(s)ds$ where $\displaystyle
a:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is non-decreasing, right
continuous, with $\displaystyle a(0)=0$, $\displaystyle a(t)>0$, for all
$\displaystyle t>0$ and $\displaystyle a(t)\rightarrow\infty$ as
$\displaystyle t\rightarrow\infty$. The conjugate N-function of $\displaystyle
A$ is defined by
$\displaystyle\overline{A}(t)=\displaystyle\int_{0}^{t}\overline{a}(s)ds$,
where $\displaystyle\overline{a}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is
given by $\displaystyle\overline{a}(t)=\sup\left\\{s:a(s)\leqslant
t\right\\}$.
Evidently we have

$st\leqslant A(t)+\overline{A}(s),$ (2)

which is known Young’s inequality. Equality holds in (2) if and only if either
$\displaystyle t=\overline{a}(s)$ or $\displaystyle s=a(t)$.

We say that the N-function $\displaystyle A$ satisfy the global
$\displaystyle\Delta_{2}$-condition if, for some $\displaystyle k>0,$

$\displaystyle A(2t)\leqslant kA(t)\text{ , }\forall t\geqslant 0.$

When this inequality holds only for $\displaystyle t\geqslant t_{0}>0$,
$\displaystyle M$ is said to satisfy the $\displaystyle\Delta_{2}$-condition
near infinity.
We call the pair $\displaystyle(A,\Omega)$ is $\displaystyle\Delta$-regular if
either :
(a) $\displaystyle A$ satisfies a global $\displaystyle\Delta_{2}$-condition,
or
(b) $\displaystyle A$ satisfies a $\displaystyle\Delta_{2}$-condition near
infinity and $\displaystyle\Omega$ has finite volume.

According to [1], $\displaystyle A$ satisfies the global
$\displaystyle\Delta_{2}$-condition if and only if

$\displaystyle(a_{1})\hskip 113.81102pt1\leqslant\inf_{s\geqslant
0}\dfrac{sa(s)}{A(s)}<\sup_{s\geqslant 0}\dfrac{sa(s)}{A(s)}<+\infty.\hskip
142.26378pt\text{.}$

We assume that :

$\displaystyle(a_{2})\hskip 85.35826pt\text{ the function }t\mapsto
A(\sqrt{t})\text{ is convex for all }t\geqslant 0.\hskip 142.26378pt\text{.}$

The Orlicz class $\displaystyle K_{A}(\Omega)$ (resp. the Orlicz space
$\displaystyle L_{A}(\Omega)$) is defined as the set of (equivalence classes
of) real-valued measurable functions $\displaystyle u$ on
$\displaystyle\Omega$ such that

$\int_{\Omega}A(u(x))dx<\infty\hskip 14.22636pt\text{ (resp.
}\int_{\Omega}A(\lambda u(x))dx<\infty\text{ for some }\lambda>0).$ (3)

$\displaystyle L_{A}(\Omega)$ is a Banach space under the Luxemburg-norm

$||u||_{A}=\inf\Bigg{\\{}\lambda>0:\int_{\Omega}A\left(\dfrac{u(x)}{\lambda}\right)dx\leqslant
1\Bigg{\\}},$ (4)

and $\displaystyle K_{A}(\Omega)$ is a convex subset of $\displaystyle
L_{A}(\Omega)$. The closure in $\displaystyle L_{A}(\Omega)$ of the set of
bounded measurable functions on $\displaystyle\Omega$ with compact support in
$\displaystyle\overline{\Omega}$ is denoted by $\displaystyle E_{A}(\Omega)$.
The equality $\displaystyle E_{A}(\Omega)=L_{A}(\Omega)$ holds if and only if
$\displaystyle A$ is satisfies the global $\displaystyle\Delta_{2}$-condition.

Using the Young’s inequality, it is possible to prove a Hölder type
inequality, that is,

$\left|\int_{\Omega}uvdx\right|\leqslant 2||u||_{A}||v||_{\overline{A}}\hskip
14.22636pt\text{ }\forall u\in L_{A}(\Omega)\text{ and }v\in
L_{\overline{A}}(\Omega).$ (5)

Now, we defined the fractional Orlicz-Sobolev space $\displaystyle
W^{s}L_{A}(\Omega)$ as follows

$\displaystyle W^{s}{L_{A}}(\Omega)=\Bigg{\\{}u\in
L_{A}(\Omega):\int_{\Omega}\int_{\Omega}A\left(\dfrac{\lambda(u(x)-u(y))}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy<\infty,\text{
for some }\lambda>0\Bigg{\\}}.$

This space is equipped with the norm,

$||u||_{s,A}=||u||_{A}+[u]_{s,A},$ (6)

where $\displaystyle[.]_{s,A}$ is the Gagliardo seminorm, defined by

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

We define $\displaystyle W^{s}_{0}L_{A}(\Omega)$ as the set of the functions
$\displaystyle u$ in $\displaystyle W^{s}L_{A}(\Omega)$ such that
$\displaystyle u=0$ in $\displaystyle\mathbb{R}^{N}\setminus\Omega$. By [6],
$\displaystyle W^{s}L_{A}(\Omega)$ and $\displaystyle W^{s}_{0}L_{A}(\Omega)$
are separable Banach spaces, (resp. reflexive spaces) if and only if
$\displaystyle A$ satisfies the global $\displaystyle\Delta_{2}$-condition
(resp. $\displaystyle A$ and $\displaystyle\overline{A}$ are satisfy the
global $\displaystyle\Delta_{2}$-condition).

Let $\displaystyle A$ be a given N-function, satisfying the following
conditions :

$\int_{0}^{1}\dfrac{A^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau<\infty,$ (7)

$\int_{1}^{\infty}\dfrac{A^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau=\infty.$ (8)

For instance if $\displaystyle A(t)=\frac{1}{p}t^{p}$, then (7) holds
precisely when $\displaystyle sp<N$.
If (8) is satisfied, we define the inverse Sobolev conjugate N-function of
$\displaystyle A$ as follows,

$A_{*}^{-1}(t)=\int_{0}^{t}\dfrac{A^{-1}(\tau)}{\tau^{\frac{N+s}{N}}}d\tau.$
(9)

######  Theorem 2.1.

(cf. [6]) Let $\displaystyle A$ be an $\displaystyle N$-function and
$\displaystyle s\in(0,1)$. Let $\displaystyle\Omega$ be a bounded open subset
of $\displaystyle\mathbb{R}^{N}$ with $\displaystyle C^{0,1}$-regularity and
bounded boundary. If $\displaystyle(\ref{5})$ and $\displaystyle(\ref{6.})$
hold, then

$W^{s}{L_{A}}(\Omega)\hookrightarrow L_{A_{*}}(\Omega).$ (10)

######  Theorem 2.2.

(cf. [6]) Let $\displaystyle s\in(0,1)$ and $\displaystyle A$ be an
$\displaystyle N$-function. Let $\displaystyle\Omega$ be a bounded open subset
of $\displaystyle\mathbb{R}^{N}$ and $\displaystyle C^{0,1}$-regularity with
bounded boundary. If $\displaystyle(\ref{5})$ and $\displaystyle(\ref{6.})$
hold, then the embedding

$W^{s}{L_{A}}(\Omega)\hookrightarrow L_{B}(\Omega),$ (11)

is compact for all $\displaystyle B\prec\prec A_{*}$.

By fixing the fractional exponent $\displaystyle s\in(0,1)$ and for any
$\displaystyle p\in[1,\infty)$, we define the fractional Sobolev space
$\displaystyle W^{s,p}(\Omega)$ as follows,

$\displaystyle W^{s,p}(\Omega)=\Bigg{\\{}u\in L^{p}(\Omega)\text{ :
}\dfrac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in
L^{p}(\Omega\times\Omega)\Bigg{\\}};$

that is, an intermediary Banach space between, endowed with the natural norm

$\displaystyle||u||_{s,p}=\Bigg{(}\int_{\Omega}|u|^{p}dx+\int_{\Omega}\int_{\Omega}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{sp+N}}dxdy\Bigg{)}^{\frac{1}{p}}.$

######  Theorem 2.3.

(cf. [30]). Let $\displaystyle s\in(0,1)$ and let $\displaystyle
p\in[1,+\infty)$ such that $\displaystyle sp<N$. Let $\displaystyle\Omega$ be
an open subset of $\displaystyle\mathbb{R}^{N}$ with $\displaystyle
C^{0,1}$-regularity and bounded boundary. So there exists a positive constant
$\displaystyle C=C(N,s,p,\Omega)$ such that, for all $\displaystyle f\in
W^{s,p}(\Omega)$ we have

$\displaystyle||f||_{L^{q}(\Omega)}\leqslant C||f||_{W^{s,p}(\Omega)}\text{
}\forall q\in[p,p^{*}],$

that is,

$\displaystyle W^{s,p}(\Omega)\hookrightarrow L^{q}(\Omega)\text{ }\forall
q\in[p,p^{*}],$

where $\displaystyle p^{*}=\frac{Np}{N-sp}$.
If, in addition, $\displaystyle\Omega$ is bounded, then the space
$\displaystyle W^{s,p}(\Omega)$ is continuously embedded in $\displaystyle
L^{q}(\Omega)$ for any $\displaystyle q\in[1,p^{*}]$.

######  Theorem 2.4.

(cf. [18]). Let $\displaystyle s\in(0,1)$ and let $\displaystyle
p\in[1,+\infty)$ such that $\displaystyle sp<N$. Let $\displaystyle\Omega$ be
a bounded open subset of $\displaystyle\mathbb{R}^{N}$ with $\displaystyle
C^{0,1}$-regularity and bounded boundary. Then the embedding

$\displaystyle W^{s,p}(\Omega)\hookrightarrow L^{q}(\Omega)\text{ }\forall
q\in[1,p^{*}),$

is compact.

######  Theorem 2.5.

(cf. [35]) Suppose $\displaystyle X$ is a reflexive Banach space with norm
$\displaystyle||.||$ and let $\displaystyle V\subset X$ be a weakly closed
subset of $\displaystyle X$. Suppose $\displaystyle
E:V\longrightarrow\mathbb{R}\cup\left\\{+\infty\right\\}$ is coercive and
(sequentially) weakly lower semi-continuous on $\displaystyle V$ with respect
to $\displaystyle X$ , that is, suppose the following conditions are
fulfilled:

(1) $\displaystyle E(u)\rightarrow\infty$ as
$\displaystyle||u||\rightarrow\infty$, $\displaystyle u\in V$.

(2) For any $\displaystyle u\in V$, any sequence
$\displaystyle\left\\{u_{n}\right\\}$ in $\displaystyle V$ such that
$\displaystyle u_{n}\rightharpoonup u$ weakly in $\displaystyle X$ there
holds:

$\displaystyle E(u)\leqslant\liminf_{n\rightarrow\infty}E(u_{n}).$

Then $\displaystyle E$ is bounded from below on $\displaystyle V$ and attains
its infimum in $\displaystyle V$.

##  3\. Variational framework

One of the aims of this paper is to study nonlocal problems driven by
$\displaystyle(-\Delta)^{s}_{A}$ (or its generalization
$\displaystyle\mathcal{L}^{K}_{A}$) with Dirichlet boundary data via
variational methods. For this purpose, we need to work in a suitable
fractional Orlicz-Sobolev space: for this, we consider a functional analytical
setting that is inspired by (but not equivalent to) the fractional Orlicz-
Sobolev spaces in order to correctly encode the Dirichlet boundary datum in
the variational formulation.

In this section, we show some basic results that will be used in the next
section.

######  Definition 3.1.

Let $\displaystyle\Omega$ be an open subset of $\displaystyle\mathbb{R}^{N}$,
we denote $\displaystyle Q=\mathbb{R}^{N}\setminus\mathcal{O}$, where
$\displaystyle\mathcal{O}=(\mathbb{R}^{N}\setminus\Omega)\times(\mathbb{R}^{N}\setminus\Omega)$,
and let $\displaystyle A$ be an $\displaystyle N$-function and $\displaystyle
K$ as defined in $\displaystyle(K_{0})$. We define the space $\displaystyle
W^{K}L_{A}(\Omega)$ since the set of Lebesgue measurable functions from
$\displaystyle\mathbb{R}^{N}$ to $\displaystyle\mathbb{R}$ such that
$\displaystyle u_{|_{\Omega}}$ in $\displaystyle L_{A}(\Omega)$ and

$\displaystyle(u(x)-u(y))K(x,y)\in L_{A}(Q).$

The space $\displaystyle W^{K}L_{A}$ is equipped with the norm

$||u||_{K,A}=||u||_{A}+[u]_{K,A}$ (12)

where $\displaystyle[.]_{K,A}$ is the Gagliardo seminorm, defined by :

$\displaystyle[u]_{K,A}=\inf\Bigg{\\{}\lambda>0:\int_{Q}A\left(\dfrac{u(x)-u(y)}{\lambda}K(x,y)\right)dxdy\leqslant
1\Bigg{\\}}.$

$\displaystyle||u||_{K,A}$ is a norm in $\displaystyle W^{K}L_{A}(\Omega)$.
Indeed, let $\displaystyle u\in W^{K}L_{A}(\Omega)$ such that $\displaystyle
u=0$ then for all $\displaystyle\lambda>0$, we get

$\displaystyle||u||_{A}=0\text{ and
}\int_{Q}A\left(\dfrac{u(x)-u(y)}{\lambda}K(x,y)\right)dxdy=0,$

this implies that

$\displaystyle||u||_{A}=0\text{ and }[u]_{K,A}=0.$

Conversely, if $\displaystyle||u||_{K,A}=0$, this implies that

$\displaystyle u=0\text{ a.e in }\Omega\text{ and
}\int_{Q}A\left(\dfrac{u(x)-u(y)}{\lambda}K(x,y)\right)dxdy\leqslant 1$

for all $\displaystyle\lambda>0$. Let $\displaystyle 0<h\leqslant 1$ then

$\displaystyle\int_{Q}A\left(\frac{h}{h}(u(x)-u(y))K(x,y)\right)dxdy\leqslant
h\int_{Q}A\left(\frac{1}{h}(u(x)-u(y))K(x,y)\right)dxdy\leqslant h$

so for $\displaystyle h\rightarrow 0^{+}$ we have

$\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy=0$

then $\displaystyle u(x)=u(y)$ a.e in $\displaystyle Q$ therefore
$\displaystyle u=c\in\mathbb{R}$ a.e. in $\displaystyle\mathbb{R}^{N}$, since
$\displaystyle u=0$ a.e in $\displaystyle\Omega$, we get $\displaystyle u=0$
in $\displaystyle\mathbb{R}^{N}$.

We shall work in the closed linear subspace

$\displaystyle W^{K}_{0}L_{A}(\Omega)=\left\\{u\in W^{K}L_{A}(\Omega)\text{ :
}u=0\text{ in }\mathbb{R}^{N}\setminus\Omega\right\\}.$

######  Lemma 3.1.

$\displaystyle C^{2}_{0}(\Omega)\subset W^{K}_{0}L_{A}(\Omega)$.

Proof. Let $\displaystyle u\in C^{2}_{0}(\Omega)$, since $\displaystyle u$
vanishes outside $\displaystyle\Omega$, we only need that to check that

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}A\left(\lambda(u(x)-u(y))K(x,y)\right)dxdy<\infty\text{
for some }\lambda>0.$

Indeed,

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}A\left((u(x)-u(y))K(x,y)\right)dxdy$
$\displaystyle=\int_{\Omega}\int_{\Omega}A\left((u(x)-u(y))K(x,y)\right)dxdy$
$\displaystyle+2\int_{\Omega}\int_{\mathbb{R}^{N}\setminus\Omega}A\left((u(x)-u(y))K(x,y)\right)dxdy$
$\displaystyle\leqslant
2\int_{\Omega}\int_{\mathbb{R}^{N}}A\left((u(x)-u(y))K(x,y)\right)dxdy.$

Now we notice that

$\displaystyle|u(x)-u(y)|\leqslant||\nabla
u||_{L^{\infty}(\mathbb{R}^{N})}|x-y|\text{ and }|u(x)-u(y)|\leqslant
2||u||_{L^{\infty}(\mathbb{R}^{N})}.$

Accordingly, we get

$\displaystyle|u(x)-u(y)|\leqslant
2||u||_{C^{1}(\mathbb{R}^{N})}\min\left\\{1,|x-y|\right\\}:=\alpha\delta(x,y).$

with $\displaystyle\alpha=2||u||_{C^{1}(\mathbb{R}^{N})}$ and since
$\displaystyle\delta K\in L_{A}(\mathbb{R}^{N}\times\mathbb{R}^{N})$. There
exist $\displaystyle\lambda>0$, such that,

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}A\left(\dfrac{\lambda}{\alpha}(u(x)-u(y))K(x,y)\right)dxdy\leqslant\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}A\left(\lambda\delta(x,y)K(x,y)\right)dxdy<\infty$

this implies that $\displaystyle(u(x)-u(y))K(x,y)\in
L_{A}(\mathbb{R}^{N}\times\mathbb{R}^{N})$.

######  Remark 3.1.

A trivial consequence of lemma 3.1, $\displaystyle W^{K}L_{A}(\Omega)$ and
$\displaystyle W^{K}_{A}L(\Omega)$ are non-empty.

######  Lemma 3.2.

Let $\displaystyle s\in(0,1)$ and let $\displaystyle
K:\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow(0,\infty)$ satisfy assumption
$\displaystyle(K_{0})$, then the following assertion hold :

1) $\displaystyle W^{K}L_{A}(\Omega)\hookrightarrow W^{s}L_{A}(\Omega)$,

2) $\displaystyle W_{0}^{K}L_{A}(\Omega)\hookrightarrow
W^{s}L_{A}(\mathbb{R}^{N})$.

Proof.

1) Let $\displaystyle u\in W^{K}L_{A}(\Omega)$, and $\displaystyle\lambda>0$,
by $\displaystyle(K_{0})$ we get

$\displaystyle\int_{\Omega}\int_{\Omega}A\left(\frac{k_{0}}{\lambda}\dfrac{u(x)-u(y)}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy\leqslant\int_{Q}A\left(\dfrac{u(x)-u(y)}{\lambda}K(x,y)\right)dxdy,$

then $\displaystyle u\in W^{s}L_{A}(\Omega)$ and

$\displaystyle[u]_{s,A}\leqslant\frac{1}{k_{0}}[u]_{K,A},$

this implies that

$\displaystyle||u||_{s,A}\leqslant\sup\left\\{1,\frac{1}{k_{0}}\right\\}||u||_{K,A}.$

2) Let $\displaystyle u\in W^{K}_{0}L_{A}(\Omega)$, so $\displaystyle u=0$ in
$\displaystyle\mathbb{R}^{N}\setminus\Omega$ and

$\displaystyle||u||_{L_{A}(\mathbb{R}^{N})}=||u||_{L_{A}(\Omega)}<\infty.$

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

$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}A\left(\frac{k_{0}}{\lambda}\dfrac{u(x)-u(y)}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy$
$\displaystyle=\int_{Q}A\left(\frac{k_{0}}{\lambda}\dfrac{u(x)-u(y)}{|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy$
$\displaystyle\leqslant\int_{Q}A\left(\dfrac{u(x)-u(y)}{\lambda}K(x,y)\right)dxdy,$

then $\displaystyle u\in W^{s}L_{A}(\mathbb{R}^{N})$ and

$\displaystyle[u]_{W^{s}L_{A}(\mathbb{R}^{N})}\leqslant\frac{1}{k_{0}}[u]_{K,A}$

this implies that

$\displaystyle||u||_{W^{s}L_{A}(\mathbb{R}^{N})}\leqslant\sup\left\\{1,\frac{1}{k_{0}}\right\\}||u||_{K,A}.$

######  Theorem 3.1.

(Generalized Poincaré inequality). Let $\displaystyle\Omega$ be a bounded open
subset of $\displaystyle\mathbb{R}^{N}$, let $\displaystyle K$ as defined in
$\displaystyle(K_{0})$, and let $\displaystyle A$ be an $\displaystyle
N$-function. Then there exist a positive constant $\displaystyle\mu$ such that

$\displaystyle||u||_{A}\leqslant\mu[u]_{K,A}\text{ for all }u\in
W^{K}_{0}L_{M}(\Omega).$

To prove Theorem 3.1, we need to the following lemma.

######  Lemma 3.3.

Let $\displaystyle\Omega$ be a bounded open subset of
$\displaystyle\mathbb{R}^{N},$ and let $\displaystyle s\in(0,1)$. Let
$\displaystyle A$ be an N-function. Then there exists a positive constants
$\displaystyle\alpha$ and $\displaystyle\beta$ such that,

$\int_{\Omega}A\left(\dfrac{u}{\alpha\lambda}\right)dx\leqslant\beta\int_{\Omega}\int_{\Omega}A\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy,$
(13)

for all $\displaystyle u\in W^{s}_{0}L_{A}(\Omega)$ and all
$\displaystyle\lambda>0$. In particular

$\displaystyle||u||_{A}\leqslant\mu[u]_{s,A}\text{ for all }u\in
W_{0}^{s}L_{A}(\Omega)$

where $\displaystyle\mu=\alpha\beta$.

Proof of Lemma 3.3. Let $\displaystyle u\in W^{s}_{0}L_{A}(\Omega)$ and
$\displaystyle B_{R}\subset\mathbb{R}^{N}\setminus\Omega$, i.e, the ball of
radius $\displaystyle R>0$ in the complement of $\displaystyle\Omega$. Then
for all $\displaystyle x\in\Omega$, $\displaystyle y\in B_{R}$ and all
$\displaystyle\lambda>0$ we have,

$\displaystyle
A(u(x))=A\left(\dfrac{u(x)-u(y)}{|x-y|^{s}A^{-1}(|x-y|^{N})}|x-y|^{s}A^{-1}(|x-y|^{N})\right),$

this implies that,

$\displaystyle A(u(x))\leqslant
A\left(\dfrac{u(x)-u(y)}{|x-y|^{s}A^{-1}(|x-y|^{N})}diam(\Omega\cup
B_{R})^{s}(A^{-1}(diam(\Omega\cup B_{R})^{N}))\right),$

we suppose $\displaystyle\alpha=diam(\Omega\cup
B_{R})^{s}(A^{-1}(diam(\Omega\cup B_{R})^{N}))$, we get

$\displaystyle|B_{R}|A\left(\dfrac{u(x)}{\alpha\lambda}\right)\leqslant\int_{B_{R}}A\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dy,$

then

$\displaystyle\int_{\Omega}A(\dfrac{u(x)}{\alpha\lambda})dx\leqslant\beta\int_{\Omega}\int_{\Omega}A\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy,$

where $\displaystyle\beta=\dfrac{1}{|B_{R}|}$. Therefore, for
$\displaystyle\mu=\alpha\beta$ we get

$\displaystyle||u||_{A}\leqslant\mu[u]_{s,A}\text{ for all }u\in
W_{0}^{s}L_{M}(\Omega).$

Proof of Theorem 3.1. Let $\displaystyle u\in W^{K}_{0}L_{A}(\Omega)$, by
Lemma 3.2 we get $\displaystyle u\in W^{s}_{0}L_{A}(\Omega)$, so by Lemma 3.3,
there exists $\displaystyle\alpha,\beta>0$ such that

$\displaystyle\int_{\Omega}A\left(\dfrac{u(x)}{\alpha\lambda}\right)dx\leqslant\beta\int_{\Omega}\int_{\Omega}A\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A^{-1}(|x-y|^{N})}\right)dxdy,$

for all $\displaystyle\lambda>0$. On the other hand by condition
$\displaystyle(K_{0})$, we have

$\displaystyle\int_{\Omega}A\left(\dfrac{k_{0}u(x)}{\alpha\lambda}\right)dx\leqslant\beta\int_{Q}A\left(\dfrac{u(x)-u(y)}{\lambda}K(x,y)\right)dxdy,$

this implies that

$\displaystyle||u||_{A}\leqslant\mu[u]_{K,A}$

where $\displaystyle\mu=\dfrac{\alpha\beta}{k_{0}}$.

As a consequence of Theorem 3.1 $\displaystyle[.]_{K,M}$ is a norm on
$\displaystyle W^{K}_{0}L_{M}(\Omega)$ equivalent to
$\displaystyle||.||_{K,M}.$ Then
$\displaystyle\left(W^{K}_{0}L_{M},[.]_{K,A}\right)$ is Banach space.

Now, we introduce the following notations :

$p_{0}:=\inf_{s\geqslant 0}\dfrac{sa(s)}{A(s)}\text{ , }\hskip
28.45274ptp^{0}:=\sup_{s\geqslant 0}\dfrac{sa(s)}{A(s)}.$ (14)

######  Proposition 3.1.

Assume that condition $\displaystyle(a_{1})$ is satisfied. Then the following
relations hods true

$[u]^{p_{0}}_{K,A}\leqslant\phi(u)\leqslant[u]^{p^{0}}_{K,A}\text{ , }\forall
u\in W^{K}_{0}L_{A}(\Omega)\text{ with }[u]_{K,A}>1,$ (15)
$[u]^{p^{0}}_{K,A}\leqslant\phi(u)\leqslant[u]^{p_{0}}_{K,A}\text{ , }\forall
u\in W^{K}_{0}L_{A}(\Omega)\text{ with }[u]_{K,A}<1.$ (16)

where
$\displaystyle\phi(u):=\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy$.

Proof. First we show that
$\displaystyle\phi(u)\leqslant[u]^{p^{0}}_{K,A}\text{ for all }u\in
W^{K}_{0}L_{A}(\Omega)\text{ with }[u]_{K,A}>1$. Indeed, since $\displaystyle
p^{0}\geqslant\dfrac{ta(t)}{A(t)}$ for all $\displaystyle t>0$ it follows that
for all $\displaystyle\sigma>1,$ we have

$\displaystyle\log(A(\sigma t))-\log(A(t))=\int_{t}^{\sigma
t}\dfrac{a(\tau)}{A(\tau)}d\tau\leqslant\int_{t}^{\sigma
t}\dfrac{p^{0}}{\tau}d\tau=\log(\sigma^{p^{0}}).$

Thus, we deduce

$A(\sigma t)\leqslant\sigma^{p^{0}}A(t)\text{ for all }t>0\text{ and
}\sigma>1.$ (17)

Let now $\displaystyle u\in W_{0}^{K}L_{A}(\Omega)$ with
$\displaystyle[u]_{K,A}>1$. Using the definition of Luxemburg norm and the
relation $\displaystyle(\ref{34})$, we deduce

$\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy$
$\displaystyle=\int_{Q}A\left([u]_{K,A}\dfrac{(u(x)-u(y))K(x,y)}{[u]_{K,A}}\right)dxdy$
$\displaystyle\leqslant[u]^{p^{0}}_{K,A}\int_{Q}A\left(\dfrac{(u(x)-u(y))K(x,y)}{[u]_{K,A}}\right)dxdy$
$\displaystyle\leqslant[u]^{p^{0}}_{K,A}.$

Now, we show that $\displaystyle\phi(u)\geqslant[u]^{p_{0}}_{K,A}\text{ for
all }u\in W^{K}_{0}L_{A}(\Omega)\text{ with }[u]_{K,A}>1$. Indeed, since

$\displaystyle p_{0}\leqslant\dfrac{ta(t)}{A(t)}$

for all $\displaystyle t\geqslant 0$, it follows that for all
$\displaystyle\sigma>1$, we have

$\displaystyle\log(A(\sigma t))-\log(A(t))=\int_{t}^{\sigma
t}\dfrac{a(\tau)}{A(\tau)}d\tau\geqslant\int_{t}^{\sigma
t}\dfrac{p_{0}}{\tau}d\tau=\log(\sigma^{p_{0}}).$

Hence, we deduce

$A(\sigma t)\geqslant\sigma^{p_{0}}A(t)\text{ for all }t>0\text{ and
}\sigma>1.$ (18)

Let $\displaystyle u\in W_{0}^{K}L_{A}(\Omega)$ with
$\displaystyle[u]_{K,A}>1$, we consider $\displaystyle\beta\in(1,[u]_{K,A})$,
since $\displaystyle\beta<[u]_{K,A}$, so by definition of Luxemburg norm, it
follows that

$\displaystyle\int_{Q}A\left(\dfrac{(u(x)-u(y))K(x,y)}{\beta}\right)dxdy>1,$

the above inequality implies that

$\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy$
$\displaystyle=\int_{Q}A\left(\beta\dfrac{(u(x)-u(y))K(x,y)}{\beta}\right)dxdy$
$\displaystyle\geqslant\beta^{p_{0}}\int_{Q}A\left(\dfrac{(u(x)-u(y))K(x,y)}{\beta}\right)dxdy$
$\displaystyle\geqslant\beta^{p_{0}},$

letting $\displaystyle\beta\nearrow[u]_{K,A}$, we deduce that relation
$\displaystyle(\ref{32})$ hold true.

Next, we show that $\displaystyle\phi(u)\leqslant[u]^{p_{0}}_{K,A}\text{ for
all }u\in W^{K}_{0}L_{A}(\Omega)\text{ with }[u]_{K,A}<1$. By the same
argument in the proof of $\displaystyle(\ref{34})$ and
$\displaystyle(\ref{35})$, we have

$A(t)\leqslant\tau^{p_{0}}A\left(\dfrac{t}{\tau}\right)\text{ for all
}t>0\text{ , }\tau\in(0,1).$ (19)

Let $\displaystyle u\in W_{0}^{K}L_{A}(\Omega)$ with
$\displaystyle[u]_{K,A}<1$. Using the definition of Luxemburg-norm and the
relation $\displaystyle(\ref{36})$, we deduce

$\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy$
$\displaystyle=\int_{Q}A\left([u]_{K,A}\dfrac{(u(x)-u(y))K(x,y)}{[u]_{K,A}}\right)dxdy$
$\displaystyle\leqslant[u]^{p_{0}}_{K,A}\int_{Q}A\left(\dfrac{(u(x)-u(y))K(x,y)}{[u]_{K,A}}\right)dxdy$
$\displaystyle\leqslant[u]^{p_{0}}_{K,A}.$

Finally, we show that $\displaystyle\phi(u)\geqslant[u]^{p_{0}}_{K,A}\text{
for all }u\in W^{K}_{0}L_{A}(\Omega)\text{ with }[u]_{K,A}<1$. Similar
techniques as those used in the proof of relation $\displaystyle(\ref{34})$
and $\displaystyle(\ref{35})$, we have

$A(t)\geqslant\tau^{p^{0}}A\left(\dfrac{t}{\tau}\right)\text{ for all
}t>0\text{ , }\tau\in(0,1).$ (20)

Let $\displaystyle u\in W_{0}^{K}L_{A}(\Omega)$ with
$\displaystyle[u]_{K,A}<1$ and $\displaystyle\beta\in(1,[u]_{K,A})$, so by
$\displaystyle(\ref{37})$ we find

$\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy\geqslant\beta^{p^{0}}\int_{Q}A\left(\dfrac{(u(x)-u(y))K(x,y)}{\beta}\right)dxdy.$
(21)

We define $\displaystyle v(x)=\dfrac{u(x)}{\beta}$ for all $\displaystyle
x\in\Omega$, we have $\displaystyle[v]_{K,A}=\dfrac{[u]_{K,A}}{\beta}>1$.
Using the relation $\displaystyle(\ref{32})$ we find

$\displaystyle\int_{Q}A\left((v(x)-v(y))K(x,y)\right)dxdy>[v]^{p_{0}}_{K,A}>1,$
(22)

by $\displaystyle(\ref{38})$ and $\displaystyle(\ref{39})$ we obtain

$\displaystyle\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy\geqslant\beta^{p^{0}}.$

Letting $\displaystyle\beta\nearrow[u]_{K,A}$, we deduce that relation
$\displaystyle(\ref{33})$ hold true.

######  Theorem 3.2.

Let $\displaystyle\Omega$ be an open subset of $\displaystyle\mathbb{R}^{N}$,
and let $\displaystyle K$ as defined in $\displaystyle(K_{0})$. The space
$\displaystyle W^{K}_{0}L_{A}(\Omega)$ is a Banach space and a separable
(resp. reflexive) space if and only if $\displaystyle(A,\Omega)$ is
$\displaystyle\Delta$-regular (resp. $\displaystyle(A,\Omega)$ and
$\displaystyle(\overline{A},\Omega)$ are $\displaystyle\Delta$-regular).
Furthermore if $\displaystyle(A,\Omega)$ is $\displaystyle\Delta$-regular and
$\displaystyle A(\sqrt{t})$ is convex, then the space $\displaystyle
W^{K}_{0}L_{A}(\Omega)$ is uniformly convex.

Proof. Let $\displaystyle\left\\{u_{n}\right\\}$ be a Cauchy sequence in
$\displaystyle W^{K}_{0}L_{A}(\Omega)$. Thus, for any
$\displaystyle\varepsilon>0,$ there exist $\displaystyle\mu_{\varepsilon}>$
such that $\displaystyle n,m\geqslant\mu_{\varepsilon}$ :

$||u_{n}-u||_{A}\leqslant||u_{n}-u||_{K,A}<\varepsilon,$ (23)

by the completeness of $\displaystyle L_{A}(\Omega)$, there exist
$\displaystyle u\in L_{A}(\Omega)$ such that $\displaystyle
u_{n}\longrightarrow u$ strongly in $\displaystyle L_{A}(\Omega)$. Since
$\displaystyle u_{n}=0$ in $\displaystyle\mathbb{R}^{N}\setminus\Omega$, we
define $\displaystyle u=0$ in $\displaystyle\mathbb{R}^{N}\setminus\Omega$,
then $\displaystyle u_{n}\longrightarrow u$ strongly in $\displaystyle
L_{A}(\mathbb{R}^{N})$. So there exist a subsequence
$\displaystyle\left\\{u_{n_{k}}\right\\}$ in $\displaystyle
W^{K}_{0}L_{A}(\Omega)$ such that $\displaystyle u_{n_{k}}\longrightarrow u$
a.e. in $\displaystyle\mathbb{R}^{N}$, then by Fatou lemma we get

$\displaystyle\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy\leqslant\liminf\int_{Q}A\left((u_{n_{k}}(x)-u_{n_{k}}(y))K(x,y)\right)dxdy<\infty.$

Thus $\displaystyle u\in W^{K}_{0}L_{A}(\Omega)$. Let $\displaystyle
n\geqslant\mu_{\varepsilon}$, by the second inequality in
$\displaystyle(\ref{17})$ and the Fatou lemma we get :

$\displaystyle||u_{n}-u||_{K,A}\leqslant\liminf||u_{n}-u_{n_{k}}||_{K,A}\leqslant\varepsilon,$

that is $\displaystyle u_{n}\longrightarrow u$ strongly in $\displaystyle
W^{K}_{0}L_{A}(\Omega)$.
Next, we prove that $\displaystyle W^{K}_{0}L_{A}(\Omega)$ is a separable,
reflexive space and uniformly convex.

We consider the operator

$\displaystyle T:W^{K}_{0}L_{A}(\Omega)\longrightarrow L_{A}(Q,dxdy).$

Clearly, T is an isometry. Since $\displaystyle L_{A}(Q,dxdy)$ is a reflexive
separable and uniformly convex space (see [1, 28]), then $\displaystyle
W_{0}^{K}L_{A}(\Omega)$ is also a reflexive separable and uniformly convex
space.

######  Theorem 3.3.

Let $\displaystyle A$ be an $\displaystyle N$-function, and $\displaystyle K$
as defined in $\displaystyle(K_{0})$. Let $\displaystyle\Omega$ be a bounded
open subset of $\displaystyle\mathbb{R}^{N}$ with $\displaystyle
C^{0,1}$-regularity and bounded boundary. If $\displaystyle(\ref{5})$ and
$\displaystyle(\ref{6.})$ hold, then

$W^{K}{L_{A}}(\Omega)\hookrightarrow L_{A_{*}}(\Omega),$ (24)

and the embedding

$W^{K}{L_{A}}(\Omega)\hookrightarrow L_{B}(\Omega),$ (25)

is compact for all $\displaystyle B\prec\prec A_{*}$.

Proof. Let $\displaystyle u\in W^{K}L_{A}(\Omega)$, so by Lemma 3.2, we get
$\displaystyle u\in W^{s}L_{A}(\Omega)$ and

$\displaystyle||u||_{s,A}\leqslant C_{1}||u||_{K,A}.$

On the other hand by Theorem 2.1 we have

$\displaystyle||u||_{A_{*}}\leqslant C_{2}||u||_{s,A},$

this implies that

$\displaystyle||u||_{A_{*}}\leqslant C||u||_{K,A}$

where $\displaystyle C=C(s,N,k_{0},\Omega)$.
By lemma 3.2 and Theorem 2.2, we have

$\displaystyle W^{K}{L_{A}}(\Omega)\hookrightarrow
W^{s}L_{A}(\Omega)\hookrightarrow L_{B}(\Omega).$

The latter embedding being compact, so we have the desired result.

Now, let $\displaystyle A$ be an N-function with $\displaystyle(A,\Omega)$ is
$\displaystyle\Delta$-regular. Since $\displaystyle\lim\limits_{t\rightarrow
0}\frac{A(t)}{t}=0$, so there exists $\displaystyle\alpha>0$ such that
$\displaystyle A(t)\leqslant t$ for all $\displaystyle t\leqslant\alpha$. We
set $\displaystyle\beta=\min\left\\{1,\alpha\right\\}$, and we introduce the
function $\displaystyle A_{1}$ as,

$A_{1}(t)=\left\\{\begin{array}[]{clclc}\frac{A(\beta)}{\beta}t&if&t\leqslant\beta,\\\
\\\ A(t)&if&t>\beta.\end{array}\right.$ (26)

$\displaystyle A_{1}$ is a convex, continuous, nondecreasing, finite valued
function, with $\displaystyle A_{1}(0)=0$ and
$\displaystyle\lim\limits_{t\rightarrow+\infty}A_{1}(t)=+\infty$.
$\displaystyle A_{1}$ is called a Young function (see. [31] ).
For a given domain $\displaystyle\Omega$ in $\displaystyle\mathbb{R}^{N}$, we
define the space $\displaystyle L_{A_{1}}(\Omega)$ as follows,

$\displaystyle
L_{A_{1}}(\Omega)=\Bigg{\\{}u:\Omega\rightarrow\mathbb{R}:\int_{\Omega}A_{1}(\lambda
u(x))dx<\infty\text{ for some }\lambda>0\Bigg{\\}},$

this space is equipped with the norm,

$||u||_{A_{1}}=\inf\Bigg{\\{}\lambda>0:\int_{\Omega}A_{1}\left(\dfrac{u(x)}{\lambda}\right)dx\leqslant
1\Bigg{\\}}.$ (27)

######  Lemma 3.4.

(cf. [6]) Let $\displaystyle\Omega$ be a bounded open subset of
$\displaystyle\mathbb{R}^{N}$ and let $\displaystyle s\in(0,1)$. Let
$\displaystyle A$ be an N-function and $\displaystyle A_{1}$ as defined in
$\displaystyle(\ref{22.})$. Then,

  1. (1)

$\displaystyle L_{A_{1}}(\Omega)=L_{A}(\Omega)$.

  2. (2)

The norm $\displaystyle||.||_{A}$ and $\displaystyle||.||_{A_{1}}$ are
equivalent.

######  Remark 3.2.

Let $\displaystyle\Omega$ be an open subset of $\displaystyle\mathbb{R}^{N}$
and let $\displaystyle s\in(0,1)$. Let $\displaystyle A_{1}$ as defined in
$\displaystyle(\ref{22.})$. Then we define the space $\displaystyle
W^{s}L_{A_{1}}(\Omega)$ by,

$W^{s}{L_{A_{1}}}(\Omega)=\Bigg{\\{}u\in
L_{A_{1}}(\Omega):\int_{\Omega}\int_{\Omega}A_{1}\left(\dfrac{\lambda(u(x)-u(y))}{|x-y|^{s}A_{1}^{-1}(|x-y|^{N})}\right)dxdy<\infty\text{
for some }\lambda>0\Bigg{\\}},$ (28)

which equipped with the norm

$\displaystyle||u||_{s,A_{1}}=||u||_{A_{1}}+[u]_{s,A_{1}}$

where,

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

If $\displaystyle\Omega$ is a bounded open subset of
$\displaystyle\mathbb{R}^{N}$, then by lemma 3.4, we have $\displaystyle
W^{s}L_{A_{1}}(\Omega)=W^{s}L_{A}(\Omega)$ and the norm
$\displaystyle||.||_{s,A}$ and $\displaystyle||.||_{s,A_{1}}$ are equivalent.

######  Lemma 3.5.

Let $\displaystyle\Omega$ be a bounded open subset of
$\displaystyle\mathbb{R}^{N}$ and let $\displaystyle s\in(0,1)$. Let
$\displaystyle A_{1}$ as defined in $\displaystyle(\ref{22.})$. Then the space
$\displaystyle W^{s}L_{A_{1}}(\Omega)$ continuously embedded in $\displaystyle
W^{s,p_{0}}(\Omega)$ where $\displaystyle p_{0}>1$ is as defined in
$\displaystyle(\ref{12})$. Therefore $\displaystyle W^{s}L_{A}(\Omega)$
continuously embedded in $\displaystyle W^{s,p_{0}}(\Omega)$.

Proof. By definition of function $\displaystyle A_{1}$ we get

$A(t)\leqslant\gamma A_{1}(t)\text{ for all }t>0,$ (29)

where $\displaystyle\gamma=\max\left\\{1,\frac{\beta}{A(\beta)}\right\\}$, and
since $\displaystyle A$ satisfies the $\displaystyle\Delta_{2}$-condition we
have

$\displaystyle p_{0}:=\inf_{s\geqslant 0}\dfrac{sa(s)}{A(s)}>1$

this fact implies that

$\displaystyle\dfrac{p_{0}}{t}\leqslant\dfrac{a(t)}{A(t)}\hskip
28.45274pt\text{ for all }t>0,$

so

$\displaystyle p_{0}(\log(t))^{\prime}\leqslant(\log(A(t)))^{\prime}\hskip
28.45274pt\text{ for all }t>0$

therefore

$|t|^{p_{0}}\leqslant C_{1}A(t)\hskip 28.45274pt\text{ for all }t>0$ (30)

we combining $\displaystyle(\ref{23})$ and $\displaystyle(\ref{24})$ we obtain

$|t|^{p_{0}}\leqslant C_{1}A_{1}(t)\hskip 28.45274pt\text{ for all }t>0.$ (31)

Let $\displaystyle u\in W^{s}L_{A}(\Omega)$, by $\displaystyle(\ref{25})$ we
have

$\displaystyle\int_{\Omega}|u|^{p_{0}}dx\leqslant
C_{1}\int_{\Omega}A_{1}(u)dx$

then

$||u||_{L^{p_{0}}}\leqslant C_{1}||u||_{A_{1}}.$ (32)

where $\displaystyle C_{1}>0$ and it is possibly different step by step. On
the other hand

$\displaystyle\int_{\Omega}\int_{\Omega}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{p_{0}s+N}}dxdy$
$\displaystyle=\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}\leqslant\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{p_{0}s+N}}dxdy$
$\displaystyle\hskip
28.45274pt+\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}>\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{p_{0}s+N}}dxdy$
$\displaystyle=I_{1}+I_{2}.$

By definition of $\displaystyle A_{1}$ and estimation
$\displaystyle(\ref{25})$, we have

$\displaystyle
I_{1}=\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}\leqslant\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{p_{0}s+N}}$
$\displaystyle
dxdy=\frac{A(\beta)}{\beta}\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}\leqslant\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{sp_{0}}A_{1}^{-1}(|x-y|^{N})}dxdy$
(33)
$\displaystyle\leqslant\left(\frac{A(\beta)}{\beta}\right)^{p_{0}}\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}\leqslant\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{sp_{0}}(A_{1}^{-1}(|x-y|^{N}))^{p_{0}}}dxdy$
$\displaystyle\leqslant
C_{1}\left(\frac{A(\beta)}{\beta}\right)^{p_{0}}\int_{\Omega}\int_{\Omega}A_{1}\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A_{1}^{-1}(|x-y|^{N})}\right)dxdy$

and

$\displaystyle I_{2}$
$\displaystyle=\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}>\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{sp_{0}+N}}dxdy$
$\displaystyle=\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|^{N}>\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{sp_{0}}(A_{1}^{-1}(|x-y|^{N}))^{p_{0}}}\dfrac{(A_{1}^{-1}(|x-y|^{N}))^{p_{0}}}{|x-y|^{N}}dxdy$
$\displaystyle\leqslant\sup_{\Omega\times\Omega\cap\left\\{|x-y|^{N}>\beta\right\\}}\dfrac{(A_{1}^{-1}(|x-y|^{N}))^{p_{0}}}{|x-y|^{N}}\int_{\Omega}\int_{\Omega\cap\left\\{|x-y|>\beta\right\\}}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{sp_{0}}(A_{1}^{-1}(|x-y|^{N}))^{p_{0}}}dxdy,$
$\displaystyle\leqslant\sup_{\Omega\times\Omega\cap\left\\{|x-y|^{N}>\beta\right\\}}\dfrac{(A_{1}^{-1}(|x-y|^{N}))^{p_{0}}}{|x-y|^{N}}C_{1}\int_{\Omega}\int_{\Omega}A_{1}\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A_{1}^{-1}(|x-y|^{N})}\right)dxdy.$

Since $\displaystyle A^{-1}_{1}(t)$ is continuous for all $\displaystyle
t>\beta$ and $\displaystyle\Omega$ is bounded so,

$\displaystyle\sup_{\Omega\times\Omega\cap\left\\{|x-y|^{N}>\beta\right\\}}\dfrac{A_{1}^{-1}(|x-y|^{N})}{|x-y|^{N}}=C_{2}<\infty,$

therefore

$I_{2}\leqslant
C_{1}C_{2}\int_{\Omega}\int_{\Omega}A_{1}\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A_{1}^{-1}(|x-y|^{N})}\right)dxdy,$
(34)

by combining $\displaystyle(\ref{27.})$ and $\displaystyle(\ref{28})$ we
obtain that

$\displaystyle\int_{\Omega}\int_{\Omega}\dfrac{|u(x)-u(y)|^{p_{0}}}{|x-y|^{p_{0}s+N}}dxdy\leqslant
C\int_{\Omega}\int_{\Omega}A_{1}\left(\dfrac{u(x)-u(y)}{\lambda|x-y|^{s}A_{1}^{-1}(|x-y|^{N})}\right)dxdy,$

where $\displaystyle
C_{3}=C_{1}\left(\dfrac{A(\beta)}{\beta}\right)^{p_{0}}+C_{1}C_{2}$. Then

$[u]_{s,p_{0}}\leqslant C_{3}[u]_{s,A_{1}}.$ (35)

Combining $\displaystyle(\ref{26})$ and $\displaystyle(\ref{29})$, we have

$\displaystyle||u||_{s,p_{0}}\leqslant C||u||_{s,A_{1}},$

Where $\displaystyle C=C_{1}+C_{3}$.

######  Corollary 3.1.

Let $\displaystyle\Omega$ be a bounded open subset of
$\displaystyle\mathbb{R}^{N}$, let $\displaystyle s\in(0,1)$ and let
$\displaystyle A$ be an $\displaystyle N$-function satisfies the global
$\displaystyle\Delta_{2}$-condition. Let $\displaystyle K$ as defined in
$\displaystyle(K_{0})$.
If $\displaystyle sp_{0}<N$, then

$\displaystyle W^{K}L_{A}(\Omega)\hookrightarrow L^{p^{*}_{0}}(\Omega),$

where $\displaystyle p^{*}_{0}=\frac{Np_{0}}{N-sp_{0}}.$ And the embedding

$\displaystyle W^{K}L_{A}(\Omega)\hookrightarrow L^{q}(\Omega),$

is compact for all $\displaystyle q\in[1,p_{0}^{*})$.
If $\displaystyle sp_{0}=N$, then embedding

$\displaystyle W^{K}L_{A}(\Omega)\hookrightarrow L^{q}(\Omega),$

is compact, for all $\displaystyle q\in[1,+\infty)$.
If $\displaystyle sp>N$, then the embedding

$\displaystyle W^{K}L_{A}(\Omega)\hookrightarrow L^{\infty}(\Omega),$

is compact.

##  4\. Existence results

In this section, we prove the existence of a weak solutions for a fractional
Kirchhoff type problem in fractional Orlicz-Sobolev spaces, by means of the
direct method in calculus variations. For this, we suppose that the Kirchhoff
function $\displaystyle M:[0,\infty)\longrightarrow(0,\infty)$ is a continuous
function satisfying the following condition:

There exists $\displaystyle m_{0}>0$ such that :

$\displaystyle(M_{0})\hskip 119.50148ptM(t)\geqslant m_{0}\text{ for all
}t\in[0,\infty).\hskip 199.16928pt$

Also, we assume that $\displaystyle
f:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ is a Carathéodory function
satisfying the following conditions :
there exists $\displaystyle\theta_{1}>0$ and $\displaystyle 1<q<p_{0}^{*}$
such that

$\displaystyle(f_{1})\hskip
119.50148pt|f(x,t)|\leqslant\theta_{1}(1+|t|^{q-1})\text{ a.e.
}(x,t)\in\Omega\times\mathbb{R}^{N},\hskip 199.16928pt$

there exist $\displaystyle\theta_{2}>0$ and an open bounded set
$\displaystyle\Omega_{0}\subset\Omega$ such that

$\displaystyle(f_{2})\hskip
119.50148pt|f(x,t)|\geqslant\theta_{2}|t|^{q-1}\text{ a.e.
}(x,t)\in\Omega_{0}\times\mathbb{R}^{N}.\hskip 199.16928pt$

######  Definition 4.1.

We say that $\displaystyle u\in W^{K}_{0}L_{A}(\Omega)$ is a weak solution of
problem $\displaystyle(D_{K,A})$ if

$\displaystyle
M\left(\int_{\mathbb{R}^{N}}A(h_{x,y}(u))dxdy\right)\int_{\mathbb{R}^{2N}}a(h_{x,y}(u)\left(h_{x,y}(v)\right)dxdy=\int_{\Omega}f(x,u)vdx,$

for all $\displaystyle v\in W^{K}_{0}L_{A}(\Omega)$, where $\displaystyle
h_{x,y}(u)=(u(x)-u(y))K(x,y)$.

######  Theorem 4.1.

Let $\displaystyle
K:\mathbb{R}^{N}\times\mathbb{R}^{N}\longrightarrow(0,\infty)$ be a function
satisfy $\displaystyle(K_{0})$, and $\displaystyle A$ be an $\displaystyle
N$\- function satisfies $\displaystyle(a_{1})$ and $\displaystyle(a_{2})$,
suppose that $\displaystyle M$ satisfy $\displaystyle(M_{0})$, and
$\displaystyle f$ satisfies $\displaystyle(f_{1})$ and $\displaystyle(f_{2})$
if $\displaystyle 1<q<p_{0}$, then the problem $\displaystyle(D_{K,A})$ has a
nontrivial weak solution in $\displaystyle W^{K}_{0}L_{A}(\Omega)$.

######  Corollary 4.1.

Let $\displaystyle
K:\mathbb{R}^{N}\times\mathbb{R}^{N}\longrightarrow(0,\infty)$ be a function
satisfies $\displaystyle(K_{0})$, and $\displaystyle A$ be an $\displaystyle
N$\- function satisfies $\displaystyle(a_{1})$ and $\displaystyle(a_{2})$,
suppose that $\displaystyle M$ satisfies $\displaystyle(M_{0})$, and
$\displaystyle f$ satisfy $\displaystyle(f_{1})$ and $\displaystyle(f_{2})$,
if $\displaystyle q=p_{0}$ and $\displaystyle\theta_{1}<(m_{0}\lambda_{1})/2$,
where $\displaystyle\lambda_{1}$ is the first eigenvalue of
$\displaystyle\mathcal{L}^{K}_{A}u$ define by

$\displaystyle\lambda_{1}=\inf\limits_{u\in
W^{k}_{0}L_{A}(\Omega)\setminus\left\\{0\right\\}}\dfrac{||u||^{p_{0}}_{K,A}}{||u||_{A}^{p_{0}}},$

then the problem $\displaystyle(D_{K,A})$ has a nontrivial weak solution in
$\displaystyle W^{K}_{0}L_{A}(\Omega)$.

For $\displaystyle u\in W^{K}_{0}L_{A}(\Omega)$, we define

$\displaystyle J(u)=\widehat{M}\left(\int_{Q}A((u(x)-u(y))K(x,y))dxdy\right),$
$\displaystyle H(u)=\int_{\Omega}F(x,u)dx\text{ and }I(u)=J(u)-H(u),$

where $\displaystyle\widehat{M}(t)=\displaystyle\int_{0}^{t}M(\tau)d\tau$ and
$\displaystyle F(x,t)=\displaystyle\int_{0}^{t}f(x,t)d\tau$. Obviously the
energy functional $\displaystyle
I:W^{K}_{0}L_{A}(\Omega)\longrightarrow\mathbb{R}$ associated with problem
$\displaystyle(D_{K,A})$ is well defined.

######  Lemma 4.1.

If $\displaystyle f$ satisfies assumption $\displaystyle(f_{1})$, then the
functional $\displaystyle H\in C^{1}(W^{K}_{0}L_{A},\mathbb{R})$ and

$\displaystyle<H^{\prime}(u),v>=\int_{\Omega}f(x,u)vdx\text{ for all }u,v\in
W^{K}_{0}L_{A}(\Omega).$

Proof. By corollary 3.1, the proof of this lemma is similar to proof of lemma
3.1 in [36].

######  Lemma 4.2.

Let $\displaystyle(M_{0})$ hold, then the function $\displaystyle J\in
C^{1}(W^{K}_{0}L_{A},\mathbb{R})$ and

$\displaystyle<J^{\prime}(u),v>=M\left(\int_{Q}A(h_{x,y}(u))dxdy\right)\int_{Q}a(h_{x,y}(u))h_{x,y}(v)dxdy$

for all $\displaystyle u,v\in W^{K}_{0}L_{A}(\Omega)$. Moreover, for each
$\displaystyle u\in W^{K}_{0}L_{A}(\Omega)$, $\displaystyle
J^{\prime}(u)\in(W^{K}_{0}L_{A}(\Omega))^{*}$ where
$\displaystyle(W^{K}_{0}L_{A}(\Omega))^{*}$, denotes the dual space of
$\displaystyle W^{K}_{0}L_{A}(\Omega)$.

Proof. First, it is easy to see that

$<J^{\prime}(u),v>=M(\int_{Q}A(h_{x,y}(u)dxdy)\int_{Q}a(h_{x,y}(u))h_{x,y}(v)dxdy$
(36)

for all $\displaystyle u,v\in W^{K}_{0}L_{A}(\Omega)$. It follows from
$\displaystyle(\ref{30})$ that for each $\displaystyle u\in
W^{K}_{0}L_{A}(\Omega)$, $\displaystyle
J^{\prime}(u)\in(W^{K}_{0}L_{A}(\Omega))^{*}$.

Next, we prove that $\displaystyle J\in
C^{1}(W^{K}_{0}L_{A}(\Omega),\mathbb{R})$. Let
$\displaystyle\left\\{u_{n}\right\\}\subset W^{K}_{0}L_{A}(\Omega)$ with
$\displaystyle u_{n}\longrightarrow u$ strongly in $\displaystyle
W^{K}_{0}L_{A}(\Omega)$, for $\displaystyle v\in W^{K}_{0}L_{A}(\Omega)$ we
have $\displaystyle h_{x,y}(v)\in L_{A}(Q,dxdy)$ and by Hölder’s inequality,

$\displaystyle\left|\int_{Q}(a(h_{x,y}(u_{n}))-a(h_{x,y}(u)))h_{x,y}(v)dxdy\right|$
$\displaystyle\leqslant
2\left|\left|a(h_{x,y}(u_{n}))-a(h_{x,y}(u))\right|\right|_{L_{\overline{A}}}\left|\left|h_{x,y}(v)\right|\right|_{L_{A}}$
$\displaystyle\leqslant
2\left|\left|a(h_{x,y}(u_{n}))-a(h_{x,y}(u))\right|\right|_{L_{\overline{A}}}\left|\left|h_{x,y}(v)\right|\right|_{K,A}.$

On the other hand, $\displaystyle u_{n}\rightarrow u$ in $\displaystyle
W^{K}_{0}L_{A}(\Omega)$, then $\displaystyle
v_{n}:=h_{x,y}(u_{n})\longrightarrow v:=h_{x,y}(u)$ in $\displaystyle
L_{A}(Q)$, so by dominated convergence theorem, there exists a subsequence
$\displaystyle\left\\{v_{n_{k}}\right\\}$ and a function $\displaystyle h$ in
$\displaystyle L_{A}(Q)$ such that

$\displaystyle|a(h_{x,y}(u_{n_{k}}))|\leqslant|a(h)|\in
L_{\overline{A}}(Q)\text{ a.e in }Q,$

and

$\displaystyle a(h_{x,y}(u_{n_{k}}))\longrightarrow a(h_{x,y}(u))\text{ a.e in
}Q.$

Then by dominated convergence theorem we obtain that

$\displaystyle\sup_{||v||_{K,A}\leqslant
1}\Bigg{|}\int_{Q}(a(h_{x,y}(u_{n}))-a(h_{x,y}(u)))h_{x,y}(v)dxdy\Bigg{|}\longrightarrow
0.$

Moreover, by the continuity of $\displaystyle M$, we have

$\displaystyle
M\left(\int_{\mathbb{R}^{N}}A(h_{x,y}(u_{n}))dxdy\right)\longrightarrow
M\left(\int_{\mathbb{R}^{N}}A(h_{x,y}(u))dxdy\right).$

Combining lemma 4.1 and lemma 4.2, we get $\displaystyle I\in
C^{1}(W^{K}_{0}L_{A}(\Omega),\mathbb{R})$ and

$\displaystyle<I^{\prime}(u),v>=M\left(\int_{Q}A(h_{x,y}(u))dxdy\right)\int_{Q}a(h_{x,y}(u)\left(h_{x,y}(v)\right)dxdy-\int_{\Omega}f(x,u)vdx$

for all $\displaystyle u,v\in W^{K}_{0}L_{A}(\Omega)$.

######  Lemma 4.3.

Let $\displaystyle(M_{0})$ and $\displaystyle(f_{1})$ be satisfied, then the
functional $\displaystyle I\in C^{1}(W^{K}_{0}L_{A}(\Omega),\mathbb{R})$ is
weakly lower semi-continuous.

Proof. First, note that the map :

$\displaystyle u\longmapsto\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy$

is lower semi-continuous for the weak topology of $\displaystyle
W^{K}_{0}L_{A}(\Omega)$. Indeed, we define a functional
$\displaystyle\psi:W^{K}_{0}L_{A}(\Omega)\longrightarrow\mathbb{R}$ as

$\displaystyle\psi(u)=\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy,$

similar to Lemma 4.2, we obtain $\displaystyle\psi\in
C^{1}(W^{K}_{0}L_{A}(\Omega),\mathbb{R}^{N})$ and

$\displaystyle<\psi^{\prime}(u),v>=\int_{Q}a\left((u(x)-u(y))K(x,y)\right)(v(x)-v(y))K(x,y)dxdy$

for all $\displaystyle u,v\in W^{K}_{0}L_{A}(\Omega)$.
On the other hand, since $\displaystyle A$ is convex so $\displaystyle\psi$ is
also convex. Now, let $\displaystyle\left\\{u_{n}\right\\}\subset
W^{K}_{0}L_{A}(\Omega)$ with $\displaystyle u_{n}\rightharpoonup u$ weakly in
$\displaystyle W^{K}_{0}L_{A}(\Omega)$, then by convexity of
$\displaystyle\psi$ we have

$\displaystyle\psi(u_{n})-\psi(u)\geqslant<\psi^{\prime}(u),u_{n}-u>,$

hence, we obtain $\displaystyle\psi(u)\leqslant\liminf\psi(u_{n})$, that is,
the map

$\displaystyle u\longmapsto\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy$

is lower semi-continuous.

Let $\displaystyle u_{n}\rightharpoonup u$ weakly in $\displaystyle
W^{k}_{0}L_{A}(\Omega)$, so by corollary 3.1, $\displaystyle
u_{n}\longrightarrow u$ in $\displaystyle L^{q}(\Omega)$ for all
$\displaystyle q\in(p_{0},p_{0}^{*})$. Without loss of generality, we assume
that $\displaystyle u_{n}\longrightarrow u$ a.e. in $\displaystyle\Omega$.
Assumption $\displaystyle(f_{1})$ implies that

$\displaystyle F(x,t)\leqslant 2\theta_{1}(|t|^{q}+1).$

Thus, for any measurable subset $\displaystyle U\subset\Omega$,

$\displaystyle\int_{U}|F(x,u_{n})|dx\leqslant
2\theta_{1}\int_{U}|u_{n}|^{q}dx+2\theta_{1}|U|.$

By Hölder inequality and corollary 3.1 , we have,

$\displaystyle\int_{U}|F(x,u_{n})|dx$ $\displaystyle\leqslant
2\theta_{1}||u_{n}^{q}||_{L^{\frac{p^{*}_{0}}{q}}}||1||_{L^{\frac{p^{*}_{0}}{p_{0}^{*}-q}}}+2\theta_{1}|U|$
(37) $\displaystyle\leqslant
2\theta_{1}C||u_{n}||^{q}_{K,A}|U|^{\frac{p_{0}^{*}-q}{p_{0}^{*}}}+2\theta_{1}|U|.$

It follows from $\displaystyle(\ref{37.})$ that the sequence
$\displaystyle\left\\{|F(x,u_{n})-F(x,u)|\right\\}$ is uniformly bounded and
equi-integrable in $\displaystyle L^{1}(\Omega)$. The Vitali Convergence
Theorem (see [34]) implies

$\displaystyle\lim\limits_{n\rightarrow
n}\int_{\Omega}|F(x,u_{n})-F(x,u)|dx=0,$

so

$\displaystyle\lim\limits_{n\rightarrow\infty}\int_{\Omega}F(x,u_{n})dx=\int_{\Omega}F(x,u)dx.$

Thus, the functional $\displaystyle H$ is weakly continuous. Further, we get
that $\displaystyle I$ is weakly lower semi-continuous.
Proof of theorem 4.1. From assumptions $\displaystyle(M_{0})$,
$\displaystyle(f_{1})$ and proposition 3.1, we have

$\displaystyle I(u)$
$\displaystyle=\widehat{M}\left(\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy\right)-\int_{\Omega}F(x,u)dx$
$\displaystyle\geqslant
m_{0}\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy-2\theta_{1}\int_{\Omega}|u|^{q}dx-2\theta_{1}|\Omega|$
$\displaystyle\geqslant
m_{0}[u]_{K,A}^{p_{0}}-2\theta_{1}C[u]_{K,A}^{q}-2\theta_{1}|\Omega|,$

since $\displaystyle p_{0}>q$, so we have $\displaystyle
I(u)\longrightarrow\infty$ as $\displaystyle[u]_{K,A}\longrightarrow\infty$,
by Lemma 4.3 $\displaystyle I$ is weakly lower semi-continuous on
$\displaystyle W^{K}_{0}L_{A}(\Omega)$, then by theorem 2.5 functional I has a
minimum point $\displaystyle u_{0}$ in $\displaystyle W^{K}_{0}L_{A}(\Omega)$
and $\displaystyle u_{0}$ is a weakly solution of problem
$\displaystyle(D_{K,A})$.

Next we need to verify that $\displaystyle u_{0}$ is nontrivial. Let
$\displaystyle x_{0}\in\Omega_{0}$, $\displaystyle 0<R<1$ satisfy
$\displaystyle B_{2R}(x_{0})\subset\Omega_{0}$, where $\displaystyle
B_{2R}(x_{0})$ is the ball of radius $\displaystyle 2R$ with center at the
point $\displaystyle x_{0}$ in $\displaystyle\mathbb{R}^{N}$. Let
$\displaystyle\varphi\in C_{0}^{\infty}(B_{2R}(x_{0}))$ satisfies
$\displaystyle 0\leqslant\varphi\leqslant 1$ and $\displaystyle\varphi\equiv
1$ in $\displaystyle B_{2R}(x_{0})$. Lemma 3.1 implies that

$\displaystyle||u||_{K,A}<\infty.$

Then for $\displaystyle 0<t<1$, by the mean value theorem and
$\displaystyle(f_{2})$, we have

$\displaystyle I(t\varphi)$
$\displaystyle=\widehat{M}\left(\int_{Q}A\left((t\varphi(x)-t\varphi(y))K(x,y)\right)dxdy\right)-\int_{\Omega}F(x,t\varphi)dx$
$\displaystyle\leqslant\widehat{M}\left(||t\varphi||^{p^{0}}_{K,A}\right)-\int_{\Omega}F(x,t\varphi)dx$
$\displaystyle\leqslant\int_{0}^{||t\varphi||^{p^{0}}_{K,A}}M(\tau)d\tau-\int_{\Omega_{0}}\dfrac{\theta_{2}}{q}|t\varphi|^{q}dx$
$\displaystyle\leqslant
M(v)||\varphi||^{p^{0}}_{K,A}t^{p^{0}}-\dfrac{\theta_{2}}{q}t^{q}\int_{\Omega_{0}}|\varphi|^{q}dx$
$\displaystyle\leqslant
Ct^{p^{0}}-\dfrac{\theta_{2}}{q}t^{q}\int_{\Omega_{0}}|\varphi|^{q}dx,$

where $\displaystyle v\in[0,||\varphi||^{p^{0}}_{k,A})$ and $\displaystyle C$
is a positive constant. Since $\displaystyle p^{0}>q$ and
$\displaystyle\displaystyle\int_{\Omega_{0}}|\varphi|^{q}dx>0,$ we have
$\displaystyle I(t_{0}\psi)<0$ for $\displaystyle t_{0}\in(0,t)$ sufficiently
small. Hence, the critical point $\displaystyle u_{0}$ of functional
$\displaystyle I$ satisfies $\displaystyle I(u_{0})\leqslant
I(t_{0}\psi)<0=I(0)$, that is $\displaystyle u_{0}\neq 0$.

Proof of corollary . In view of the proof of theorem 4.1, we only need to
check that $\displaystyle I(u)\longrightarrow\infty$ as
$\displaystyle||u||_{K,A}\rightarrow\infty$. Since $\displaystyle p=q$ and
$\displaystyle\theta_{1}<(m_{0}\lambda_{1})/(2)$, by assumption
$\displaystyle(f_{1})$ and the definition of first eigenvalue of
$\displaystyle\mathcal{L}^{K}_{A}u$ , we have

$\displaystyle I(u)$
$\displaystyle=\widehat{M}\left(\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy\right)-\int_{\Omega}F(x,u)dx$
$\displaystyle\geqslant
m_{0}\int_{Q}A\left((u(x)-u(y))K(x,y)\right)dxdy-2\theta_{1}\int_{\Omega}|u|^{p_{0}}dx-2\theta_{1}|\Omega|$
$\displaystyle\geqslant
m_{0}[u]_{K,A}^{p_{0}}-2\theta_{1}\frac{1}{\lambda_{1}}[u]_{K,A}^{p_{0}}-2\theta_{1}|\Omega|,$
$\displaystyle=\left(m_{0}-2\theta_{1}\frac{1}{\lambda_{1}}\right)[u]_{K,A}^{p_{0}}-2\theta_{1}|\Omega|.$

So we have $\displaystyle I(u)\longrightarrow\infty$ as
$\displaystyle[u]_{K,A}\longrightarrow\infty$.

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