Keywords: Espaces de Sobolev fractionnaires, Espaces d'Orlicz-Sobolev fractionnaires, Espaces de Musielak-Sobolev fractionnaires, Opérateurs intégro-différentiels non locaux, p-Laplacien fractionnaire Fractional Sobolev spaces, Fractional Orlicz-Sobolev spaces, Fractional Musielak-Sobolev spaces, Nonlocal integro-differential operators, Fractional p-Laplace operator, Fractional a(.)-Laplace operator, Fractional a(x, .)-Laplace operator, nonlocal problems, Dirichlet problems, Critical point theorems, Variational methods, Eigenvalue problems, Calcul of varaitions

Les espaces d'Orlicz sont des espaces de fonctions qui généralisent les espaces de Lebesgue classiques L p , où la fonction puissance pème est remplacée par une fonction croissante convexe. Ces espaces de Lebesgue généralisés, introduits par W. Orlicz et Z. W. Birnbaum en 1931 et étudiés plus tard, ainsi que les espaces de Sobolev associés, ont trouvé des applications dans divers domaines d'Analyse Mathématique, citons à titre d'exemple : les équations aux dérivées partielles, les probabilités, les statistiques, la théorie du potentiel, le traitement d'image et l'analyse harmonique. Notre objectif dans cette thèse est d'introduire la version fractionnaire des espaces d'Orlicz-Sobolev, à savoir, les espaces de Sobolev fractionnaires construits à partir d'un espace d'Orlicz au lieu d'un espace de Lebesgue classique et qui seront nommés : Espaces d'Orlicz-Sobolev fractionnaires. Nous étudions aussi les propriétés fonctionnelles des espaces d'Orlicz-Sobolev fractionnaires. Nous nous sommes intéressés aussi dans cette thèse à l'introduction des espaces, encore plus généraux que ceux de Sobolev-Orlicz fractionnaire, à savoir les espaces de Musielak-Sobolev fractionnaires, pour lesquels nous établissons quelques propriétés qualitatives. Par ailleurs, nous étudions certains problèmes faisant intervenir des opérateurs intégro-différentiels dans les espaces de Sobolev fractionnaires, les espaces d'Orlicz-Sobolev fractionnaires et dans les espaces de Musielak-Sobolev fractionnaires.

List of Symbols

D α ∂ α 1 +...+α N ∂ α 1 x 1 , ..., ∂ α N x N , α = {α 1 , ..., α N } , |α| = α 1 + ... + α N div f ∑ i=N i=1 ∂ f ∂ x i C(Ω)
the spaces of continuous functions on Ω C ∞ (Ω) the spaces of infinitely differentiable functions on Ω C ∞ 0 (Ω) infinitely differentiable functions with compact support on Ω (denoted also by D(Ω))

a.e. almost everywhere → strong convergence weak convergence → continuous embedding → → compact embedding supp f support of function u X arbitrary Banach space

Introduction

Historical

When we introduce the notion of derivative, we quickly realize that we can apply the concept of derivative to the derivative function itself, and by the same concept we can introduce the second derivative, then the successive derivatives of integer order. Integration, is an inverse operation of the derivative, can possibly be considered as a derivative of the order minus one. We can also wonder if these derivatives of successive orders have a fractional-order equivalent. Fractional derivation theory is a subject almost as old as classical calculus as we know it today, its origins date back to the late 17th century, the time when Newton and Leibniz developed the foundations of calculus differential and integral.

The idea of generalizing the notion of differentiation d n f (x) dx n to noninteger orders of n appeared at the birth of the differential calculus itself. The first attempt to discuss such an idea recorded in history was contained in the correspondence of Leibniz. In one of his letters to Leibniz concerning the theorem on the differentiation of a product of functions, Bernoulli asked about the meaning of this theorem in the case of noninteger order of differentiation. Leibniz in his letters to L'Höpital (1695) made some remarks on the possibility of considering differentials and derivatives of order 1/2. In 1820 Lacroix showed an exact formula for the evaluation of the derivative d

1 2 f (x) dx 1 2
.

After that, the fractional derivation was developed intensely, therefore, different examples of such operators are present in literature. Among these ones, Riemann-Liouville and Caputo fractional derivatives are the most exploited in the one-dimensional applications. Given a sufficiently smooth function u on an interval (a, b) and s ∈ (0, 1), the left and right Riemann-Liouville s-fractional derivatives of u are defined as

D s a + [u](x) := d dx 1 Γ(1 -s) x a u(t) (x -t) s dt, D s b -[u](x) := - d dx 1 Γ(1 -s) b x u(t) (t -x) s dt,
respectively, where Γ is Euler's Gamma function. On the other hand, the left and right Caputo s-fractional derivatives of u are set to be

c D s a + [u](x) := 1 Γ(1 -s) x a u (t) (x -t) s dt, xv c D s b -[u](x) := - 1 Γ(1 -s) x a u (t) (t -x) s dt.
The fractional derivation is of particular interest in several fields of application. However, a simple example of fluid mechanics shows how the derivative of order one half appears quite naturally when we want to explain a heat flow coming out laterally from a fluid flow as a function of the temporal evolution of the internal source. Another particular interest in the fractional derivation is linked to the mechanical modeling of gums and rubbers, in short, all kinds of materials that retain the memory of past deformations and whose behavior is said to be viscoelastic. Indeed, the fractional derivation is introduced there naturally.

After the construction of these different types of fractional derivatives, It was quite natural to ask for spaces W s,p with s fractional, filling the gaps between L p , W 1,p , W 2,p , . . . . Several proposals were made in the fifties and these attempts culminated around 1960. Let us first consider defining the Holder spaces C s with 0 < s = integer (filling the gaps between the spaces C k with k ∈ N * ). Let 0 < ρ < 1, then we introduce the norm

|| f || C ρ := || f || C + sup | f (x) -f (y) |x -y| ρ
where the supremum is taken over all x ∈ R N and y ∈ R N with x = y. Let s ∈ R, then we put

s = [s] + {s}
where [s] it is the integer part of s and 0 < {s} < 1. Then for 0 < s = integer, the Hölder space is defined as

C s = f ∈ C : || f || C s = || f || C [s] + ∑ |α=[s] ||D α f || C {s} < ∞ .
Hölder spaces have been employed by J. Schauder and other mathematicians since the midthirties in connection with boundary value problems for second order elliptic differential equations.

The final step in this direction is due to C. Miranda, see his book [START_REF] Miranda | Partial Differential Equations of Elliptic Type[END_REF] where one finds also many references. S. Agmon, A. Douglis and L. Nirenberg extended these considerations to boundary value problems for higher order elliptic differential equations, see [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF].

The spaces introduced by S.M. Nikol'skij in 1951, see [START_REF] Nikol | skij, Inequalities for entire functions of finite order and their application in the theory of differentiable functions of several variables (Russian)[END_REF], can be obtained from the Hölder space if one replaces there the C-norm by an L p norm : Let 0 < s = integer and 1 < p < ∞, then B s p,∞ is the collection of all f ∈ L p such that

|| f || B s p,∞ = || f || W [s] p + ∑ |α=[s] sup R N h =0 |h| -{s} ||∆ h D α f || L p (1)
is finite.

N. Aronszajn [9], L.N. Slobodeckij [START_REF] Slobodeckii | Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations (Russian)[END_REF] and E. Gagliardo [START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF] suggested in 1955-1958 to replace the sup-norm in (1) with respect to h by an L p -norm : Let 0 < s = integer and 1 < p < ∞ xvi then B s p,p s is the collection of all f ∈ L p such that

|| f || B s p,p = || f || W [s] p + ∑ |α=[s] R N |h| -{s}p ||∆ h D α f || p L p dh |h| N 1 p
. is finite.

Let, for example, 0 < s < 1, then we get the fractional Sobolev space W s,p defined as the collection of all f ∈ L p such that

|| f || B s p,p = || f || L p + R N R N | f (x) -f (y)| p |x -y| sp+N dxdy 1 p
.

which makes clear that we replaced the sup-norm in the Hölder spaces, by an L p -norm.

Motivation

The fractional Sobolev spaces have been a classical topic in functional and harmonic analysis all along, and some important books, such as [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF][START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF] treat the topic in detail. Recently, great attention has been focused on the study of fractional spaces, and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operator arises in a quite natural way in many different contexts, such as, among others, the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultrarelativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, chemical reactions of liquids, population dynamics, geophysical fluid dynamics, and mathematical finance (American options). The fractional Laplacian also provides a simple model to describe certain jump Lévy processes in probability theory. In all these cases, the nonlocal effect is modeled by the singularity at infinity. For more details and applications, see [7,[START_REF] Binlin | Superlinear nonlocal fractional problems with infinitely many solutions[END_REF][START_REF] Bucur | Nonlocal diffusion and applications[END_REF][START_REF] Caffarelli | Nonlocal equations, drifts and games[END_REF][START_REF] Cont | Financial modelling with jump processes[END_REF][START_REF] Majda | A two-dimensional model for quasigeostrophic flow : comparision with the two-dimensional Euler flow[END_REF][START_REF] Valdinoci | From the long jump random walk to the fractional Laplacian[END_REF] and the references therein. From a physical point of view, nonlocal operators play a crucial rule in describing several phenomena. As a general reference in this topic, we cite the recent paper of Vázquez [START_REF] Vã Ązquez | Nonlinear diffusion with fractional laplacian operators[END_REF]. In that paper, the author describes two models of flow in porous media, including nonlocal (long-range) diffusion effects, providing a long list of references related to physical phenomena and nonlocal operators. The first model is based on Darcy's law, and the pressure is related to the density by an inverse fractional Laplacian operator. The second model is more in the spirit of fractional Laplacian flows but nonlinear : contrary to the usual porous medium flows, it has infinite speed of propagation.

Even with this considerable interest in (fractional and classical) Lebesgue spaces, but in certain equations, precisely in the case of nonhomogeneous operators when we try to consider certain conditions on these operators, the problem cannot be formulated with the classical spaces of Lebesgue and Sobolev. As an example of application in the field of image processing, let us consider the restoration model with the following additive noise :

f = u + bruit (2)
xvii

The energy function associated with the equation (2) is then :

J(u) := Ω ( f -u) 2 dx, (3)
with Ω a bounded open of R 2 . As this is a poorly conditioned problem, then we need to add an energy regularization term, which is usually written in the form :

Ω A(|∇u|)dx,
this term represents diffusion. The energy function then takes the following form :

J(u) := λ 2 ( f -u) 2 dx + Ω A(|∇u|)dx.
The problems of minimization with sublinear regularization terms of this kind are in general badly posed in spaces with bounded variation (BV). On the other hand, thanks to the Orlicz-Sobolev spaces, this model of restoration, not only is well-posed but also the existence and the uniqueness of a minimum are guaranteed (see [START_REF] Samar | Méthodes variationnelles : applications á l'analyse d'image et au modèle de frenkel-kontorova[END_REF]Theorem 3.1.1]).

Hence the interest of the Orlicz spaces. This functional framework is a generalization of the classical Lebesgue spaces L p . They were first introduced by Birnbaum-Orlicz in [START_REF] Birnbaum | Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen[END_REF] and Orlicz in [START_REF] Orlicz | Über eine gewisse klasse von räumen vom typus[END_REF]. The problems of these spaces were later developed by some schools : of H. Nakano (Japan), of M. A. Krasnosel'skii, Yz. B. Rutickii (USSR), of A.c. Zaanen and W. A. J. Luxemburg (Netherlands), of W. Orlicz and J. Musielak (Poland), Ren and Reo (China).

Objective

In this sense, our objective in this thesis is to introduce the fractional version of the Orlicz-Sobolev spaces, namely a fractional Sobolev space constructed from an Orlicz space instead of L p , who will be nomed : the fractional Orlicz-Sobolev space. The first proposals to define this space are given by Salort et al [START_REF] Bonder | Fractional order Orlicz-Soblev spaces[END_REF] and Azroul et al [START_REF] Azroul | Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces[END_REF][START_REF] Azroul | Introduction to fractional Orlicz-Sobolev spaces[END_REF] in 2017-2019, before focusing on the following definition :

Let Ω un open of R N , s ∈ (0, 1) and A be an N-function. Then the fractional Orlicz-Sobolev space W s L A (Ω) given as sequence :

W s L A (Ω) = u ∈ L A (Ω) : Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞ for some λ > 0 .
Observe that in the case A(t) = t p , this space coincides with the fractional Sobolev space W s,p (Ω). Also, we extend the operator p-Laplacian fractional (-∆) s p :

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x)
|u(x)u(y)| p-2 (u(x)u(y)) |x -y| N+sp dy, namely a fractional-order integrodifferential operator named : a(.)-fractional Laplacian (-∆) s a(.) :

(-∆) s a(.) u(x) = 2 lim xviii This operator possesses more complicated non linearities than the farctional p-Laplace operator, due to the fact that (-∆) s a(.) is not homogeneous. These integro-differential operators are part of a larger class of pseudo-differential operators.

ε 0 R N \B ε (x)
For a long time, interest in these operators has grown steadily in recent years. Nonlocal operators such as (-∆) s (ie : (-∆) s p=2 ) appear naturally in continuum mechanics, phase transition phenomena, dynamics of population and game theory, see for example Caffarelli [START_REF] Caffarelli | Nonlocal equations, drifts and games[END_REF] and his references.

In the works of Metzler and Klafter [START_REF] Metzler | The random walk's guide to anomalous diffusion : a fractional dynamics approach[END_REF][START_REF] Metzler | The restaurant at the random walk : recent developments in the description of anomalous transport by fractional dynamics[END_REF], we study the description of anomalous diffusion by fractional dynamics and various fractional partial differential equations are derived from Lévy random walk models, by extending the Brownian Walk models of natural way. In particular, in Laskin's paper [START_REF] Laskin | Fractional Schrödinger equation[END_REF], a fractional Schrödinger equation was obtained, which extends to a Lévy frame, the classical result, which the integral path on the trajectories Brownians leads to the Schrödinger equation. Fractional traders are also involved in financial mathematics, since Lévy deals with jumps revealed as more appropriate stock price models, compared to Brownians used in the famous Black and Sholes option pricing model (see Applebaum [7]).

Plan

This work consists of five chapters :

-In the first chapter, we present all the necessary ingredients thereafter (definitions, theorem, lemmas ...).

-In the second chapter, we first extend the fractional Sobolev spaces W s,p to include the general case W s L A where A is a N -function and s ∈ (0, 1). We are interested in some qualitative properties of the space W s L A (completeness, reflexivity and separability, extension results...).

Furthermore, Moreover, we prove a continuous and compact embedding theorem of these spaces into Orlicz and Lebesgue spaces.

-In the third chapter, we start to study some problems in the special case where A(t) = t p , i.e. we study nonlocal problems involving the fractional p-Laplacian operator in fractional Sobolev space.

-In the fourth chapter, we base on the theory of fractional Orlicz-Sobolev spaces and by different critical point theorems, we study some nonlocal problems involving the fractional a(.)-Laplacian operator in fractional Orlicz-Sobolev space.

-In chapter five, we integrate both the functional structures of fractional Sobolev spaces with variable exponent and fractional Orlicz-Sobolev spaces. Therefore we introduce fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces (completeness, reflexivity and separability, generalized Poincaré inequality, embedding results ...).

As an application, we use variational calculus to study the existence solution for a class of fractional type problems with Dirichlet boundary data.

Table des matières

Introduction xv 1 Preliminaries

This chapter is devoted to recalling some definitions and classical results that will play a role in this thesis.

Orlicz Spaces

First, we briefly recall the definitions and some elementary properties of the Orlicz-Sobolev spaces. We refer the reader to [1,[START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF][START_REF] Rao | Theory of Orlicz Spaces[END_REF] for further reference and for some of the proofs of the results in this section.

N-Functions

We start by recalling the definition of the well-known N-functions.

Definition 1.1.1. Let A : R + → R + is said to be an N-function, if

• A is continuous, convex,

• A(t) > 0 for t > 0 and A(0) = 0, 1 2 Chapitre 1. Preliminaries

• A(t) t → 0 as t → 0, • A(t) t → ∞ as t → ∞.
Equivalently, A admits the representation :

A(t) = t 0 a(s)ds
where a : R + → R + is non-decreasing, right continuous, with a(0) = 0, a(t) > 0 ∀t > 0 and a(t) → ∞ as t → ∞. The conjugate N-function of A is defined by

A(t) = t 0 a(s)ds,
where a : R + → R + is given by a(t) = sup {s : a(s) t}. Evidently we have st A(t) + A(s),

which is known as Young inequality. Equality holds in (1.1) if and only if either t = a(s) or s = a(t).

We will extend these N-functions into even functions on all R.

Example 1.

1) A(t) = t p p , A(t) = t q q , 1 < p < ∞, wih

1 p + 1 q = 1,
2) A(t) = e tt -1, A(t) = (1 + t) log(1 + s)s.

Definition 1.1.2. The N-function A is said to satisfy the global ∆ 2 -condition if, for some k > 0, A(2t) kA(t) , ∀t 0.

When this inequality holds only for t t 0 > 0, A is said to satisfy the ∆ 2 -condition near infinity.

Definition 1.1.3. We call the pair (A, Ω) is ∆-regular if either :

• A satisfies a global ∆ 2 -condition, or

• A satisfies a ∆ 2 -condition near infinity and Ω has finite volume. where c > 0.

Definition 1.1.4. Let A, B be two N-functions. A is stronger (resp essentially stronger) than B, A B (resp A B) in symbols, if B(x) A(ax), , x x 0 0, for some (resp for each) a > 0 and x 0 (depending on a).

Remark 1. (see. [1,Section 8.5]). A B is equivalent to the condition

lim x→∞ B(λ x) A(x) = 0,
for all λ > 0.

Orlicz spaces

Let Ω be an open subset of R N . The Orlicz class K A (Ω) (resp. the Orlicz space L A (Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that

Ω A(|u(x)|)dx < ∞ (resp. Ω A(λ |u(x)|)dx < ∞ for some λ > 0). (1.4)
L A (Ω) is a Banach space under the Lexumburg norm

||u|| A = inf λ > 0 : Ω A |u(x)| λ dx 1 , (1.5)
and K A (Ω) is a convex subset of L A (Ω). The closure in L A (Ω) of the set of bounded measurable functions on Ω with compact support in Ω is denoted by E A (Ω).

Remark 2.

Let Ω be an open subset of R N .

1) E A (Ω) ⊂ K A (Ω) ⊂ L A (Ω),

2) E A (Ω) = L A (Ω) if and only if (A, Ω) is ∆-regular,

3) E A (Ω) is separable.

||u|| p + A Ω A(|u|)dx ||u|| p - A , ∀u ∈ L A (Ω) with ||u|| A < 1.
Theorem 1.1.2. (see. [1,Theorem 8.25]) Let Ω be an open subset of R N which has a finite volume, and suppose A, B two N-functions such that B ≺≺ A.Then any bounded subset S of L A (Ω) which is precompact in L 1 (Ω), is also precompact in L B (Ω).

Fractional Sobolev spaces

This section is devoted to the definition of fractional Sobolev spaces, and we recall some result of regarding continuous and compact embedding of these spaces. We refer the reader to [START_REF] Demengel | Demengel Functional Spaces for the Theory of Elliptic Partial Differential Equations[END_REF][START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] for further reference and for some of the proofs of these results. -If 1 p < ∞ then L p (Ω) is separable.

-If 1 < p < ∞ then L p (Ω) is reflexive.

-If 1 < p < ∞ then L p (Ω) is uniformly convex.

Fractional Sobolev spaces

We start by fixing the fractional exponent s ∈ (0, 1). For any p ∈ [1, ∞), we define the fractional Sobolev space W s,p (Ω) as follows : .

Theorem 1.2.3. (cf. [START_REF] Demengel | Demengel Functional Spaces for the Theory of Elliptic Partial Differential Equations[END_REF]) Let Ω be an open subset in R N , s ∈ (0, 1) and p ∈ [1, ∞), we have

• W s,p (Ω) is a Banach space.

• If 1 p < ∞ then W s,p (Ω) is separable.

• If 1 < p < ∞ then W s,p (Ω) is reflexive.

• If 1 < p < ∞ then W s,p (Ω) is uniformly convex.

Theorem 1.2.4. (cf. [START_REF] Demengel | Demengel Functional Spaces for the Theory of Elliptic Partial Differential Equations[END_REF]). Let Ω = R N , let s ∈ (0, 1) and p ∈ (1, +∞) be such that sp < N. Then there exists a positive constant c = c(N, s, p), such that,

||u|| p p * s c R N R N |u(x) -u(y)| p |x -y| N+sp dxdy,
for all u ∈ W s,p (R N ), where p * s = N p Nsp is the so-called fractional critical exponent. Consequentially, the space W s,p (R N ) is continuously embedded in L q (R N ) for any q ∈ [p, p * s ]. Moreover the embedding W s,p (R N ) → L q loc (R N ) is compact for all q ∈ [p, p * s ).

Theorem 1.2.5. (see : [START_REF] Demengel | Demengel Functional Spaces for the Theory of Elliptic Partial Differential Equations[END_REF]Corollary 4.53], [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]) Let s ∈ (0, 1), p ∈ [1, +∞) and let Ω be an open subset of R N with C 0,1 -regularity and bounded boundary. Then there exists a constant c = c(N, s, p, Ω) such that, for all f ∈ W s,p (Ω), we have

|| f || L q (Ω) c|| f || W s,p (Ω) for all q ∈ [p, p * s ],
that is, W s,p (Ω) → L q (Ω) for all q ∈ [p, p * s ],

where

p * s =            N p N-sp if N > sp ∞ if N sp.
If, in addition, Ω is bounded, then the space W s,p (Ω) is continuously embedded in L q (Ω) for any q ∈ [1, p * s ].

Theorem 1.2.6. (see : [START_REF] Demengel | Demengel Functional Spaces for the Theory of Elliptic Partial Differential Equations[END_REF]Theorem 4.58]). Let s ∈ (0, 1), p ∈ [1, +∞) and let Ω be a bounded open subset of R N with C 0,1 -regularity and bounded boundary. Then

• if sp < N, then the embedding W s,p (Ω) → L q (Ω) is compact for all q ∈ [1, p * s ),

• if sp = N, then the embedding W s,p (Ω) → L q (Ω) is compact for all q ∈ [1, ∞),

• if sp > N, then the embedding W s,p (Ω) → L ∞ (Ω) is compact.

1.3. Variable exponent Lebesgue spaces 7

Variable exponent Lebesgue spaces

Next, we recall some useful properties of variable exponent spaces. For more details we refer the reader to [START_REF] Fan | On the Spaces L p(x) (Ω) and W m,p(x) (Ω)[END_REF][START_REF] Kováčik | On Spaces L p(x) (Ω) and W m,p(x) (Ω)[END_REF], and the references therein.

Consider the set C + (Ω) = q ∈ C(Ω) : q(x) > 1 for all x ∈ Ω .

For all q ∈ C + (Ω), we define q + = sup x∈Ω q(x) and q -= inf x∈Ω q(x).

For any q ∈ C + (Ω), we define the variable exponent Lebesgue space as L q(x) (Ω) = u : Ω -→ R measurable :

Ω |u(x)| q(x) dx < +∞ .
This vector space endowed with the Luxemburg norm, which is defined by

u L q(x) (Ω) = inf λ > 0 : Ω u(x) λ q(x)
dx 1 is a separable reflexive Banach space.

Let q ∈ C + (Ω) be the conjugate exponent of q, that is, 1 q(x) + 1 q(x) = 1. Then we have the following Hölder-type inequality.

Lemma 1.3.1. (Hölder inequality) If u ∈ L q(x) (Ω) and v ∈ L q(x) (Ω), then

Ω uvdx 1 q -+ 1 q-u L q(x) (Ω) v L q(x) (Ω) 2 u L q(x) (Ω) v L q(x) (Ω) .
A very important role in manipulating the generalized Lebesgue spaces with variable exponent is played by the modular of the L q(x) (Ω) space, which defined by ρ q(.) : L q(x) (Ω) -→ R u -→ ρ q(.) (u) = Ω |u(x)| q(x) dx.

Proposition 1.3.1. Let u ∈ L q(x) (Ω), then we have 1. u L q(x) (Ω) < 1 (resp. = 1, > 1) ⇔ ρ q(.) (u) < 1 (resp. = 1, > 1), Chapitre 1. Preliminaries 2. u L q(x) (Ω) < 1 ⇒ u q+ L q(x) (Ω) ρ q(.) (u) u q- L q(x) (Ω) , 3. u L q(x) (Ω) > 1 ⇒ u q- L q(x) (Ω) ρ q(.) (u) u q+ L q(x) (Ω) .

Proposition 1.3.2. If u, u k ∈ L q(x) (Ω) and k ∈ N, then the following assertions are equivalent 1. lim k→+∞ u ku L q(x) (Ω) = 0, 2. lim k→+∞ ρ q(.) (u ku) = 0, 3. u k -→ u in measure in Ω and lim k→+∞ ρ q(.) (u k ) = ρ q(.) (u).

Some important results

Lemma 1.4.1. (Fatou's Lemma). Let { f n } n be a sequence of measurable functions which are nonnegative almost everywhere on Ω. Then

lim inf n→∞ f n (x)
and

Ω lim inf n→∞ f n (x)dx lim inf n→∞ Ω f n (x)dx.
Theorem 1.4.1. (Dominated Convergence Theorem) Let { f n } n be a sequence of real-valued measurable functions in a measure space (Ω, Σ, µ). Suppose that the sequence converges almost everywhere to a function f and is dominated by some integrable function g in the sense that :

| f n (x)| g(x),
for all n ∈ N and for almost everywhere in Ω. Then, f is integrable and

lim n→∞ Ω | f n -f |dµ = 0.
Which also implies

lim n→∞ Ω f n dµ = lim n→∞ Ω f dµ.
Theorem 1.4.2. Let { f n } n be a sequence of L p (Ω) and f ∈ L p (Ω), such that lim n→∞ f nf p = 0.

Then there exists a subsequence { f n k } n such that a) f n k (x) → f (x) a.e in Ω, b) | f n k (x)| f (x) for all k and a.e in Ω, with h ∈ L p (Ω).

1.5. Calculus of Variations

Calculus of Variations

The Calculus of Variations continues to be an area of very rapid growth. Variational methods are indispensable as a tool in mathematical physics and geometry. Results on Ginzburg-Landau type variational problems inspire research on the related Seiberg-Witten functional on a Kähler surface and invite speculations about possible applications in topology (Ding-Jost-Li-Peng-Wang [START_REF] Ding | Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials[END_REF]).

Many problems in analysis can be cast into the form of functional equations F(u) = 0, the solution u being sought among a class of admissible functions belonging to some Banach space V . Typically, these equations are nonlinear ; for instance, if the class of admissible functions is restricted by some (nonlinear) constraint. A particular class of functional equations is the class of Euler-Lagrange equations

DE(u) = 0
for a functional E on V , which is Fréchet-differentiable with derivative DE. We say such equations are of variational form. For equations of variational form an extensive theory has been developed, and variational principles play an important role in mathematical physics and differential geometry, optimal control and numerical anlysis.

In this section, we review some basic methods for proving the existence of relative minimizers.

Somewhat imprecisely we summarily refer to these methods as the direct methods in the calculus of variations. Moreover, we recall Ekeland's variational principle, Mountain Pass lemma, Fountain theorem and three critical point theorem.

At firstly, we give sufficient conditions for a function to be bounded from below and to attain its infimum. The discussion can be made largely independent of any differentiability assumptions on E or structure assumptions on the underlying space of admissible functions M. In fact, we have the following classical result.

Theorem 1.5.1. Let M be a topological Hausdorff space, and suppose E : M -→ R∪{+∞} satisfies the condition of bounded compactness :

For any α ∈ R the set

K α = {u ∈ M; E(u) α} is compact. (1.7)
Then E is uniformly bounded from below on M and attains its infimum. The conclusion remains valid if instead of (1.7) we suppose that any sub-level set K α is sequentially compact.

Chapitre 1. Preliminaries

Note that if E : M -→ R satisfies (1.7), then for any α ∈ R the set

{u ∈ M, E(u) > α}
is open, that is, E is lower semi-continuous. (Respectively, if each K α is sequentially compact, then E will be sequentially lower semi-continuous). Conversely, if E is (sequentially) lower semicontinuous and for some α the set K α is (sequentially) compact, then K α will be (sequentially) compact for all α α and again the conclusion of Theorem 1.5.1 will be valid.

Note that the lower semi-continuity condition can be more easily fulfilled the finer the topology on M. In contrast, the condition of compactness of the sub-level sets K α , α ∈ R, calls for a coarse topology and both conditions are competing. In practice, there is often a natural weak Sobolev space topology where both conditions can be simultaneously satisfied. However, there are many interesting cases where condition (1.7) cannot hold in any reasonable topology (even though relative minimizers may exist). In applications, the conditions of the following special case of Theorem 1.5.1 can often be checked more easily.

Theorem 1.5.2. (see :[154, Theorem 1.2]) Suppose that X is a reflexive Banach space with norm ||.|| and let V ⊂ X be a weakly closed subset of X. Suppose E : V -→ R ∪ {+∞} is coercive and (sequentially) weakly lower semi-continuous on V with respect to X, that is, suppose the following conditions are fulfilled :

• E(u) → ∞ as ||u|| → ∞, u ∈ V .
• For any u ∈ V , any sequence {u n } in V such that u n u weakly in X there holds :

E(u) lim inf n→∞ E(u n ).
Then E is bounded from below on V and attains its infimum in V .

In general it is not clear that a bounded and lower semi-continuous functional E actually attains its infimum. The analytic function f (x) = arctan x, for example, neither attains its infimum nor its supremum on the real line. A variant due to Ekeland [START_REF] Ekeland | On the variational principle[END_REF] of Dirichlet's principle, however, permits one to construct minimizing sequences for such functionals E whose elements u m each minimize a functional E m , for a sequence of functionals E m converging locally uniformly to E.

Theorem 1.5.3. (see : [START_REF] Ekeland | On the variational principle[END_REF]) Let V be a complete metric space and F : V -→ R ∪ {+∞} be a lower semicontinuous functional on V , that is bounded below and not identically equal to +∞. Fix ε > 0 1.5. Calculus of Variations and a point u ∈ V such that

F(u) ε + inf x∈V F(x).
Then for every λ > 0, there exists some point v ∈ V such that :

F(v) F(u), d(u, v) λ and for all w = v F(w) > F(v) - ε λ d(v, w).
In the preceding theorems, we have seen that (weak sequential) lower semicontinuity and (weak sequential) compactness of the sub-level sets of a functional E on a Banach space V suffice to guarantee the existence of a minimizer of E. To prove the existence of saddle points we will now strengthen the regularity hypothesis on E and in general require E to be of class C 1 (V ), that is continuously Fréchet differentiable. The minimax principle and its variants essentially cover all possibilities how existence results for saddle points can be drawn from information about the topology of the sub-level sets of a functional E. However, unless the domain of E itself has a rich topology, finding the right notion of flow-invariant family may be quite tiresome. Fortunately, there are existence results for saddle points tailor-made for applications. These are the famous (infinite dimensional) mountain pass lemma and its variants, due to Ambrosetti and Rabinowitz [6]. The simplest form of these results reads as follows.

Definition 1.5.1. Let X be a real Banach space and I ∈ C 1 (X, R). We say that a sequence {u n } ⊂ X is a Palais-Smale (PS) c sequence of I if

I(u n ) -→ c in R and I(u n ) -→ 0 in (X) * .
Theorem 1.5.4. (see : [6]) Let X be a real Banach space and I ∈ C 1 (X, R) satisfies the (PS) c with I(0) = 0. Suppose that the following conditions hold :

(G 1 ) There exists ρ > 0 and r > 0 such that I(u) r for ||u|| = ρ.

(G 2 ) There exists e ∈ X with ||e|| > ρ such that I(e) 0.

Chapitre 1. Preliminaries Let Γ = {γ ∈ C([0, 1], X); γ(0) = 0, γ(1) = e} . Then c := inf γ∈Γ max t∈[0,1] I(γ(t)) > r
is a critical value of I.

Theorem 1.5.5. (see : [START_REF] Struwe | Variational Methods : Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems[END_REF]) Let X be a real Banach space and I ∈ C 1 (X, R) with I(0) = 0. Suppose that the following conditions hold :

(G 1 ) There exists ρ > 0 and r > 0 such that I(u) r for ||u|| = ρ.

(G 2 )
There exists e ∈ X with ||e|| > ρ such that I(e) 0.

Let c := inf γ∈Γ max t∈[0,1] I(γ(t)) with Γ = {γ ∈ C([0, 1], X); γ(0) = 0, γ(1) = e} .
Then there exists a sequence {u n } in X such that

I(u n ) → c and I (u n ) → 0.
The preceding theorems allow us to obtain the existence of at least one solution. Fountain theorems and their dual form were established by Bartsch in [START_REF] Bartsch | Infinitely many solutions of a symmetric Dirichlet problem[END_REF] and by Bartsch-Willem in [START_REF] Bartsch | On an elliptic equation with concave and convex nonlinearities[END_REF] (see also [START_REF] Willem | Minimax Theorems[END_REF]) respectively. They are effective tools in studying the existence of infinitely many solutions. It should be noted that a decomposition of the Banach space plays an important role in proving these theorems. The decomposition allows one to apply Borsuk-Ulam theorem to establish a proper intersection lemma. Let X be a reflexive and separable Banach space and X * its dual space, then from [START_REF] Zhao | Structure theory of Banach spaces[END_REF] there are {φ n } n∈N ⊂ X and {φ * n } n∈N ⊂ X * such that

X = span {φ n , n ∈ N} and X * = span {φ * n , n ∈ N} and φ n , φ m =            1 n = m 0 n = m. For k = 1, 2, ..., let Y k = span {φ 1 , ..., φ k } and Z k = span {φ k , φ k+1 ...}.
Theorem 1.5.6. [START_REF] Bartsch | Infinitely many solutions of a symmetric Dirichlet problem[END_REF] Assume that the even functional I ∈ C 1 (X, R) satisfies the (PS) c condition and for almost every k ∈ N, there exists ρ k , r k >0 such that

1.5. Calculus of Variations (a) b k := inf u∈Z k ,||u||=r k I(u) -→ ∞ as k → ∞, (b) a k := max u∈Y k ,||u||=ρ k I(u) 0.
Then I has a sequence of critical points {u n } such that I(u k ) -→ ∞ as k → ∞.

The last theorem that we will recall in this section is the three critical points theorem, given by Recciri. This theorem is a way to obtain the existence of at least three solutions. His proof was based on the variational principle of the classical mountain pass theorem. At firstly, we give the following definition.

Definition 1.5.2. Let X be a real Banach space, we denote by W X the class of functional A : X → R possessing the following property : if {u n } is a sequence in X weakly converging to u ∈ X and lim inf n→∞ A(u n ) A(u), then {u n } has a subsequence strongly converging to u.

Theorem 1.5.7. (see. [START_REF] Ricceri | A further three critical points theorem[END_REF]) Let X be a separable and reflexive real Banach space with norm ||.||, let Ψ : X -→ R be a coercive, sequentially weakly lower semicontinuous C 1 functional, belonging to W X , bounded on each bounded subset of X and whose derivative admits a continuous inverse on X * , and let J : X -→ R be a C 1 functional with compact derivative. Assume that Ψ has a strict local minimum x 0 , with Ψ(x 0 ) = J(x 0 ) = 0. Finally, assume that

max lim sup ||x||→+∞ J(x) Ψ(x) , lim sup x→x 0 J(x) Ψ(x) 0
and that

sup x∈X min {Ψ(x), J(x)} > 0. Let θ * := inf Ψ(x) J(x) : x ∈ X, min {Ψ(x), J(x)} > 0 .
Then, for each compact interval Λ ⊂ (θ * , +∞), there exists a number δ > 0 with the following property : for every λ ∈ Λ and every C 1 functional Γ : X -→ R with compact derivative, there exists β * > 0 such that for each β ∈ [0, β * ], the equation

Ψ (x) = λ J (x) + β Γ (x)
has at least three solutions whose norms are less than δ .

Fractional Orlicz-Sobolev spaces

Introduction

The study of nonlinear elliptic equations involving quasilinear homogeneous type operators is based on the theory of Sobolev spaces and fractional Sobolov spaces W s,p (Ω) 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-Sobolev spaces. These spaces consists of functions that have weak derivatives and satisfy certain integrability conditions. Many properties of Orlicz-Sobolev spaces were proved in [1,[START_REF] Donaldson | Orlicz-Sobolev spaces and embedding theorems[END_REF][START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF][START_REF] Rao | Theory of Orlicz Spaces[END_REF]. 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,[START_REF] Azroul | Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces[END_REF][START_REF] Bonanno | Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz-Sobolev spaces[END_REF][START_REF] Bonanno | Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces[END_REF][START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF]). they have been extensively studied in the monograph of Krasnoselśkii and Rutickii [START_REF] Krasnosel | Convex functions and Orlicz spaces[END_REF] as well as in Luxemburg's doctoral thesis [START_REF] Luxemburg | Banach function spaces[END_REF].

If the role played by L p (Ω) in the definition of fractional Sobolev spaces W s,p (Ω) is replaced by an Orlicz L A (Ω) space, the resulting space W s L A (Ω) is exactly a fractional Orlicz-Sobolev space, that is,

W s L A (Ω) = u ∈ L A (Ω) : Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞ for some λ > 0 .
Observe that in the case A(t) = t p , these spaces coincide with the fractional Sobolev space W s,p (Ω).

In this chapter, we are concerned with some qualitative properties of the fractional Orlicz-Sobolev spaces 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 A (Ω) may be extended to a function in

W s L A (R N ), with Ω ⊂ R N is a bounded domain of class C 0,1 .

Fractional Orlicz-Sobolev spaces

First, we define the fractional Orlicz-Sobolev spaces, and we will present some important results of them. Definition 2.2.1. Let A be an N-function. For a given domain Ω in R N and 0 < s < 1, we define the fractional Orlicz-Sobolev space W s L A (Ω) as follows :

W s L A (Ω) = u ∈ L A (Ω) : Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞ for some λ > 0 .
This space is equipped with the norm,

||u|| s,A = ||u|| A + [u] s,A , (2.1)
where [.] s,A is the Gagliardo seminorm, defined by

[u] s,A = inf λ > 0 : Ω Ω A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N 1 .
(2.2) Definition 2.2.2. Let A be an N-function. For a given domain Ω in R N and 0 < s < 1, we define, the space W s E A (Ω) as follows :

W s E A (Ω) = u ∈ E A (Ω) : Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞ . (2.3) Remark 3. -W s E A (Ω) ⊂ W s L A (Ω). -W s E A (Ω) coincides with W s L A (Ω) if and only if (A, Ω) is ∆-regular. -If A(t) = t p , where 1 < p < ∞, then W s L A (Ω) = W s E A (Ω) = W s,p (Ω).
To simplify the notation, we put

D s u := u(x) -u(y) |x -y| s . Theorem 2.2.1.
Let Ω be an open subset of R N , and let s ∈ (0, 1).

dµ := |x -y| -N dxdy.
It therefore also converges to an element of L A (Ω × Ω, dµ). Let us extract a subsequence u σ (n) of {u n } that converges almost everywhere to u. We note that D s u σ (n) converges, for almost every pair (x, y) to D s u. Applying Fatou's lemma, we obtain, for some λ (note that λ exists since

u σ (n) ⊂ W s L A (Ω)), Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N lim inf n→∞ Ω Ω A λ |u ϕ(n) (x) -u ϕ(n) (y)| |x -y| s dxdy |x -y| N < ∞. Hence u ∈ W s L A (Ω).
On the other hand, since D s u n converges in L A (Ω × Ω, dµ), then by dominated convergence theo-

[u n -u] s,A -→ 0. Finally u n → u in W s L A (Ω).
To establish the reflexivity and separation of the fractional Orlicz-Sobolev spaces, we define the operator T :

W s L A (Ω) → L A (Ω) × L A (Ω × Ω, dµ) by T (u) = u(x), |u(x) -u(y)| |x -y| s .
Clearly, T is an isometry. Since L A (Ω) is a reflexive, separable space and uniformly convex (see [1,[START_REF] Mihäilescu | Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces[END_REF]), then W s L A (Ω) is also a reflexive, separable space and uniformly convex.

Let W s 0 L A (Ω) denote the closure of C ∞ 0 (Ω) in the norm ||.|| s,A defined in (2.1). Then we have the following result.

Theorem 2.2.2. (Generalized Poincaré inequality). Let Ω be a bounded open subset of R N , and let s ∈ (0, 1). Let A be an N-function. Then there exists a positive constant µ such that,

||u|| A µ[u] s,A for all u ∈ W s 0 L A (Ω).
Therefore, if Ω is bounded and A be an

N-function, then [.] s,A is a norm of W s 0 L A (Ω) equivalent to ||.|| s,A . Proof. Since W s 0 L A (Ω) is the closure of C ∞ 0 (Ω) in W s L A (Ω)
, then it is enough to prove that there exists a positive constant µ such that,

||u|| A µ[u] s,A for all u ∈ C ∞ 0 (Ω).
Indeed, let u ∈ C ∞ 0 (Ω) and B R ⊂ R N \ Ω, that is, the ball of radius R in the complement of Ω. Then for all x ∈ Ω, y ∈ B R and all λ > 0 we have,

A |u(x)| λ = A |u(x) -u(y)| λ |x -y| s |x -y| s |x -y| N |x -y| N , this implies that, A |u(x)| λ A |u(x) -u(y)| λ |x -y| s diam(Ω ∪ B R ) s diam(Ω ∪ B R ) N |x -y| N , we suppose α = diam(Ω ∪ B R ) s , we get A |u(x)| αλ A |u(x) -u(y)| λ |x -y| s diam(Ω ∪ B R ) N |x -y| N , therefore |B R |A |u(x)| αλ diam(Ω ∪ B R ) N B R A |u(x) -u(y)| λ |x -y| s dy |x -y| N , then Ω A |u(x)| αλ dx diam(Ω ∪ B R ) N |B R | Ω B R A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N , so, ||u|| A µ[u] s,A for all u ∈ C ∞ 0 (Ω), where µ = diam(Ω ∪ B R ) N α |B R |
. By passing to the limit, the desired result is obtained.

Corollary 2.2.1. Let Ω be a bounded open subset of R N , and let s ∈ (0, 1). Let A be an N-function, we define the space W s 0 L A (Ω) as follows :

W s 0 L A (Ω) = u ∈ W s L A (R N ) : u = 0 a.e in R N \ Ω .
Then there exists a positive constant µ such that,

||u|| A µ[u] s,A for all u ∈ W s 0 L A (Ω).
Proof of this corollary is similar to proof of Theorem 2.2.2.

Theorem 2.2.3. Let Ω be a bounded open subset of R N , let 0 < s < 1 and let A be an N-function.

Chapitre 2. Fractional Orlicz-Sobolev spaces Then,

C 2 0 (Ω) ⊂ W s 0 L A (Ω).
Lemma 2.2.1. Let A be an N-function. Then

δ (x) |x| s ∈ L A (R N , |x| -N dx) with δ (x) = min {1, |x|} .
Proof of Lemma 2.2.1. We put

Ω 1 = x ∈ R N : |x| > 1 and Ω 2 = x ∈ R N : |x| 1 . Then, we have R N A δ (x) |x| s dx |x| N = Ω 1 A δ (x) |x| s dx |x| N + Ω 2 A δ (x) |x| s dx |x| N = Ω 1 A 1 |x| s dx |x| N + Ω 2 A |x| |x| s dx |x| N A (1)
Ω 1 dx |x| N+s + A (1) Ω 2 dx |x| N+s-1 ,
note that the last integrals are finite since N + s > N and N + s -1 < N respectively. Therefore

R N A δ (x) |x| s dx |x| N < ∞.
The proof of Lemma 2.2.1 is completed Proof of Theorem 2.2.3. Let u ∈ C 2 0 (Ω), we only need that to check that

R N R N A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞ for some λ > 0.
Indeed, since u vanishes outside Ω, we have

R N R N A |u(x) -u(y)| |x -y| s dxdy |x -y| N = Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N + 2 Ω R N \Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N 2 Ω R N A |u(x) -u(y)| |x -y| s dxdy |x -y| N . Now we notice that |u(x) -u(y)| ||∇u|| L ∞ (R N ) |x -y| and |u(x) -u(y)| 2||u|| L ∞ (R N ) .

Fractional Orlicz-Sobolev spaces

21

Accordingly, we get

|u(x) -u(y)| 2||u|| C 1 (R N ) min {1, |x -y|} := αδ (x -y), with α = 2||u|| C 1 (R N ) and since δ (x) |x| s ∈ L A (R N , |x| -N dx).
There exists λ > 0, such that,

R N R N A λ |u(x) -u(y)| α|x -y| s dxdy |x -y| N 2 Ω R N A λ δ (x -y) |x -y| s dxdy |x -y| N = 2|Ω| R N A λ δ (ξ ) |ξ | s dξ |ξ | N < ∞, this implies that u ∈ W s 0 L A (Ω). Remark 4. A trivial consequence of Theorem 2.2.3, W s L A (Ω) is non-empty. Lemma 2.2.2. Assume condition (1.
2) is satisfied. Then the following relations holds true A(σt) σ p + A(t) for all t > 0 and σ > 1.

(2.4)

A(σt) σ p -A(t) for all t > 0 and σ > 1.

(2.5)

A(τt) τ p + A(t) for all t > 0 and τ ∈ (0, 1).

(2.6)

A(t) τ p - A t
τ for all t > 0 and τ ∈ (0, 1).

(2.7)

A(t) τ p + A t
τ for all t > 0 and τ ∈ (0, 1).

(2.8)

Proof. Since p + ta(t) A(t) for all t > 0 it follows that for letting σ > 1 we have

log(A(σt)) -log(A(t)) = σt t a(τ) A(τ) dτ σt t p + τ dτ = log(σ p + ).

Thus, we deduce

A(σt) σ p + A(t) for all t > 0 and σ > 1.

Next, since p -ta(t) A(t) for all t > 0 it follows that for letting σ > 1 we have

log(A(σt)) -log(A(t)) = σt t a(τ) A(τ) dτ σt t p - τ dτ = log(σ p -).
[u] p - s,A Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p + s,A ∀u ∈ W s L A (Ω) with [u] s,A > 1,
(2.9)

[u] p + s,A Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p - s,A ∀u ∈ W s L A (Ω) with [u] s,A < 1.
(2.10)

Proof. First we show that

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p + s,A for all u ∈ W s L A (Ω) with [u] s,A > 1. Indeed, let now u ∈ W s L A (Ω) with [u] s,A > 1.
Using the definition of the Luxemburg norm and the relation (2.4), we deduce

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N = Ω Ω A [u] s,A |u(x) -u(y)| [u] s,A |x -y| s dxdy |x -y| N [u] p + s,A Ω Ω A |u(x) -u(y)| [u] s,A |x -y| s dxdy |x -y| N [u] p + s,A .

Now, we show that

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p - s,A for all u ∈ W s L A (Ω) with [u] s,A > 1. Let u ∈ W s L A (Ω) with [u] s,A > 1, we consider β ∈ (1, [u] s,A ), since β < [u] s,A , so by definition of Luxemburg norm, it follows that Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N > 1,
the above consideration and (2.5) implies that

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N = Ω Ω A β |u(x) -u(y)| β |x -y| s dxdy |x -y| N β p - Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N β p -, letting β [u] s,A
, we deduce that relation (2.9) hold true.

Next, we show that

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p - s,A for all u ∈ W s L A (Ω) with [u] s,A < 1. Let u ∈ W s L A (Ω) with [u] s,A < 1.
Using the definition of the Luxemburg norm and the relation (2.7), we deduce

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p - s,A Ω Ω A |u(x) -u(y)| [u] s,A |x -y| s dxdy |x -y| N [u] p - s,A .
Finally, we show that

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N [u] p + s,A for all u ∈ W s L A (Ω) with [u] s,A < 1. Let u ∈ W s L A (Ω) with [u] s,A < 1 and β ∈ (0, [u] s,A
), so by (2.8) we find

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N β p + Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N . (2.11) We define v(x) = u(x) β for all x ∈ Ω, we have [v] s,A = [u] s,A β > 1.
Using the relation (2.9) we find

Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N = Ω Ω A |v(x) -v(y)| |x -y| s dxdy |x -y| N > [v] p - s,A > 1, (2.12
) by (2.11) and (2.12) we obtain

Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N β p + .

Letting β

[u] s,A , we deduce that relation (2.10) hold true.

Chapitre 2. Fractional Orlicz-Sobolev spaces

We put

Ψ(u) := Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω A (|u(x)|) dx. Proposition 2.2.2. On W s L A (Ω) the following norm ||u|| s,A = ||u|| A + [u] s,A , ||u|| max = max {||u|| A , [u] s,A } , ||u|| = inf λ > 0 : Ψ |u| λ 1 , are equivalent.
Proof. Next, we remark that

Ω Ω A |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + Ω A |u(x)| ||u|| dx 1.
Using the above relation, we obtain

Ω Ω A |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N 1 and Ω A |u(x)| ||u|| dx 1.

So, [u]

It following that :

Ω Ω A |u(x) -u(y)| 2||u|| max |x -y| s dxdy |x -y| N + Ω A |u(x)| 2||u|| max dx 1 2 Ω Ω A |u(x) -u(y)| ||u|| max |x -y| s dxdy |x -y| N + Ω A |u(x)| ||u|| max dx 1 2 Ω Ω A |u(x) -u(y)| [u] s,A |x -y| s dxdy |x -y| N + Ω A |u(x)| ||u|| A dx 1.
Then, by the above relation and (2.

Ψ (u) ||u|| p -∀u ∈ W s L A (Ω) , ||u|| > 1, (2.16
)

Ψ (u) ||u|| p + ∀u ∈ W s L A (Ω) , ||u|| < 1.
(2.17)

Proof. First, assume that ||u|| > 1. Let β ∈ (1, ||u||), by relation (2.5), we have

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω A (|u(x)|) dx β p - Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N + Ω A |u(x)| β dx .
Since β < ||u||, we find

Ω Ω A |u(x) -u(y)| β |x -y| s dxdy |x -y| N + Ω A |u(x)| β dx > 1.
Thus, we find

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω A (|u(x)|) dx β p -.

Letting β

||u||, we deduce that (2.16) holds true.

Chapitre 2. Fractional Orlicz-Sobolev spaces

Next, assume that ||u|| < 1, let ξ ∈ (0, ||u||). By relation (2.6), we obtain

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω A (|u(x)|) dx ξ p + Ω Ω A |u(x) -u(y)| ξ |x -y| s dxdy |x -y| N + Ω A |u(x)| ξ dx .
(2.18)

Defining v(x) = u(x) ξ for all x ∈ Ω, we define ||v|| = ||u|| ξ > 1.
Using relation (2.16), we find

Ω Ω A |v(x) -v(y)| |x -y| s dxdy |x -y| N + Ω A (|v(x)|) dx ||v|| p -> 1.
(

Ω Ω A |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω A (|u(x)|) dx ξ p -.

Letting ξ

||u|| in the above inequality we obtain that relation (2.17) holds true.

Some embeddings results

The embeddings results obtained in the fractional Sobolev space W s,p (Ω) can also be formulated for the fractional Orlicz-Sobolev spaces. For this results, we follow the approach of Donaldson and Trudinger in [START_REF] Donaldson | Orlicz-Sobolev spaces and embedding theorems[END_REF].

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 A (Ω) is continuously embedded in W s 2 L A (Ω)
is satisfied, then W s 1 L A (Ω) continuously embedded in W s 2 L A (Ω).
Proof. First for all λ > 0, we have

Ω Ω∩{|x-y| 1} A |u(x)| λ |x -y| s 2 dxdy |x -y| N = Ω Ω∩{|z| 1} A |u(x)| λ |z| s 2 dxdz |z| N Ω Ω∩{|z| 1} A |u(x)| λ |z| s 2 dxdz |z| N Ω∩{|z| 1} 1 |z| s 2 +N dz Ω A |u(x)| λ dx = c (N, s 2 , Ω) Ω A |u(x)| λ dx, (2.20)
and we used the fact that the kernel 1 |z| s 2 +N is integrable since s 2 + N > N. Taking into account the above estimate, it follows

Ω Ω∩{|x-y| 1} A |u(x) -u(y)| λ |x -y| s 2 dxdy |x -y| N Ω Ω∩{|x-y| 1} A 2|u(x)| + 2|u(y)| 2λ |x -y| s 2 dxdy |x -y| N 2 p + 2 Ω Ω∩{|x-y| 1} A |u(x)| λ |x -y| s 2 dxdy |x -y| N + Ω Ω∩{|x-y| 1} A |u(y)| λ |x -y| s 2 dxdy |x -y| N 2 p + c (N, s 2 , Ω) Ω A |u(x)| λ dx.
(2.21)

On the other hand, for all λ > 0 we have

Ω Ω∩{|x-y| 1} A |u(x) -u(y)| λ |x -y| s 2 dxdy |x -y| N Ω Ω∩{|x-y| 1} A |u(x) -u(y)| λ |x -y| s 1 dxdy |x -y| N . (2.22)
Thus, By combining (2.21) and (2.22), we get for all λ > 0,

Ω Ω A |u(x) -u(y)| λ |x -y| s 2 dxdy |x -y| N + Ω A |u(x)| λ dx (2 + 2 p + c (N, s 2 , Ω)) Ω A |u(x)| λ dx + Ω Ω A |u(x) -u(y)| λ |x -y| s 1 dxdy |x -y| N . Therefore, ||u|| s 2 ,A c||u|| s 1 ,A ,
where c = (2 + 2 p + c (N, s 2 , Ω)). Let 0 < s < 1 and let A be a given N-function, satisfying the following conditions :

1 0 A -1 (τ) τ N+s N dτ < ∞, (2.23) ∞ 1 A -1 (τ) τ N+s N dτ = ∞. ( 2
(p -) * s =            N p - N-sp - if N > sp - ∞ if N sp -.
Indeed ) is an N-function. Let A * be defined by (2.25). Then for all s ∈ (0, s), the following conclusions may be drawn.

[A * (t)]

N-s N is an N-function, in particular, A * is an N-function.

2. For every ε > 0, there exists a constant K ε > 0 such that for every t,

[A * (t)] N-s N 1 2ε A * (t) + K ε ε t. (2.31) Chapitre 2. Fractional Orlicz-Sobolev spaces Proof. (1) Let Q(t) = [A * (t)] N-s N , Noting that B -1 (t) = [A -1 (t)] p , we get (Q -1 ) (t) = d dt A -1 * (t N N-s ) = N N -s t N N-s -1 A -1 (t N N-s ) [t N N-s ] N+s N = N N -s A -1 (t N N-s ) t N+s-s N-s = N N -s B -1 (t N N-s ) t N N-s 1 p t -µ , where µ = N + s -s N -s - N N -s 1 p = N(p -1) + (s -s )p (N -s )p 0.
Being the inverse of an N-function,

B -1 satisfies lim t→0 + B -1 (t) t = ∞ and lim t→∞ B -1 (t) t = 0, Moreover, B -1 is concave, so for 0 < r < σ we have, B -1 (r) B -1 (σ ) > r σ . Hence, if 0 < t 1 < t 2 , then we get, (Q -1 ) (t 1 ) (Q -1 ) (t 2 ) ( t 2 t 1 ) -µ > 1.
It follows that (Q -1 ) is positive and decreases monotonically from ∞ to 0 as t increases from 0 to ∞, so that Q is an N-function.

(

K ε = ε sup 0 t t 0 h(t), then [A * (t)] N-s N 1 2ε A * (t) + K ε ε t.
The proof of this Lemma is completed.

Lemma 2.3.2.
Let Ω be an open subset of R N , and 0 < s < 1. Let f satisfies a Lipschitz-condition on R and f (0) = 0, then, 1. For every u ∈ W s,1 loc (Ω), g ∈ W s,1 loc (Ω) where g(x) = f (|u(x)|).

For every u

∈ W s L A (Ω), g ∈ W s L A (Ω) where g(x) = f (|u(x)|).
In particular, for every u ∈ W s,1 (Ω), g ∈ W s,1 (Ω) where g(x) = f (|u(x)|).

Proof. 1) Let K be a compact subset of Ω, follows that 1 K g ∈ W s,1 (Ω). Since f (0) = 0, then we have,

Ω |1 K(x) g(x)|dx = Ω |1 K(x) ( f (u(x)) -f (0))|dx c Ω |1 K(x) u(x)|dx < ∞,
where c is the Lipschitz constant of f . On the other hand,

Ω Ω |1 K(x) g(x) -1 K(y) g(y)| |x -y| N+s dxdy = K K |g(x) -g(y)| |x -y| N+s dxdy + 2 Ω\K K |1 K(x) g(x)| |x -y| N+s dxdy + Ω\K Ω\K |1 K(x) g(x) -1 K(y) g(y)| |x -y| N+s dxdy, (2.32)
where the third term in the right hand-side of (2.32) is null, and since f satisfies the Lipschitzcondition, then,

Ω Ω |1 K(x) g(x) -1 K(y) g(y)| |x -y| N+s dxdy c K K |u(x) -u(y)| |x -y| N+s dxdy + 2 Ω\K K |g(x)| |x -y| N+s dxdy, (2.33)
where the first term in the right hand-side of (2.33) is finite since u ∈ W s,1 loc (Ω) and 2

K Ω\K |g(x)| |x -y| N+s dxdy 2 K |g(x)|dx Ω\K 1 d(y, ∂ K) N+s dy < ∞.
Note that due to the fact that K is a compact subset, then dis(y, ∂ K) N+s > 0 for all y ∈ R N \ K and we have N + s > N. Therefore,

Ω Ω |1 K(x) g(x) -1 K(y) g(y)| |x -y| N+s dxdy < ∞.
2) Let u ∈ W s L A (Ω) then there exists λ > 0 such that,

Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞.
Let c > 0 denotes the Lipschitz constant of f . If |c| 1, then we have,

Ω Ω A λ |g(x) -g(y)| |x -y| s dxdy |x -y| N = Ω Ω A λ | f • u(x) -f • u(y)| |x -y| s dxdy |x -y| N Ω Ω A |c|λ |u(x) -u(y)| |x -y| s dxdy |x -y| N Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞. Chapitre 2. Fractional Orlicz-Sobolev spaces If |c| > 1 for λ 1 = λ |c| we get, Ω Ω A λ 1 |g(x) -g(y)| |x -y| s dxdy |x -y| N = Ω Ω A λ 1 | f • u(x) -f • u(y)| |x -y| s dxdy |x -y| N Ω Ω A |c|λ |u(x) -u(y)| |c||x -y| s dxdy |x -y| N Ω Ω A λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞, this implies that g ∈ W s L A (Ω).
< s < s < 1. Assume condition (1.2) is satisfied, then the space W s L A (Ω) is continuously embedded in W s ,q (Ω) for all q ∈ [1, p -].
Remark 6. If s = s , the conclusion of Lemma 2.3.3 will be incorrect. Indeed, if we take

A(t) = t p p ,
then, we have p -= p and the embedding W s,p (Ω) into W s,q (Ω) for all q ∈ [1, p) is not satisfied (see [START_REF] Mironescu | A Sobolev non embedding[END_REF]).

Proof of Theorem 2.3.

1. Let 0 < s < s < 1, σ (t) = [A * (t)] N-s N
and u ∈ W s L A (Ω), we suppose for the moment that u is bounded on Ω and not equal to zero in L A (Ω), then

Ω A * |u(x)| λ dx
decreases continuously from infinity to zero as λ increases from zero to infinity and according, assumes the value unity for some positive value k of λ , thus

Ω A * |u(x)| k dx = 1 , k = ||u|| A * . (2.34) Let f (x) = σ u(x) k . By Lemma 2.3.3 u ∈ W s ,1 (Ω),
W s ,1 (Ω) → L N N-s (Ω). So || f || L N N-s k 1 || f || L 1 + [ f ] s ,1 ,

Some embeddings results

and by (2.34),

1 = Ω A * |u(x)| k dx N-s N = || f || L N N-s , this implies that, 1 k 1 || f || L 1 + [ f ] s ,1 = k 1 Ω σ u(x) k dx + Ω Ω | f (x) -f (y)| |x -y| N+s dxdy = k 1    Ω σ u(x) k dx + Ω Ω |σ ( u(x) k ) -σ ( u(y) k )| |x -y| N+s dxdy    = k 1 I 1 + k 1 I 2 .
(2.35) By (2.31) we have for ε = k 1 ,

k 1 I 1 1 2 Ω A * |u(x)| k dx + k ε k Ω |u(x)|dx 1 2 + k ε k ||u|| A , (2.36)
where k ε = 2k ε ||1|| A since Ω has a finite volume.

On the other hand, since σ is Lipschitz, then there exists c 1 > 0 such that,

k 1 I 2 c 1 k Ω Ω |u(x) -u(y)| |x -y| N+s dxdy.
But by Lemma 2.3.3, we have

Ω Ω |u(x) -u(y)| |x -y| N+s dxdy c 2 [u] s,A , (2.37)
and

k 1 I 2 c 1 k c 2 [u] s,A . (2.38)
We pose

k 3 = c 1 k 1 c 2 . Combining (2.36)-(2.38) we obtain 1 1 2 + k ε k ||u|| A + k 3 k [u] s,A , this implies that, k 2 k ε ||u|| A + k 3 [u] s,A .
Chapitre 2. Fractional Orlicz-Sobolev spaces

So we obtain,

||u|| A * k 4 ||u|| s,A ,
where k 4 = max {2k ε , 2k 3 }.

If u ∈ W s L A (Ω) arbitrary, we define

u n (x) =      u(x) if |u(x)| n, n sgn u(x) if |u(x)| > n,
{u
=            N p - N-s p - if N > s p - ∞ if N s p -.
• If s p -< N, then W s L A (Ω) → L q (Ω), for all q ∈ [1, p * s ] and the embedding W s L A (Ω) → L q (Ω) is compact for all q ∈ [1, p * s ).

• If s p -= N, then W s L A (Ω) → L q (Ω), for all q ∈ [1, ∞] and the embedding W s L A (Ω) → L q (Ω) is compact for all q ∈ [1, ∞).

• If s p -> N, then the embedding W s L A (Ω) → L ∞ (Ω) is compact.

Extending a W s L A (Ω)

As well known, when s ∈ (0, 1), p ∈ [1, ∞) and Ω is an open subset of R N , with C 0,1 -regularity, any function in fractional Sobolev space W s,p (Ω) may be extended to a function in W s,p (R N ) (see [START_REF] Biler | Critical nonlinearity exponent and self-similar asymptotics for LÂt'evy conservation laws[END_REF]). Extension results are quite important in applications and for improve certain embeddings theorems, in the classic case as well as in the fractional case. The problem of extensibility was studied in the classical Sobolev spaces, in this direction we mention, in particular, the works of Sobolev [START_REF] Sobolev | Nekotorye Primeneniya Funkcional nogo Analiza v MatematiOcesko. Fizike, Izdat[END_REF], [START_REF] Sobolev | Applications of Functional Analysis in Mathematical Physics[END_REF], Deny and Lions [START_REF] Deny | Les espaces du type de Beppo Levi[END_REF] and Gagliardo [START_REF] Gagliardo | Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili[END_REF] ; see also [8,[START_REF] Biler | Critical nonlinearity exponent and self-similar asymptotics for LÂt'evy conservation laws[END_REF][START_REF] Calderón | Lebesgue spaces of differentiable functions and distributions[END_REF][START_REF] Diening | Lebesgue and Sobolev Spaces with Variable Exponents[END_REF][START_REF] Lizorkin | L p r (Ω) spaces, continuation and imbedding theorems[END_REF][START_REF] Peter | Quasiconformal mappings and extendability of functions in Sobolev spaces[END_REF][START_REF] Rao | Theory of Orlicz Spaces[END_REF][START_REF] Slobodeckii | Sobolev's spaces of fractional order and their application to boundary pr oblems for partial differential equations[END_REF] and the references therein. All these previous results are held under certain crucial regularity assumptions on the domain Ω. we cite [START_REF] Peter | Quasiconformal mappings and extendability of functions in Sobolev spaces[END_REF] for a complete characterization in the special case when s = 1, A(t) = t 2 and N = 2, and we refer the interested reader to the recent book by Leoni [START_REF] Leoni | A first course in Sobolev spaces[END_REF], in which this problem is very well discussed (see, in particular, Chapter 11 and Chapter 12).

As usual, for any k ∈ N and α ∈ (0, 1], we say that Ω is of class C k,α if there exists M > 0 such that for any x ∈ ∂ Ω there exists a ball B = B r (x), r > 0, and an isomorphism T : Q → B such that

T ∈ C k,α (Q), T -1 ∈ C k,α (B), T (Q + ) = B ∩ Ω, T (Q 0 ) = B ∩ ∂ Ω (2.39)
and

||T || C k,α (Q) + ||T -1 || C k,α (B) M,
where

Q := x = (x , x N ) ∈ R N-1 × R : |x | < 1 and |x N | < 1 , Q + := x = (x , x N ) ∈ R N-1 × R : |x | < 1 and 0 < x N < 1 ,
and

Q 0 := {x ∈ Q : x N = 0} .
In this section, we will show that any open set Ω of class C 0,1 with bounded boundary is an extension domain for W s L A .

We start with some preliminary lemmas, in which we will construct the extension to the whole of R N of a function u defined on Ω in two separated cases : when the function u is identically zero in a neighborhood of the boundary ∂ Ω and when Ω coincides with the half-space R N + .

Lemma 2.4.1. Let Ω be an open subset of R N , and u a function in W s L A (Ω). Assume condition

(1.
2) is satisfied, if there exists a compact subset K ⊂ Ω such that u ≡ 0 in Ω \ K, then the extension function u defined as

u(x) =            u(x) i f x ∈ Ω, 0 i f x ∈ R N \ Ω, belongs to W s L A (R N ) and || u|| W s L A (R N ) c||u|| W s L A (Ω) ,
where c is a positive constant depending on N, s, K and Ω.

Proof. Clearly u ∈ L A (R N ). On the other hand for all λ > 0, we have

R N R N A | u(x) -u(y)| λ |x -y| s dxdy |x -y| N = Ω Ω A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N + 2 R N \Ω Ω A |u(x)| λ |x -y| s dxdy |x -y| N ,
where the first term in the right hand-side is finite, and for any y ∈ R N \ K we have

R N \Ω Ω A |u(x)| λ |x -y| s dxdy |x -y| N R N \Ω Ω A |u(x)| λ dxdy |x -y| sp+N R N \Ω Ω A 1 K |u(x)| λ dxdy dis(y, ∂ K) sp+N Ω A |u(x)| λ dx R N \Ω 1 dis(y, ∂ K) sp+N dy, 2.4. Extending a W s L A (Ω)

37

where p = 1 or p + . Then

R N R N A | u(x) -u(y)| λ |x -y| s dxdy |x -y| N c Ω A |u(x)| λ dx + Ω Ω A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N where c = 1 + R N \K 1 dis(y, ∂ K) sp+N dy, (2.40)
note that the integral in (2.40) is finite since dis(∂ Ω, ∂ K) > 0 and N + sp > N. Therefore,

|| u|| W s L A (R N ) c||u|| W s L A (Ω) .
The proof of this Lemma is completed.

Lemma 2.4.2. Let Ω be an open subset of R N , symmetric with respect to the coordinate x N , and consider the sets

Ω + = {x ∈ Ω : x N > 0} and Ω -= {x ∈ Ω : x N 0}. Let u be a function in W s L A (Ω + ), we define u(x) =            u(x , x N ) i f x n 0, u(x , -x N ) i f x n < 0.
Then u belongs to W s L A (Ω) and

|| u|| W s L A (Ω) 4||u|| W s L A (Ω + ) .
Proof. By splitting the integrals and changing variable x = (x , -x N ), we get for all λ > 0,

Ω A | u(x)| λ dx = Ω + A |u(x , x N )| λ dx + Ω - A |u(x , -x N )| λ dx = Ω + A |u(x)| λ dx + Ω + A |u( x , x N )| λ dx = 2 Ω + A |u(x)| λ dx, therefore || u|| L A (Ω) 2||u|| L A (Ω + ) . (2.41)
On the other hand, we have for all λ > 0, Chapitre 2. Fractional Orlicz-Sobolev spaces

Ω Ω A | u(x) -u(y)| λ |x -y| s dxdy |x -y| N = R N \Ω + R N \Ω + A |u(x , -x N ) -u(y , -y N )| λ |x -y| s dxdy |x -y| N + 2 Ω + R N \Ω + A |u(x) -u(y , -y N )| λ |x -y| s dxdy |x -y| N + Ω + Ω + A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N ,
by changing variable x = (x , -x N ) and y = (y , -y N ) we get

Ω Ω A | u(x) -u(y)| λ |x -y| s dxdy |x -y| N = 4 Ω + Ω + A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N , therefore [ u] W s L A (Ω) 4[u] W s L A (Ω + ) .
(2.42)

This concludes the proof.

Lemma 2.4.3. Let Ω be an open subset of R N , let u ∈ W s L A (Ω) and ψ ∈ C 0,1 (Ω) such that 0

ψ 1. Assume condition (1.2) is satisfied. Then ψu ∈ W s L A (Ω) and ||ψu|| W s L A (Ω) c||u|| W s L A (Ω) ,
where c = c(N, s, Ω) > 0.

Proof. It is clear that ψu ∈ L A (Ω) and ||ψu|| L A (Ω) ||u|| L A (Ω) since |ψ| 1. On the other hand, adding and subtracting the factor ψ(x)u(y), we get

Ω Ω A |ψ(x)u(x) -ψ(y)u(y)| λ |x -y| s dxdy |x -y| N Ω Ω A 2|ψ(x)u(x) -ψ(x)u(y)| 2λ |x -y| s + 2|ψ(x)u(y) -ψ(y)u(y)| 2λ |x -y| s dxdy |x -y| N 2 p + 2 Ω Ω A |ψ(x)u(x) -ψ(x)u(y)| λ |x -y| s dxdy |x -y| N + Ω Ω A |ψ(x)u(y) -ψ(y)u(y)| λ |x -y| s dxdy |x -y| N 2 p + -1 Ω Ω A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N + Ω Ω A |u(y)(ψ(x) -ψ(y))| λ |x -y| s dxdy |x -y| N .
Since ψ belongs to C 0,1 (Ω), we have

Ω Ω A |u(y)(ψ(x) -ψ(y))| λ |x -y| s dxdy |x -y| N = Ω Ω∩{|x-y| 1} A |u(y)(ψ(x) -ψ(y))| λ |x -y| s dxdy |x -y| N + Ω Ω∩{|x-y| 1} A u(y)(ψ(x) -ψ(y)) λ |x -y| s dxdy |x -y| N Ω Ω∩{|x-y| 1} A |u(y)| λ |x -y| s dxdy |x -y| N + Ω Ω∩{|x-y| 1} A L|u(y)||x -y| λ |x -y| s dxdy |x -y| N c Ω A |u(y)| λ dy,
where L denotes the Lipschitz constant of ψ and

c = Ω∩{|z| 1} 1 |z| s+N dz + Ω∩{|z| 1} L p |z| (s-1)+N dz. with p = 1 if L 1 and p = p + if L > 1. Therefore ψu ∈ W s L A (Ω) and ||ψu|| W s L A (Ω) c||u|| W s L A (Ω) .
The proof of this Lemma is completed. Now, we are ready to prove the main theorem of this section, that states that every open Lipschitz subset Ω with bounded boundary is an extension domain for W s L A (Ω).

Theorem 2.4.1. Let Ω be an open subset of R N , with C 0,1 -regularity and bounded boundary. Assume condition (1.2) is satisfied. Then W s L A (Ω) is continuously embedded in W s L A (R N ), namely for any u ∈ W s L A (Ω) there exists u ∈ W s L A (R N ) such that u| Ω = u and

|| u|| W s L A (R N ) c||u|| W s L A (Ω) ,
where c = c(N, s, p + , Ω) > 0.

Proof of Theorem 2.4.1. Since ∂ Ω is compact, we can find a finite number of balls B j such that ∂ Ω ⊂ ∪ k j=1 B j and so we can write

R N = ∪ k j=1 B j ∪ (R N \ ∂ Ω).
If we consider this covering, there exists a partition of unity related to it, that is, there exist k + 1 smooth functions ψ 0 , ψ 1 , ..., ψ k such that supp(ψ 0 ) ⊂ R N , supp(ψ j ) ⊂ B j for any j ∈ {0, ..., k} and 40

Chapitre 2. Fractional Orlicz-Sobolev spaces k ∑ j=0 ψ j = 1. Clearly, u = k ∑ j=0 ψ j u.
By Lemma 2.4.3, we know that ψ 0 u belongs to W s L A (Ω). Furthermore, since ψ 0 u = 0 in a neighborhood of ∂ Ω, we can extend it to the whole of R N by setting

ψ 0 u(x) =            ψ 0 u i f x ∈ Ω, 0 i f x ∈ R N \ Ω and ψ 0 u ∈ W s L A (Ω). Precisely || ψ 0 u|| W s L A (R N ) c||ψu|| W s L A (Ω) c||u|| W s L A (Ω) (2.43)
where c = c(N, s, Ω) > 0 (possibly different step by step). For any j ∈ {1, ..., k}, let us consider u| B j ∩Ω and set v j (y) := u(T j (y)) for any y ∈ Q + .

where T j : Q → B j is the isomorphism of class C 0,1 (Ω) defined in (2.39). Note that such a T j exists because Ω is an open subset of class C 0,1 . Now, we state that v j ∈ W s L A (Q + ). Indeed, using the standard changing variable formula by setting x = T j ( x) we have

Q + Q + A |v j ( x) -v j ( y)| λ | x -y| s d xd y | x -y| N = Q + Q + A |u(T j ( x)) -u(T j ( y))| λ | x -y| s d xd y | x -y| N = B j ∩Ω B j ∩Ω A |u(x) -u(y)| λ |T -1 j (x) -T -1 j (y)| s det(T -1 ) dxdy |T -1 j (x) -T -1 j (y)| N c p+1 B j ∩Ω B j ∩Ω A |u(x) -u(y)| λ |x -y| s dxdy |x -y| N , (2.44)
where (2.44) follows from the fact that T j is bi-Lipschitz. Hence v j ∈ W s L A (Q + ). Moreover, using Lemma 2.4.2, we cane extend v j to all Q so that the extension v j belongs to W s L A (Q) and

|| v j || W s L A (Q) 4||v j || W s L A (Q + ) .
We set

w j (x) := v j (T -1 j (x)) for any x ∈ B j 2.5. Complemented subspaces in W s L A (Ω)
41 by arguing as above it follows that w j ∈ W s L A (B j ). Note that w j ≡ u (and consequently ψ j w j ≡ ψ j u) on B j ∩ Ω. By definition ψ j w j has compact support in B j and therefore, as done for ψ 0 u, we can consider the extension ψ j w j to all R N in such a way that ψ j w j ∈ W s L A (R N ). Also, using Lemma

|| ψ j w j || W s L A (R N ) c||ψ j w j || W s L A (B j ) c||w j || W s L A (B j ) c|| v j || W s L A (Q) c||v j || W s L A (Q + ) c||u|| W s L A (Ω∩B j ) (2.45)
where c = c(N, s, p, Ω) and it is possibly different step by step.

Finally, let

u = ψ 0 u + k ∑ j=1 ψ j w j
be the extension of u defined on all R N . By construction, it is clear that u| Ω = u and combining (2.43) with (2.45), we get

|| u|| W s L A (R N ) c||u|| W s L A (Ω)
where c = c(N, s, p + , Ω).

Complemented subspaces in W s L A (Ω)

Denoting the trace operator by T F = F| E . The above construction applies, in particular, to the

Sobolev space A = W s L A (R N ). A closed subspace Y of a Banach space X is complemented if there is another closed subspace Z of X such that X = Y ⊕ Z.
That is, Y ∩ Z = 0 and every element x ∈ X can be written as x = y + z, with y ∈ Y and z ∈ Z. Let us consider the trace operator

T : W s L A (R N ) -→ W s L A (Ω),
defined by T u = u| Ω . Our next result relates to the complemented subspace problem. More precisely, we have the following theorem, Chapitre 2. Fractional Orlicz-Sobolev spaces Theorem 2.5.1. Suppose that Ω is of class C 0,1 . Then

W s L A (R N ) = ker T ⊕ ε(W s L A (Ω)),
where ε is a continuous linear extension operator defined by,

ε : W s L A (Ω) -→ W s L A (R N ),
such that εu| Ω = u for each u ∈ W s L A (Ω).

Proof. From the theorem 2.4.1 we know that there exists a linear operator ε :

W s L A (Ω) -→ W s L A (R N ), such that 1. εu| Ω = u, 2. ||εu|| W s L A (R N ) c||u|| W s L A (R N ) . Note that ε(W s L A (Ω)) ⊂ W s L A (R N
) is a closed subspace. On the other hand for every element u in

W s L A (R N ) can be written as u = u -ε(T (u)) + ε(T (u)), since u -ε(T (u)) ∈ ker T , ε(T (u)) ∈ ε(W s L A (Ω)) and ker T ∩ ε(W s L A (Ω)) = 0, we conclude that W s L A (R N ) = ker T ⊕ ε(W s L A (Ω)).
On some nonlocal problems in fractional Sobolev spaces

Introduction

This chapter, devoted to the study of some problems involving nonlocal operators in fractional Sobolev spaces W s,p (Ω), which is the special case of the fractional Orlicz-Sobolev spaces, when we take A(t) = t p . In applied PDE, fractional spaces and the corresponding nonlocal equations, are now experiencing impressive applications in different subjects, such as, among others, the thin obstacle problem [START_REF] Milakis | Regularity for the nonlinear Signorini problem[END_REF], finance [START_REF] Cont | Financial modelling with jump processes[END_REF], phase transitions [START_REF] Cabre | Layer solutions in a half-space for boundary reactions[END_REF], stratified materials [START_REF] Chermisi | A symmetry result for a general class of divergence form PDEs in fibered media[END_REF], crystal dislocation [START_REF] Biler | Nonlinear diffusion of dislocation density and self-similar solutions[END_REF], soft thin films [START_REF] Kurzke | A nonlocal singular perturbation problem with periodic well potential[END_REF], semipermeable membranes and flame propagation [START_REF] Caffarelli | Traveling waves for a boundary reaction-diffusion equation[END_REF], conservation laws [START_REF] Biler | Critical nonlinearity exponent and self-similar asymptotics for LÂt'evy conservation laws[END_REF], ultra-relativistic limits of quantum mechanics [START_REF] Fefferman | Relativistic stability of matter. I[END_REF], quasi-geostrophic flows [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasigeostrophic equation[END_REF], multiple scattering [START_REF] Duistermaat | The spectrum of positive elliptic operators and periodic bicharacteristics[END_REF], minimal surfaces [START_REF] Caffarelli | Uniform estimates and limiting arguments for nonlocal minimal surfaces[END_REF] , materials science [START_REF] Bates | On some nonlocal evolution equations arising in materials science[END_REF], water waves [START_REF] Whitham | Linear and nonlinear waves[END_REF], gradient potential theory [START_REF] Mingione | Gradient potential estimates[END_REF] and singular set of minima of variational functionals [START_REF] Mingione | The singular set of solutions to non-differentiable elliptic systems[END_REF]. See also [START_REF] Silvestre | Regularity of the obstacle problem for a fractional power of the Laplace operator[END_REF] for further motivation.

44

Chapitre 3. On some nonlocal problems in fractional Sobolev spaces Also, 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 [START_REF] Chen | Variable exponent linear growth functionals in image processing[END_REF][START_REF] Diening | Theorical and numerical results for electrorheological fluids[END_REF][START_REF] Halsey | Electrorheological fluids[END_REF]).

The problems in this chapter involving nonlocal intgro-differential operators named by fractional p-laplacian operator (-∆) s p and generalized farctional K-laplacian operator L K p defined as following :

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y))
|x -y| N+sp dy and

L K p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y))K(x -y)dy, along all u ∈ C ∞ 0 (R N ) and K : R N \ {0} → R + is a measurable function.
One typical feature of these operators is the nonlocality, in the sense that the value of (-∆) s p at any point x ∈ Ω depends not only on the values of u on the whole Ω, but actually on the whole R N , since u(x) represents the expected value of a random variable tied to a process randomly jumping arbitrarily far from the point x. While in the classical case, by the continuity properties of the Brownian motion, at the exit time from Ω one necessarily is on ∂ Ω, due to the jumping nature of the process, at the exit time one could end up anywhere outside Ω. In this sense, the natural nonhomogeneous Dirichlet boundary condition consists in assigning the values of u in R N \ Ω rather than mererly on ∂ Ω. Then, it is reasonable to search for solution in the space of functions u ∈ W s,p (R N ) vanishing on the outside of Ω. It should be pointed out that, in a bounded domain, this is not the only possible way of providing a formulation of the problem.

The problems of this chapter are related to the stationary version of the Kirchhoff equation

ρ ∂ 2 u ∂t 2 -   P 0 h + E 2L L 0 ∂ u ∂ x 2 dx   ∂ 2 u ∂ x 2 (3.1)
presented by Kirchhoff [92] 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 (3.1), L is the length of string, h is the area of the cross section, E is the Young modulus of the material, ρ is the mass density, and 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 [START_REF] Chipot | Some remarks on non local elliptic and parabolic problems[END_REF]. On the other hand, Kirchhoff-type boundary value problems model several physical and biological systems where u describes a process which depend on the average of itself, as for example, the population density. We refer the reader to [5,[START_REF] Graef | A variational approach to a Kirchhoff-type problem involving two parameters[END_REF][START_REF] Ricceri | On an elliptic Kirchhoff-type problem depending on two parameters[END_REF] for some related works.

In recent years, many authors have been studied some Kirchhoff type problems. In a very recent paper, Fiscella and Valdinoci in [START_REF] Fiscella | A critical Kirchhoff type problem involving a nonlocal operator[END_REF] proposed the following fractional Kirchhoff type model

               -M R 2N |u(x) -u(y)| 2 K(x -y)dxdy L K u = λ f (x, u) + |u| 2 * -2 u in Ω u = 0 in R N \ Ω, (3.2)
where N > 2s with s ∈ (0, 1),

2 * = 2N/(N -2s), λ is a positive parameter Ω ⊂ R N , is an open boun-
ded set, M and f are two continuous functions and L K is a nonlocal operator defined as follows :

L K u(x) = 1 2 R N (u(x + y) -u(x -y))K(y)dy,
where the kernel K : R N {0} -→ (0, ∞) is a measurable function. This problem considers the nonlocal aspect of the tension of the string. The authors first provided a detailed discussion about the physical meaning underlying the fractional Kirchhoff models and their applications. Consequently, they obtained the existence of non-negative solutions for problem (3.2) using a truncation argument and the mountain pass theorem. In particular, in order to obtain the compactness of the bounded (PS) sequence, they performed the fractional version of the principle of concentration compactness of Palatucci and Pisante [START_REF] Palatucci | Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces[END_REF]. Mishra and Sreenadh in [START_REF] Pk | Existence of solutions for a fractional p-Kirchhoff equations with critical nonlinearities[END_REF] have extended the main result of [START_REF] Fiscella | A critical Kirchhoff type problem involving a nonlocal operator[END_REF] to the setting of fractional p-Laplacian using the same method.

For p = 2, Xiang et al in [START_REF] Xiang | A critical Kirchhoff type problem involving the fractional Laplacian in[END_REF] proposed an extend in several directions the recent results of Fiscella, and Valdinoci with the following problem

               -M R 2N |u(x) -u(y)| p K(x -y)dxdy L p K u = λ f (x, u) + |u| p * s -2 u in Ω u = 0 in R N \ Ω, where 0 < s < 1 < p < ∞ with sp < N, f : R N × R → R is a Carathéodory functions satisfying the Ambrosetti-Rabinowitz type condition, λ is a positive real parameter, M : [0, ∞) → (0, ∞) is a
Chapitre 3. On some nonlocal problems in fractional Sobolev spaces nondecreasing continuous function and L K p is the nonlocal integro-differential operator of elliptic type defined as :

L K p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y))K(x -y)dy.
Under some suitable assumptions, the authors showed the existence of nontrivial solutions for above problem by applying the mountain pass theorem.

For the problems involving fractional Kirchhoff type, we refer the reader to the works [START_REF] Azroul | Multipe solutions for a nonlocal fractional (p, q)-Schrodinger-Kirchhoff system[END_REF][START_REF] Xiang | Superlinear Schrödinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent[END_REF][START_REF] Xiang | Multiplicity results for variable-order fractional Laplacian equations with variable growth[END_REF][START_REF] Xiang | Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities[END_REF][START_REF] Xiang | Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian[END_REF]. they use different methods to establish the existence of solutions.

In this chapter, we will study some nonlocal elliptic problems in the fractional Sobolev spaces by using three critical point theorem, Ekeland's variational principle and direct variational approach.

• In section 3.2, we are concerned with a class of nonlocal problems in fractional Sobolev spaces of the following form

(D s,p )            M [u] p s,p (-∆) s p u = λ f (x, u) + µg(x, u) in Ω, u = 0 in R N \ Ω,
using the three critical points theorem which introduced by B. Ricceri [START_REF] Ricceri | A further three critical points theorem[END_REF], we obtain the existence of a three weak solutions.

• In section 3.3, we investigate the existence of three solutions for an elliptic Schrödinger-Kirchhoff type equation involving nonlocal fractional integro deferential operator. More precisely, we consider the following equation

(D K,p ) M R N R N |u(x) -u(y)| p K(x -y)dxdy + R N V (x)|u| p dx L K p u +V (x)|u| p-2 u = λ f (x, u) + µg(x, u) in R N .
The technical approach is mainly based on a three critical points theorem.

• In section 3.4, we are interested to study the existence of weak solution of the following problem

(P s )            M(||u|| p s,p ) (-∆) s p u + |u| p-2 u = λ f (x, u) in Ω u = 0 in R N \ Ω.

Three solutions for a Kirchhoff type problem involving nonlocal fractional p-Laplacian

In this section, we are concerned with a class of nonlocal problems in fractional Sobolev spaces of the following form

(D s,p )            M [u] p s,p (-∆) s p u = λ f (x, u) + µg(x, u) in Ω, u = 0 in R N \ Ω,
where

Ω is an open bounded subset in R N , N 1, with Lipschitz boundary ∂ Ω, 0 < s < 1 < p < ∞, M : [0, ∞) -→ (0, ∞) is a nondecreasing continuous function, f , g : Ω × R -→ R are two
Carathéodory functions, λ , µ are two real parameters and (-∆) s p is the nonlocal integro-differential operator of elliptic type defined as :

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp dy.
This problem is motivated by some problems, in the local case (s = 1), which involves the p-laplace operator :

(D p )            -M ||u|| p p ∆ p u = λ f (x, u) + µg(x, u) in Ω, u = 0 in ∂ Ω,
where ∆ p = div |∇u| p-2 ∇u . In recent years, problem (D p ) has been studied in many papers, we refer to [START_REF] Bonanno | Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces[END_REF][START_REF] Chen | The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions[END_REF][START_REF] Corrêa | On a p-Kirchhoff equation via Krasnoselskiiś genus[END_REF][START_REF] Ma | Remarks on an elliptic equation of Kirchhoff type[END_REF][START_REF] Ricceri | On an elliptic Kirchhoff-type problem depending on two parameters[END_REF], in which the authors have used different methods to get the existence of solutions for (D p ). In the case when p(.) is a continuous function, problem (D p ) has also been studied by many authors, see for examples [START_REF] Cammaroto | Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator[END_REF][START_REF] Chung | Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities[END_REF][START_REF] Chung | Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities[END_REF][START_REF] Fan | On the Spaces L p(x) (Ω) and W m,p(x) (Ω)[END_REF]. To our knowledge, this is the first contribution to studying of nonlocal problems in this class of fractional Sobolev spaces.

Assumptions and mains Results

To prove the existence of three weak solutions for the problem (D s,p ), we give the following definition.

Definition 3.2.1. If sp N, we denote by A the class of all Carathéodory function f :

Ω × R -→ R such that sup (x,t)∈Ω×R | f (x,t)| 1 + |t| q-1 < ∞ (3.3) for all q ∈ [1, p * s ).
While when N < sp, we denote by A the class of all Carathéodory function f :

Ω × R -→ R such that for each δ > 0, the function x -→ sup |t| δ | f (x,t)| belongs to L 1 (Ω).
For f ∈ A , we put :

F(x,t) = t 0 f (x, s)ds, (x,t) ∈ Ω × R and M(t) = t 0 M(s)ds, t 0,
we assume that :

    
there exist two constants m 0 > 0 and 1

< α < p * s p , such that M(t) m 0 t α-1 for all t 0, ( M 1 )
sup

u∈W s,p 0 (Ω) Ω F(x, u)dx > 0, ( F 1 )
lim sup

t→0 sup x∈Ω F(x,t) |t| α p 0, (F 2 ) lim sup |t|→∞ sup x∈Ω F(x,t) |t| α p 0. ( F 3 )
Under such hypothesis, we set

θ * = inf        1 p M([u] p s,p ) Ω F(x, u)dx : u ∈ W s,p (Ω), Ω F(x, u)dx > 0        .
To deal with the problem (D s,p ), , we work in closed space W s,p 0 (Ω) which can be equivalently renormed by setting . := [.] s,p . The dual space of W s,p 0 (Ω), . is denoted by

(W s,p 0 (Ω)) * , . * . Definition 3.2.2. We say that u ∈ W s,p 0 (Ω) is a weak solution of problem (D s,p ), if M([u] p s,p ) Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy = λ Ω f (x, u)vdx + µ Ω g(x, u)vdx,
Chapitre 3. On some nonlocal problems in fractional Sobolev spaces for all v ∈ W s,p 0 (Ω).

Theorem 3.2.1. For f ∈ A , suppose that (F 1 )-(F 2 )-(F 3 ) and (M 1 ) holds true. Then for each compact interval Λ ⊂ (θ * , ∞), there exists a number δ > 0 with the following property : for every λ ∈ Λ and every g ∈ A there exists µ * > 0 such that, for each µ ∈ [0, µ * ], problem (D s,p ) has at least three weak solutions whose norms are less than δ .

Proofs of mains results

The proof will be carried out in a several lemmas. We first prove the following useful result, which helps us to apply theorem 1.5.7. For this, we define the functionals Ψ, J : W s,p 0 (Ω) -→ R by

J(u) = Ω F(x, u)dx, Ψ(u) = 1 p M([u] p s,p ). Lemma 3.2.1. Let f ∈ A , then the functional J ∈ C 1 (W s,p 0 (Ω), R
) with the derivative is given by

J (u), v = Ω f (x, u)vdx, (3.4)
for all u, v ∈ W s,p 0 (Ω). Moreover J : W s,p 0 (Ω) -→ (W s,p 0 (Ω)) * is compact.

Proof. First, observe that by the definition of A and the embedding theorem, J is well-defined on W s,p 0 (Ω). Usual arguments shown that J is Gâteaux-differentiable on W s,p (Ω) with the derivative is given by (3.4). We distinguish two cases, according as sp N or not.

Assume first that sp N, we claim that the operator J is continuous. Actually, let {u n } ⊂ W s,p 0 (Ω) be a sequence converging strongly to u ∈ W s,p 0 (Ω). Since W s,p 0 (Ω) is compactly embedded in L q (Ω), then {u n } converges strongly to u in L q (Ω). So there exists a subsequence of {u n }, still denoted by {u n }, and a function u ∈ L q (Ω) such that {u n } converges to u almost everywhere in Ω and |u n | |u| for all n ∈ N and almost everywhere in Ω. Since f ∈ A , we have for all measurable functions u : Ω -→ R, the operator defined by u -→ f (., u(.)) maps L q (Ω) into L q (Ω). Fix v ∈ W s,p 0 (Ω) with ||v|| 1, we using the Hölder inequality and the embedding of W s,p 0 (Ω) into L q (Ω),

we have

| J (u n ) -J (u), v | = Ω ( f (x, u n (x)) -f (x, u(x))) v(x)dx , || f (x, u n (x)) -f (x, u(x))|| q ||v|| q , c 1 || f (x, u n (x)) -f (x, u(x))|| q ||v||, 3.2.
Three solutions for a Kirchhoff type problem involving nonlocal fractional p-Laplacian 51 for some c 1 > 0. Thus, passing to the supremum for ||v|| 1, we get

||J (u n ) -J (u)|| * c 1 || f (., u n (.)) -f (., u(.))|| q . Since f ∈ A , we deduce f (x, u n (x)) -f (x, u(x)) -→ 0 as n → ∞ and | f (x, u n (x)) -f (x, u(x))| c 2 (2 + |u(x)| q-1 + |u(x)| q-1 ),
for almost everywhere x ∈ Ω. Note that the majorant function in the previous relation is in L q (Ω), hence, by applying the dominate convergence theorem we get that

|| f (x, u n (x)) -f (x, u(x))|| q → 0 as n → ∞.
This proves that J is continuous. Now, in order to verify the compactness of J , take

{u n } ⊂ W s,p 0 (Ω) bounded.
Then there exists a subsequence of {u n } (still denoted by {u n }) weakly convergent in W s,p 0 (Ω) and strongly convergent in L q (Ω). Arguing in the same way as above, we deduce that {J (u n )} is strongly convergent and the operator J is compact.

The proof in the case sp > N is carried out in a similar fashion. In this case, W s,p 0 (Ω) is compactly embedded into L ∞ (Ω) and f (., u(.)) map L ∞ (Ω) in L 1 (Ω). So, as before, fixing v ∈ W s,p 0 (Ω)

with ||v|| 1, we get

||J (u n ) -J (u)|| * c 3 || f (., u n (.)) -f (., u(.))|| L 1 (Ω) . Since f ∈ A we deduce f (., u n (.)) -f (., u(.)) -→ 0 as n → ∞ and | f (x, u n (x)) -f (x, u(x))| sup |t| ||u n || L ∞ (Ω) | f (x,t)| + sup |t| ||u|| L ∞ (Ω) | f (x,t)|,
for almost everywhere x ∈ Ω. Note that the majorant function in the previous relation is in L 1 (Ω). By applying the dominate convergence theorem we draw the conclusion that

|| f (x, u n (x))-f (x, u(x))|| L 1 (Ω) →
0 as n → ∞ and hence the entire sequence goes to 0.

52

Chapitre 3. On some nonlocal problems in fractional Sobolev spaces Lemma 3.2.2. Let (M 1 ) hold true, then the functional Ψ ∈ C 1 (W s,p 0 (Ω), R) and

Ψ (u), v = M([u] p s,p ) Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy for all u, v ∈ W s,p 0 (Ω). Moreover, for each u ∈ W s,p 0 (Ω), Ψ (u) ∈ (W s,p 0 (Ω)) * .
Proof. First, it is easy to see that

Ψ (u), v = M([u] p s,p ) Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy (3.5) for all u, v ∈ W s,p 0 (Ω). It follows from (3.5) that for each u ∈ W s,p 0 (Ω), Ψ (u) ∈ (W s,p 0 (Ω)) * .
Next, we prove that

Ψ ∈ C 1 (W s,p 0 (Ω), R). Let {u n } ⊂ W s,p 0 (Ω) with u n -→ u strongly in W s,p 0 (Ω),
without loss of generality, we assume that u n -→ u a.e. in Ω. Then the sequence

|u n (x) -u n (y)| p-2 (u n (x) -u n (y)) |x -y| (N+sp)/p is bounded in L p (Ω × Ω)
as well as a.e. in Ω × Ω,

U n (x, y) := |u n (x) -u n (y)| p-2 (u n (x) -u n (y)) |x -y| (N+sp)/p -→ U(x, y) := |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| (N+sp)/p .
Thus, the Brezis-Lieb lemma (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext[END_REF]) implies

lim n→∞ Ω Ω |U n (x, y) -U(x, y)| p dxdy = lim n→∞ Ω Ω |u n (x) -u n (y)| p |x -y| N+sp - |u(x) -u(y)| p |x -y| N+sp dxdy.
The fact that u n → u strongly in W s,p 0 (Ω) yields that

lim n→∞ Ω Ω |u n (x) -u n (y)| p |x -y| N+sp - |u(x) -u(y)| p |x -y| N+sp dxdy = 0. (3.6)
Moreover, the continuity of M implies that

M([u n ] p s,p ) -→ M([u] p s,p ). ( 3
||Ψ (u n ) -Ψ (u)|| * = sup v∈W s,p 0 (Ω),||v|| 1 | Ψ (u n ) -Ψ (u), v | -→ 0.
The proof of Lemma 3.2.2 is completed.

Lemma 3.2.3. The following proprieties hold true :

(i) The functional Ψ is sequentially weakly lower semicontinuous.

(ii) The functional Ψ belongs to the class W W s,p 0 (Ω) .

Proof. (i) First, notice that the map u → [u] p s,p is lower semicontinuous for the weak topology of W s,p 0 (Ω) and since M is nondecreasing and continuous function, so the map u → M([u] p s,p ) is lower semicontinuous for the weak topology of W s,p 0 (Ω). Indeed, we define a functional φ :

W s,p 0 (Ω) -→ R as φ (u) = Ω Ω |u(x) -u(y)| p |x -y| N+sp dxdy.
By an argument similar to Lemma 3.2.2, we obtain φ ∈ C 1 (W s,p 0 (Ω), R) and

φ (u), v = p Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy for all u, v ∈ W s,p 0 (Ω). Notice that φ u + v 2 = Ω Ω 1 2 p |(u + v)(x) -(u + v)(y)| p |x -y| N+sp dxdy Ω Ω 1 2 |u(x) -u(y)| p |x -y| N+sp dxdy + Ω Ω 1 2 |v(x) -v(y)| p |x -y| N+sp dxdy = 1 2 φ (u) + 1 2 φ (v).
Thus, φ is a convex functional on W s,p 0 (Ω). Now, let {u n } ⊂ W s,p 0 (Ω) with u n u weakly in W s,p 0 (Ω), then by convexity of φ , we have

φ (u n ) -φ (u) φ (u), u n -u , hence, we obtain φ (u) lim inf n→∞ φ (u n ), that is, the map u → [u] p s,
Ψ(u n ) = lim inf n→∞ 1 p M([u n ] p s,p ) 1 p M(lim inf n→∞ [u n ] p s,p ) 1 p M([u] p s,p ).
Thus, the functional Ψ is sequentially weakly lower semicontinuous.

(ii) Since M is continuous and strictly increasing, it suffices to show that φ ∈ W W s,p 0 (Ω) . Then, let {u n } be a sequence weakly converging to u in W s,p 0 (Ω) and let lim inf

n→∞ φ (u n ) φ (u).
Since the functional φ is sequentially weakly lower semicontinuous, there exists a subsequence of {u n } , still

denoted by {u n } such that lim inf n→∞ φ (u n ) φ (u), so : lim n→∞ φ (u n ) = φ (u), that is, lim n→∞ ||u n || = ||u||.
Since W s,p 0 (Ω) is uniformly convex space, we deduce that u n -→ u strongly in W s,p 0 (Ω).

Lemma 3.2.4. Assume that the sequence {u n } converges weakly to u in W s,p 0 (Ω) and

lim sup n→∞ (-∆) s p u n , u n -u 0. (3.8)
Then the sequence {u n } is convergence strongly to u in W s,p 0 (Ω).

Proof. First for u, v ∈ W s,p 0 (Ω), we have

(-∆) s p u -(-∆) s p v, u -v [u] p-1 s,p -[v] p-1 s,p ([u] s,p -[v] s,p ) . (3.9)
Indeed, by Hölder inequality, we have

(-∆) s p u -(-∆) s p v, u -v = (-∆) s p u, u -v -(-∆) s p v, u -v , = (-∆) s p u, u + (-∆) s p v, v -(-∆) s p u, v -(-∆) s p v, u , [u] p s,p + [v] p s,p -[u] p-1 s,p [v] s,p -[v] p-1 s,p [u] s,p , = [u] p-1 s,p -[v] p-1 s,p ([u] s,p -[v] s,p
(-∆) s p u n -(-∆) s p u, u n -u = lim n→∞ (-∆) s p u n , u n -u -lim n→∞ (-∆) s p u, u n -u = 0.
By (3.9), we have

(-∆) s p u n -(-∆) s p u, u n -u [u n ] p-1 s,p -[u] p-1 s,p ([u n ] s,p -[u] s,p ) .
Hence, ||u n || -→ ||u||. Since W s,p 0 (Ω) is uniformly convex space, we deduce that u n -→ u strongly in W s,p 0 (Ω).

Lemma 3.2.5. Let (M 1 ) hold, then the operator

Ψ : W s,p 0 (Ω) -→ (W s,p 0 (Ω)) * is invertible and Ψ -1
is continuous.

Proof. First, we assume that the operator

Ψ : W s,p 0 (Ω) -→ (W s,p 0 (Ω)) * is invertible on W s,p 0 (Ω).
By Minty-Browder theorem (see. [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext[END_REF][START_REF] Zeidler | Nonlinear Functional Analysis and Applications[END_REF]), it suffices to prove that Ψ is strictly monotone, hemicontinuous and coercive in the sense of monotone operators.

So, let u, v ∈ W s,p 0 (Ω), with u = v and δ , β ∈ [0, 1] with δ + β = 1. By (3.9), we have φ (u) -φ (v), u -v = p (-∆) s p u -(-∆) s p v, u -v , p [u] p-1 s,p -[v] p-1 s,p ([u] s,p -[v] s,p ) .
So, φ : W s,p 0 (Ω) -→ (W s,p 0 (Ω)) * is strictly monotone, so by [START_REF] Zeidler | Nonlinear Functional Analysis and Applications[END_REF]Proposition 25.10], φ is strictly convex. Moreover, since M is nondecreasing, the function M is convex in R + , thus we have

M(φ (δ u + β v)) < M(δ φ (u) + β φ (v)) δ M(φ (u)) + β M(φ (v)).
This shows that Ψ is strictly convex and is already said, that Ψ is strictly monotone.

Let u ∈ W s,p 0 (Ω), by (M 1 ), we have

Ψ (u), u ||u|| = M(||u|| p ) (-∆) s p u, u ||u|| m 0 ||u|| α p ||u|| p ||u|| = m 0 ||u|| (α+1)p-1 . Thus lim ||u||→∞ Ψ (u), u ||u|| = ∞.
Hence, Ψ is coercive. Now, by Lemma 3.2.2 we have Ψ ∈ C 1 (W s,p 0 (Ω), R), then Ψ is hemicontinuous. Thus, in view of Minty-Browder Theorem, there exists Ψ -1 : (W s,p 0 (Ω)) * -→ W s,p 0 (Ω) and it is bounded.

Let us prove that Ψ -1 is continuous by showing that its is sequentially continuous. Let {u n } ⊂ (W s,p 0 (Ω)) * be a sequence strongly converge to u ∈ (W s,p 0 (Ω)) * and let

v n = Ψ -1 (u n ) and v = Ψ -1 (u).
Then, {v n } bounded in W s,p 0 (Ω), so, we can assume that it converges weakly to a certain v 0 ∈ W s,p 0 (Ω). Since u n converges strongly to u, we have

lim n→∞ Ψ (v n ), v n -v 0 = lim n→∞ u n , v n -v 0 = 0, that is, lim n→∞ M([v n ] p s,p ) (-∆) s p v n , v n -v 0 = 0. (3.10) Since {v n } is bounded in W s,p 0 (Ω), then φ (v n ) is also bounded, therefore ||v n || p -→ t 0 0 as n → ∞.
If t 0 = 0, then, we get {v n } converges strongly to v 0 in W s,p 0 (Ω), by the continuity and injectivity of Ψ -1 we obtain the desired result.

If t 0 > 0, it follows from the continuity of the function M that

M(||v n || p ) -→ M(t 0 ) as n → ∞.
Thus, by (M 1 ), for sufficiently large n, we get

M(||v n || p ) c 4 > 0. ( 3
lim n→∞ (-∆) s p v n , v n -v 0 = 0. (3.12)
From (3.12) and since v n converges weakly to v 0 in W s,p 0 (Ω), we can apply Lemma 3.2.4, then we 3.2. Three solutions for a Kirchhoff type problem involving nonlocal fractional p-Laplacian 57 deduce that v n converge strongly to v 0 in W s,p 0 (Ω). The continuity and injectivity of Ψ imply that v n converges strongly to v, so Ψ 1 is continuous.

Proof of Theorem 3.2.1. We want to apply Theorem 1.5.7, taking X = W s,p 0 (Ω), Ψ and J are as before, by Lemma 3.2.1 J is C 1 -functional with compact derivative. Moreover by Lemma 3.2.3, Ψ is a sequentially weakly lower continuous and C 1 -functional belongs to W W s,p 0 (Ω) , also by Lemma 3.2.5, the operator Ψ admits a continuous inverse on (W s,p 0 (Ω)) * .

On the other hand, we show that Ψ is coercive. Indeed, let u ∈ W s,p 0 (Ω), by (M 1 ), we have

Ψ(u) = 1 p M(||u|| p ), m 0 pα ||u|| α p ,
from which we have the coercivity of Ψ.

It is evident that u 0 = 0 is global minimum of Ψ and that Ψ(u 0 ) = J(u 0 ) = 0. Moreover, Ψ is bounded on each bounded subset of W s,p 0 (Ω). Indeed, if ||u|| c 5 , then Ψ(u) = 1 p M(||u|| p ) 1 p M(c p 5 ).
Now, by the assumptions (F 2 ) and (F 3 ), for all ε > 0, there exit compact. Then for some positive constants c 7 and c 8 , one has for all u ∈ W s,p 0 (Ω)

η 2 > η 1 > 0 such that |F(x,t)| ε|t| α p (3.13) for each x ∈ Ω and t ∈ R \ [-η 2 , -η 1 ] ∪ [η 1 , η 2 ]. Then, since F(x, .) is bounded on [-η 2 , -η 1 ] ∪ [η 1 , η 2 ] for all x ∈ Ω, we can choose c 6 > 0 and r > α p (with r < p * if sp N), such that |F(x,t)| c 6 |t| r (3.14) for each (x,t) ∈ Ω × [-η 2 , -η 1 ] ∪ [η 1 , η 2 ], combining (
J(u) ε||u|| α p L α p (Ω) + c 6 ||u|| r L r (Ω) εc 7 ||u|| α p + c 6 c 8 ||u|| r .
Or by (M 1 ), we have ||u|| α p pα m 0 Ψ(u), then

J(u) εc 7 pα m 0 Ψ(u) + c 6 c 8 pα m 0 Ψ(u) r α p .
Consequently, since r > α p and the fact that Ψ(u) → 0 as u → 0, we have lim sup u→0 J(u)

Ψ(u) εc 7 pα m 0 . (3.15)
By (F 3 ), for all ε > 0, there exists η 2 > 0 such that

|F(x,t)| ε|t| α p (3.16)
for all x ∈ Ω and |t| > η 2 . By Theorem 1.2.4, we have

J(u) Ψ(u) = p J(u) M(||u|| p ) p {x∈Ω:|u(x)| η 2 } F(x, u)dx m 0 α ||u|| α p + p {x∈Ω:|u(x)|>η 2 } F(x, u)dx m 0 α ||u|| α p , p |Ω| sup Ω×[-η 2 ,η 2 ] F m 0 α ||u|| α p + p ε||u|| α p L α p (Ω) m 0 α ||u|| α p , p |Ω| sup Ω×[-η 2 ,η 2 ] F m 0 α ||u|| α p + c 9 ε. So, lim sup ||u||→∞ J(u) Ψ(u) εc 9 . (3.17)
Since ε > 0 is arbitrary, relations (3.15) and (3.17) implies that max lim sup ||u||→+∞ J(u)

Ψ(u) , lim sup u→0 J(u) Ψ(u) 0.
Hence, all assumptions of Theorem 1.5.7 are satisfied. So, for each compact interval Λ ⊂ (θ * , +∞),

Γ(u) = Ω G(x, u)dx, G(x,t) = t 0 g(x, s)ds
for all u ∈ W s,p 0 (Ω). Then, Γ is a C 1 functional on W s,p 0 (Ω) with compact derivative. So, there exists µ * > 0 such that, for each µ ∈ [0, µ * ], the equation

Ψ (u) = λ J (u) + µΓ (u)
has at least three solutions whose norms are less than δ . But the solutions in W s,p 0 (Ω) of the above equation are exactly the weak solutions of problem (D s,p ).

Examples

We present in this section an example of functions that satisfies the conditions of Theorem 3.2.1.

Let N 1 and 0

< s < 1 < p < ∞ such that p < α p < N p N -sp with 1 < α < N N -sp . Let q ∈ (p, α p) and m 0 > 0, we consider f : R -→ R and M : [0, ∞) -→ (0, ∞) be the functions defined by f (t) = |t| q-2 t -|t| p-2 t and M(t) = m 0 t α-1 . (3.18)
So, from (3.18), we have

F(x,t) = F(t) = |t| q q - |t| p p and M(t) = m 0 α t α .
We will next show that all the hypotheses of Theorem 3.2.1 are satisfied.

For each t ∈ R, we claim that f ∈ A . Actually, if sp N, then since p < q, we have

||t| q-2 t -|t| p-2 t| c(1 + |t| q-1 ),
for some c > 0 and for all t ∈ R,

if sp > N the function x → sup |t| δ | f (t)| is obviously in L 1 (Ω) for
all δ > 0, being Ω bounded. On the other hand, we have :

lim |t|→0 |t| q q - |t| p p |t| α p = -∞ and lim |t|→∞ |t| q q - |t| p p |t| α p = 0.
Select a compact subset V of Ω, large enough and let v ∈ W s,p 0 (Ω) such that v(x) = t 0 in V and 1 v(x) t 0 in Ω \V , where t 0 > 1 is chosen such that

t q 0 q - t p 0 p > 0.
One has

Ω F(v(x))dx = 1 q Ω |v(x)| q dx - 1 p Ω |v(x)| p dx 1 q V |v(x)| q dx - 1 p V |v(x)| p dx - 1 p Ω\V |v(x)| p dx |V | t q 0 q - t p 0 p - t p 0 |Ω \V | p > 0, hence sup u∈W s,p 0 (Ω) Ω |u(x)| q q - |u(x)| p p dx > 0.
Which means that (F 1 ), (F 2 ) and (F 3 ) are verified, we set

θ * = inf        m 0 pα [u] α p s,p Ω |u(x)| q q - |u(x)| p p dx : u ∈ W s,p 0 (Ω), Ω |u(x)| q q - |u(x)| p p dx > 0        .
Then, it follows from Theorem 3.2.1 that, for each compact interval Λ ⊂ (θ * , +∞), there exists a number δ > 0 for every λ ∈ Λ and all g ∈ A there exists µ * > 0 such that for all µ ∈ [0, µ * ], the following equation

           m 0 [u] p(α-1) s,p (-∆) s p u = λ |u| q-2 u -|u| p-2 u + µg(x, u) in Ω u = 0 in R N \ Ω,
has at least three weak solutions whose norms are less than δ .

3.3. Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 61

Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators

In this section, we investigate the existence of three solutions for an elliptic Schrödinger-Kirchhoff type equation involving nonlocal fractional integro deferential operator. More precisely, we consider the following equation

(D K,p ) M R N R N |u(x) -u(y)| p K(x -y)dxdy + R N V (x)|u| p dx L K p u +V (x)|u| p-2 u = λ f (x, u) + µg(x, u) in R N ,
where

N 1, 0 < s < 1 < p < ∞ with sp < N, f , g : R N × R → R are two Carathéodory functions, λ , µ are two real parameters, M : [0, ∞) → (0, ∞) is a nondecreasing continuous function and L K p
is the nonlocal integro-differential operator of elliptic type defined as :

L K p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y))K(x -y)dy, along all u ∈ C ∞ 0 (R N
) and K : R N \ {0} → R + is a measurable function with the proprieties :

(K 1 ) K(x) = K(-x).
(K 1 ) There exists k 0 > 0 such that K(x) k 0 |x| -N-sp , for any x ∈ R N \ {0} .

(K 1 ) ρK ∈ L 1 (R N ), where ρ(x) = min {|x| p , 1} .
Also, the potential function V satisfies the following condition :

V ∈ C(R N ), inf R N V (x) V 0 > 0 and |x ∈ R N : V (x) M| < ∞ ∀M > 0. (V 1 )
A typical example of the function K is given by K

(x) = |x| -N-sp . In this case, problem (D K,p ) becomes M R N R N |u(x) -u(y)| p |x -y| sp+N dxdy + R N V (x)|u| p dx (-∆) s p u +V (x)|u| p-2 u = λ f (x, u) + µg(x, u) in R N ,
where (-∆) s p is the fractional p-laplace operator which (up to normalization factors) may be defined as

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp dy.
Chapitre 3. On some nonlocal problems in fractional Sobolev spaces When p = 2, g ≡ 0, M, λ ≡ 1, and K(x) = |x| -N-sp , problem (D K,p ) reduces to the fractional Laplacian problem

(-∆) s u +V (x)u = f (x, u) in R N . (3.19) Equation (3.19
) is the so-called fractional Shcrödinger equation. In the context of fractional quantum mechanics, nonlinear fractional Schrödinger equation has been proposed by Laskin [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF][START_REF] Laskin | Fractional Schrödinger equation[END_REF] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. In the last years, there has been a great interest in the study of the fractional Schrödinger equation. For standing wave solutions of fractional Schrödinger equations in R N , see, for instance, [START_REF] Chang | Ground state of scalar field equations involving a fractional a fractional Laplacian with general nonlinearity[END_REF][START_REF] Felmer | Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian[END_REF] and the references therein. Models governed by the fractional p-Laplacian and unbounded potentials were investigated in Ref. [START_REF] Secchi | Ground state solutions for the fractional Schrödinger in R N[END_REF].

Variational framework

We consider space X s defined by

X s := u ∈ W s,p (R N ) : R N V (x)|u| p dx < ∞ , endowed with the norm ||u|| X s = R N R N |u(x) -u(y)| p |x -y| N+sp dxdy + R N V (x)|u| p dx 1 p
.

From (V 1 ), Hölder's inequality and Theorem 1.2.5, we have the space X s is continuously embedded in L q (R N ) for any q ∈ [p, p * s ]. Moreover the following compactness result holds.

Lemma 3.3.1. (cf. [100]) Let 0 < s < 1 < p < ∞ such that sp < N, suppose that (V 1 ) hold. Then the embedding X s → L q (R N ) is continuous for all q ∈ [p, p * s ]. Moreover the embedding X s → L q (R N )
is compact for all q ∈ [p, p * s ).

Now we introduce the main space (W s , ||.||) naturally associated with (D K,p ). We put

L p (R N ,V ) = u : R N → R is measurable : R N V (x)|u| p dx < ∞ ;
and set :

W = u ∈ L p (R N ,V ) : R N R N |u(x) -u(y)| p K(x -y)dxdy < ∞ ,
||u|| := ||u|| W = [u] p K,p + ||u|| p p,V 1 p ,
where

[u] K,p = R N R N |u(x) -u(y)| p K(x -y)dxdy 1 p , and
||u|| p,V = R N V (x)|u| p dx 1 p
.

By [135, Lemma A.4], we known that (L p (R N ,V ), ||.|| p,V
) is a separable and reflexive Banach space and uniformly convex space for all p ∈ (1, ∞).

Lemma 3.3.2. The space W is a separable and reflexive Banach space and uniformly convex space

for all p ∈ (1, ∞). Actually C ∞ 0 (R N ) is dense in W .
Proof of this Lemma is similar to [135, Lemma A.5 and Lemma A.6].

Lemma 3.3.3. Let 0 < s < 1 < p < ∞ such that sp < N, suppose that (V 1 ) hold. The embedding W → L q (R N ) is continuous for all q ∈ [p, p * s ]. Moreover The embedding W → L q (R N ) is compact for all q ∈ [p, p * s ).
Proof. For all u ∈ W , we have

||u|| X s max 1, k -1 0 ||u||,
so by using Lemma 3.3.1, the desired result is obtained.

The dual space of (W, . ) is denoted by (W * , . * ) .

Assumptions and mains Results

In this subsection, we prove the existence of three weak solutions for the problem (D K,p ) in fractional Sobolev spaces applying the theorem 1.5.7. At firstly, we give the following definition.

Definition 3.3.1. Let 0 < s < 1 < p < ∞ with sp < N, we denote by A the class of all Carathéodory function f : R N × R → R such that sup (x,t)∈R N ×R | f (x,t)| |t| q-1 < ∞ (3.20)
for all q ∈ [p, p * s ).

Chapitre 3. On some nonlocal problems in fractional Sobolev spaces

For f ∈ A , we put :

F(x,t) = t 0 f (x, s)ds, ∀(x,t) ∈ R N × R and M(t) = t 0 M(s)ds, ∀t 0.
We assume that : which satisfies the condition :

    
there exist two constants m 0 > 0 and 1 < α < p * p , such that M(t) m 0 t α-1 for all t 0.

(M 1 )

sup u∈W R N F(x, u)dx > 0, (F 1 ) lim sup t→0 sup x∈R N F(x,t) |t| α p 0, (F 2 ) lim sup |t|→∞ sup x∈R N F(x,t) |t| α p 0. ( F 3 )
Under such hypothesis, we set

θ * = inf      1 p M(||u|| p ) R N F(x, u)dx : u ∈ W, R N F(x, u)dx > 0      . Definition 3.3.2. We say that u ∈ W is a weak solution of problem (D K,p ) , if M(||u|| p ) R N R N |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y))K(x -y)dxdy + R N V (x)|u| p-2 uvdx = λ R N f (x, u)vdx + µ R N g(x, u)vdx, for all v ∈ W . Theorem 3.3.1. For f ∈ A , suppose that (F 1 )-(F 2 )-(F 3 )-(V 1
) and (M 1 ) holds true. Then for each compact interval Λ ⊂ (θ * , ∞), there exists a number δ > 0 with the following property : for every λ ∈ Λ and every g ∈ A there exists µ * > 0 such that, for each µ ∈ [0, µ * ], problem (D K,p ) has at least three weak solutions whose norms are less than δ .

3.3. Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 65

Proofs of mains results

The proof will be carried out in several lemmas. We first prove the following useful result, which helps us to apply theorem 1.5.7. For this, we define the functionals Ψ, J : W → R by

J(u) = R N F(x, u)dx, Ψ(u) = 1 p M(||u|| p ). Lemma 3.3.4. Let f ∈ A , then the functional J ∈ C 1 (W, R)
with the derivative is given by

J (u), v = R N f (x, u)vdx, (3.21)
for all u, v ∈ W . Moreover J : W → W * is compact.

Proof. First, observe that by the definition of A and the embedding theorem, J is well-defined on W . Usual arguments shown that J is Gâteaux-differentiable on W with the derivative is given by (3.21). Actually, let {u n } ⊂ W be a sequence converging strongly to u ∈ W . Since W is compactly embedded in L q (R N ), then {u n } converges strongly to u in L q (R N ). So there exists a subsequence of {u n }, still denoted by {u n }, and a function u ∈ L q (R N ) such that {u n } converges to u almost everywhere in R N and |u n | |u| for all n ∈ N and almost everywhere in Ω. Since f ∈ A , we have for all measurable functions u : R N → R, the operator defined by u -→ f (., u(.)) maps L q (R N ) into L q (R N ). Fix v ∈ W with ||v|| 1, we using the Hölder's inequality and the embedding of W into L q (R N ), we have

| J (u n ) -J (u), v | = R N ( f (x, u n (x)) -f (x, u(x))) v(x)dx , || f (x, u n (x)) -f (x, u(x))|| q ||v|| q , c 1 || f (x, u n (x)) -f (x, u(x))|| q ||v||,
for some c 1 > 0. Thus, passing to the supremum for ||v|| 1, we get

||J (u n ) -J (u)|| W * c 1 || f (., u n (.)) -f (., u(.))|| q .
Since f ∈ A , we deduce

f (., u n (.)) -f (., u(.)) → 0 as n → ∞ and | f (x, u n (x)) -f (x, u(x))| c 2 (|u(x)| q-1 + |u(x)| q-1 ).
for almost everywhere x ∈ R N . Note that the majorant function in the previous relation is in L q (R N ) ; hence, by applying the dominate convergence theorem, we get that

|| f (x, u n (x)) -f (x, u(x))|| q → 0 as n → ∞.
This proves that J is continuous. Now, in order to verify the compactness of J , take

{u n } ⊂ W bounded.
Then there exists a subsequence of {u n } (still denoted by {u n }) weakly convergent in W and strongly convergent in L q (R N ). Arguing in the same way as above, we deduce that {J (u n )} is strongly convergent and the operator J is compact.

Lemma 3.3.5. Let (M 1 )-(V 1 ) hold true, then the functional Ψ ∈ C 1 (W, R) and Ψ (u), v = M(||u|| p ) R N R N |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y))K(x -y)dxdy + R N V (x)|u| p-2 uvdx , (3.22)
for all u, v ∈ W . Moreover, for each u ∈ W , Ψ (u) ∈ W * .

Proof. First, it is easy to see that

Ψ (u), v = M(||u|| p ) R N R N |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y))K(x -y)dxdy + R N V (x)|u| p-2 uvdx , for all u, v ∈ W . It follows from (3.22) that for each u ∈ W , Ψ (u) ∈ W * .
Next, we prove that Ψ ∈ C 1 (W, R). Let {u n } ⊂ W with u n → u strongly in W , without loss of generality, we assume that u n → u a.e. in R N . Then the sequence

|u n (x) -u n (y)| p-2 (u n (x) -u n (y))K(x -y) 1 p is bounded in L p (R N × R N ) 3.3.
× R N , U n (x, y) := |u n (x) -u n (y)| p-2 (u n (x) -u n (y))K(x -y) 1 p -→ U(x, y) := |u(x) -u(y)| p-2 (u(x) -u(y))K(x -y) 1 p .
Thus, the Brezis-Lieb lemma (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext[END_REF]) implies

lim n→∞ R N R N |U n (x, y) -U(x, y)| p dxdy = lim n→∞ R N R N (|u n (x) -u n (y)| p K(x -y) -|u(x) -u(y)| p K(x -y)) dxdy.
The fact that u n → u strongly in W yields that

lim n→∞ R N R N (|u n (x) -u n (y)| p K(x -y) -|u(x) -u(y)| p K(x -y)) dxdy = 0. (3.23) Similarly, lim n→∞ R N V (x)||u n (x)| p-2 u n (x) -|u(x)| p-2 u(x)| p dx = 0. (3.24)
Moreover, the continuity of M implies that

M(||u n || p ) → M(||u|| p ). ( 3
||Ψ (u n ) -Ψ (u)|| * = sup v∈W,||v|| 1 | Ψ (u n ) -Ψ (u), v | -→ 0.
The proof of Lemma 3.3.5 is completed.

Lemma 3.3.6. The following proprieties hold true :

(i) The functional Ψ is sequentially weakly lower semicontinuous.

(ii) The functional Ψ belongs to the class W W .

Proof. (i) First, notice that the map u → ||u|| p is lower semicontinuous for the weak topology of W and since M is nondecreasing and continuous function, so the map u → M(||u|| p ) is lower semicontinuous for the weak topology of W . Indeed, we define a functional φ : W → R as

φ (u) = 1 p R N R N |u(x) -u(y)| p K(x -y)dxdy + R N V (x)|u| p dx .
Chapitre 3. On some nonlocal problems in fractional Sobolev spaces

By an argument similar to Lemma 3.3.5, we obtain φ ∈ C 1 (W, R) and

φ (u), v = R N R N |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y))K(x -y)dxdy + R N V (x)|u| p-2 uvdx , for all u, v ∈ W . Notice that φ u + v 2 = R N R N 1 2 p |(u + v)(x) -(u + v)(y)| p K(x -y)dxdy + R N 1 2 p V (x)|u + v| p dx 1 2 R N R N |u(x) -u(y)| p K(x -y)dxdy + R N V (x)|u| p dx + 1 2 R N R N |v(x) -v(y)| p K(x -y)dxdy + R N V (x)|v| p dx = 1 2 φ (u) + 1 2 φ (v).
Thus, φ is a convex functional on W . Now, let {u n } ⊂ W with u n u weakly in W , then by convexity of φ , we have

φ (u n ) -φ (u) φ (u), u n -u ,
hence, we obtain φ (u) lim inf n→∞ φ (u n ), that is, the map u → ||u|| p is lower semi-continuous. On the other hand, by the continuity and monotonicity of the function t → M(t), we get

lim inf n→∞ Ψ(u n ) = lim inf n→∞ 1 p M(||u n || p ) 1 p M(lim inf n→∞ ||u n || p ) 1 p M(||u|| p ) = Ψ(u).
Thus, the functional Ψ is sequentially weakly lower semicontinuous.

(ii) Since M is continuous and strictly increasing, it suffices to show that φ ∈ W W . Then, let {u n } be a sequence weakly converging to u in W and let lim inf

n→∞ φ (u n ) φ (u).
Since the functional φ is sequentially weakly lower semicontinuous, there exists a subsequence of {u n } , still denoted by Since W is uniformly convex space, we deduce that u n → u strongly in W .

{u n } such that lim inf n→∞ φ (u n ) φ (u), so : lim n→∞ φ (u n ) = φ (u),
Lemma 3.3.7. i) Assume that the sequence {u n } converges weakly to u in W and

lim sup n→∞ φ (u n ), u n -u 0. (3.26)
Then the sequence {u n } is convergence strongly to u in W .

ii) Let u, v ∈ W, it holds that

φ (u) -φ (v), u -v (||u|| p-1 -||v|| p-1 )(||u|| -||v||). (3.27)
Proof of this Lemma is similar to [100, Lemma 2.2 and Lemma 2.3] (see also Lemma 3.2.4).

Lemma 3.3.8. Let (M 1 ) hold, then the operator Ψ : W → W * is invertible and Ψ -1 is continuous.

Proof. First, we assume that the operator Ψ : W → W * is invertible on W . By Minty-Browder theorem (see. [START_REF] Zeidler | Nonlinear Functional Analysis and Applications[END_REF]), it suffices to prove that Ψ is strictly monotone, hemicontinuous and coercive in the sense of monotone operators.

M(φ (δ u + β v)) < M(δ φ (u) + β φ (v)) δ M(φ (u)) + β M(φ (v)).
This shows that Ψ is strictly convex and is already said, that Ψ is strictly monotone.

Let u ∈ W , by (M 1 )-(V 1 ), we have Ψ (u), u ||u|| = M(||u|| p )||u|| p ||u|| m 0 ||u|| p(α-1) ||u|| p ||u|| = m 0 ||u|| α p-1 .
Thus,

lim ||u||→∞ Ψ (u), u ||u|| = ∞.
Hence, Ψ is coercive. Now, by Lemma 3.3.5 we have Ψ ∈ C 1 (W, R), then Ψ is hemicontinuous. Thus, in view of Minty-Browder Theorem, there exists Ψ -1 : W * → W and it is bounded.

Let us prove that Ψ -1 is continuous by showing that its is sequentially continuous. Let {u n } ⊂ W * be a sequence strongly converge to u ∈ W * and let v n = Ψ -1 (u n ) and v = Ψ -1 (u). Then, {v n } bounded in W , so, we can assume that it converges weakly to a certain v 0 ∈ W . Since u n converges strongly to u, we have

lim n→∞ Ψ (v n ), v n -v 0 = lim n→∞ u n , v n -v 0 = 0, that is, lim n→∞ M(||v n || p ) φ (v n ), v n -v 0 = 0. (3.28) Since {v n } is bounded in W , then φ (v n ) is also bounded, therefore ||v n || p -→ t 0 0 as n → ∞.
If t 0 = 0, then, we get {v n } converges strongly to v 0 in W , by the continuity and injectivity of Ψ -1

we obtain the desired result.

If t 0 > 0, it follows from the continuity of the function M that

M(||v n || p ) → M(t 0 ) as n → ∞.
Thus, by (M 1 ), for sufficiently large n, we get

M(||v n || p ) c 3 > 0. (3.29)
By (3.28) and (3.29), we have :

lim n→∞ φ (v n ), v n -v 0 = 0. (3.30)
From (3.30) and since v n converges weakly to v 0 in W , we can apply Lemma 3.3.7, then we deduce that v n converge strongly to v 0 in W . The continuity and injectivity of Ψ imply that v n converges strongly to v, so Ψ 1 is continuous. On the other hand, we show that Ψ is coercive. Indeed, let u ∈ W , by (M 1 ), we have

Ψ(u) = 1 p M(||u|| p ), m 0 pα ||u|| α p ,
from which we have the coercivity of Ψ.

It is evident that u 0 = 0 is global minimum of Ψ and that Ψ(u 0 ) = J(u 0 ) = 0. Moreover, Ψ is bounded on each bounded subset of W . Indeed, if ||u|| c 4 , then

Ψ(u) = 1 p M(||u|| p ) 1 p M(c p 4 ).
Now, by the assumptions (F 2 ), for all ε > 0, there exists η 1 > 0 such that

|F(x,t)| ε|t| α p (3.31) for each x ∈ R N and t ∈ [-η 1 , η 1 ]. Since f ∈ A , then for r ∈ (α p, p * s ), we can choose c 5 > 0 such that |F(x,t)| c 5 |t| r (3.32) for each (x,t) ∈ R N × (R \ [-η 1 , η 1 ]), combining (3.31) and (3.32), we get |F(x,t)| ε|t| α p + c 5 |t| r for each (x,t) ∈ R N × R. So by Lemma 3.3.3, the embedding W in L α p (R N ) and L r (R N ) is compact.
Then for some positive constants c 6 and c 7 , one has for all u ∈ W J(u)

ε||u|| α p L α p (R N ) + c 5 ||u|| r L r (R N ) εc 6 ||u|| α p + c 5 c 7 ||u|| r .
Or by (M 1 ), we have ||u|| α p pα m 0 Ψ(u), then

J(u) εc 6 pα m 0 Ψ(u) + c 5 c 7 pα m 0 Ψ(u) r α p .
Consequently, since r > α p and the fact that Ψ(u) → 0 as u → 0, we have lim sup u→0 J(u)

Ψ(u) εc 6 pα m 0 . ( 3
Ψ(u) = p J(u) M(||u|| p ) p {x∈R N :|u(x)| η 2 } F(x, u)dx m 0 α ||u|| α p + p {x∈R N :|u(x)|>η 2 } F(x, u)dx m 0 α ||u|| α p , p c 8 ||u|| r L r (R N ) m 0 α ||u|| α p + p ε||u|| α p L α p (R N ) m 0 α ||u|| α p , p c 8 c 6 ||u|| r m 0 α ||u|| α p + p εc 7 ||u|| α p m 0 α ||u|| α p , p c 8 c 6 ||u|| r m 0 α ||u|| α p + c 9 ε. Since α p > r,
Ψ(u) , lim sup u→0 J(u) Ψ(u) 0.
Hence, all assumptions of Theorem 1.5.7 are satisfied. So, for each compact interval Λ ⊂ (θ * , +∞),

Γ(u) = R N G(x, u)dx, G(x,t) = t 0 g(x, s)ds
for all u ∈ W . Then, Γ is a C 1 functional on W with compact derivative. So, there exists µ * > 0 such that, for each µ ∈ [0, µ * ], the equation

Ψ (u) = λ J (u) + µΓ (u)
has at least three solutions whose norms are less than δ . But the solutions in W of the above equation are exactly the weak solutions of problem (D K,p ).

Examples

We present in this subsection an example of functions that satisfies the conditions of theorem

3.3.1. Let s ∈ (0, 1), p ∈ [2, N), b > max {2, α p} where 1 < α < N N -sp
and let a 1 > 0, a 2 0 we consider

f (t) = b cos(t) sin(t)| sin(t)| b-2 ∀t ∈ R and M(t) = a 1 + a 2 t α-1 ∀t 0. (3.37)
So, from (3.37), we have

F(x,t) = F(t) = | sin(t)| b and M(t) = a 1 t + a 2 α t α .
We will next show that all the hypotheses of Theorem 3.3.1 are satisfied.

For each t ∈ R, we claim that f ∈ A . Indeed, we have

sup t∈R | f (t)| |t| q-1 < ∞,
holds for any p < q < b. On the other hand, we have

lim |t|→0 | sin(t)| b |t| α p = 0 and lim |t|→∞ | sin(t)| b |t| α p = 0. Select a compact set V ⊂ R N of positive measure and v ∈ W such that v(x) = π 2 in V and 0 v(x) π 2 in R N \V . We obtain R N | sin(v(x))| b dx = |V | + R N \V | sin(v(x))| b dx > 0,
which means that (F 1 ), (F 2 ) and (F 3 ) are verified. Also, for m 0 = a 1 the condition (M 1 ) is verified, we set

θ * = inf      a 1 p ||u|| + a 2 pα ||u|| α p R N |sin(u(x))| b dx : u ∈ W, R N |sin(u(x))| b dx > 0      .
Then, it follows from Theorem 3.3.1 That, for each compact interval Λ ⊂ (θ * , +∞), there exists a number δ > 0 such that, for every λ ∈ Λ and all g ∈ A there exists µ * > 0 such that for all µ ∈ [0, µ * ], the following equation

a 1 + a 2 ||u|| p(α-1) L K p u +V (x)|u| p-2 u = λ b cos(u) sin(u)| sin(u)| b-2 + µg(x, u) in R N ,
has at least three weak solutions whose norms are less than δ . In this section, we are interested to study the existence of weak solution of the following problem

(P s )            M(||u|| p s,p ) (-∆) s p u + |u| p-2 u = λ f (x, u) in Ω u = 0 in R N \ Ω,
where

Ω is an open bounded subset in R N with Lipschitz boundary ∂ Ω, 0 < s < 1, f : Ω×R -→ R is a Carathéodory function, λ is a positive constant, (-∆) s p is the nonlocal integro-differential operator
of elliptic type defined as :

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp dy. ∀x ∈ R N ,
and M : R + → R * + is a nondecreasing continuous function which satisfies the condition :

there exists m 0 > 0 such that M(t) m 0 for all t ∈ R + .

(M 1 )

• A typical prototype for M, due to Kirchhoff in 1883, is given by

M(t) = a + bt α-1 , a, b 0, a + b > 0, t 0, (3.38)
and

     α ∈ (1, +∞) i f b > 0, α = 1 i f b = 0.
When M(t) > 0 for all t 0, Kirchhoff problems are said to be nondegenerate and this happens for example if a > 0 and b 0 in the model case (3.38). Otherwise, if M(0) = 0 and M(t) > 0 for all t > 0, the Kirchhoff problems are called degenerate and this occurs in the model case (3.38) when a = 0 and b > 0.

When M ≡ 1, λ = 1 and p = 2 problem (P s ) reduces to the following problem

           (-∆) s u + u = f (x, u) in Ω, u = 0 in R N \ Ω, (3.39)
The functional framework that takes into account problem (3.39) with Dirichlet boundary condition was introduced in [START_REF] Servadei | Mountain pass solutions for non-local elliptic operators[END_REF][START_REF] Servadei | Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators[END_REF]. We refer also to [START_REF] Franzina | Fractional p-eigenvalues[END_REF][START_REF] Lindgren | Fractional eigenvalues[END_REF][START_REF] Zhen | The Nehari manifold for fractional pLaplacian system involving concave-convex nonlinearities and sign-changing weight functions[END_REF] for further details on the functional framework and its applications to the existence of solutions for problem (3.39).

Motivated by the above papers, the aim of this section is to study the existence of a weak solutions for problem (P s ) by means of Ekeland's variational principle and direct variational approach.

Assumptions and mains results

Now, we prove the existence of a weak solution in fractional Sobolev spaces. For this, we suppose that f : Ω × R → R is a Carathéodory function satisfying :

| f (x,t)| c 1 |t| q-1 , ( F 1 )
c 2 |t| q F(x,t) := t 0 f (x, τ)dτ, ( F 2 )
for all x ∈ Ω and all t ∈ R, where c 1 and c 2 are positive constants and q ∈ [1, p).

Example 2. We point out certain examples of functions f and F which satisfy hypotheses (F 1 ) and

(F 2 ).
-f (x,t) = q|t| q-2 t and F(x,t) = |t| q where 2 q < p * s .

-f (x,t) = q|t| q-2 t +(q-2) log(1+t 2 )|t| q-4 t + t 1 + t 2 |t| q-2 and F(x,t) = |t| q +log(1+t 2 )|t| q-2 , where 4 q < p * s .

-f (x,t) = q|t| q-2 t +(q-1) sin(sint)×|t| q-3 t cos(sint) cost|t| q-1 and F(x,t) = |t| q +sin(sint)|t| q-1 , where 3 q < p * s .

To deal with the problem (P s ), we work in closed space W s s,p (Ω) and we denote . := . s,p .

The dual space of (W s,p (Ω), . ) is denoted by ((W s,p (Ω)) * , . * ) .

Now, we give the definition of weak solution for problem (P s ).

Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy + Ω |u| p-2 uvdx -λ Ω f (x, u)vdx = 0, (3.40)
for all v ∈ W s,p (Ω).

The main result of this section is given by the follows theorem. We define the functionals

I i : W s,p (Ω) -→ R i = 1, 2 by I 1 (u) = 1 p M(||u|| p ) and I 2 (u) = Ω F(x, u)dx,
where M(t) = t 0 M(τ)dτ, and for each λ > 0, the energy functional J λ : W s,p (Ω) -→ R is given by

J λ (u) = 1 p M(||u|| p ) -λ Ω F(x, u)dx. (3.41)
Remark 8. We note that the functional J λ : W s,p (Ω) -→ R in (3.41) is well defined. Indeed, if u ∈ W s,p (Ω), then, we have u ∈ L q (Ω) for all q ∈ [1, p * s ). Hence, by the condition (F 1 ),

|F(x, u)| u 0 | f (x,t)|dt = c 1 |u| q
and thus,

Ω |F(x, u)|dx < ∞.

Proofs of mains results

We first establish some basis properties of J λ .

Proposition 3.4.1. For each λ > 0, J λ ∈ C 1 (W s,p (Ω), R) with the derivative given by

J λ (u), v = M(||u|| p ) Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy + Ω |u| p-2 uvdx -λ Ω f (x, u)vdx, for all u, v ∈ W s,p (Ω).
Proof of this Proposition is similar to Lemma 3.3.4 and Lemma 3.3.5.

Lemma 3.4.1. The functional J λ is weakly lower semi continuous.

Proof. First, note that the map :

Ψ(u) := Ω Ω |u(x) -u(y)| p |x -y| N+sp dxdy + Ω |u(x)| p dx
is lower semi-continuous in the weak topology of W s,p (Ω). Indeed, similar to Lemma 3.3.5, we obtain Ψ ∈ C 1 (W s,p (Ω), R) and

Ψ (u), v = Ω Ω |u(x) -u(y)| p-2 (u(x) -u(y))(v(x) -v(y)) |x -y| N+sp dxdy + Ω |u| p-2 uvdx
for all u, v ∈ W s,p (Ω). Notice that

Ψ u + v 2 = Ω Ω 1 2 p |(u + v)(x) -(u + v)(y)| p |x -y| N+sp dxdy + Ω 1 2 p |u + v| p dx 1 2 Ω Ω |u(x) -u(y)| p |x -y| N+sp dxdy + Ω |u(x)| p dx + 1 2 Ω Ω |v(x) -v(y)| p |x -y| N+sp dxdy + Ω |v(x)| p dx = 1 2 Ψ(u) + 1 2 Ψ(v).
Thus, Ψ is a convex functional on W s,p (Ω). Now, let {u n } ⊂ W s,p (Ω) with u n u weakly in W s,p (Ω), then by convexity of Ψ, we have

Ψ(u n ) -Ψ(u) Ψ (u), u n -u , hence, we obtain Ψ(u) lim inf n→∞ Ψ(u n ),
that is, the map u → ||u|| p is lower semi-continuous. On the other hand, by the continuity and monotonicity of the function t → M(t), we get

lim inf n→∞ I 1 (u n ) = lim inf n→∞ 1 p M(||u n || p ) 1 p M(lim inf n→∞ ||u n || p ) 1 p M(||u|| p ) = I 1 (u).
(3.42)

Thus, the functional I 1 is sequentially weakly lower semicontinuous. Combining the above result with (3.42), we find

J λ (u) lim inf n→∞ J λ (u n ).
Therefore, J λ is weakly lower semi continuous and Lemma 3.4.1 is verified.

Lemma 3.4.2. Assume that the sequence {u n } converges weakly to u in W s,p (Ω) and

lim sup n→∞ Ψ (u n ), u n -u 0. (3.43)
Then the sequence {u n } is convergence strongly to u in W s,p (Ω).

Proof. First for u, v ∈ W s,p (Ω), we have

Ψ (u) -Ψ (v), u -v ||u|| p-1 -||v|| p-1 (||u|| -||v||) . (3.44)
Indeed, by Hölder inequality, we have

Ψ (u) -Ψ (v), u -v = (-∆) s p u + |u| p-2 u, u -v -(-∆) s p v + |v| p-2 v, u -v , = (-∆) s p u + |u| p-2 u, u + (-∆) s p v + |v| p-2 v, v -(-∆) s p u + |u| p-2 u, v -(-∆) s p v + |v| p-2 v, u , ||u|| p + ||v|| p -||u|| p-1 ||v|| -||v|| p-1 ||u||, = ||u|| p-1 -||v|| p-1 (||u|| -||v||) . So, if u n u in W s,p (Ω), we get lim n→∞ Ψ (u n ) -Ψ (u), u n -u = lim n→∞ Ψ (u n ), u n -u -lim n→∞ Ψ (u), u n -u = 0.
By (3.44), we have

Ψ (u n ) -Ψ (u), u n -u ||u n || p-1 -||u|| p-1 (||u n || -||u||) .
Hence, ||u n || -→ ||u||. Since W s,p (Ω) is uniformly convex space, we deduce that u n -→ u strongly in W s,p (Ω). Proof. By Theorem 1.2.6, it follows that W s,p (Ω) is continuously embedded in L q (Ω) for q ∈ [1, p), so there exists a positive constant c 4 > 0 such that

||u|| q c 4 ||u|| , ∀u ∈ W s,p (Ω). (3.45)
We fix ρ ∈ (0, 1) such that ρ < 1 c 4 . Combining relation (3.45) with conditions (F 1 ) and (M 1 ), we deduce that for any u ∈ W s,p (Ω) with ||u|| = ρ :

J λ (u) = 1 p M(||u|| p ) -λ Ω F(x, u)dx m 0 p ||u|| p -λ c 1 Ω |u| q dx m 0 p ||u|| p -λ c 1 c q 4 ||u|| q = ρ q m 0 p ρ p-q -λ c q 4 c 1 .
By the above inequality, we remark if we define

λ * = m 0 ρ p-q 2pc 1 c q 4 . (3.46)
Then for any u ∈ W s,p (Ω) with ||u|| = ρ, there exists α = ρ p 2 > 0 such that

J λ (u) α > 0.
The proof of Lemma 3.4.3 is completed. such that θ > 0 and J λ (tθ ) < 0 for t > 0 small enough.

Proof. Let Ω 0 ⊂⊂ Ω, for x 0 ∈ Ω 0 , 0 < R < 1 satisfy B 2R (x 0 ) ⊂ Ω 0 , where B 2R (x 0 ) is the ball of radius 2R with center at the point x 0 in R N . Let θ ∈ C ∞ 0 (B 2R (x 0 )) satisfies 0 θ 1 and θ ≡ 1 in B R (x 0 ). Then for 0 < t < 1, by the mean value theorem and (F 2 ), we have

J λ (tθ ) = 1 p M(||tθ || p ) -λ Ω F(x,tθ )dx 1 p ||tθ || p 0 M(τ)dτ -λ c 2 Ω 0 |tθ | q dx 1 p M(v)|t| p ||θ || p -λ c 2 |t| q Ω 0 |θ | q dx,
where v ∈ [0, ||tθ || p ). Since p > q and Ω 0 |θ | q dx > 0 we have J λ (t 0 θ ) < 0 for t 0 ∈ (0,t) sufficiently small.

Lemma 3.4.5. Assume the hypotheses of Theorem 3.4.1 are fulfilled. Then for any λ > 0 the functional J λ is coercive.

Proof. For each u ∈ W s,p (Ω) and λ > 0, relation (3.45) and the conditions (F 1 ) and (M 1 ) imply

J λ (u) = 1 p M(||u|| p ) -λ Ω F(x, u)dx m 0 p ||u|| p -λ c 1 Ω |u| q dx m 0 p ||u|| p -λ c 1 c q 4 ||u|| q .
Since p > q the above inequality implies that J λ (u) -→ ∞ as ||u|| → ∞, that is, J λ is coercive.

Proof of Theorem 3.4.1. Let λ * > 0 be defined as in (3.46) and λ ∈ (0, λ * ). By Lemma 3.4.3 it follows that on the boundary of the ball centered in the origin and of radius ρ in W s,p (Ω), denoted by B ρ (0), we have inf

∂ B ρ (0) J λ > 0.
On the other hand, by Lemma 3.4.4, there exists θ ∈ W s,p (Ω) such that J λ (tθ ) < 0 for all t > 0 small enough. Moreover for any u ∈ B ρ (0), we have

J λ (u) m 0 p ||u|| p -λ c 1 c q 4 ||u|| q .
It follows that

-∞ < c := inf B ρ (0) J λ < 0.
We let now 0 < ε < inf

∂ B ρ (0) J λ -inf B ρ (0)
J λ . Applying Theorem 1.5.3 to the functional

J λ : B ρ (0) -→ R, 82
Chapitre 3. On some nonlocal problems in fractional Sobolev spaces

we find u ε ∈ B ρ (0) such that              J λ (u ε ) < inf B ρ (0) J λ + ε, J λ (u ε ) < J λ (u) + ε||u -u ε ||, u = u ε . Since J λ (u ε ) inf B ρ (0) J λ + ε inf B ρ (0) J λ + ε < inf ∂ B ρ (0)
J λ , we deduce u ε ∈ B ρ (0).

Now, we define

Λ λ : B ρ (0) -→ R by Λ λ (u) = J λ (u) + ε||u -u ε ||.
It's clear that u ε is a minimum point of Λ λ and then

Λ λ (u ε + tv) -Λ λ (u ε ) t 0
for small t > 0, and any v ∈ B ρ (0). The above relation yields

J λ (u ε + tv) -J λ (u ε ) t + ε||v|| 0.
Letting t → 0 it follows that J λ (u ε ), v + ε||v|| > 0 and we infer that ||J λ (u ε )|| * ε. We deduce that there exists a sequence {v n } ⊂ B ρ (0) such that

J λ (v n ) -→ c and J λ (v n ) -→ 0. (3.47)
It is clear that {v n } is bounded in W s,p (Ω). Thus, there exists v ∈ W s,p (Ω), such that up to a subsequence {v n } converges weakly to v in W s,p (Ω). Since by Theorem 1.2.6, W s,p (Ω) is compactly embedded in L q (Ω). The above information combined with condition (F 1 ) and Hölder's inequality implies

Ω f (x, v n )(v n -v)dx c 1 Ω |v n | q-1 |v n -v| dx c 1 |v n | q-1 q q-1 ||v n -v|| q -→ 0. (3.48)
On the other hand, by (3.47) we have

lim n→∞ J λ (v n ), v n -v = 0. ( 3
(v n ), v n -v = 0, that is, lim n→∞ M(||v n || p ) Ψ (v n ), v n -v = 0. (3.50)
Since {v n } is bounded in W s,p (Ω), then

||v n || p -→ t 0 0 as n → ∞.
If t 0 = 0, then, we get {v n } converges strongly to 0 in W s,p (Ω). If t 0 > 0, it follows from the continuity of the function M that

M(||v n || p ) -→ M(t 0 ) as n → ∞.
Thus, by (M 1 ), for sufficiently large n, we get

M(||v n || p ) m 0 > 0. ( 3
lim n→∞ Ψ (v n ), v n -v = 0. (3.52)
From (3.52) and since v n converges weakly to v in W s,p (Ω), we can apply Lemma 3.4.2 we find that {v n } converges strongly to v in W s,p (Ω), so by (3.47) :

J λ (v) = c < 0 and J λ (v) = 0.
We conclude that v is a nontrivial weak solution for problem (P s ) for any λ ∈ (0, λ * ).

Next, by Lemma 3.4.5 and Proposition 3.4.1 we infer that J λ is coercive and weakly lower semi continuous in W s,p (Ω) for all λ > 0. Then Theorem 1.5.2 implies that there exists u λ ∈ W s,p (Ω) a global minimized of J λ and thus a weak solution of problem (P s ).

Now, we show that u λ is non trivial. Indeed, letting t 0 > 1 be fixed real and u 0 (x) = t 0 for all Chapitre 3. On some nonlocal problems in fractional Sobolev spaces

x ∈ Ω, we have u 0 ∈ W s,p (Ω) and by the mean value theorem and (F 2 ), we have

J λ (u 0 ) = 1 p M(||t 0 || p ) -λ Ω F(x,t 0 )dx = 1 p ||t 0 || p 0 M(τ)dτ -λ Ω F(x,t 0 )dx 1 p M(t 1 )|Ω||t 0 | p -λ c 2 Ω |t 0 | q dx = 1 p M(t 1 )|Ω||t 0 | p -λ c 2 |t 0 | q |Ω|,
where 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-theological fluids was proposed by Rajagopal and Ruzicka (we refer to [START_REF] Rajagopal | Mathematical modeling of electrorheological materials[END_REF][START_REF] Ruzicka | Electrorheological fluids : modeling and mathematical theory[END_REF] for more details). Another important application is related to image processing [START_REF] Malik | Scale-space and edge detection using anisotropic diffusion[END_REF] where this kind of diffusion operator is used to underline the borders of the distorted image and to eliminate the noise. From 86 Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces a mathematical standpoint, it is a hard task to show the existence of classical solutions, i.e., solutions which are continuously differentiable as many times as the order of the differential equations under consideration. However, the concept of weak solution is not enough to give a formulation to all problems and may not provide existence or stability properties.

t 1 ∈ [0, |t 0 | p ) is a positive constant. Since p > q, thus, for λ * > 0 large enough, J λ (u 0 ) < 0 for any λ ∈ [λ * , ∞). It follows that J λ (u λ ) < 0 for any λ ∈ [λ * , ∞)
In all the problems of this chapter, we assume that a : R -→ R is such that : ϕ : R -→ R defined by :

ϕ(t) =            a(|t|)t for t = 0, 0 for t = 0, (4.1)
is increasing homeomorphism from R onto itself. Let

Φ(t) = t 0 ϕ(τ)dτ. (4.2)
Then, Φ, is N-function.

Furthermore, we assume that Φ satisfies the following conditions

1 < ϕ -:= inf t 0 tϕ(t) Φ(t) ϕ + := sup t 0 tϕ(t) Φ(t) < +∞ (4.3) and the function [0, ∞) t → Φ( √ t) is convex. (4.4)
Remark 9. We point out certain examples of functions ϕ : R → R which are odd, increasing homeomorphisms from R into R and satisfy conditions (4.3) (see [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF]).

1)

Let

ϕ(t) = p|t| p-2 t ∀t ∈ R,
with p > 1, For this function it can be proved that

ϕ -= ϕ + = p.
Furthermore, in this particular case the corresponding Orlicz space L Φ (Ω) is the classical Lebesgue space L p (Ω) while the fractional Orlicz-Sobolev space W s L Φ (Ω) is the fractional Sobolev space in this particular case.

2) Consider

ϕ(t) = log(1 + |t|)|t| p-2 t, ∀t ∈ R with p > 1.
In this case it can be proved that

ϕ -= p, ϕ + = p + 1.
3) Let

ϕ(t) = |t| p-2 t log(1 + |t|) , if t = 0, ϕ(0) = 0 with p > 2.
In this case we have

ϕ -= p -1, ϕ + = p.
The problems in this chapter involving nonlocal intgro-differential operators named by fractional a(.)-laplacian operator (-∆) s a(.) and defined as following :

(-∆) s a(.) u(x) = 2 lim ε 0 R N \B ε (x) a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s , (4.5)
for all x ∈ R N .

The most canonical and important example of nonlocal operator is given by the fractional Laplacian :

(-∆) s u(x) = p.v. R N \B ε (x) (u(x) -u(y)) |x -y| N+2s dy.
which can be obtained by minimizing the Gagliardo seminorm of the fracional Sobolev space H s , s ∈ (0, 1) (see for instance [START_REF] Caffarelli | Traveling waves for a boundary reaction-diffusion equation[END_REF]) ; from this point of view, it is important to point out that when a(t) = 1 in (4.5), then (-∆) s a(.) becomes (a multiple of) (-∆) s . More generally, when a(t) = t p-2 in (P a ), 1 < p < ∞, we get the eigenvalue problem for the so called fractional p-Laplacian

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp dy.
However, the fractional a(.)-Laplace operator (-∆) s a(.) possesses more complicated non linearities than the farctional p-Laplace operator, due to the fact that (-∆) s a(.) is not homogeneous. These kind of operator appear in several problems in Mathematical physics, for example in Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces nonlinear elasticity [START_REF] Fukagai | Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on R N . Funkcial[END_REF] when

Φ(t) = (1 + t 2 ) α -1, α ∈ (1, N N -2
).

In plasticity [START_REF] Fukagai | Quasilinear elliptic equations with slowly growing principal part and critical Orlicz-Sobolev nonlinear term[END_REF] when

Φ(t) = t p ln(1 + t), where p ∈ -1 + √ 1 + 4N 2 , N -1 and N ≥ 3.
In biophysics and physics of plasmas [START_REF] He | The existence of a nontrivial solution to the p&q-Laplacian problem with nonlinearity asymptotic to u p-1 at infinity in R N[END_REF] when

Φ(t) = 1 p |t| p + 1 q |t| q
, where 1 < p < q < N and q ∈ (p, p * ).

Recently, we have seen a tremendous development of the theory of nonlocal operators ; such operators arise naturally in the context of stochastic Lévy processes with jumps and have been studied

thoroughly both from the point of view of Probability and Analysis as they proved to be accurate models to describe different phenomena in Physics, Finance, Image processing, or Ecology, see for instance [START_REF] Biler | Nonlinear diffusion of dislocation density and self-similar solutions[END_REF][START_REF] Bucur | Nonlocal diffusion and applications[END_REF][START_REF] Caffarelli | Traveling waves for a boundary reaction-diffusion equation[END_REF][START_REF] Bisci | Variational Methods for Nonlocal Fractional Problems[END_REF] and references therein.

In this chapter, we will study some nonlocal elliptic problems in the fractional Orlicz-Sobolev spaces by using some different methods.

• In section 4.2, by using the direct method in calculus variations, we obtain the existence of a weak solution for the following problem

           (-∆) s a(.) u = f (x, u) in Ω, u = 0 in R N \ Ω.
• In section 4.3, by using the mountain pass theorem, we obtain the existence of a weak solution for the following problem

           M Ω×Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N (-∆) s a(.) u = f (x, u) in Ω u = 0 in R N \ Ω.
• In section 4.4, we are interested to study the existence of weak solutions of the following

problem            (-∆) s a(.) u + a(|u|)u = λ f (x, u) in Ω u = 0 in R N \ Ω,
by means of Ekeland's variational principle and direct variational approach, we obtain the existence of λ * > λ * > 0 such that for any λ ∈ (0, λ * ) ∪ [λ * , ∞), problem (P a ) has a nontrivial weak solutions.

• In section 4.5, we are concerned with a class of nonlocal problems in fractional Orlicz-Sobolev spaces of the form

                         M Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N (-∆) s a(.) u = λ f (x, u) + µg(x, u) in Ω, u = 0 in R N \ Ω,
by using the three critical points theorem, we obtain the existence of a three weak solutions of this problem.

• In section 4.6, the aim goal is to study the existence of weak solutions of the following nonlocal problem

       (-∆) s a 1 (.) u -λ a 2 (|u|)u = f (x, u) + g(x)|u| q(x)-2 u in Ω u = 0 in R N \ Ω,
by using the mountain pass theorem and Fountain Theorem in critical point theory, we obtain the existence and multiplicity of nontrivial solutions of this problem.

• In section 4.7, the main goal is to study the eigenvalue problem

           M 1 (Ψ 1 (u))(-∆) s 1 a 1 (.) u + M 2 (Ψ 2 (u))(-∆) s 2 a 2 (.) u = λ |u| q(x)-2 u in Ω u = 0 in R N \ Ω,
we establish the existence of two positive constants λ 0 and λ 1 with λ 0 λ 1 such that any

λ ∈ [λ 1 , ∞
) is an eigenvalue, while any λ ∈ (0, λ 0 ) is not an eigenvalue for this problem.

• In section 4.8, we will study the existence of two non-negative weak solutions of the following nonlocal problem

(P a )        (-∆) s a 1 (.) u + a 1 (|u|)u = λ β 2 (x)a 2 (|u|)u + β 3 (x)a 3 (|u|)u in Ω u = 0 in R N \ Ω,
we introduce the Nehari manifold associated with (P a ) and we study its behavior by carefully analyzing the three parts corresponding to local minima, local maxima and points of inflection Finally, by using the Nehari manifold approach, we obtain the existence of two non-negative solutions of problem (P a ).

To simplify the notation, we put

D s u := u(x) -u(y) |x -y| s and dµ := dxdy |x -y| N .
dµ is a regular Borel measure on the set Ω × Ω. In this section, we establish the existence of a weak solution for the following Dirichlet type problem

(P a )            (-∆) s a(.) u = f (x, u) in Ω, u = 0 in R N \ Ω,
where

Ω is an open bounded subset in R N with Lipschitz boundary ∂ Ω, 0 < s < 1, f : Ω × R -→ R
is a Carathéodory function and (-∆) s a(.) is the fractional a(.)-Laplacian operator defined as

(-∆) s a(.) u(x) = 2 lim ε 0 R N \B ε (x) a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s .
When a(t) = t p-2 , problem (P a ) reduces to the fractional p-Laplacian problem

(P p )            (-∆) s p u = f (x, u) in Ω u = 0 in R N \ Ω,
where (-∆) s p is the fractional p-Laplacian operator which, up to normalization, may be defined as

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp dy.
In recent years, problem (P p ) has been studied in many papers, we refer to [START_REF] Azroul | Three solutions for a nonlocal fractional p-Kirchhoff Type elliptic system[END_REF][START_REF] Franzina | Fractional p-eigenvalues[END_REF][START_REF] Lindgren | Fractional eigenvalues[END_REF], in which the authors have used different methods to get existence of solutions for (P p ).

Assumptions and mains results

The aim goal is to prove the existence of a weak solution for problem (P a ) in fractional Orlicz Sobolev spaces, by means of the direct method in calculus of variations. For this, we suppose that f : Ω × R → R is a Carathéodory function satisfying the following conditions :

| f (x,t)| c 1 (1 + g(|t|)) a.e. (x,t) ∈ Ω × R N , (C 1 ) | f (x,t)| c 2 g(|t|) a.e. (x,t) ∈ Ω 0 × R N . (C 2 )
where g : R + → R + is nondecreasing, right continuous function, with g(0) = 0, g(t) > 0, ∀t > 0 and g(t) → ∞ as t → ∞ and c 1 , c 2 are two nonnegative constants and Ω 0 ⊂ Ω is an open bounded set. By setting

G(s) = s 0 g(t)dt, G(s) = s 0 g(t)dt,
where g(t) = sup {s : g(s) t}, we obtain complementary N-function which define corresponding

Orlicz spaces L G and L G . We will also the following conditions :

1 < q -= inf t>0 tg(t) G(t) q + = sup t>0 tg(t) G(t) < +∞. (C 3 ) 1 0 Φ -1 (τ) τ N+s N dτ < ∞ and ∞ 1 Φ -1 (τ) τ N+s N dτ = ∞. (C 4 ) lim t→∞ G(kt) Φ * (t) = 0 for all k > 0. ( C 5 )
Remark 10. Condition (C 5 ) implies that G ≺≺ A * . So by Theorem 2.3.2, the following embedding :

W s 0 L Φ (Ω) → L G (Ω)
is compact.

To deal with the problem (P a ), we work in closed space W s 0 L Φ (Ω) which can be equivalently renormed by setting . := [.] s,A . The dual space of (W s 0 L Φ (Ω), . ) is denoted by Remark 11. We note that the functional I :

((W s 0 L Φ (Ω)) * , . * ) . Definition 4.2.1. We say that u ∈ W s 0 L Φ (Ω) is a weak solution of problem (P a ) if Ω Ω a (|D s u|) D s uD s vdµ - Ω f (x, u)vdx = 0, for all v ∈ W s 0 L Φ (Ω).
W s 0 L Φ (Ω) -→ R in (4.6) is well defined. Indeed, if u ∈ W s 0 L Φ (Ω)
, then by condition (C 5 ), we have that u ∈ L G (Ω) and thus u ∈ L 1 (Ω). Hence, by the condition (C 1 ),

|F(x, u)| u 0 | f (x,t)|dt = c 1 (|u| + G(|u|))
and thus,

Ω |F(x, u)|dx < ∞.

Proofs of mains results

Lemma 4.2.1. Assume assumption (C 1 ) is satisfied. Then the functional H ∈ C 1 (W s 0 L Φ (Ω), R) and

H (u), v = Ω f (x, u)vdx for all u, v ∈ W s 0 L Φ (Ω). (4.7)
Proof. First, observe that by Remark 11, H is well-defined on W s 0 L Φ (Ω). Usual arguments show that H is Gâteaux-differentiable on W s 0 L A (Ω) with the derivative is given by (4.7). Actually, let

{u n } ⊂ W s 0 L Φ (Ω) be a sequence converging strongly to u ∈ W s 0 L Φ (Ω). Since W s 0 L Φ (Ω) is compactly embedded in L G (Ω), then {u n } converges strongly to u in L q (Ω). So
| H (u n ) -H (u), v | = Ω ( f (x, u n (x)) -f (x, u(x))) v(x)dx , || f (x, u n (x)) -f (x, u(x))|| G ||v|| G , c 3 || f (x, u n (x)) -f (x, u(x))|| G ||v||,
for some c 3 > 0. Thus, passing to the supremum for ||v|| 1, we get

||H (u n ) -H (u)|| * c 3 || f (., u n (.)) -f (., u(.))|| G .
Since f is a continuous function, then

f (x, u n (x)) -f (x, u(x)) -→ 0 as n → ∞ and | f (x, u n (x)) -f (x, u(x))| c 1 (2 + g(|u(x)|) + g(|u(x)|),
for almost everywhere x ∈ Ω. Note that the majorant function in the previous relation is in L G (Ω).

Hence, by applying the dominate convergence theorem we get that || f (x,

u n (x)) -f (x, u(x))|| G → 0 as n → ∞. This proves that H is continuous. Lemma 4.2.2. The function J ∈ C 1 (W s 0 L Φ (Ω), R) and J (u), v = Ω Ω a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s v(x) -v(y) |x -y| s dxdx |x -y| N for all u, v ∈ W s 0 L Φ (Ω). Moreover, for each u ∈ W s 0 L Φ (Ω), J (u) ∈ (W s 0 L Φ (Ω)) * .
Proof. First, it is easy to see that

J (u), v = Ω Ω a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s v(x) -v(y) |x -y| s dxdx |x -y| N (4.8) for all u, v ∈ W s 0 L Φ (Ω). It follows from (4.8) that J (u) ∈ (W s 0 L Φ (Ω)) * for each u ∈ W s 0 L Φ (Ω).
Next, we prove that So, for w ∈ W s 0 L Φ (Ω) we have D s w ∈ L Φ (Ω × Ω, dµ) and by Hölder's inequality,

J ∈ C 1 (W s 0 L Φ (Ω), R). Let {u n } ⊂ W s 0 L Φ (Ω) with u n -→ u strongly in W s 0 L Φ (Ω), then D s u n -→ D s u in L Φ (Ω × Ω,
Ω Ω (a(|D s u n k |)D s u n k -a(|D s u|)D s u)D s wdµ 2 a(|D s u n k |)D s u n k -a(|D s u|)D s u s,Φ [w] s,Φ 2 a(|D s u n k |)D s u n k -a(|D s u|)D s u s,Φ ||w|| .
By dominated convergence theorem we obtain that sup

||w|| 1 Ω Ω (a(|D s u n k |)D s u n k -a(|D s u|)D s u)D s wdµ -→ 0. (4.9)
Then, we have

||J(u n ) -J(u)|| * = sup ||v|| 1 | J(u n ) -J(u), v | -→ 0.
The proof of Lemma 4.2.2, is completed.

Combining Lemma 4.2.1 and Lemma 4.2.2, we get I ∈ C 1 (W s 0 L Φ (Ω), R) and Proof. First, note that the map :

I (u), v = Ω Ω a (|D s u|) D s uD s vdµ - Ω f (x, u)vdx for all u, v ∈ W s 0 L Φ (Ω).
u -→ Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N
is lower semicontinuous for the weak topology of W s 0 L A (Ω). Indeed, by Lemma 4.2.2, we have Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

J ∈ C 1 (W s 0 L Φ (Ω), R N ) and J (u), v = Ω Ω a (|D s u|) D s uD s vdµ
for all u, v ∈ W s 0 L Φ (Ω). On the other hand, since Φ is convex so J is also convex. Now, let {u n } ⊂ W s 0 L Φ (Ω) with u n u weakly in W s 0 L Φ (Ω). Then by convexity of J, we have

J(u n ) -J(u) J (u), u n -u ,
and hence, we obtain J(u) lim inf n→∞ J(u n ), that is, the map : Next we need to verify that u 0 is nontrivial. Let x 0 ∈ Ω 0 , 0 < R < 1 satisfy B 2R (x 0 ) ⊂ Ω 0 , where

u -→ Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N is lower semicontinuous. Let u n u weakly in W s 0 L Φ (Ω),
I(u) = Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N - Ω F(x, u)dx Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N -c 1 Ω G(|u|)dx -c 1 Ω |u|dx ||u|| ϕ --c 1 c 4 ||u|| q+ -c 1 c 5 ||u||, since ϕ -> q + > 1, so we have I(u) -→ ∞ as ||u|| -→ ∞, by
B 2R (x 0 )
I(tθ ) = Ω Ω Φ |tθ (x) -tθ (y)| |x -y| s dxdy |x -y| N - Ω F(x,tθ )dx ||tθ || ϕ --c 2 Ω 0 G(|tθ |)dx ||tθ || ϕ --c 2 ||tθ || q + G t p -||θ || ϕ --c 2 t q + ||θ || q + G .
Since ϕ -> q + and ||θ || q + G > 0, we have I(t 0 θ ) < 0 for t 0 ∈ (0,t) sufficiently small. Hence, the critical point u 0 of functional I satisfies I(u 0 ) I(t 0 θ ) < 0 = I(0), that is u 0 = 0.

Remark 12. In [START_REF] Boumazourh | Leray-Schauder's solution for a nonlocal problem in a fractional Orlicz-Sobolev space[END_REF], we have considered the same problem (P a ), and via the nonlinear Leray-Schauder alternative, we obtained the existence of a weak solution with different conditions.

Mountain pass type solution for a nonlacal fractional a-Kirchhoff type problem

In this section, we are interested to study the existence of weak solution of the following problem

(P a )            M Ω×Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N (-∆) s a(.) u = f (x, u) in Ω u = 0 in R N \ Ω,
where

Ω is an open bounded subset in R N , N 1, with Lipschitz boundary ∂ Ω, 0 < s < 1, M : [0, ∞) → (0, ∞) is a continuous function. f : Ω × R -→ R is a Carathéodory function and (-∆) s a(.)
is the nonlocal integro-differential operator of elliptic type defined as :

(-∆) s a(.) u(x) = 2 lim ε 0 R N \B ε (x) a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s , for all x ∈ R N .
A typical prototype for M, due to Kirchhoff in 1883, is given by

M(t) = a + bt α-1 , a, b 0, a + b > 0, t 0, (4.10) and      α ∈ (1, +∞) i f b > 0, α = 1 i f b = 0,
when M(t) > 0 for all t 0, Kirchhoff problems are said to be nondegenerate and this happens for example if a > 0 and b 0 in the model case (4.10). Otherwise, if M(0) = 0 and M(t) > 0 for all t > 0, the Kirchhoff problems are called degenerate and this occurs in the model case (4.10) when a = 0 and b > 0.

Notice that if a(t) = t p-2 , the problem (P a ) reduces to the fractional p-Laplacian problem

(P p )            M([u] s,p )(-∆) s p u = f (x, u) in Ω u = 0 in R N \ Ω,
where (-∆) s p is the fractional p-Laplacian operator, define by

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp dy.
In recent years, problem (P p ) has been studied in many papers, we refer to [START_REF] Pei | Ground State Solutions for Fractional p-Kirchhoff Equation with Subcritical and Critical Exponential Growth[END_REF][START_REF] Xiang | Multiplicity results for variable-order fractional Laplacian equations with variable growth[END_REF][START_REF] Xiang | Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian[END_REF], in which the authors have used different methods to get the existence of solutions for (P p ).

Motivated by the above papers, the aim of this section to study the existence of a weak solution for problem (P a ).

Assumptions and mains Results

By applying the mountain pass theorem, we prove the existence of a weak solution in fractional Orlicz-Sobolev spaces. For this, we suppose the following assumptions :

• M : [0, ∞) → (0, ∞
) is a continuous, nondecreasing function satisfying the following conditions :

there exists m 0 > 0 such that M(t) m 0 for all t 0, (

there exists θ > 0 such that M(t) θ M(t)t for all t 0 (M 2 )

where M(t) = t 0 M(τ)dτ.

• f : Ω × R → R is a Carathéodory function satisfying :

| f (x,t)| c 1 (1 + g(|t|)). (C 1 )
where g : R + → R + is nondecreasing, right continuous function, with g(0) = 0, g(t) > 0, ∀t > 0 and g(t) → ∞ as t → ∞ and c 1 is a nonnegative constant. By setting :

G(s) = s 0 g(t)dt, G(s) = s 0 g(t)dt,
where g(t) = sup {s : g(s) t}, we obtain complementar N-functions which define corresponding Orlicz spaces L G and L G . We will also assume that

1 < q -= inf t 0 tg(t) G(t) q + = sup t 0 tg(t) G(t) < +∞. (C 2 ) 1 0 Φ -1 (τ) τ N+s N dτ < ∞ and ∞ 1 Φ -1 (τ) τ N+s N dτ = ∞. (C 3 ) lim t→∞ G(kt) Φ * (t) = 0 for all k > 0. (C 4 )
There exist β > ϕ + θ and r > 0 such that for all |t| r and a.e. x ∈ Ω

t f (x,t) β F(x,t) 0 (C 5 )
where F(x,t) := t 0 f (x, τ)dτ and θ is given in assumption (M 2 ).

lim sup t→0 F(x,t) Φ(t) < 1 λ 1 uniformly for a.e. x ∈ Ω, ( C 6 )
where λ 1 is as in (4.13).

Note that assumption (C 5 ), is not the usual Ambrosetti-Rabinowitz condition, since here we suppose that β > ϕ + θ . This difference is caused by the function M in problem (P a ).

To deal with the problem (P a ), we work in closed space W s 0 L Φ (Ω) which can be equivalently renormed by setting . := [.] s,A . The dual space of (W s 0 L Φ (Ω), . ) is denoted by ((W s 0 L Φ (Ω)) * , . * ) .

To simplify the notation, we ask

Ψ(u) := Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N .
Now, we give the definition of weak solution for problem (P a ).

Definition 4.3.1. u ∈ W s 0 L Φ (Ω) is called a weak solution of problem (P a ) if, M (Ψ(u)) Ω Ω a(|D s u|)D s uD s vdµ = Ω f (x, u)vdx, (4.11
)

for all v ∈ W s 0 L Φ (Ω).
The main result of this section is as follows. Remark 13. We note that the functional I : W s 0 L Φ (Ω) -→ R in (4.12) is well defined. Indeed, if u ∈ W s 0 L Φ (Ω), then by condition (C 4 ), we have that u ∈ L G (Ω) and thus u ∈ L 1 (Ω). Hence, by the condition (C 1 ),

|F(x, u)| u 0 | f (x,t)|dt = c 1 (|u| + G(|u|))
and thus, for all u ∈ W s 0 L Φ (Ω).

Ω |F(x, u)|dx < ∞.

Main results and proofs

Proof. Let u ∈ W s 0 L Φ (Ω) and B R ⊂ R N \ Ω, that is, the ball of radius R > 0 in the complement of Ω. Then for all x ∈ Ω, y ∈ B R and all λ > 0 we have,

Φ(|u(x)|) = Φ |u(x) -u(y)| |x -y| s |x -y| s |x -y| N |x -y| N ,
this implies that,

Φ(|u(x)|) Φ |u(x) -u(y)| |x -y| s diam(Ω ∪ B R ) s diam(Ω ∪ B R ) N |x -y| N ,
we suppose α = diam(Ω ∪ B R ) s and we using the estimation (2.4) , we get

Φ(|u(x)|) max α, α ϕ + Φ |u(x) -u(y)| |x -y| s diam(Ω ∪ B R ) N |x -y| N ,
Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

therefore |B R |Φ(|u(x)|) max α, α ϕ + diam(Ω ∪ B R ) N B R Φ |u(x) -u(y)| |x -y| s dy |x -y| N . Then Ω Φ(|u(x)|)dx λ 1 Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N , where λ 1 = max α, α ϕ + diam(Ω ∪ B R ) N |B R | . 4.
I 2 (u), v = Ω f (x, u(x))v(x)dx (4.14)
for all u, v ∈ W s 0 L Φ (Ω). Proof. First, it is easy to see that

I 1 (u), v = M(Ψ(u)) Ω Ω a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s v(x) -v(y) |x -y| s dxdy |x -y| N , (4.15) for all u, v ∈ W s 0 L Φ (Ω). It follows from (4.15) that I 1 (u) ∈ (W s 0 L Φ (Ω)) * for each u ∈ W s 0 L Φ (Ω).
Next, we prove that

I 1 ∈ C 1 (W s 0 L Φ (Ω), R). Let {u n } ⊂ W s 0 L Φ (Ω) with u n -→ u strongly in W s 0 L Φ (Ω)
, then by Lemma 4.2.2, we have

Ψ(u n ) -→ Ψ(u).
M (Ψ(u n )) -→ M (Ψ(u)) . (4.16)
The proof of Lemma 4.3.3 is completed.

Next, we show an important lemma, namely that if the functional I of (4.12) satisfies the conclusion of Theorem 1.5.5, then it has a critical point.

I(u n ) -→ c 2 > 0 , ||I (u n )|| * -→ 0. (4.17)
Then there exists u ∈ W s 0 L Φ (Ω), such that

I(u) = c 2 , I (u) = 0.
Proof. It follows from (4.17 0 < tg(t) q + G(t) for all t > 0 (4. [START_REF] Azroul | Existence Results for Fractional p(x, .)-Laplacian Problem Via the Nehari Manifold Approach[END_REF] and

Ω∩{|u n | r} (F(x, u n ) -β -1 f (x, u n )u n )dx c 1 (1 + β -1 )r + (1 + β -1 q + )G(r) c 4 . (4.20) 104
Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces Thus, by (M 1 ), (M 2 ), and (4.18)-(4.20), we get

c 3 + c 3 ||u n || I(u n ) - 1 β I (u n ), u n M(Ψ(u n )) - 1 β M(Ψ(u n )) Ω Ω ϕ (h(u n )) h(u n ) dxdy |x -y| N - Ω∩{|u n | r} (F(x, u n ) -β -1 f (x, u n )u n )dx M(Ψ(u n )) - ϕ + β M(Ψ(u n ))Ψ(u n ) -c 4 1 - ϕ + θ β M(Ψ(u n )) -c 4 m 0 1 - ϕ + θ β ||u n || ϕ ∓ -c 4 . (4.21)
Hence, {u n } is bounded in W s 0 L Φ (Ω). Since W s 0 L Φ (Ω) is a reflexive space, we may assume that u n converges weakly to u in W s 0 L Φ (Ω). Further, since the embedding of

W s 0 L Φ (Ω) into L G (Ω) is compact, we obtain that u n -→ u in L G (Ω). It follows from Lemma 4.3.2 that lim n→∞ I 2 (u n ) = I 2 (u) and lim n→∞ I 2 (u n ) = I 2 (u) in (W s 0 L Φ (Ω)
) * and as

I (u n ) → 0 in (W s 0 L Φ (Ω)) * , then I 1 (u n ) -→ I 2 (u) , in (W s 0 L Φ (Ω)) * , (4.22)
since Ψ is a convex function and M is a convex non-decreasing function. So, I 1 is convex, then we have

I 1 (u n ) I 1 (u) + I 1 (u n ), u n -u
and therefore, using (4.22), we may deduce that lim sup

n→∞ I 1 (u n ) I 1 (u).
It further follows from the convexity of I 1 that it is weakly lower semicontinuous and hence lim inf

n→∞ I 1 (u n ) I 1 (u)
which implies that We finally show that I (u) = 0. The convexity of I 1 implies that I 1 is monotone and hence

lim n→∞ I 1 (u n ) = I 1 (u)
I 1 (u n ), u n -v I 1 (v), u n -v , for all v ∈ W s 0 L Φ (Ω)
by (4.22), we have

I 2 (u) -I 1 (v), u -v 0, for all v ∈ W s 0 L Φ (Ω), setting v = u -th, h ∈ W s 0 L Φ (Ω), t ∈ R + we get I 2 (u) -I 1 (u -th), h 0 for all h ∈ W s 0 L Φ (Ω)
, and hence letting t → 0 and using that h is arbitrary in W s 0 L Φ (Ω), we find that

I (u) = I 1 (u) -I 2 (u) = 0
and therefore u is a critical point of I.

On the geometry of the functional I

In this subsection we will show that under the conditions we have imposed on the functions a and f , the geometric conditions (G 1 ) and (G 2 ) of Theorem 1.5.5 will hold. Proof. For all u ∈ W s 0 L Φ (Ω) \ {0}, the functional I is satisfied :

I(u) = M(Ψ(u)) - Ω F(x, u)dx m 0 Ψ(u) - Ω F(x, u)dx = m 0 Ψ(u)   1 -Ω F(x, u)dx m 0 Ψ(u)    . (4.23)
Using the condition (C 6 ), we have that there exist ε ∈ (0, 1) and t 0 > 0 such that Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces F(x,t) 1ε λ 1 Φ(t) for all |t| t 0 and all x ∈ Ω.

We pose Ω 0 := {x ∈ Ω : |u(x)| t 0 }, then we have

Ω F(x, u(x))dx 1 -ε λ 1 Ω\Ω 0 Φ(u(x))dx + Ω 0 F(x, u(x))dx. (4.24)
By (4.13), we have

(1 -ε) Ω\Ω 0 Φ(|u(x)|)dx λ 1 Ψ (u) 1 -ε. (4.25)
Next, from (C 1 ) we have

F(x,t) c 1 (|t| + G(|t|)),
for all |t| t 0 and for almost every x ∈ Ω. Then

Ω 0 F(x, u)dx c 1 ||u|| L 1 + Ω G(|u|)dx c 1 ||u|| L 1 + ||u|| q - G + ||u|| q + G .
By, the embedding W s 0 L Φ → L G and W s 0 L Φ → L 1 , we have

Ω 0 F(x, u(x))dx c 5 c 1 (||u|| + ||u|| q -+ ||u|| q + ). (4.26)
Then, for ||u|| 1 we find

Ω 0 F(x, u(x))dx 3c 5 c 1 ||u||, (4.27)
where c 5 denote various positive constants. By Proposition 2.2.1, we have Which is a weak solutions to our problem (P a ).

Ω 0 F(x, u)dx m 0 Ψ(u) 3c 5 c 1 m 0 ||u|| 1-ϕ + . ( 4
I(u) m 0 Ψ(u) ε - 3c 5 c 1 m 0 ||u|| 1-ϕ + ε 2 Ψ(u)

Example

We present in this subsection an example of functions that satisfies the conditions of Theorem 4.3.1. For take

M(t) = a + bt α-1 (4.33) ϕ(t) = log(1 + |t|)|t| p-2 t (4.34) f (x, s) = f (s) = |s| δ -1 s (4.35)
where p ∈ (1, N -1), α 1 and δ + 1 > α(p + 1). We considerer the problem

(P log )            a + b(Ψ(u)) α-1 (-∆) s log u = |u| δ -1 u in Ω u = 0 in R N \ Ω,
where

(-∆) s log u = 2 p.v R N log(1 + |D s u|)|D
M(t) = at + b α t α and F(x, s) = F(s) = |s| δ +1 δ + 1 . (4.37)
We will next show that all the hypotheses of Theorem 4.3.2 are satisfied.

• First, we verify that (4. • On the other hand, it is easy to see that

M(t) = a + bt α-1 a > 0 for all t 0 and M(t) = t 0 M(τ)dτ 1 α M(t)t for all t 0.
So, for m 0 = a and θ = 1 α , (M 1 ) and (M 2 ) holds true.

• By L' Hôpital' s rule we have

lim t→0 Φ(t) t p+1 = lim t→0 ϕ(t) (p + 1)t p = 1 p + 1 lim t→0 log(1 + t) t = 1 p + 1 lim t→0 1 1 + t = 1 p + 1 .
We deduce that Φ is equivalent to t p+1 near zero. Using that fact and the remarks on p.248 in [1] we infer that the first integral in condition (C 3 ) holds true if and only if

1 0 τ 1 p+1 τ s+N N dτ < ∞, or s(p + 1) < N. (4.39)
Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

The last inequality holds since p < N -1. On the other hand, by the change of variable τ = Φ(t)

we have

t 1 Φ -1 (τ) τ N+s N dτ = Φ -1 (t) Φ -1 (1)
tϕ(t)

Φ(t) (Φ(t)) -s/N dt
Φ(t) = lim t→∞ 1 p log(1 + t)t p     1 - |t| 0 τ p 1 + τ dτ log(1 + t)|t| p     = ∞.
[log(1 + τ)] s/N τ sp/N 1 τ s(p+1)/N . Since s(p + 1) < N, we find ∞ Φ -1 (1) dτ τ s(p+1)/N = ∞, we conclude that ∞ 1 Φ -1 (τ) τ N+s N dτ = ∞.
So, the second integral in (C 3 ) is satisfied.

• We will next check that for any given δ > 0, conditions (C 1 ) and (C 5 ) are satisfied. Indeed, (C 1 ) is trivially satisfied with c 1 = 1 and g(t) = |t| δ -1 t. Also, since t f (t)

F(t) = δ +1 > α(p+1) = ϕ + θ , ( C
(δ + 1) lim sup t→∞ Φ -1 (t) t 1 δ +1 + s N , setting τ = Φ -1 (t) we obtain lim sup t→∞ Φ -1 * (kt) t 1 1+δ (δ + 1) lim sup τ→∞ τ Φ(τ) 1 δ +1 + s N . Now, since lim t→∞ t N(δ +1) N+s(δ +1) Φ(t) = lim t→∞ t N(δ +1) N+s(δ +1) log(1 + t)t p   1 - t 0 τ p 1 + τ dτ log(1 + t)t p    -1 , ( 4

Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space

In this section, we are interested to study the existence of weak solutions of the following problem

(P a )            (-∆) s a(.) u + a(|u|)u = λ f (x, u) in Ω u = 0 in R N \ Ω,
where and(-∆) s a(.) is the nonlocal integro-differential operator of elliptic type defined as :

Ω is an open bounded subset in R N , N 1, with Lipschitz boundary ∂ Ω, 0 < s < 1, f : Ω × R -→ R is a Carathéodory function,
(-∆) s a(.) u(x) = 2 lim ε 0 R N \B ε (x) a |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s , for all x ∈ R N .
When a(t) = t p-2 , problem (P a ) reduces to the fractional eigenvalue p-Laplacian problem

(P p )            (-∆) s p u + |u| p-2 u = λ f (x, u) in Ω u = 0 in R N \ Ω,
where (-∆) s p is the fractional p-Laplacian operator which, up to normalization, may be defined as

(-∆) s p u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N+sp

Assumptions and Main results

The main goal is to study the existence of a weak solutions in fractional Orlicz-Sobolev spaces.

For this, we suppose that f : Ω × R → R is a Carathéodory function satisfying :

| f (x,t)| c 1 |t| q-1 , (C 1 ) c 2 |t| q F(x,t) := t 0 f (x, τ)dτ, ( C 2 )
for all x ∈ Ω and all t ∈ R, where c 1 and c 2 are positive constants and q ∈ [1, ϕ -).

To deal with the problem (P a ), we work in closed space W s L Φ (Ω) and we denote . := . s,Φ .

The dual space of (W s L Φ (Ω), ||.||) is denoted by ((W s L Φ (Ω)) * , ||.|| * ) .

Example 3. We point out certain examples of functions f and F which satisfy hypotheses (C 1 ) and

(C 2 ) :
-f (x,t) = q|t| q-2 t and F(x,t) = |t| q where 2 q < ϕ -.

-f (x,t) = q|t| q-2 t +(q-2) log(1+t 2 )|t| q-4 t + t 1 + t 2 |t| q-2 and F(x,t) = |t| q +log(1+t 2 )|t| q-2 , where 4 q < ϕ -.

-f (x,t) = q|t| q-2 t +(q-1) sin(sint)×|t| q-3 t cos(sint) cost|t| q-1 and F(x,t) = |t| q +sin(sint)|t| q-1 , where 3 q < ϕ -. Now, we give the definition of weak solutions for problem (P a ).

Definition 4.4.1. u ∈ W s L Φ (Ω) is called a weak solution of problem (P a ) if, Ω Ω a(|D s u|)D s uD s vdµ + Ω a(|u|)uvdx -λ Ω f (x, u)vdx = 0, (4.49)
for all v ∈ W s L Φ (Ω).

The main result of this section is given by the follows theorem. For each λ > 0, we define the energy functional J λ : Remark 14. We note that the functional J λ : W s L Φ (Ω) -→ R in (4.50) is well defined. Indeed, if u ∈ W s L Φ (Ω), then, we have u ∈ L q (Ω) for all q ∈ [1, ϕ -). Hence, by the condition (C 1 ),

W s L Φ (Ω) -→ R by J λ (u) = Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ (|u(x)|) dx -λ Ω F(x, u)dx.
|F(x, u)| u 0 | f (x,t)|dt = c 1 |u| q
and thus,

Ω |F(x, u)|dx < ∞.

Proofs of main results

We first establish some basis properties of J λ .

Proposition 4.4.1. Assume condition (C 1 ) is satisfied. Then, for each λ > 0,

J λ ∈ C 1 (W s L Φ (Ω), R)
with the derivative given by

J λ (u), v = Ω Ω a(|D s u|)D s uD s vdµ + Ω a(|u|)uvdx -λ Ω f (x, u)vdx for all u, v ∈ W s L Φ (Ω).
To prove Proposition 4.4.1, we define the functionals

I i : W s L Φ (Ω) -→ R i = 1, 2 by I 1 (u) = Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ (|u(x)|) dx and I 2 (u) = Ω F(x, u)dx. Lemma 4.4.1. The functional I 1 : W s L Φ (Ω) -→ R is of class C 1 and I 1 (u), v = Ω Ω a (|D s u|) D s uD s vdµ + Ω a(|u|)uvdx, for all u, v ∈ W s L Φ (Ω).
Proof. First, it is easy to see that

I 1 (u), v = Ω Ω a (|D s u|) D s uD s vdµ + Ω a(|u|)uvdx (4.51) for all u, v ∈ W s L Φ (Ω). It follows from (4.51) that I 1 (u) ∈ (W s L Φ (Ω)) * for each u ∈ W s L Φ (Ω).
Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

Next, we prove that

I 1 ∈ C 1 (W s L Φ (Ω), R). Let {u n } ⊂ W s L Φ (Ω) with u n -→ u strongly in W s L Φ (Ω), then u n -→ u in L Φ (Ω) v n := D s u n -→ v := D s u in L Φ (Ω × Ω, dµ).
So by dominated convergence theorem, there exist a subsequence {u n k } (resp. {v n k }) and a function h 1 (resp. h 2 ) in

L Φ (Ω) (resp. L Φ (Ω × Ω, dµ)) such that, a(|u n k |)u n k -→ a(|u|)u a.e. x ∈ Ω a(|v n k |)v n k -→ a(|v|)v a.e. (x, y) ∈ Ω × Ω
and by Lemma 1.1.1, we have

|a(|u n k |)u n k | |a(|h 1 |)h 1 | ∈ L Φ (Ω) a.e in Ω |a(|v n k |)v n k | |a(|h 2 |)h 2 | ∈ L Φ (Ω × Ω, dµ) a.e in Ω × Ω.
So, for w ∈ W s L Φ (Ω) we have D s w ∈ L Φ (Ω × Ω, dµ) and by Hölder's inequality,

Ω Ω (a(|u n k |)u n k -a(|u|)u)wdx 2 ||a(|u n k |)u n k -a(|u|)u|| L Φ ||w|| Φ 2 ||a(|u n k |)u n k -a(|u|)u|| L Φ ||w|| and Ω Ω (a(|v n k |)v n k -a(|v|)v)D s wdµ 2 [a(|v n k |)v n k -a(|v|)v] s,Φ [w] s,Φ 2 [a(|v n k |)v n k -a(|v|)v] s,Φ ||w|| .
Then by dominated convergence theorem we obtain that sup with derivative given by

||w|| 1 Ω Ω (a(|u n k |)u n k -a(|u|)u)wdx -→ 0 (4.
||I 1 (u n ) -I 1 (u)|| * = sup ||v|| 1 | I 1 (u n ) -I 1 (u), v | -→ 0.
I 2 (u), v = Ω f (x, u(x))v(x)dx (4.54)
for all u, v ∈ W s L Φ (Ω).

Proof. First, observe that by the assumption of (C 1 ) and the embedding theorem, I 2 is well-defined on W s L Φ (Ω). Usual arguments shown that I 2 is Gâteaux-differentiable on W s L Φ (Ω) with the derivative is given by (4.54). Actually, let {u n } ⊂ W s L Φ (Ω) be a sequence converging strongly to u ∈ W s L Φ (Ω). Since W s L Φ (Ω) is compactly embedded in L q (Ω), then {u n } converges strongly to u in L q (Ω). So there exists a subsequence of {u n }, still denoted by {u n }, and a function u ∈ L q (Ω) such that {u n } converges to u almost everywhere in Ω and |u n | |u| for all n ∈ N and almost everywhere in Ω. Since f satisfies the assumption (C 1 ), we have for all measurable functions u : Ω -→ R, the operator defined by u -→ f (., u(.)) maps L q (Ω) into L q (Ω), where

1 q + 1 q = 1. Fix v ∈ W s L Φ (Ω)
with ||v|| 1, we using the Hölder's inequality and the embedding of W s L Φ (Ω) into L q (Ω), we have

| I 2 (u n ) -I 2 (u), v | = Ω ( f (x, u n (x)) -f (x, u(x))) v(x)dx , || f (x, u n (x)) -f (x, u(x))|| q ||v|| q , c|| f (x, u n (x)) -f (x, u(x))|| q ||v||,
for some c > 0. Thus, passing to the supremum for ||v|| 1, we get

||I 2 (u n ) -I 2 (u)|| * || f (., u n (.)) -f (., u(.))|| q .
By (C 1 ), we deduce

f (x, u n (x)) -f (x, u(x)) -→ 0 as n → ∞ and | f (x, u n (x)) -f (x, u(x))| c 1 (|u(x)| q-1 + |u(x)| q-1 ),
for almost everywhere x ∈ Ω. Note that the majorant function in the previous relation is in L q (Ω), hence, by applying the dominate convergence theorem we get that || f (x, u n (x))f (x, u(x))|| q → 0 as n → ∞. This proves that I 2 is continuous. Proof. First, note that the map :

u -→ Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ (|u(x)|) dx
is lower semi-continuous in the weak topology of W s L Φ (Ω). Indeed, similar to Lemma 4.4.1, we obtain I 1 ∈ C 1 (W s L Φ (Ω), R) and

I 1 (u), v = Ω Ω a (|D s u|) D s uD s vdµ + Ω a(|u|)uvdx
for all u, v ∈ W s L Φ (Ω). On the other hand, since Φ is a convex function so I 1 is also convex. Now, let {u n } ⊂ W s L Φ (Ω) with u n u weakly in W s L Φ (Ω), then by convexity of I 1 we have

I 1 (u n ) -I 1 (u) I 1 (u), u n -u ,
and hence, we obtain I 1 (u) lim inf Proof. We show that J λ is weakly lower semi continuous. Let {u n } ⊂ W s L Φ (Ω) be a sequence which convergences weakly to u in W s L Φ (Ω). By Lemma 4.4.3, we have I 1 is weakly lower semi continuous, that is,

I 1 (u) lim inf n→∞ I 1 (u n ).
On the other hand, using condition (C 1 ) and by Lemma 4.4.2, we have

lim n→∞ Ω F(x, u n )dx = Ω F(x, u)dx.
Thus, we find Then the sequence {u n } is convergence strongly to u in W s L Φ (Ω).

J λ (u) lim inf n→∞ J λ (u n ).
Proof

I 1 (u n ) -→ c.
Or since I 1 is weak lower semi continuous, we get

I 1 (u) lim inf n→∞ I 1 (u n ) = c.
On the other hand, by the convexity of I 1 , we have

I 1 (u) I 1 (u n ) + I 1 (u n ), u n -u .
Next, by the hypothesis (4.55), we conclude that

I 1 (u) = c.
Since u n + u 2 converges weakly to u in W s L Φ (Ω), so since I 1 is sequentially weakly lower semicontinuous :

c = I 1 (u) lim inf n→∞ I 1 u n + u 2 . (4.56)
We assume by contradiction that {u n } does not converge to u in W s L Φ (Ω). Hence, there exist a subsequence of {u n }, still denoted by {u n } and there exits ε 0 > 0 such that

u n -u 2 ε 0 2 ,
by Proposition 2.2.3, we have

I 1 u n -u 2 max ε ϕ - 0 , ε ϕ + 0 .
On the other hand, by the conditions (4.3) and (4.4), we can apply [96, Lemma 2.1] in order to obtain 1 2

I 1 (u n ) + 1 2 I 1 (u) -I 1 u n + u 2 I 1 u n -u 2 max ε ϕ - 0 , ε ϕ + 0 . (4.57)
It follows from (4.57) that Proof. By Theorem 2.3.1, it follows that W s L Φ (Ω) is continuously embedded in L q (Ω) for q ∈

I 1 (u) -max ε ϕ - 0 , ε ϕ + 0 lim sup n→∞ I 1 u n + u 2 , ( 4
[1, ϕ * s ). So there exists a positive constant c > 0 such that

||u|| q c||u|| ∀u ∈ W s L Φ (Ω). (4.59)
We fix ρ ∈ (0, 1) such that ρ < 1 c . Then relation (4.59) implies that for any u ∈ W s L Φ (Ω) with ||u|| = ρ :

J λ (u) ||u|| ϕ + -λ c 2 c q ||u|| q = ρ q ρ ϕ + -q -λ c q c 2 .
By the above inequality, we remark if we define

λ * = ρ ϕ + -q 2c 2 c q . (4.60)
Then for any u ∈ W s L Φ (Ω) with ||u|| = ρ, there exists α = ρ ϕ + 2 > 0 such that such that θ > 0 and J λ (tθ ) < 0 for t > 0 small enough.

J λ (u) α > 0.
Proof. Let Ω 0 ⊂⊂ Ω, for x 0 ∈ Ω 0 , 0 < R < 1 satisfy B 2R (x 0 ) ⊂ Ω 0 , where B 2R (x 0 ) is the ball of radius 2R with center at the point

x 0 in R N . Let θ ∈ C ∞ 0 (B 2R (x 0 )) satisfies 0 θ 1 and θ ≡ 1 in B R (x 0 ). Theorem 2.2.3 implies that ||θ || < ∞. Then for 0 < t < 1, by (C 2 ), we have J λ (tθ ) = Ω Ω Φ |tθ (x) -tθ (y)| |x -y| s dxdy |x -y| N + Ω Φ(|tθ |)dx -λ Ω F(x,tθ )dx ||tθ || ϕ --λ c 2 Ω 0 |tθ | q dx t ϕ -||θ || ϕ --λ c 2 t q Ω 0 |θ | q dx.
Since ϕ -> q + and Ω 0 |θ | q dx > 0 we have J λ (t 0 θ ) < 0 for t 0 ∈ (0,t) sufficiently small. Proof. For each u ∈ W s L Φ (Ω) with ||u|| > 1 and λ > 0, relations (2.16), (4.59) and the condition

(C 1 ) imply J λ (u) = Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ(|u|)dx -λ Ω F(x, u)dx ||u|| ϕ --λ c 1 Ω |u| q dx ||u|| ϕ --λ c 1 c||u|| q Since ϕ -> q the above inequality implies that J λ (u) -→ ∞ as ||u|| → ∞, that is, J λ is coercive.
Proof of Theorem 4.4.1. Let λ * > 0 be defined as in (4.60) and λ ∈ (0, λ * ). By Lemma 4.4.5 it follows that on the boundary oh the ball centered in the origin and of radius ρ in W s L Φ (Ω), denoted by B ρ (0), we have inf

∂ B ρ (0) J λ > 0.
On the other hand, by Lemma 4.4.6, there exists θ ∈ W s L Φ (Ω) such that J λ (tθ ) < 0 for all t > 0 small enough. Moreover for any u ∈ B ρ (0), we have

J λ (u) ||u|| ϕ + -λ c 1 c||u|| q . It follows that -∞ < c := inf B ρ (0) J λ < 0. We let now 0 < ε < inf ∂ B ρ (0) J λ -inf B ρ (0) J λ . Applying Theorem 1.5.3 to the functional J λ : B ρ (0) -→ R, we find u ε ∈ B ρ (0) such that              J λ (u ε ) < inf B ρ (0) J λ + ε, J λ (u ε ) < J λ (u) + ε||u -u ε ||, u = u ε . Since J λ (u ε ) inf B ρ (0) J λ + ε inf B ρ (0) J λ + ε inf ∂ B ρ (0)
J λ , we deduce u ε ∈ B ρ (0).

Now, we define

Λ λ : B ρ (0) -→ R by Λ λ (u) = J λ (u) + ε||u -u ε ||.
It's clear that u ε is a minimum point of Λ λ and then

Λ λ (u ε + tv) -Λ λ (u ε ) t 0
for small t > 0, and any v ∈ B ρ (0). The above relation yields

J λ (u ε + tv) -J λ (u ε ) t + ε||v|| 0.
Letting t → it follows that J λ (u ε ), v + ε||v|| > 0 and we infer that ||J λ (u ε )|| * ε. We deduce that there exists a sequence {v n } ⊂ B ρ (0) such that

J λ (v n ) -→ c and J λ (v n ) -→ 0. (4.61)
It is clear that {v n } is bounded in W s L Φ (Ω). Thus, there exists v ∈ W s L Φ (Ω), such that up to a subsequence {v n } converges weakly to v in W s L Φ (Ω). Since by Theorem 2.3.1, W s L Φ (Ω) is a compactly embedded in L q (Ω). The above information combined with condition (C 1 ) and Hölder's

Ω f (x, v n )(v n -v)dx c 1 Ω |v n | q-1 |v n -v| dx c 1 |v n | q-1 q q-1 ||v n -v|| q -→ 0. (4.62)
On the other hand, by (4.61) we have

lim n→∞ J λ (v n ), v n -v = 0. (4.63)
Relations (4.62) and (4.63) imply

lim n→∞ I 1 (v n ), v n -v = 0.
Thus, by Lemma 4.4.4 we find that {v n } converges strongly to v in W s L Φ (Ω), so by (4.61) :

J λ (v) = c < 0 and J λ (v) = 0.
We conclude that v is a nontrivial weak solution for problem (P a ) for any λ ∈ (0, λ * ).

Next, by Lemma 4.4.7 and Proposition 4.4.2 we infer that J λ is coercive and weakly lower semi continuous in W s L Φ (Ω) for all λ > 0. Then Theorem 1.5.2 implies that there exists

u λ ∈ W s L Φ (Ω)
a global minimized of J λ and thus a weak solution of problem (P a ). Now, we show that u λ is non trivial. Indeed, letting t 0 > 1 be fixed real and u 0 (x) = t 0 for all

x ∈ Ω, we have u 0 ∈ W s L Φ (Ω) and

J λ (u 0 ) = I 1 (u 0 ) -λ Ω F(x, u 0 )dx = Ω Φ(t 0 )dx -λ Ω F(x,t 0 )dx Ω Φ(t 0 )dx -λ c 2 Ω |t 0 | q dx = L -λ c|t 0 | q |Ω|,
where L is a positive constant. Thus, for λ * > 0 large enough, J λ (u 0 ) < 0 for any λ ∈ [λ * , ∞). It follows that J λ (u λ ) < 0 for any λ ∈ [λ * , ∞) and thus u λ is a nontrivial weak solution of problem (P a ) for any λ ∈ [λ * , ∞). Therefore, problem (P a ) has a nontrivial weak solution for all λ ∈ (0, λ * ) ∪

[λ * , ∞).

Multiple solutions for a nonlocal Kirchhoff problem in Fractional

Orlicz-Sobolev spaces

In this section, we are concerned with a class of nonlocal problems in fractional Orlicz-Sobolev spaces of the form

(P a )                          M Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N (-∆) s a(.) u = λ f (x, u) + µg(x, u) in Ω, u = 0 in R N \ Ω,
where The problem (P a ) is motivated by the class of problems on the form :

Ω is an open bounded subset in R N , (N 1) with Lipschitz boundary ∂ Ω, 0 < s < 1, M : [0, ∞) -→ (0, ∞) is a nondecreasing continuous function, f , g : Ω × R -→ R
(P)            Au = λ f (x, u) + µg(x, u) in Ω, u = 0 in ∂ Ω,
where Ω is an open subset of R N , f , g : R N × R -→ R are two Carathéodory functions and λ , µ are two real parameters. For Au = -∆ p = -div |∇u| p-2 ∇u , the problem (P) has been studied in many papers, we refer to [START_REF] Bonanno | Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces[END_REF][START_REF] Chen | The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions[END_REF][START_REF] Corrêa | On a p-Kirchhoff equation via Krasnoselskiiś genus[END_REF][START_REF] Ma | Remarks on an elliptic equation of Kirchhoff type[END_REF][START_REF] Ricceri | On an elliptic Kirchhoff-type problem depending on two parameters[END_REF], in which the authors have used different methods to get the existence of solutions for (P). In the case when Au = -∆ p(.) = -div |∇u| p(.)-2 ∇u , where p(.) is a continuous function, problem (P) has also been studied [START_REF] Cammaroto | Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator[END_REF][START_REF] Chung | Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities[END_REF][START_REF] Chung | Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities[END_REF][START_REF] Fan | On the Spaces L p(x) (Ω) and W m,p(x) (Ω)[END_REF]. On the other hand,

(P Φ )            -M Ω Φ(|∇u|)dx div(a(|∇u|∇u)) = λ f (x, u) + µg(x, u) in Ω, u = 0 in ∂ Ω,
where M : [0, ∞) -→ (0, ∞) is a nondecreasing continuous Kirchhoff function. Under some suitable conditions, the author obtained the existence of three weak solutions of (P Φ ), by using the three critical point theorem. For M ≡ 1 in the problem (P Φ ), Cammaroto and Vilasti in [START_REF] Cammaroto | Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator[END_REF], by the same theorem, they showed the existence of three weak solutions.

Using the ideas first presented in articles [10,[START_REF] Cammaroto | Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator[END_REF][START_REF] Chung | Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities[END_REF]. Our result in this section generalizes special cases, in which we will consider the problem (P a ) with M(t) = 1 or M(t) = 1 and Φ(t) = 1 p t p (see section 3.2).

Assumptions and mains results

By using three critical point theorem 1.5.7, we prove the existence of three weak solutions in fractional Orlicz-Sobolev spaces. First we give the following definition.

Ω × R -→ R such that sup (x,t)∈Ω×R | f (x,t)| 1 + |t| q-1 < ∞ (4.64)
for all q ∈ [1, ϕ * s ).

While when N < s ϕ -, we denote by A the class of all Carathéodory functions f :

Ω × R -→ R such that for each C > 0, the function x -→ sup |t| C | f (x,t)| belongs to L 1 (Ω).
Now, we suppose that the Kirchhoff function M : [0, ∞) -→ (0, ∞) is a continuous and nondecreasing function satisfying the following condition :

there exists m 0 > 0 such that M(t) m 0 for all t 0. (M 1 )

For f ∈ A , we assume that sup

u∈W s 0 L Φ (Ω) Ω F(x, u)dx > 0, (F 1 ) lim sup t→0 sup x∈Ω F(x,t) |t| ϕ + 0, (F 2 ) lim sup |t|→∞ sup x∈Ω F(x,t) |t| ϕ - 0 (F 3 )
where F(x,t) = t 0 f (x, τ)dτ.

Under such hypothesis, we set

Ψ(u) = Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N and θ * = inf      M(Ψ(u)) Ω F(x, u)dx : u ∈ W s 0 L Φ (Ω), Ω F(x, u)dx > 0      .
To deal with the problem (P a ), we work in closed space W s 0 L Φ (Ω) which can be equivalently renormed by setting . := [.] s,A . The dual space of (W s 0 L Φ (Ω), . ) is denoted by

((W s 0 L Φ (Ω)) * , . * ) . Definition 4.5.2. We say that u ∈ W s 0 L Φ (Ω) is a weak solution of problem (P a ) if M(Ψ(u)) R N ×R N a(|D s u|)D s uD s vdµ = λ Ω f (x, u)vdx + µ Ω g(x, u)vdx, for all v ∈ W s 0 L Φ (Ω).
Theorem 4.5.1. Suppose that M satisfy (M 1 ) and for f ∈ A , we suppose that (F 1 ), (F 2 ) and (F 3 ) hold true. If ϕ + < ϕ * s , then for each compact interval Λ ⊂ (θ * , ∞), there exists a number δ > 0 with the following property : for every λ ∈ Λ and every g ∈ A there exists µ * > 0 such that, for each µ ∈ (0, µ * ), problem (P a ) has at least three weak solutions whose norms are less than δ .

Proofs of mains results

We first prove the following useful result, which helps us to apply Theorem 1.5.7. For this, we define the functionals J, I :

W s 0 L Φ (Ω) -→ R by J(u) = Ω F(x, u)dx, and
I(u) = M (Ψ(u)) ,
where M(t) = t 0 M(τ)dτ.

, Ψ ∈ C 1 (W s 0 L Φ (Ω), R)
with derivative given by

J (u), v = Ω f (x, u)vdx and
I (u), v = M(Ψ(u)) Ω×Ω a(|D s u|)D s uD s vdµ for all u, v ∈ W s 0 L Φ (Ω). Moreover J : W s 0 L Φ (Ω) -→ (W s 0 L Φ (Ω)) * is compact.
Lemma 4.5.1. The following proprieties hold true :

(i) The functional I is sequentially weakly lower semi continuous.

(ii) The functional I belongs to the class W W s 0 L Φ (Ω) .

Proof. First, by Lemma 4.2.3 the map :

u -→ Ω Ω Φ |u(x) -u(y)| |x -y| s dxdy |x -y| N is lower semi-continuous in the weak topology of W s 0 L Φ (Ω).
M(lim inf n→∞ Ψ(u n )) M(Ψ(u)) = I(u).
Thus, the functional I is sequentially weakly lower semicontinuous.

(ii) Since M is continuous and strictly increasing, it suffices to show that Ψ ∈ W W s 0 L Φ (Ω) . Then, let {u n } be a sequence weakly converging to in W s 0 L Φ (Ω) and let lim inf n→∞ Ψ(u n ) Ψ(u). Since the functional Ψ is sequentially weakly lower semicontinuous, there exists a subsequence of {u n } , still denoted by {u n } such that :

lim n→∞ Ψ(u n ) = Ψ(u). ( 4
Ψ(u n ) + 1 2 Ψ(u) -Ψ u n + u 2 Ψ u n -u 2 max ε p - 0 , ε p + 0 . (4.67)
It follows from (4.67) that Then the sequence {u n } is convergence strongly to u in W s 0 L Φ (Ω).

Ψ(u) -max ε p - 0 , ε p + 0 lim sup n→∞ Ψ u n + u 2 , ( 4
Proof. Since u n converges weakly to u in W s 0 L Φ (Ω), then {||u n ||} is a bounded sequence of real numbers, that fact and Proposition 2.2.1, implies that the {Ψ(u n )} is bounded, then for a subsequence, we deduce that,

Ψ(u n ) -→ c 1 .
Or since Ψ is weak lower semi continuous, we get On the other hand, by the convexity of Ψ, we have

Ψ(u) lim inf n→∞ Ψ(u n ) = c 1 .
Ψ(u) Ψ(u n ) + Ψ (u n ), u n -u .
Next, by the hypothesis (4.69), we conclude that

Ψ(u) = c 1 . Since u n + u 2 converges weakly to u in W s 0 L Φ (Ω)
, so since Ψ is sequentially weakly lower semicontinuous :

c 1 = Ψ(u) lim inf n→∞ Ψ u n + u 2 .
Seminary to proof of Lemma 4.5.1, we assume by contradiction that u n is converge strongly to u in

W s 0 L Φ (Ω).
Lemma 4.5.3. Let (M 1 ) hold Then the operator I :

W s 0 L Φ (Ω) -→ (W s 0 L Φ (Ω))
* is invertible and

I -1 is continuous.
Proof. First, we assume that the operator I :

W s 0 L Φ (Ω) -→ (W s 0 L Φ (Ω)) * is invertible on W s 0 L Φ (Ω).
By the Minty-Browder theorem (see [START_REF] Zeidler | Nonlinear Functional Analysis and Applications[END_REF]), it suffices to prove that I is strictly monotone, hemicontinuous and coercive in the sense of monotone operators.

So, let u, v ∈ W s 0 L Φ (Ω), with u = v and let δ , β ∈ [0, 1] with δ + β = 1. Since a(|t|)t is increasing, then :

Ψ (u) -Ψ (v), u -v = Ω Ω (a(|D s u|)D s u -a(|D s u|)D s v) (D s u -D s v) dµ > 0. So, I : W s 0 L Φ (Ω) -→ (W s 0 L Φ (Ω)) * is strictly monotone, so by [164, Proposition 25.10], Ψ is strictly convex. Moreover, since M is nondecreasing the function M is convex in R + . Thus M(Ψ(δ u + β v)) < M(δ Ψ(u) + β Ψ(v)) δ M(Ψ(u)) + β M(Ψ(v)).
This shows that I is strictly convex and already said, that I is strictly monotone.

Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

Let u ∈ W s 0 L Φ (Ω), with ||u|| > 1, by (M 1 ) and Proposition 2.2.1, we have

I (u), u ||u| = M(Ψ(u)) Ψ (u), u ||u|| m 0 ϕ -Ψ(u) ||u|| m 0 ϕ -||u|| ϕ --1 .
Thus

lim ||u||→∞ I (u), u ||u|| = ∞, that is, I is coercive. Now, since I ∈ C 1 (W s 0 L Φ (Ω), R
), then I is hemicontinuous. Thus, in view of the Minty-Browder theorem, there exists

I -1 : (W s 0 L Φ (Ω)) * -→ W s 0 L Φ (Ω)
and it is bounded.

Let us prove that I -1 is continuous by showing that its is sequentially continuous. Let {u n } ⊂

(W s 0 L Φ (Ω)) * be a sequence strongly converging to u ∈ (W s 0 L Φ (Ω)) * and let v n = I -1 (u n ) and v = I -1 (u). Then, {v n } bounded in W s 0 L Φ (Ω)
, then, we can assume that it converges weakly to a certain

v 0 ∈ W s 0 L Φ (Ω).
Since u n converges strongly to u, we have

lim n→∞ I (v n ), v n -v 0 = lim n→∞ u n , v n -v 0 = 0, i.e. lim n→∞ M(Ψ(v n )) Ω Ω a(|D s v n |)D s v n (D s v n -D s v 0 ) dµ = 0. (4.70) Since {v n } is bounded in W s 0 L Φ (Ω), then by Proposition 2.2.1, Ψ(v n ) is also bounded, then Ψ(v n ) -→ t 0 0 as n → ∞.
If t 0 = 0, then using Proposition 2.2.1, we get {v n } that strongly converges to v 0 in W s 0 L Φ (Ω), by the continuity and injectivity of I -1 we obtain the desired result. If t 0 > 0, it follows from the continuity of the function M that

M(Ψ(v n )) -→ M(t 0 ) as n → ∞.
Thus, by (M 1 ), for sufficiently large n, we get

M(Ψ(v n )) c 2 > 0, ( 4
Ω Ω a(|D s v n |)D s v n (D s v n -D s v 0 ) dµ = 0. (4.72)
From (4.72) and since v n converges weakly to v 0 in W s 0 L Φ (Ω), we can apply Lemma 4.5.2, in order to deduce that v n converge strongly to v 0 in W s 0 L Φ (Ω). The continuity and injectivity of I imply that v n converges strongly to v, so I 1 is continuous.

Proof of Theorem 1.5.7. We wish to apply Theorem 1.5.4 taking X = W s 0 L Φ (Ω), I and J are as before, we have J is C 1 -functional with compact derivative. Moreover by Lemma 4.5.1, I is a sequentially weakly lower continuous and C 1 -functional belongs to the class W W s 0 L Φ (Ω) , also by Lemma 4.5.3, the operator I admits a continuous inverse on (W s 0 L Φ (Ω)) * .

On the other hand, we show that I is coercive. In fact, if ||u|| > 1, by (M 1 ) and Proposition 2.2.1, we have

I(u) = M(Ψ(u)) m 0 Ψ(u) m 0 ||u|| ϕ -.
From which we have the coercivety of I.

It is evident that u 0 = 0 is the global minimum of I and that I(u 0 ) = J(u 0 ) = 0. Moreover, I is bounded on each bounded subset of W s 0 L Φ (Ω). Indeed, if ||u|| c 3 , then

I(u) = M(Ψ(u)),            M(c ϕ - 3 ) if ||u|| > 1, M(1) if ||u|| 1. So I(u) max M(1), M(c ϕ - 3 ) .
Now, by the assumptions (F 2 ) for all ε > 0, there exit η 1 > 0 such that

|F(x,t)| ε|t| ϕ + (4.73)
for each x ∈ Ω and |t| η 1 . Since ϕ + < ϕ * s , so by Theorem 2.3.1, the embedding W s 0 L Φ (Ω) in

L ϕ + (Ω)
J(u) ε||u|| ϕ + L ϕ + εc 4 ||u|| ϕ + εc 4 Ψ(u).
Or by (M 1 ), we have Ψ(u)

1 m 0 I(u), then J(u) εc 4 1 m 0 I(u).
Consequently, we have

lim sup u→0 J(u) I(u) εc 4 1 m 0 . (4.74)
By (F 3 ), for all ε > 0, there exists η 2 > 0 such that

|F(x,t)| ε|t| ϕ - (4.75)
for all x ∈ Ω and |t| > η 2 .

For ||u|| > 1 large enough, from (4.75), Proposition 2.2.1 and Theorem 2.3.1, we have

J(u) I(u) = J(u) M(Ψ(u)) {x∈Ω:|u| η 2 } F(x, u)dx m 0 ||u|| ϕ - + {x∈Ω:|u|>η 2 } F(x, u)dx m 0 ||u|| ϕ - , |Ω| sup Ω×[-η 2 ,η 2 ] F m 0 ||u|| ϕ -+ ε||u|| ϕ - L ϕ -(Ω) m 0 ||u|| ϕ -, |Ω| sup Ω×[-η 2 ,η 2 ] F m 0 ||u|| ϕ -+ c 5 ε. So, lim sup ||u||→∞ J(u) I(u) εc 5 . (4.76)
Since ε > 0 is arbitrary, relations (4.74) and (4.76) implies that max lim sup

||x||→+∞ J(x) I(x) , lim sup x→x 0 J(x) I(x) 0.
Hence, all assumptions of Theorem 1.5.7 are satisfied. So, for each compact interval Λ ⊂ (θ * , +∞), there exists a number δ > 0 with the property described in the conclusion of Theorem 1.5.7. Fix λ ∈ Λ and g ∈ A . Put

Γ(u) = Ω G(x, u)dx, G(x,t) = t 0 g(x, s)ds for all u ∈ W s 0 L Φ (Ω).
Then, Γ is a C 1 functional on W s 0 L Φ (Ω) with compact derivative. So, there exists µ * > 0 such that, for each µ ∈ [0, µ * ], the equation

I (x) = λ J (x) + µΓ (x)
has at least three solutions whose norms are less than δ . But the solutions in W s 0 L Φ (Ω) of the above equation are exactly the weak solutions of problem (P a ) and thus, the proof of Theorem 4.5.1 is completed.

Example

We present in this subsection an example of functions that satisfies the conditions of Theorem 4.5.1. For take

ϕ(t) = log(1 + |t|)|t| p-2 t (4.77)
where p ∈ [2, N). Let b > max {2, ϕ + }, a > 0, b 0 and α 1 we consider

f (t) = b cos(t) sin(t)| sin(t)| b-2 ∀t ∈ R (4.78)
M(t) = a + bt α-1 ∀t 0. (4.79)
So, from (4.77), (4.78) and (4.79), we have

Ψ(s) = 1 p log(1 + |t|)|t| p - 1 p |t| 0 t p 1 + t dt , M(t) = at + b α t α and F(x,t) = F(t) = | sin(t)| b .
We will next show that all the hypotheses of Theorem 4.5.1 are satisfied.

By [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF]Example 2] it follows that

ϕ + = p + 1 and ϕ -= p. (4.80)
On the other hand, we point out that trivial computations imply that

d 2 Φ( √ t) dt 2 = 1 4 1 1 + | √ t| + (p -2) log(1 + | √ t|) 0
for all t ∈ R and thus, relations (4.3)-(4.4) are satisfied.

• For each t ∈ R, we claim that f ∈ A . Actually, the inequality

sup t∈R | f (t)| 1 + |t| q-1 < b < ∞,
holds for any 1 < q < ϕ * s . and the other hand, we have

lim |t|→0 | sin(t)| b |t| ϕ + = 0 and lim |t|→∞ | sin(t)| b |t| ϕ -= 0. Select a compact set V ⊂ Ω of positive measure and v ∈ W s 0 L Φ (Ω) such that v(x) = π 2 in V and 0 v(x) π 2 in Ω \V . We obtain Ω | sin(v(x))| b dx = |V | + Ω\V | sin(v(x))| b dx > 0.
Which means that (F 1 ), (F 2 ) and (F 3 ) are verified. Also, for m 0 = a the condition (M 1 ) is satisfied, we set

θ * = inf      aΨ(u) + b α (Ψ(u)) α Ω |sin(u(x))| b dx : u ∈ W s 0 L Φ (Ω), Ω |sin(u(x))| b dx > 0      .
Then, for a bounded domain Ω in R N of class C 0,1 , it follows from Theorem 4.5.1 that for each compact interval Λ ⊂ (θ * , +∞), there exist a number δ > 0 and µ * > 0 such that, for every λ ∈ Λ such that for all µ ∈ [0, µ * ], and all g ∈ A the following problem

           a + b(Ψ(u)) α-1 (-∆) s log u = λ b cos(u) sin(u)| sin(u)| b-2 + µg(x, u) in Ω u = 0 in R N \ Ω,
where

(-∆) s log u = 2 p.v R N log(1 + |D s u|)|D s u| p-2 D s udµ
has at least three weak solutions whose norms are less than δ .

4.6. Infinitely solutions for a fractional a(.)-laplace problem with sign-changing weight function 135

Infinitely solutions for a fractional a(.)-laplace problem with signchanging weight function

The aim of this section is to study the existence of weak solutions of the following nonlocal problem

(P a )        (-∆) s a 1 (.) u -λ a 2 (|u|)u = f (x, u) + g(x)|u| q(x)-2 u in Ω u = 0 in R N \ Ω,
where

Ω is a Lipschitz open bounded subset of R N , N 1, q : Ω → (1, +∞) is a bounded continuous function, s ∈ (0, 1), λ is a positive real parameter, f : Ω × R → R is a Carathéodory functions with
a subcritical growth conditions and (-∆) s a 1 (.) is a nonlocal integro-differential operator of elliptic type defined as

(-∆) s a 1 u(x) = 2 lim ε 0 R N \B ε (x) a 1 |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s ,
for all x ∈ R N and for any i = 1, 2, a i : R -→ R are defined as in (4.1).

When q(x) = q ∀x ∈ Ω, with q a positive constant and a 1 (t) = t p 1 -2 and a 2 (t) = t p 2 -2 , the problem (P a ) is the well known as the fractional p 1 -Laplacian problem

(P p )        (-∆) s p 1 u -λ |u| p 2 -2 u = f (x, u) + g(x)|u| q-2 u in Ω u = 0 in R N \ Ω,
where (-∆) s p 1 is the fractional p-Laplacian operator defined as

(-∆) s p 1 u(x) = 2 lim ε 0 R N \B ε (x) |u(x) -u(y)| p 1 -2 (u(x) -u(y)) |x -y| N+sp 1 dy.
Problem (P p ) have been extensively investigated in recent years, and many existence results have been obtained under general hypotheses [START_REF] Bartolo | Asymptotically linear fractional p-Laplacian equations[END_REF][START_REF] Bueno | Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition[END_REF][START_REF] Huang | Asymmetric critical fractional p-Laplacian problems[END_REF][START_REF] Ledesma | Existence and symmetry result for fractional p-Laplacian in R N[END_REF][START_REF] Massar | On a class of p-fractional Laplacian equations with potential depending on parameter[END_REF][START_REF] Mosconi | The Brezis-Nirenberg problem for the fractional p-Laplacian[END_REF]. In [START_REF] Bartolo | Asymptotically linear fractional p-Laplacian equations[END_REF] Bartolo and Bisci have studied the multiplicity of weak solutions to problem (P p ) with g(x) = 0. In [START_REF] Mosconi | The Brezis-Nirenberg problem for the fractional p-Laplacian[END_REF] Mosconi et. all.

have studied the existence of weak solution of (P p ) with g(x) = 0 and f (x, u) = |u| p * s -2 u. Furthermore, we also mention the recent work [START_REF] Bueno | Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition[END_REF] where Bueno et. all. have considered the existence and multiplicity of solution for problem (P p ) with exponential growth.

Assumptions and mains results

Before stating our results let us introduce the main ingredients involved in our approach. Let

Φ i (t) = t 0 ϕ i (τ)dτ for all t ∈ R, i = 1, 2.
We assume the following conditions :

1 0 Φ -1 1 (τ) τ N+s N dτ < ∞, ∞ 1 Φ -1 1 (τ) τ N+s N dτ = ∞ (H 1 )
and

lim t→∞ Φ 2 (kt) (Φ 1 ) * (t) = 0 ∀k > 0. ( H 2 )
Regarding the non-linearity f we assume that f : Ω × R → R is such that :

f : Ω × R → R is a Carathéodory function satisfying : f (x,t) = o(a 1 (|t|)t) t → 0 uniformly in x, f (x,t) = o(a 2 (|t|)t) t → ∞ uniformly in x. ( H 3 )
There exists β ∈ (ϕ + 1 , ϕ - 2 ) such that t f (x,t) β F(x,t) > 0 for all |t| 0 and a.e. x ∈ Ω (H 4 )

where F(x,t)

:= t 0 f (x, τ)dτ. f (x, -t) = -f (x,t) for all (x,t) ∈ Ω × R. (H 5 ) 0 g ∈ L ∞ (Ω). (H 6 ) Definition 4.6.1. So we say that u ∈ W s 0 L Φ 1 (Ω) is a weak solution of problem (P a ) if, Ω Ω a 1 (|D s u|)D s uD s vdµ -λ Ω a 2 (|u)uvdx - Ω f (x, u)vdx - Ω g(x)|u| q(x)-2 uv = 0, for all v ∈ W s 0 L Φ 1 (Ω).
Now we are in position to states our main results.

4.6. Infinitely solutions for a fractional a(.)-laplace problem with sign-changing weight function 137

Theorem 4.6.1. If λ = 0. Suppose that (H 1 )-(H 4 ) and (H 6 ) hold. Then there exists λ 0 > 0 such that if ||g|| q(x) < λ 0 , the problem (P a ) has at lest one nontrivial solution.

Theorem 4.6.2. If g(x) = 0. Suppose that (H 1 )-(H 5 ) hold. Then the problem (P a ) has infinitely many nontrivial solutions.

Proof of main results

Now, we are going to prove Theorem 4.6.1 and Theorem 4.6.2. We start our analysis with the following Remark :

Remark 15.

1. By (H 2 ), we can apply Theorem 2.3.2 and we obtain that

W s 0 L Φ 1 (Ω) is com- pactly embedded in L Φ 2 (Ω), that is, ||u|| Φ 2 c 1 ||u|| (4.81)
where c 1 > 0. On the other hand, since q + ϕ - 2 , that is,

q + tϕ 2 (t) Φ 2 (t) ∀t > 0
this implies for t > t 0 > 0,

|t| q + cΦ 2 (t) ∀t > t 0
then we obtain that W s 0 L Φ 1 (Ω) is compactly embedded in L q + (Ω) on particular in L q(x) (Ω) , that is,

||u|| q(x) c 2 ||u|| (4.82)
where c 2 > 0. Thus, a solution for a problem of type (P a ) will be sought in W s 0 L Φ 1 (Ω) which can be equivalently renormed by setting .

:= [.] s,Φ 1 . The dual space of (W s 0 L Φ 1 (Ω), ||.||) is denoted by ((W s 0 L Φ 1 (Ω)) * , ||.|| * ) .
In order to prove our main results, we introduce the following functional J in W s 0 L Φ 1 (Ω) by

J λ (u) = Ψ(u) -λ Ω Φ 2 (|u|)dx - Ω F(x, u)dx - 1 q(x) Ω g(x)|u| q(x) dx with Ψ(u) = Ω Ω Φ 1 |u(x) -u(y)| |x -
) to show that J λ ∈ C 1 (W s 0 L Φ 1 (Ω), R) and J λ (u), v = Ω Ω a 1 (|D s u|)D s uD s vdµ -λ Ω a 2 (|u)uvdx - Ω f (x, u)vdx - Ω g(x)|u| q(x)-2 uvdx,
for all u, v ∈ W s 0 L Φ 1 (Ω). Therefore, the critical points of J are weak solution of problem (P a ).

Lemma 4.6.1. Suppose that (H 1 )-(H 3 ) hold, let u n u weakly in W s 0 L Φ 1 (Ω). Then, up to a subsequence, we have

Ω a 2 (|u n |)u n (u n -u)dx -→ 0, (4.83)
Ω g(x)|u n | q(x)-2 u n (u n -u)dx -→ 0, (4.84
)

Ω f (x, u n )(u n -u)dx -→ 0. (4.85)
Proof. From Remark 15, up to a subsequence, we see

u n -→ u strongly in L Φ 2 (Ω)
and by dominated convergence theorem, there exist a subsequence, still denoted by u n , and h 1 ∈ L Φ 2 (Ω), such that, for almost everywhere x on Ω

|u n (x)| h 1 (x) ∀n ∈ N and u n (x) -→ u(x) ∀n ∈ N.
From the Hölder inequality and Lemma 1.1.1, we have

Ω a 2 (|u n |)u n (u n -u)dx Ω |a 2 (|h 1 |)h 1 ||u n -u|dx ||a 2 (|h 1 |)h 1 || Φ 2 ||u n -u|| Φ 1 → 0.
Next, we have W s 0 L Φ 1 (Ω) is compactly embedded in L q(x) (Ω) passing the a subsequence if necessary, to see

u n -→ u strongly in L q(x) (Ω)
and by dominated convergence theorem, there exist a subsequence, still denoted bu u n , and h 2 ∈ 4.6. Infinitely solutions for a fractional a(.)-laplace problem with sign-changing weight function 139

L q(x) (Ω), such that, for almost everywhere on Ω

|u n (x)| h 2 (x) ∀n ∈ N and u n (x) -→ u(x) ∀n ∈ N.
From the Hölder inequality, we have

Ω g(x)|u n | q(x)-2 u n |u n -u|dx Ω g(x)|h 2 | q(x)-1 |u n -u|dx ||g|| ∞ Ω |h 2 | q(x)-1 |u n -u|dx ||g|| ∞ ||h 2 || q(x)-1 q(x) ||u n -u|| q(x) → 0.
On the other hand, from, (H 3 ), given ε > 0 there exists c ε > 0 such that

| f (x,t)| εϕ 1 (t) + c ε ϕ 2 (t) ∀(x,t) ∈ Ω × R. Since W s 0 L Φ 1 (Ω) is compactly embedded in L Φ 1 (Ω) and in L Φ 2 (Ω). Then u n → u strongly in L Φ i (Ω)
for any i = 1, 2, so by dominated convergence theorem, for any i = 1, 2, there exist h i ∈ L Φ i (Ω) such that, for almost everywhere on Ω

|u n (x)| h i (x) ∀n ∈ N i = 1, 2 and u n (x) -→ u(x) ∀n ∈ N.
So, we have

Ω f (x, u n )(u n -u)dx ε Ω ϕ 1 (u n )(u n -u)dx + c ε Ω ϕ 2 (u n )(u n -u)dx ε Ω ϕ 1 (h 1 )|u n -u|dx + c ε Ω ϕ 2 (h 2 )|u n -u|dx ε||ϕ 1 (h 1 )|| Φ 1 ||u n -u|| Φ 1 + c ε ||ϕ 2 (h 2 )|| Φ 2 ||u n -u|| Φ 2 -→ 0.
This completes the proof.

Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces Lemma 4.6.2. There exist positive numbers ρ, δ , λ 0 such that J 0 (u) δ with ||u|| = ρ for all g satisfying ||g|| ∞ < λ 0 .

Proof. From, (H 3 ), given ε > 0 there exists c ε > 0 such that

|F(x,t)| εΦ 1 (t) + c ε Φ 2 (t) ∀(x,t) ∈ Ω × R. (4.86)
Then, by (4.13) and (4.86), we have

J 0 (u) = Ψ(u) - Ω F(x, u)dx - 1 q(x) Ω g(x)|u| q(x) dx Ψ(u) -ε Ω Φ 1 (|u|) -c ε Ω Φ 2 (|u|)dx - 1 q - Ω g(x)|u| q(x) dx (1 -ελ 1 )Ψ(u) -c ε Ω Φ 2 (|u|)dx - 1 q - Ω g(x)|u| q(x) dx (1 -ελ 1 )||u|| ϕ ∓ 1 -cε||u|| ϕ ∓ 2 Φ 2 - 1 q -||g|| ∞ ||u|| q ∓ q(x) (1 -ελ 1 )||u|| ϕ ∓ 1 -cε(c 1 ||u||) ϕ ∓ 2 - 1 q -||g|| ∞ (c 2 ||u||) q ∓ .
For ρ > 0 sufficiently small such that ρ = ||u|| < min 1,

1 c 1 , 1 c 2 , we have J 0 (u) (1 -ελ 1 )||u|| ϕ - 1 -cεc ϕ + 2 1 ||u|| ϕ + 2 - c q + 2 q -||g|| ∞ ||u|| q + = ||u|| q + (1 -ελ 1 )||u|| ϕ - 1 -q + -c ε c ϕ + 2 1 ||u|| ϕ + 2 -q + - c q + 2 q -||g|| ∞ .
Note that ε may be chosen small enough and 1 < q + < ϕ - 1 < ϕ + 2 , and we easily obtain that there exist ρ, δ 0 > 0 small enough such that :

(1 -ελ 1 )ρ ϕ - 1 -q + -c ε c ϕ + 2 1 ρ ϕ + 2 -q + δ 0 . Take λ 0 = q -δ 0 2c q + 2
, then we have

J 0 (u) ρ q + δ 0 - c q + 2 q -||g|| ∞ ρ q + δ 0 - c q + 2 q -λ 0 = δ 0 2 ρ q + , 4.6.
Infinitely solutions for a fractional a(.)-laplace problem with sign-changing weight function 141

with ||u|| = ρ. Therefore, we can choose δ = δ 0 2 ρ q + such that the conclusion holds.

Lemma 4.6.3. There exists e ∈ W s 0 L Φ 1 (Ω) with ||e|| > ρ such that J 0 (e) < 0, where ρ is given in Lemma 4.6.2.

Proof. From (H 4 ), we have

F(x,tu) t β F(x, u) for all t 1 and all u ∈ W s 0 L Φ 1 (Ω).
By Theorem 2.2.3, we can fix u 0 ∈ C ∞ 0 (Ω), such that ||u 0 || = 1 and let t 1, we have

J 0 (tu 0 ) = Ψ(tu 0 ) - Ω F(x,tu 0 )dx - 1 q(x) Ω g(x)|tu 0 | q(x) dx ||tu 0 || ϕ + 1 -t β Ω F(x, u 0 )dx - t q - q + Ω g(x)|u 0 | q(x) dx t ϕ + 1 -t β Ω F(x, u 0 )dx - t q - q + Ω g(x)|u 0 | q(x) dx.
Note that β > ϕ + 1 > q -> 1, so there exists t 0 > 0 large enough such that ||t 0 u|| > ρ and J 0 (t 0 u) < 0.

The proof is completed by taking e = Tu with T > 0 large enough.

Lemma 4.6.4. Suppose that {u n } ⊂ W s 0 L Φ 1 (Ω) is a (PS) c sequence of J 0 with c = 0. Then {u n } has a convergent subsequence in W s 0 L Φ 1 (Ω).

To prove this lemma, we recall the following result.

Lemma 4.6.5. (see Lemma 4.5.2) Assume that the sequence {u n } converges weakly to u in W s 0 L Φ 1 (Ω)

and

lim sup n→∞ Ψ (u n ), u n -u 0. (4.87)
Then the sequence {u n } is convergence strongly to u in W s 0 L Φ 1 (Ω).

Proof of Lemma 4.6.4.

Let {u n } ⊂ W s 0 L Φ 1 (Ω) is a (PS) c sequence of J 0 we assume that {u n } is bounded. Indeed for n large enough such that ||u n || > max 1, 1 c 2 .
From (H 4 ) we have

c + 1 + ||u n || J 0 (u n ) - 1 β J 0 (u n ), u n = Ψ(u n ) - Ω F(x, u n )dx - 1 q(x) Ω g(x)|u n | q(x) dx - Ω Ω ϕ(|D s u n |)D s u n dµ + Ω f (x, u n )u n dx + Ω g(x)|u n | q(x) dx 1 - ϕ - 1 β Ψ(u n ) - 1 q -- 1 β Ω g(x)|u n | q(x) dx 1 - ϕ - 1 β ||u n || ϕ - 1 - 1 q -- 1
β ||g|| ∞ c q + 2 ||u n || q + . (4.88)
Note that ϕ - 1 > q + > 1, so (4.88) implies that {u n } is bounded in W s 0 L Φ 1 (Ω). Thus, passing to a subsequence, we obtain that u n u in W s 0 L Φ 1 (Ω) weakly, and we have

J 0 (u n ), u n -u = Ψ (u n ), u n -u - Ω f (x, u n )(u n -u)dx - Ω g(x)|u n | q(x)-2 u n (u n -u)dx.
Note that, J 0 (u n ), u nu -→ 0 as n → ∞, from Lemma 4.6.1, we obtain

Ψ (u n ), u n -u -→ 0 as n → ∞.
Then by used Lemma 4.6.5 and the fact that u n u weakly, we have u n -→ u strongly in W s 0 L Φ 1 (Ω).

Proof of Theorem 4.6.1. From Lemma 4.6.2 -4.6.4 and by mountain pass theorem 1.5.5, J 0 has a positive critical value c, that is, there exists

u ∈ W s 0 L Φ 1 (Ω), such that J(u) = c > 0 and J 0 (u) = 0.
Thus u is a solution for (P a ). Thus completes the Proof. Now, we prove Theorem 4.6.2. Since W s 0 L Φ 1 (Ω) is a separable reflexive Banach space, then from [START_REF] Zhao | Structure theory of Banach spaces[END_REF] there are

{φ n } n∈N ⊂ W s 0 L Φ 1 (Ω) and {φ * n } n∈N ⊂ (W s 0 L Φ 1 (Ω)) * such that W s 0 L Φ 1 (Ω) = span {φ n , n ∈ N} and (W s 0 L Φ 1 (Ω)) * = span {φ * n , n ∈ N} 4.6.
Infinitely solutions for a fractional a(.)-laplace problem with sign-changing weight function 143 and

φ n , φ m =            1 n = m 0 n = m. For k = 1, 2, ..., let Y k = span {φ 1 , ..., φ k } and Z k = span {φ k , φ k+1 ...}.
We first given some preliminary lemmas.

Lemma 4.6.6. Under the assumptions of Theorem 4.6.2.

Then b k := inf u∈Z k ,||u||=r k J λ (u) -→ ∞ as k → ∞ and a k := max u∈Y k ,||u||=ρ k J λ (u) 0.
Proof. By (H 4 ), there exist

d 1 > 0, M > 0 such that F(x,t) d 1 |t| β ∀|t| M, x ∈ Ω, (4.89)
and by (H 3 ), for every ε > 0 and all |t| M, we have

|F(x,t)| εΦ 1 (t) + c ε Φ 2 (t) εΦ 1 (t) + c ε Φ 2 (M) = εΦ 1 (t) + c ε .
Then combining this with (4.89), we find

F(x,t) d 1 |t| β -εΦ 1 (t) -c ε ∀(x,t) ∈ Ω × R. (4.90)
Then, it following from (4.90) that

J λ (u) = Ψ(u) -λ Ω Φ 2 (|u|)dx - Ω F(x, u)dx ||u|| ϕ ∓ 1 + Ω c ε + εΦ 1 (|u|) -d 1 |u| β dx ||u||ϕ ∓ 1 + c ε |Ω| + ε||u|| ϕ ∓ 1 Φ 1 -d 1 ||u|| β L 1 ||u|| ϕ + 1 + c ε |Ω| + εd ϕ + 1 2 ||u|| ϕ + 1 -d 1 d β 3 ||u|| β , (4.91)
the above inequality is given because all norms are equivalent on the finite dimensional space Y k and for by take ||u|| max 1, On the other hand, let

1 d 2 , 1 d 3 . So, since β > ϕ + 1 > 1, there exists d k max 1, 1 d 2 , 1 d
B 1 (k) = sup u∈Z k ,||u||=1 ||u|| ϕ 1 and B 2 (k) = sup u∈Z k ,||u||=1 ||u|| ϕ 2 , we have B i (k) -→ 0 as k → ∞. Now for u ∈ Z k with ||u|| = r k = 1 B 1 (k) + B 2 (k)
, from (4.91), we obtain

J λ (u) = Ψ(u) -λ Ω Φ 2 (|u|)dx - Ω F(x, u)dx Ψ(u) -λ Ω Φ 2 (|u|)dx -ε Ω Φ 1 (|u|)dx -c ε Ω Φ 2 (|u|)dx ||u|| ϕ ∓ 1 -λ ||u|| ϕ ∓ 2 Φ 2 -ε||u|| ϕ ∓ 1 Φ 1 -c ε ||u|| ϕ ∓ 2 Φ 2 ||u|| ϕ ∓ 1 -λ B ϕ ∓ 2 2 (k)||u|| ϕ ∓ 2 -εB ϕ ∓ 1 1 (k)||u|| ϕ ∓ 1 -c ε B ϕ ∓ 1 1 (k)||u|| ϕ ∓ 2 r ϕ ∓ 1 k -λ -ε -c ε -→ ∞ as k → ∞. (4.93)
Hence,

b k := inf u∈Z k ,||u||=r k J λ (u) -→ ∞ as k → ∞.
Combining this and (4.92), we can take ρ k = max {d k , r k + 1} and we have

a k := max u∈Y k ,||u||=ρ k J λ (u) 0.
This completes the proof.

Lemma 4.6.7. Under the assumptions of Theorem 4.6.2, then every (PS) c sequence has a convergence of subsequence.

Proof. Let {u n } be a sequence of J λ . Then

J λ (u n ) -→ c in R and J λ (u n ) -→ 0 as n → ∞,
we claim {u n } is bounded. Indeed, note that

J λ (u n ) - 1 β J λ (u n ), u n = Ψ(u n ) - 1 β Ω×Ω ϕ 1 (D s u n )D s u n dµ -λ Ω Φ 2 (|u n |)dx + λ β Ω ϕ 2 (u n )u n dx - Ω F(x, u n )dx + 1 β f (x, u n )u n dx, consequently 4.6.
Infinitely solutions for a fractional a(.)-laplace problem with sign-changing weight function 145

λ ϕ - 2 β -1 Ω Φ 2 (|u|)dx λ β Ω ϕ 2 (u n )u n dx -λ Ω Φ 2 (|u n |)dx = J λ (u n ) - 1 β J λ (u n ), u n -Ψ(u n ) + 1 β Ω×Ω ϕ 1 (D s u n )D s u n dµ + Ω F(x, u n )dx - 1 β f (x, u n )u n dx, ||u n || + 1 + c + ϕ + 1 β -1 Ψ(u n ) ||u n || + 1 + c. So, Ω Φ 2 (|u n |)dx c(||u n || + 1).
Then, by (4.13) and (4.86) we have

Ψ(u n ) = J λ (u n ) + λ Ω Φ 2 (|u n |)dx + Ω F(x, u n )dx J λ (u n ) + λ Ω Φ 2 (|u n |)dx + ε Ω Φ 1 (|u|)dx + c ε Ω Φ 2 (|u|)dx c + o n (1) + (λ + c ε ) Ω Φ 2 (|u n |)dx + ελ 1 Ψ(u n ).
This implies that

(1 -ελ 1 )Ψ(u n ) c(1 + ||u n ||) + o n (1).
Since ε is arbitrary, then for ε sufficiency small and for n sufficiently large, we have

Ψ(u n ) c(1 + ||u n ||). If ||u n || > 1, by Proposition 2.2.1, it following that ||u n || ϕ - 1 c(1 + ||u n ||).
Using that ϕ - 1 > 1, the above inequality given that {u n } is bounded in W s 0 L Φ 1 (Ω). Above all, from Lemma 4.6.4, we obtain the desired assertion.

Proof Theorem 4.6.2. By (H 5 ) f is odd, then J is an even functional. From Lemma 4.6.6 and Lemma 4.6.7, the functional J satisfies all the conditions of Fountain theorem 1.5.6 Hence, J has an unbounded sequence of critical values, that is there exists a sequence {u n } ⊂ W s 0 L Φ 1 (Ω) such that

Eigenvalue problem associated with nonhomogeneous integro-differential operators

The main goal of this section is to study the eigenvalue problem

(P a )            M 1 (Ψ 1 (u))(-∆) s 1 a 1 (.) u + M 2 (Ψ 2 (u))(-∆) s 2 a 2 (.) u = λ |u| q(x)-2 u in Ω u = 0 in R N \ Ω,
where

Ω is a Lipschitz open bounded subset of R N , N 1, q : Ω → (1, +∞) is a bounded continuous function, 0 < s 2 < s 1 < 1, λ is a positive real parameters. • f : Ω × R → R is a
Carathéodory functions with a subcritical growth conditions.

• For any i = 1, 2, M i : R + → R * + is a nondecreasing continuous function which satisfies the condition :

there exists m i > 0 such that M i (t) m i for all t ∈ R + . ( M 1 )
• For any i = 1, 2, (-∆) s i a i (.) are two nonlocal integro-differential operators of elliptic type defined as :

(-∆) s a i (.) u(x) = 2 lim ε 0 R N \B ε (x) a i |u(x) -u(y)| |x -y| s i u(x) -u(y) |x -y| s dy |x -y| N+s i ,
for all x ∈ R N , where for any i = 1, 2, a i : R -→ R are defined as in (4.1) and the constant λ > 0.

• For any i = 1, 2, Ψ i are a modular functions associated to (-∆) s a i (.) defined by

Ψ i (u) = Ω×Ω Φ i |u(x) -u(y)| |x -y| s i dxdy |x -y| N
In the last decade, great attention has been devoted to the study of eigenvalue problems involving nonlocal operators. The archetypal example being the fractional p-Laplacian, have been widely studied and are by now fairly well understood. That is, in the works of Franzina & Palatucci [START_REF] Franzina | Fractional p-eigenvalues[END_REF] and of Lindgren & Linqvist [START_REF] Lindgren | Fractional eigenvalues[END_REF], the eigenvalue problem associated with (-∆) s p u is studied, and particularly some properties of the first eigenvalue and of the higher order (variational) eigenvalues are obtained. For nonhomogeneous problems in Orlicz spaces, i.e. the local problems have also been studied thoroughly, see for instance Clément et al [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF], Mustonen and Tienari [START_REF] Mustonen | An eigenvalue problem for generalized Laplacian in Orlic-Sobolev spaces[END_REF], Mihȃilescu & Rȃdulescu, [START_REF] Mihȃilescu | Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces[END_REF] and Mihȃilescu et al [START_REF] Mihȃilescu | On a non-homogeneous eigenvalue problem involving a potential : An Orlicz-Sobolev space setting[END_REF], these last two reference being a primary source of Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces motivation for our present work.

Assumptions and mains results

Before stating our results let us introduce the main ingredients involved in our approach. Let

Φ i (t) = t 0 ϕ i (τ)dτ for all t ∈ R, i = 1, 2.
We analyze problem (P a ) under the following basic assumptions :

1 0 Φ -1 2 (τ) τ N+s 2 N dτ < ∞, ∞ 1 Φ -1 2 (τ) τ N+s 2 N dτ = ∞, (V 1 ) 1 < ϕ - 2 ϕ + 2 < q(x) < ϕ - 1 ϕ + 1 < ∞ ∀x ∈ Ω, (V 2 ) lim t→∞ |t| q + (Φ 2 ) * (kt) = 0 ∀k > 0. ( V 3 )
We first by some auxiliary results.

Lemma 4.7.1. If the relation (V 2 ) holds true. Then we have the continuous embedding

W s 1 L Φ 1 (Ω) ⊂ W s 2 L Φ 2 (Ω).
Proof. Since ϕ + 2 < ϕ - 1 it is easy to see that Φ 1 dominates Φ 2 near infinity (see [START_REF] Mihȃilescu | Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces[END_REF]), that is, there exist k > 0 and t 0 > 0 such that

Φ 2 (t) Φ 1 (kt), ∀t t 0 . (4.94)
So, by [1, Lemma 8.10 (b)] we have

L Φ 1 (Ω) → L Φ 2 (Ω). (4.95)
Now, let u ∈ W s 1 L Φ 1 (Ω) and λ > 0, we define

Ω 1 = (x, y) ⊂ Ω × Ω |D s 1 u| λ t 0 Ω 2 = Ω × Ω \ Ω 1 .

Eigenvalue problem associated with nonhomogeneous integro-differential operators 149

Then

Ω Ω Φ 2 |D s 2 u| λ dµ = Ω Ω Φ 2 |D s 1 u| λ 1 |x -y| s 2 -s 1 dµ Ω Ω Φ 2 |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) = Ω 1 + Ω 2 Φ 2 |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) = I + II, where p = 1 if |x -y| > 1 and p = ϕ + if |x -y| 1 for all (x, y) ∈ Ω × Ω. Notice that I = Ω 1 Φ 2 |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) Φ 2 (t 0 ) Ω 1 dxdy |x -y| N+p(s 2 -s 1 ) := c,
note that the above integral is finite since N > N + p(s 2s 1 ). To estimate the second term, we invoke (4.94) and obtain

II Ω 2 Φ 1 k |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) d p(s 1 -s 2 ) Ω 2 Φ 1 k |D s 1 u| λ dxdy |x -y| N < ∞, when d = d(Ω) is the diameter in Ω. Then Ω Ω Φ 2 |D s 2 u| k[u] s 1 ,Φ 1 dµ c + d p(s 1 -s 2 ) ,
this implies that

[u] s 2 ,Φ 2 k c + d p(s 1 -s 2 ) [u] s 1 ,Φ 1 . (4.96)
The proof of Lemma 4.7.1 is completed.

Lemma 4.7.2. (see [START_REF] Mihȃilescu | Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces[END_REF]) Assume relation (V 2 ) holds true. Then, there exists c > 0 such that the following inequality holds true

c (Φ 1 (t) + Φ 2 (t)) t ϕ - 1 + t ϕ + 2 ∀t 0.
It following by Lemma 4.7.

1 that W s 1 0 L Φ 1 (Ω) is continuously embedded in W s 2 0 L Φ 2 (Ω). Thus, a
solution for a problem of type (P a ) will be sought in W s 1 0 L Φ 1 (Ω) which can be equivalently renormed by setting . := [.] s 1 ,Φ 1 and the dual space of

W s 1 0 L Φ 1 (Ω), ||.|| is denoted by (W s 1 0 L Φ 1 (Ω)) * , ||.|| * .
Definition 4.7.1. We say that λ ∈ R is an eigenvalue of problem (P a ) if there exists u

∈ W s 1 L Φ 1 (Ω) \ {0} such that M 1 (Ψ 1 (u)) Ω×Ω a 1 (|D s 1 u|)D s 1 uD s 1 vdµ + M 2 (Ψ 2 (u)) Ω×Ω a 2 (|D s 2 u|)D s 2 uD s 2 vdµ -λ Ω |u| q(x)-2 uvdx = 0 for all v ∈ W s 1 L Φ 1 (Ω).
We pont that if λ is an eigenvalue of problem (P a ) then the corresponding

u ∈ W s 1 L Φ 1 (Ω) \ {0}
is a weak solution of (P a ).

Define

λ 1 = inf u∈W s L Φ 1 (Ω)\{0} M 1 Ω×Ω Φ 1 (|D s 1 u|)dµ + M 2 Ω×Ω Φ 2 (|D s 2 u|)dµ Ω 1 q(x) |u| q(x) dx (4.97)
where for any i = 1, 2, M(t) = t 0 M(τ)dτ. Our main results is given by the following theorem.

Theorem 4.7.1. Assume that conditions (V 2 ) and (V 3 ) are fulfilled. Then λ > 0 Moreover, any

λ ∈ [λ 1 , ∞
) is an eigenvalue of problem (P a ). Furthermore, there exists a positive constant λ 0 such that λ 0 λ 1 and any λ ∈ (0, λ 0 ) is not an eigenvalue of problem (P a ).

Remark 16. By (V 3 ), we can apply Theorem 2.3.2 to obtain that W s 2 0 L Φ 2 (Ω) is compactly embedded in L q+ (Ω). That fact combined with the continuous embedding of L q + (Ω) in L q(x) (Ω) and with Lemma 4.7.1, implies that W s 1 0 L Φ 1 (Ω) is compactly embedded in L q(x) (Ω).

Proofs of mains results

In order to prove our main result, we introduce the following functionals J, I, J 1 , I 1 :

W s 1 0 L Φ 1 (Ω) → R by J(u) = M 1 Ω×Ω Φ 1 |u(x) -u(y)| |x -y| s 1 dµ + M 2 Ω×Ω Φ 2 |u(x) -u(y)| |x -y| s 2 dµ , I(u) = Ω 1 q(x) |u| q(x) dx, J 1 (u) = M 1 (Ψ 1 (u)) Ω×Ω a 1 (|D s 1 u|)|D s 1 u| 2 dµ + M 2 (Ψ 2 (u)) Ω×Ω a 2 (|D s 2 u|)|D s 2 u| 2 dµ, I 1 (u) = Ω |u| q(x) dx.
, I ∈ C 1 (W s 1 0 L Φ 1 (Ω), R), J (u), v =M 1 (Ψ 1 (u)) Ω×Ω a 1 (|D s 1 u|)D s 1 uD s 1 vdµ + M 2 (Ψ 2 (u)) Ω×Ω a 2 (|D s 2 u|)D s 2 uD s 2 vdµ, and
I (u), v = Ω |u| q(x)-2 uvdx, for all u, v ∈ W s 1 0 L Φ 1 (Ω).
We split to proof of Theorem 4.7.1 into four steps.

• Step 1. We show that λ 1 > 0.
For all u ∈ W s 1 0 L Φ 1 (Ω). By using Propositions 2.2.1, Remark 16 and Condition (M 1 ), we have

J(u) = M 1 Ω×Ω Φ 1 (|D s 1 u|)dµ + M 2 Ω×Ω Φ 2 (|D s 2 u|)dµ min {m 1 , m 2 } Ω×Ω Φ 1 (|D s 1 u|)dµ + Ω×Ω Φ 2 (|D s 2 u|)dµ min {m 1 , m 2 } ||u|| ϕ ∓ 1 1 + ||u|| ϕ ∓ 2 2 min {m 1 , m 2 } min c ϕ ∓ 1 1 , c ϕ ∓ 2 2 ||u|| ϕ ∓ 1 q(x) + ||u|| ϕ ∓ 2 q(x) = c ||u|| ϕ ∓ 1 q(x) + ||u|| ϕ ∓ 2 q(x) , (4.98)
where c = min {m 1 , m 2 } min c

ϕ ∓ 1 1 , c ϕ ∓ 2 2
. Now, by Proposition 1.3.1, if ||u|| q(x) > 1, then

J(u) c||u|| ϕ ∓ 1 q(x) c||u|| ϕ - 1 q(x) c||u|| q + q(x) c Ω |u(x)| q(x) dx.
If ||u|| q(x) 1, then

J(u) c||u|| ϕ ∓ 2 q(x) c||u|| ϕ + 2 q(x) c||u|| q - q(x) c Ω |u(x)| q(x) dx.
Hence, for all u ∈ W s 1 0 L Φ 1 (Ω),

J(u) c Ω |u(x)| q(x) dx. (4.99)
This implies that J(u) q -cI(u).

The last inequality ensures that λ 1 > 0 and thus Step 1 is verified.

Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces Remark 17. Following the definition of ϕ - i , we point out that for any i = 1, 2,

a i (t).t 2 = ϕ i (t)t ϕ - i Φ i (t) ∀t 0 i = 1, 2.
This inequality and relation (4.99) imply that

λ 0 := inf u∈W s 1 0 L Φ (Ω)\{0} J 1 (u) I 1 (u) > 0.
(4.100)

• Step 2. We show that λ 1 is an eigenvalue of problem (P a ). Proof. Since W s 1 0 L Φ 1 (Ω) is continuously embedded in L q + (Ω) and in L q -(Ω) it follows that there exist two positive constants c 1 and c 2 such that

||u|| 1 c 1 ||u|| q + ∀u ∈ W s 1 0 L Φ 1 (Ω) (4.103)
and

||u|| 1 c 2 ||u|| q -∀u ∈ W s 1 0 L Φ 1 (Ω). (4.104)
Then for all u ∈ W s

J(u) I(u) m 1 Ω×Ω Φ 1 |u(x) -u(y)| |x -y| s 1 dµ I(u) q -m 1 ||u|| ϕ - 1 1 ||u|| q + q + + ||u|| q - q - q - m 1 ||u|| ϕ - 1 1 c -q + 1 ||u|| q + 1 + c -q - 2 ||u|| q - 1 .
Since ϕ - 1 > q + q -, passing to the limit as u 1 → ∞ in the above inequality, we deduce that relation (4.101) holds true.

On the other hand, since (V 3 ) holds true, we deduce that W s 2 0 L Φ 2 (Ω) is continuously embedded in L q + (Ω) and in L q -(Ω). So, it follows that there exist two positive constants c 3 and c 4 such that

||u|| 2 c 3 ||u|| q + ∀u ∈ W s 2 0 L Φ 2 (Ω) (4.105)
and

||u|| 2 c 4 ||u|| q -∀u ∈ W s 2 0 L Φ 2 (Ω). (4.106)
Then for all u ∈ W s 1 0 L Φ 1 (Ω) with ||u|| 1 < 1 (and ||u|| 2 < 1) small enough, by Proposition 2.2.1 and relations (4.105), (4.106), we deduce

J(u) I(u) m 2 Ω×Ω Φ 2 |u(x) -u(y)| |x -y| s 2 dµ I(u) q -m 2 ||u|| ϕ + 2 2 ||u|| q + q + + ||u|| q - q - q - m 2 ||u|| ϕ + 2 2 c -q + 3 ||u|| q + 2 + c -q - 4 ||u|| q - 2 .
Since ϕ + 2 < q -q + , passing to the limit as u 1 → 0 (and thus, u 2 → 0) in the above inequality, we deduce that relation Proof.

Let {u n } ⊂ W s 1 0 L Φ 1 (Ω) \ {0} be a minimizing sequence of λ 1 , that is lim n→∞ J(u n ) I(u n ) = λ 1 > 0. (4.107) By relation (4.101) it is clear that {u n } is bounded in W s 1 0 L Φ 1 (Ω). Since W s 1 0 L Φ 1 (Ω) is reflexive, it
follows that there exists u ∈ W s 1 0 L Φ 1 (Ω) such that u n converges weakly to u in W s 1 0 L Φ 1 (Ω). On the other hand, by Lemma 4.2.3, for any i = 1, 2, we have that Ψ i are weakly lower semi continuous,

Clearly, T λ ∈ C 1 (W s 1 0 L Φ 1 (Ω), R) and T λ (u), v = J (u), v -λ I (u), v for all u, v ∈ W s 1 0 L Φ 1 (Ω).
T λ (u) M 1 Ω Ω Φ 1 |u(x) -u(y)| |x -y| s 1 dxdy |x -y| N -λ Ω 1 q(x) |u| q(x) dx m 1 ||u|| ϕ - 1 -λ 1 q- ||u|| q + q + + ||u|| q - q - m 1 ||u|| ϕ - 1 -λ c q- ||u|| q + 1 + ||u|| q - 1 .
Since ϕ -> q + q -the above inequality implies that

T λ (u) -→ ∞ as ||u|| 1 → ∞, that is, T λ is coercive.
Now, we show that T λ is weakly lower semi continuous. Let {u n } ⊂ W s 1 0 L Φ 1 (Ω) be a sequence which converges weakly to u in W s 1 0 L Φ 1 (Ω). Since J is weakly lower semi continuous, then

J(u) lim inf n→∞ J(u n ).
On the other hand, by (4.109), we have

lim n→∞ I(u n ) = I(u).
Thus, we find

T λ (u) lim inf n→∞ T λ (u n ).
Therefore, T λ is weakly lower semi continuous. Then we apply Theorem 1.5.2 in order to prove that there exists u λ ∈ W s 1 0 L Φ 1 (Ω) a global minimizing point of T λ , and thus a critical point of T λ .

Next we prove that u λ = 0. Indeed since

λ 1 = inf w∈W s 1 0 L Φ 1 (Ω)\{0}
J(w) I(w) and λ > λ 1 , then there exists T λ (w) < 0.

v λ ∈ W s 1 0 L Φ 1 (Ω) such that J(v λ ) < λ I(v λ )
So, we conclude that u λ is a nontrivial critical point of T λ , that is, λ is an eigenvalue of problem (P a ). Then, step 3 is verified.

• Step 4. We show that any λ ∈ (0, λ 0 ) is not an eigenvalue of problem (P a ), where λ 0 is given by (4.100). Indeed, assuming by contradiction that there exists λ ∈ (0, λ 0 ) an eigenvalue of problem (P a ). That is, it follows that there exists

u λ ∈ W s 1 0 L Φ 1 (Ω) \ {0} such that J (u λ ), v = λ I (u λ ), v ∀v ∈ W s 1 0 L Φ 1 (Ω).
Then, for v = u λ we have

J (u λ ), u λ = λ I (u λ ), u λ , so, J 1 (u λ ) = λ I 1 (u λ ).
The fact that u λ ∈ W s 1 0 L Φ 1 (Ω) \ {0} implies that I 1 (u λ ) > 0. Since λ < λ 0 , then

J 1 (u λ ) λ 0 I 1 (u λ ) > λ I 1 (u λ ) = J 1 (u λ ).
Clearly, the above inequality leads to a contradiction. Thus, step 4 is verified.

By steps 2 -4, we deduce that λ 0 λ 1 . The proof of Theorem 4.7.1 is now completed.

4.8 The Nehari manifold approach for Fractional a(.)-Laplacian problem

The aim of this section is to study the existence of two non-negative weak solutions of the following nonlocal problem

(P a )        (-∆) s a 1 (.) u + a 1 (|u|)u = λ β 2 (x)a 2 (|u|)u + β 3 (x)a 3 (|u|)u in Ω u = 0 in R N \ Ω,
where Ω is a Lipschitz open bounded subset of R N , s ∈ (0, 1), λ is a positive real parameter, β 2 , β 3 ∈ L ∞ (Ω) are two non-negative functions in Ω and (-∆) s a 1 (.) is a nonlocal integro-differential operator of elliptic type defined as

(-∆) s a 1 (.) u(x) = 2 lim ε 0 R N \B ε (x) a 1 |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s ,
for all x ∈ R N and for any i = 1, 2, 3, a i : R -→ R are defined as in (4.1).

The method of Nehari manifold goes back to Nehari's work [START_REF] Nehari | On a class of nonlinear second-order differential equations[END_REF][START_REF] Nehari | Characteristic values associated with a class of non-linear secondorder differential equations[END_REF], where he considered a boundary value problem for a certain nonlinear second order ordinary differential equation in an interval (a, b) and showed that it has a nontrivial solution which may be found by constrained minimization of the Euler-Lagrange functional corresponding to the problem.

In recent years, many authors have been studied some non-linear problems in modular spaces, by using the Nehari manifold approach. Recently, Azroul et al. in [START_REF] Azroul | Existence Results for Fractional p(x, .)-Laplacian Problem Via the Nehari Manifold Approach[END_REF] proposed the following fractional problem

       (-∆ p(x,.) ) s u + |u| p(x)-2 u = λ w 1 (x)|u| q(x)-2 u + w 2 (x)|u| r(x)-2 u in Ω u = 0 in R N \ Ω, (4.112)
where,

• Ω ⊂ R N is a Lipschitz bounded open domain, s ∈ (0, 1), λ is a positive real number,

• p : Q -→ (1, +∞) and q, r : Ω -→ (1, +∞) are three bounded continuous functions such that 1 < q -q + < p -p + < r -r + < +∞ (4.113)

with Q := R 2N \ (CΩ ×CΩ) and CΩ = R N \ Ω, while p -= inf (x,y)∈Q p(x, y), p + = sup (x,y)∈Q p(x, y), 4.8. The Nehari manifold approach for Fractional a(.)-Laplacian problem 159 q -= inf x∈Ω q(x), q + = sup x∈Ω q(x) and p(x) = p(x, x) for all x ∈ Ω.

• w 1 , w 2 ∈ C(Ω) are two non-negative weighted functions satisfy the following assumptions :

(W 1 ) : w 1 : Ω → (0, +∞) such that w 1 ∈ L β (x) (Ω) with N p(x) N p(x) -q(x)(N -s p(x)) < β (x) < p(x) p(x) -q(x)
for all x ∈ Ω.

(W 2 ) : w 2 : Ω → (0, +∞) such that w 2 ∈ L β (x) (Ω) with p(x) p(x) -r(x) < β (x) < N p(x) N p(x) -r(x)(N -s p(x))
for all x ∈ Ω.

• The operator (-∆ p(x,.) ) s is the fractional p(x, .)-Laplacian defined as follows

(-∆ p(x,.) ) s u (x) = 2 lim ε 0 R N \B ε (x)
|u(x)u(y)| p(x,y)-2 (u(x)u(y)) |x -y| N+sp i (x,y) dy for all x ∈ R N .

The authors obtained the existence of two non-negative weak solutions of problem (4.112) by using the Nehari manifold approach.

On the other hand, when s → 1 -, problem (4.112) reduces to the following p(x)-Laplacian

problem        -∆ p(x) u + |u| p(x)-2 u = λ w 1 (x)|u| q(x)-2 u + w 2 (x)|u| r(x)-2 u in Ω u = 0 in ∂ Ω, (4.114)
where -∆ p(x) is the p(x)-Laplacian defined as follows

-∆ p(x) = -div |u(x)| p(x)-2 u(x) .
By using the Nehari manifold approach, Problem (4.114) and the similar problems in form (4.114), have been studied by many authors. Rabil et al in [START_REF] Mashiyev | The Nehari manifold approach for Dirichlet problem involving the p(x)-Laplacian equation[END_REF] and Rasouli in [START_REF] Rasouli | On a PDE involving the variable exponent operator with nonlinear boundary conditions[END_REF][START_REF] Rasouli | The Nehari Manifold Approach for a p(x)-Laplacian Problem with Nonlinear Boundary Conditions[END_REF] for the case when w 1 and w 2 are two non-negative weighted functions and p, q, r : Ω are three bounded continuous functions such that

1 < q + = sup x∈Ω q(x) < p -= inf x∈Ω p(x) p -= sup x∈Ω p(x) < r -= inf x∈Ω r(x) < ∞
and other some different conditions, the authors showed that there exist two non-negative weak Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces solutions for the problem (4.114). In [START_REF] Ali | Multiplicity of positive solution of p(x)-Laplacian problems with sign-changing weight functions[END_REF] Ben Ali proposed the same problem (4.114), but with w 1 and w 2 are smooth functions which may change sign in Ω and by same approach it showed that there exist two non-negative weak solutions for the problem (4.114). For read other problems by this approach, we refer the reader to the works [START_REF] Afrouzi | The Nehari manifold approach for p(x)-Laplacian problem with Neumann boundary condition[END_REF][START_REF] Chen | The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities[END_REF][START_REF] Corrâa | Sign Changing Solutions for Quasilinear Superlinear Elliptic Problems[END_REF].

Inspirit by these previous works, the aim of this section is to study the existence of two nonnegative weak solutions for problem (P a ) in another functional framework, which is the fractional Orlicz-Sobolev spaces.

Assumptions and mains results

Let

Φ i (t) = t 0 ϕ i (τ)dτ. for all t ∈ R, i = 1, 2, 3.
We assume the following conditions :

1 0 Φ -1 1 (τ) τ N+s N
dτ < ∞, and

∞ 1 Φ -1 1 (τ) τ N+s N dτ = ∞, (V 1 ) lim t→∞ Φ 3 (kt) (Φ 1 ) * (t) = 0 ∀k > 0, (V 2 ) ϕ + 2 < ϕ - 1 ϕ + 1 < ϕ - 3 , (V 3 ) -1 < ϕ - i -2 := inf t 0 (ta i (t)) t (ta i (t)) ϕ + i -2 := sup t 0 (ta i (t)) t (ta i (t)) < +∞ i = 1, 2, 3. (V 4 )
Remark 18. It can be show that (V 4 ) implies that the condition

1 < ϕ - i := inf t 0 t 2 a i (t) Φ i (t) ϕ + i := sup t 0 t 2 a i (t) Φ i (t) < +∞ i = 1, 2, 3. (V 5 )
Definition 4.8.1. We say that u ∈ W s L Φ 1 (Ω) is a weak solution of problem (P a ) if,

Ω Ω a 1 (|D s u|)D s uD s vdµ + Ω a 1 (|u|)uvdx -λ Ω β 2 (x)a 2 (|u|)uvdx - Ω β 3 (x)a 3 (|u|)uvdx = 0, for all v ∈ W s L Φ 1 (Ω).
Now we are in position to states our main results. There exists λ such that for all λ ∈ (0, λ ), problem (P a ) has at least two nonnegative weak solutions.

Variational setting

To deal with this situation we introduce the weighted Orlicz spaces to investigate problem (P a ).

Let us recall the definitions and some elementary properties of these spaces.

We define the spaces L

β i Φ i (Ω)
for any i = 2, 3, by

L β i Φ i (Ω) = u : Ω -→ R mesurable : Ω β 1 (x)Φ i (λ |u(x)|)dx < ∞ for some λ > 0 .
The spaces L

β i Φ i (Ω) i = 2, 3
, are Banach spaces endowed with the Luxemburg norm

||u|| β i ,Φ i = inf λ > 0 : Ω β i (x)Φ i |u(x)| λ dx 1 .
Moreover, the weighted modular on L β i Φ i (Ω) satisfies the following result.

Proposition 4.8.1. Assume condition (V 4 ) is satisfied. Then the following relations holds true

||u|| ϕ - i β i ,Φ i Ω β i (x)Φ i (|u(x)|)dx ||u|| ϕ + i β i ,Φ i , ∀u ∈ L β i Φ i (Ω) with ||u|| β i ,Φ i > 1, , i = 2, 3, ||u|| ϕ + i β i ,Φ i Ω β i (x)Φ i (|u(x)|)dx ||u|| ϕ - i β i ,Φ i , ∀u ∈ L β i Φ i (Ω) with ||u|| β i ,Φ i < 1, i = 2, 3. Theorem 4.8.2.
Let Ω be a bounded open subset of R N and C 0,1 -regularity with bounded boundary.

If (V 1 ), (V 2 ) and (V 4 ) hold, then

W s L Φ 1 (Ω) → L β i Φ i (Ω), i = 2, 3, (4.115)
is compact.

Proof. By condition (V 2 ) and Theorem 2.3.2, we have

W s L Φ 1 (Ω) → → L Φ 3 (Ω). (4.116)
Then, it suffices to show that

L Φ 3 (Ω) → L β 3 Φ 3 (Ω). (4.117)
Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces Indeed, let u ∈ L Φ 3 (Ω), then there exists λ > 0 such that

Ω β 3 (x)Φ 3 (λ |u(x)|)dx ||β 3 || ∞ Ω Φ 3 (λ |u(x)|)dx < ∞ and ||u|| β 3 ,Φ 3 c 1 ||u|| Φ 3 ,
where

c 1 = ||β 3 || ∞ . So, W s L Φ 1 (Ω) → → L β 3 Φ 3 (Ω).
On the other hand, since ϕ + 2 < ϕ - 3 , it is easy to see that Φ 2 ≺≺ Φ 3 . Then

L Φ 3 (Ω) → L Φ 2 (Ω), (4.118)
and seminary to (4.117), we have

L Φ 2 (Ω) → L β 2 Φ 2 (Ω) (4.119)
combining (4.116) -(4.119) we obtain

W s L Φ 1 (Ω) → → L β 2 Φ 2 (Ω).
Corollary 4.8.1. Under the assumptions of Theorem 4.8.2. Let u ∈ W s L Φ 1 (Ω), then there exist two positive constants c 2 and c 3 such that the following inequalities hold

Ω β i (x)Φ i (|u(x)|)dx c i ||u|| ϕ + i i f ||u|| > 1, i = 2, 3
and

Ω β i (x)Φ i (|u(x)|)dx c i ||u|| ϕ - i i f ||u|| < 1, i = 2, 3.
Proof. Let u ∈ W s L Φ 1 (Ω), by Proposition 4.8.1 and Theorem 4.8.2 we have

Ω β i (x)Φ i (|u(x)|)dx ||u|| ϕ + i β i ,Φ i + ||u|| ϕ - i β i ,Φ i c i ||u|| ϕ + i + ||u|| ϕ - i i = 2, 3.
4.8. The Nehari manifold approach for Fractional a(.)-Laplacian problem 163 Then, if ||u|| > 1 we have

Ω β i (x)Φ i (|u(x)|)dx c i ||u|| ϕ + i i = 2, 3,
where

c i = 2c i . If ||u|| < 1 we have Ω β i (x)Φ i (|u(x)|)dx c i ||u|| ϕ - i i = 2, 3,
where c i = 2c i .

Nehari Maniford Analysis for (P a )

Let J ∈ C 1 (X, R) be the Euler functional associated with an elliptic problem on Banach space X. If J is bounded below and has a minimizer on X, then this minimizer is a critical point of J.

Hence, it is a solution of the corresponding elliptic problem. However, in many problems J is not bounded below on the whole space X, but is bounded below on an appropriate subset of X, and minimizer on this set (if it exists) may give rise to solutions of the corresponding elliptic problem.

A good candidate for an appropriate subset of X is the Nehari manifold.

If we consider our problem (P a ) then, the corresponding Euler functional is deflned by

J λ (u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx -λ Ω β 2 (x)Φ 2 (|u|)dx - Ω β 3 (x)Φ 3 (|u|)dx and
Ψ(u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx.
By a standard argument to Lemma 4.4.1, we can to show that

J λ ∈ C 1 (W s L Φ 1 (Ω), R) and J λ (u), v = Ω Ω a 1 (|D s u|)D s uD s vdµ + Ω a 1 (|u|)uvdx -λ Ω β 2 (x)a 2 (|u|)uvdx - Ω β 3 (x)a 3 (|u|)uvdx, for all u, v ∈ W s L Φ 1 (Ω). Now,
J λ (u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx -λ Ω β 2 (x)Φ 2 (|u|)dx - Ω β 3 (x)Φ 3 (|u|)dx ||u|| ϕ - 1 -λ c 2 ||u|| ϕ + 2 -c 3 ||u|| ϕ + 3 .
Since 1 < ϕ - 1 < ϕ + 3 , then J λ is not bounded below on the whole space W s L Φ 1 (Ω). However, we shall show it is bounded on the Nehari manifold N λ which is given by

N λ := u ∈ W s L Φ 1 (Ω) \ {0) : J λ (u), u = 0 .
It is clear that all critical points of J λ , must lie on N λ and local minimizers on N λ are usually critical points of J λ . Thus, u ∈ N λ if and only if Proof. Let u ∈ N λ , such that ||u|| > 1, form (4.120),

J λ (u), u = Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx -λ Ω β 2 (x)a 2 (|u|)|u| 2 dx - Ω β 3 (x)a 3 (|u|)|u| 2 dx = 0.
Ω β 3 (x)a 3 (|u|)|u| 2 dx = Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx -λ Ω β 2 (x)a 2 (|u|)|u| 2 dx.
Then, by (V 5 ), we have

Ω β 3 (x)Φ 3 (|u|)dx 1 ϕ - 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx = 1 ϕ - 3 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx -λ Ω β 2 (x)a 2 (|u|)|u| 2 dx 1 ϕ - 3 ϕ + 1 Ψ(u) -ϕ - 2 λ Ω β 2 (x)Φ 2 (|u|)dx .
4.8. The Nehari manifold approach for Fractional a(.)-Laplacian problem 165 So,

J λ (u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx -λ Ω β 2 (x)Φ 2 (|u|)dx - Ω β 3 (x)Φ 3 (|u|)dx 1 - ϕ + 1 ϕ - 3 Ψ(u) + λ (ϕ - 2 -1) Ω β 2 (x)Φ 2 (|u|)dx 1 - ϕ + 1 ϕ - 3 ||u|| ϕ - 1 + λ c 2 (ϕ - 2 -1)||u|| ϕ - 2 .
Since, ϕ -

3 > ϕ + 1 ϕ - 1 > ϕ - 2 , then J λ is coercive.
Next, we consider the functional I λ : N λ -→ R defined by

I λ (u) := J λ (u), u for all u ∈ N λ .
Hence, it is natural to split N λ into three parts : the first set corresponding to local minima, the second set corresponding to local maxima, and the third one corresponding to points of inflection which defined respectively as follows :

N + λ := u ∈ N λ : I λ (u), u > 0 , N - λ := u ∈ N λ : I λ (u), u < 0 , N 0 λ := u ∈ N λ : I λ (u), u = 0 .
We have the following Lemma.

Lemma 4.8.2. Under the assumptions of Theorem 4.8.2, there exists λ * > 0 such that for any λ ∈ (0, λ * ), we have that N 0 λ = ∅.

Proof. Suppose otherwise, that is, N 0 λ = ∅, for any λ > 0. Let u ∈ N 0 λ with ||u|| > 1, combing (4.120) with the definition of N 0 λ , we get

0 = I λ (u), u ϕ - 1 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + ϕ - 1 Ω a 1 (|u|)|u| 2 dx -λ ϕ + 2 Ω β 2 (x)a 2 (|u|)|u| 2 dx -ϕ + 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx = (ϕ - 1 -ϕ + 2 ) Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx + (ϕ + 2 -ϕ + 3 ) Ω β 3 (x)a 3 (|u|)|u| 2 dx (ϕ - 1 -ϕ + 2 )ϕ - 1 Ψ(u) + (ϕ + 2 -ϕ + 3 )ϕ + 3 Ω β 3 (x)Φ 3 (|u|)dx (ϕ - 1 -ϕ + 2 )ϕ - 1 ||u|| ϕ - 1 + (ϕ + 2 -ϕ + 3 )ϕ + 3 c 3 ||u|| ϕ + 3 .
This implies that :

(ϕ + 3 -ϕ + 2 )ϕ + 3 c 3 ||u|| ϕ + 3 (ϕ - 1 -ϕ + 2 )ϕ - 1 ||u|| ϕ - 1 , so, ||u|| c 3 (ϕ - 1 -ϕ + 2 )ϕ - 1 (ϕ + 3 -ϕ + 2 )ϕ + 3 1 ϕ + 3 -ϕ - 1 . (4.121)
Analogously,

0 = I λ (u), u ϕ + 1 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + ϕ + 1 Ω a 1 (|u|)|u| 2 dx -λ ϕ - 2 Ω β 2 (x)a 2 (|u|)|u| 2 dx -ϕ - 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx = (ϕ + 1 -ϕ - 3 ) Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx + λ (ϕ - 3 -ϕ - 2 ) Ω β 2 (x)a 2 (|u|)|u| 2 dx (ϕ + 1 -ϕ - 3 )ϕ - 1 Ψ(u) + λ (ϕ - 3 -ϕ - 2 )ϕ + 2 Ω β 2 (x)Φ 2 (|u|)dx (ϕ + 1 -ϕ - 3 )ϕ - 1 ||u|| ϕ - 1 + λ (ϕ - 3 -ϕ - 2 )ϕ + 2 c 2 ||u|| ϕ + 2 .
This implies that :

(ϕ - 3 -ϕ + 1 )ϕ - 1 ||u|| ϕ - 1 λ (ϕ - 3 -ϕ - 2 )ϕ + 2 c 2 ||u|| ϕ + 2 .
So,

||u|| c 2 λ (ϕ - 3 -ϕ - 2 )ϕ + 2 (ϕ - 3 -ϕ + 1 )ϕ - 1 1 ϕ - 1 -ϕ + 2 . (4.122)
4.8. The Nehari manifold approach for Fractional a(.)-Laplacian problem

167 For λ < c 3 c 2 ϕ - 1 -ϕ + 2 (ϕ - 1 -ϕ + 2 )ϕ - 1 (ϕ + 3 -ϕ + 2 )ϕ + 3 ϕ - 1 -ϕ + 2 ϕ + 3 -ϕ - 1 × (ϕ - 3 -ϕ + 1 )ϕ - 1 (ϕ - 3 -ϕ - 2 )ϕ + 2 := λ * , then the inequality (4.122), implies that ||u|| < c 3 (ϕ - 1 -ϕ + 2 )ϕ - 1 (ϕ + 3 -ϕ + 2 )ϕ + 3 1 ϕ + 3 -ϕ - 1 .
The above inequality contradicts with (4.121). Then, we can conclude that there exists λ * > 0 such that N 0 λ = ∅ for any λ ∈ (0, λ * ). Proof. Let u ∈ N + λ , from the definition of J λ we get

J λ (u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx -λ Ω β 2 (x)Φ 2 (|u|)dx - Ω β 3 (x)Φ 3 (|u|)dx 1 ϕ - 1 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + 1 ϕ - 1 Ω a 1 (|u|)|u| 2 dx - λ ϕ + 2 Ω β 2 (x)a 2 (|u|)|u| 2 dx - 1 ϕ + 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx. (4.123)
Since u ∈ N + λ , then we have

ϕ + 1 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + ϕ + 1 Ω a 1 (|u|)|u| 2 dx -λ ϕ - 2 Ω β 2 (x)a 2 (|u|)|u| 2 dx -ϕ - 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx > 0. (4.124)
On the other hand, by (4.120), we have

-ϕ - 2 Ω Ω a 1 (|D s u|)|D s u| 2 dµ -ϕ - 2 Ω a 1 (|u|)|u| 2 dx + ϕ - 2 λ Ω β 2 (x)a 2 (|u|)|u| 2 dx + ϕ - 2 Ω β 3 (x)a 3 (|u|)|u| 2 dx = 0, ( 4
J λ (u) 1 ϕ - 1 - 1 ϕ + 2 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx + 1 ϕ + 2 - 1 ϕ + 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx 1 ϕ - 1 - 1 ϕ + 2 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx + 1 ϕ + 2 - 1 ϕ + 3 ϕ + 1 -ϕ - 2 ϕ - 3 -ϕ - 2 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx < - (ϕ - 1 -ϕ + 2 )(ϕ + 3 -ϕ - 1 ) ϕ + 3 ϕ + 2 ϕ - 1 ϕ + 1 Ψ(u). (4.127)
Finally, we deduce that σ

+ λ = inf u∈N + λ J λ (u) < 0.
Lemma 4.8.4. Under the assumptions of Theorem 4.8.2, there exists λ * * > 0 such that σ - λ > 0, for any λ ∈ (0, λ * * ).

Proof. Let u ∈ N - λ , from definition J λ and (4.120), we have

J λ (u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx -λ Ω β 2 (x)Φ 2 (|u|)dx - Ω β 3 (x)Φ 3 (|u|)dx 1 ϕ + 1 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + 1 ϕ + 1 Ω a 1 (|u|)|u| 2 dx - λ ϕ - 2 Ω β 2 (x)a 2 (|u|)|u| 2 dx - 1 ϕ - 3 Ω β 3 (x)a 3 (|u|)|u| 2 dx. = 1 ϕ + 1 - 1 ϕ - 3 Ω Ω a 1 (|D s u|)|D s u| 2 dµ + Ω a 1 (|u|)|u| 2 dx + λ 1 ϕ - 3 - 1 ϕ - 2 Ω β 2 (x)a 2 (|u|)|u| 2 dx ϕ - 3 -ϕ + 1 ϕ + 1 ϕ - 3 ϕ - 1 Ψ(u) + λ ϕ - 2 -ϕ - 3 ϕ - 3 ϕ - 2 ϕ + 2 Ω β 2 (x)Φ 2 (|u|)dx ϕ - 3 -ϕ + 1 ϕ + 1 ϕ - 3 ϕ - 1 ||u|| ϕ - 1 + λ ϕ - 2 -ϕ - 3 ϕ - 3 ϕ - 2 ϕ + 2 c 2 ||u|| ϕ + 2 .
(4.128)

Since ϕ - 1 > ϕ + 2 , we have

J λ (u) ϕ - 3 -ϕ + 1 ϕ + 1 ϕ - 3 ϕ - 1 + λ ϕ - 2 -ϕ - 3 ϕ - 3 ϕ - 2 ϕ + 2 c 2 ||u|| ϕ - 1 .
4.8. The Nehari manifold approach for Fractional a(.)-Laplacian problem 169 Thus, il we choose λ < ϕ

- 2 (ϕ - 3 -ϕ + 1 )ϕ - 1 ϕ + 1 c 2 ϕ + 2 (ϕ - 3 -ϕ - 2 ) := λ * * , we deduce that J λ (u) > 0. It follows that σ - λ = inf u∈N - λ J λ (u) > 0.

Existence on Non-Negative Solutions

Next, we prove the existence of two non-negative solutions of Problem (P a ). For this, we first show the existence of minimizers in N + λ and N - λ for all λ ∈ (0, λ ), where λ := min {λ * , λ * * } .

Theorem 4.8.3. Under the assumptions of Theorem 4.8.2. Then for all λ ∈ (0, λ ), there exists a

minimizer u + 0 of J λ on N + λ such that J λ (u + 0 ) = σ + λ .
Proof. J λ is bounded below on N λ , in particular is bounded below on N + λ . Then there exists a minimizing sequence {u

+ n } ⊂ N + λ , such that lim n→∞ J λ (u + n ) = σ + λ . As J λ is coercive, it follows that {u + n } ⊂ N + λ is bounded in W s L Φ 1 (Ω).
Ψ(u + 0 ) < lim inf n→∞ Ψ(u + n ).
Using (4.129), we obtain

Ω β 2 (x)Φ 2 (|u + 0 |)dx = lim inf n→∞ Ω β 2 (x)Φ 2 (|u + n |)dx.
weakly in W s L Φ 1 (Ω). By Theorem 4.8.2, we get

           u - n → u - 0 strongly in L β 2 Φ 2 (Ω), u - n → u - 0 strongly in L β 3 Φ 3 (Ω). (4.130)
On the other hand, there exists t 0 > 1 such that t 0 u - 0 ∈ N - λ and J λ (u - 0 ) J λ (t 0 u - n ). Indeed, from the definition of J λ , we have

I λ (t 0 u - 0 ),t 0 u - 0 ϕ + 1 Ω Ω a 1 (|D s (t 0 u - 0 )|)|D s (t 0 u - 0 )| 2 dµ + ϕ + 1 Ω a 1 (|t 0 u - 0 |)|t 0 u - 0 | 2 dx -λ ϕ - 2 Ω β 2 (x)a 2 (|t 0 u - 0 |)|t 0 u - 0 | 2 dx -ϕ - 3 Ω β 3 (x)a 3 (|t 0 u - 0 |)|t 0 u - 0 | 2 dx (ϕ + 1 ) 2 Ψ(t 0 u + 0 ) -λ (ϕ - 2 ) 2 Ω β 2 (x)Φ 2 (|t 0 u - 0 |)dx -(ϕ - 3 ) 2 Ω β 3 (x)Φ 3 (|t 0 u - 0 |)dx (ϕ + 1 ) 2 |t 0 | ϕ + 1 Ψ(u + 0 ) -λ (ϕ - 2 ) 2 |t 0 | ϕ - 2 Ω β 2 (x)Φ 2 (|u - 0 |)dx -(ϕ - 3 ) 2 |t 0 | ϕ - 3 Ω β 3 (x)Φ 3 (|u - 0 |)dx.
Since, ϕ - 2 < ϕ + 1 < ϕ - 3 , then for t 0 large enoughe, we have

I λ (t 0 u - 0 ),t 0 u - 0 < 0.
Thus, we deduce that t 0 u + 0 ∈ N - λ . On the other hand by definition of J λ and since t 0 large enough then

J λ (u - 0 ) J λ (t 0 u - 0 ). (4.131)
Now, we aim to prove that u

- n -→ u - n in W s L Φ 1 (Ω). Suppose that u - n u - 0 in W s L Φ 1 (Ω), then Ψ(u - 0 ) < lim inf n→∞ Ψ(u - n ). (4.132)
Using (4.130), we obtain

           Ω β 2 (x)Φ 2 (|u - 0 |)dx = lim inf n→∞ Ω β 2 (x)Φ 2 (|u - n |)dx Ω β 3 (x)Φ 3 (|u - 0 |)dx = lim inf n→∞ Ω β 3 (x)Φ 3 (|u - n |)dx. (4.133)
Then, combining (4.132) with (4.133), we obtain

σ - λ = lim n→∞ J λ (u - n ) > J λ (u - 0 ).
The above inequality and (4.131) implies that

σ - λ = inf u∈N - λ J λ (u) > J λ (u - 0 ) J λ (t 0 u - 0 ). Which is a contradiction. Consequently u - n -→ u - 0 in W s L Φ 1 (Ω) and lim n→∞ J λ (u - n ) = J λ (u - 0 ) = σ - λ .
Then, we conclude that u - 0 is a minimizer of J λ on N - λ .

Proof of Theorem 4.8.1. From Theorem 4.8.3 and 4.8.4, we deduce that for any λ ∈ (0, λ ), there

exist u + 0 ∈ N + λ and u - 0 ∈ N - λ such that J λ (u - 0 ) = inf u∈N - λ J λ (u) and J λ (u + 0 ) = inf u∈N + λ J λ (u).
The Problem (P a ) has two solutions u + 0 and u - 0 in W s L Φ 1 (Ω). By Lemma 4.8.2, we have N 0 λ = ∅, it follows that N + λ ∩ N - λ = ∅. Then u + 0 = u - 0 , hence there exist two solutions distinct.

Next, we prove u + 0 and u - 0 are non-negative in Ω. For this, we consider the auxiliary function

f + (x,t) =            λ β 2 (x)a 2 (|t|)t + β 3 (x)a 3 (|t|)t if t > 0 0 if t 0.
The associated functional J + λ given by

J + λ (u) = Ω Ω Φ 1 (|D s u|)dµ + Ω Φ 1 (|u|)dx - Ω F + (x, u)dx u ∈ W s L Φ 1 (Ω)
where

F + (x,t) = t 0 f + (x, τ)dτ, x ∈ Ω, t ∈ R.
The functional J + λ is will defined and it is Gâteaux differentiable in W s L Φ 1 (Ω), such that :

J + λ (u), v = Ω Ω a 1 (|D s u|)D s uD s vdµ + Ω a 1 (|u|)uvdx - Ω f + (x, u)vdx, for all v ∈ W s L Φ 1 (Ω). Moreover, since J λ (u ± 0 ) = J λ (|u ± 0 |), then for any λ ∈ (0, λ ), u ± 0 are critical points of J + λ on W s L Φ 1 (Ω). Now we assume that u ± 0 0. Let u ∈ W s L Φ 1 (Ω), set u + = max {u, 0}
4.8. The Nehari manifold approach for Fractional a(.)-Laplacian problem and u -= min {u, 0}, so that u = u + + u -. If u a critical point of J + λ , then we have

0 Ψ(u -) 1 ϕ - 1 Ω Ω a 1 (|D s u -|)|D s u -| 2 dµ + Ω a 1 (|u -|)|u -| 2 dx = 1 ϕ - 1 λ Ω β 2 (x)Φ 2 (|u -|)dx + Ω β 3 (x)Φ 3 (|u -|)dx = 0. (4.134)
As a consequence of (4.134), has u -= 0 and so u = u + 0 in Ω. Then by taking u = u + 0 and u = u - 0 respectively, we deduce that u = u + 0 and u = u - 0 are non-negative solutions of Problem (P a ).

Fractional Musielak-Sobolev spaces

Introduction and preliminaries results

In the local case, when we try to integrate both the functional structures of variable exponent Lebesgue spaces and Orlicz spaces, we are led to the so-called Musielak-Orlicz spaces. This later functional structure was extensively studied since the 1950's by Nakano [START_REF] Nakano | Modulared Semi-Ordered Linear Spaces[END_REF] and developed by Musielak and Orlicz [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF][START_REF] Musielak | On modular spaces[END_REF]. A natural question has been asked : can we see the same generalization in the fractional case ? The answer to this question is the objective of this chapter.

In this chapter, we introduce a new fractional Musielak-Sobolev space and the associated nonlocal integro-differential operator. As an application, we are concerned with the existence of weak solutions of nonlocal problems involving the nonlocal integro-differential operator of elliptic type defined as follows

(-∆) s a (x,.) u(x) = 2 lim ε 0 R N \B ε (x) a (x,y) |u(x) -u(y)| |x -y| s u(x) -u(y) |x -y| s dy |x -y| N+s , (5.1)
for all x ∈ R N , where (x, y,t) → a (x,y) (t) := a(x, y,t) : Ω × Ω × R -→ R is such that : ϕ(., ., .) :

Musielak-Sobolev spaces (see [START_REF] Mihäilescu | Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces[END_REF]).

One typical feature of this operator (-∆) s a (x,.) is the nonlocality, in the sense that the value of (-∆) s a (x,.) u(x) at any point x ∈ Ω depends not only on the values of u on Ω, but actually on the entire space R N . Therefore, the Dirichlet datum is given in R N \ Ω (which is different from the classical case of the a(.)-Laplacian) and not simply on ∂ Ω. Hence, it is often called nonlocal problem.

This causes some mathematical difficulties which make the study of such a problem particularly interesting.

To deal with the problems involving (-∆ p(x,.) ) s , we will introduce the new fractional Musielak- assumed by a more general function Φ x . Such spaces originated with Nakano [START_REF] Nakano | Modulared Semi-Ordered Linear Spaces[END_REF] and were developed by Musielak and Orlicz in [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF].

For the function Φ x given in (5.2), we introduce the Musielak class as follows

K Φ x (Ω) = u : Ω -→ R mesurable : Ω Φ x (|u(x)|)dx < ∞ ,
and the Musielak space

L Φ x (Ω) = u : Ω -→ R mesurable : Ω Φ x (λ |u(x)|)dx < ∞ for some λ > 0 .
The space L Φ x (Ω) is a Banach space endowed with the Luxemburg norm

||u|| Φ x = inf λ > 0 : Ω Φ x |u(x)| λ dx 1 .
The conjugate function of Φ x,y is defined by Φ x,y (t) = t 0 ϕ x,y (τ)dτ for all (x, y) ∈ Ω×Ω and all t 0, where ϕ x,y : R -→ R is given by ϕ x,y (t) := ϕ(x, y,t) = sup {s : ϕ(x, y, s) t} . Furthermore, we have the following Hölder type inequality

Ω uvdx 2||u|| Φ x ||v|| Φ x for all u ∈ L Φ x (Ω) and v ∈ L Φ x (Ω). ( 5
W s L Φ x,y (Ω) = u ∈ L Φ x (Ω) : Ω Ω Φ x,y λ |u(x) -u(y)| |x -y| s dxdy |x -y| N < ∞ for some λ > 0 .
This space can be equipped with the norm

||u|| s,Φ x,y = ||u|| Φ x + [u] s,Φ x,y , (5.7)
where [.] s,Φ x,y is the Gagliardo seminorm defined by

[u] s,Φ x,y = inf λ > 0 : Ω Ω Φ x,y |u(x) -u(y)| λ |x -y| s dxdy |x -y| N 1 . Remark 22.
a)-For the case : Φ x,y (t) = Φ(t), i.e. Φ is independent of variables x, y, we say that L Φ and W s L Φ are Orlicz spaces and fractional Orlicz-Sobolev spaces respectively. b)-For the case : Φ x,y (t) = |t| p(x,y) for all (x, y) ∈ Ω × Ω, where p : Ω × Ω -→ (1, +∞) is a continuous bounded function such that

1 < p -= min (x,y)∈Ω×Ω p(x, y) p(x, y) p + = max (x,y)∈Ω×Ω p(x, y) < +∞, (5.8)
and p is symmetric, that is, p(x, y) = p(y, x) for all (x, y) ∈ Ω × Ω.

(5.9)

If denoted by p(x) = p(x, x) for all x ∈ Ω. Then, we replace L Φ x by L p(x) , and W s L Φ 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, (see [START_REF] Azroul | Eigenvalue problems involving the fractional p(x)-Laplacian operator[END_REF][START_REF] Bahrouni | On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent[END_REF][START_REF] Kaufmann | Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians[END_REF]) defined by

L p(x) (Ω) = u : Ω -→ R measurable : Ω |u(x)| p(x) dx < +∞ , and W = W s,p(x,y) (Ω) = u ∈ L p(x) (Ω) : Ω×Ω |u(x) -u(y)| p(x,y)
λ p(x,y) |x -y| sp(x,y)+N dxdy < +∞, for some λ > 0 .

with the norm

u W = u L p(x) (Ω) + [u] W ,
where [.] W is a Gagliardo seminorm with variable exponent given by

[u] W = [u] s,p(x,y) = inf λ > 0 : Ω×Ω |u(x) -u(y)| p(x,y)
λ p(x,y) |x -y| N+sp(x,y) dxdy 1 .

When p(x, y) = p = constant ∈ (1, +∞), the space W reduce to the fractional Sobolev space W s,p (Ω).

For a comprehensive introduction to the study of these spaces and the related nonlocal problems, we refer the reader to [START_REF] Azroul | Eigenvalue problems involving the fractional p(x)-Laplacian operator[END_REF][START_REF] Caffarelli | Traveling waves for a boundary reaction-diffusion equation[END_REF].

c)-Note that our new framework is a natural generalization of the above mentioned functional spaces.

W s,p (Ω) W s,p(x,y) (Ω) [START_REF] Azroul | Eigenvalue problems involving the fractional p(x)-Laplacian operator[END_REF]). Let Ω be a Lipschitz bounded domain in R N and let s ∈ (0, 1). Let p :

W s L Φ (Ω) W s L Φ x,y (Ω) Theorem 5.1.2. ( [
Ω × Ω -→ (1, +∞) be a continuous function satisfies (5.8) and (5.9) with sp + < N. Let r : Ω -→

(1, +∞) be a continuous variable exponent such that

1 < r -= min x∈Ω r(x) r(x) < p * s (x) = N p(x) N -s p(x)
for all x ∈ Ω.

Then, there exists a constant c = c(N, s, p, r, Ω) > 0 such that for any u ∈ W s,p(x,y) (Ω),

u L r(x) (Ω) c u W .
Thus, the space W is continuously embedded in L r(x) (Ω) with r(x) ∈ (1, p * s (x)), for all x ∈ Ω. Mo-5.2. Some qualitative properties of fractional Musielak-Sobolev spaces 181 reover, this embedding is compact.

This chapter is organized as follows, in Section 5.1, we are introduced the new fractional Musielak-Sobolev spaces and the associated nonlocal integro-differential operator. In Section 5.2, we establish some qualitative properties of these new spaces. Finally, we will study some nonlocal problems in fractional Musielak-Sobolv space.

Some qualitative properties of fractional Musielak-Sobolev spaces

In this section we establish some qualitative properties of the new fractional Musielak-Sobolev spaces W s L Φ x,y (Ω).

Theorem 5.2.1. Let Ω be an open subset of R N , and let s ∈ (0, 1). The space W s L Φ x,y (Ω) is a Banach space with respect to the norm (5.7), and a separable (resp. reflexive) space if and only if

Φ x,y ∈ ∆ 2 (resp. Φ x,y ∈ ∆ 2 and Φ x,y ∈ ∆ 2 ). Furthermore, if Φ x,y ∈ ∆ 2 and Φ x,y ( √ t) is convex, then the space W s L Φ x,y (
Ω) is an uniformly convex space.

Proof. Let {u n } be a Cauchy sequence in W s L Φ x,y (Ω). In particular, {u n } is a Cauchy sequence in L Φ x (Ω). Then, it converges strongly to a function u ∈ L Φ x (Ω). Moreover, the sequence {v n } defined by

v n (x, y) := u n (x) -u n (y) |x -y| s is a Cauchy sequence in L Φ x,y (Ω × Ω, dµ), that is, for any ε > 0 there exists n * ε such that if n, m n * ε , we have v n -v m L Φx,y (Ω×Ω,dµ) < ε, (5.10)
where µ is a measure on Ω × Ω which is given by dµ

:= |x -y| -N dxdy. Since L Φ x,y (Ω × Ω, dµ) is a
Banach space, then {v n } converges to an element of L Φ x,y (Ω × Ω, dµ). Let us extract a subsequence {u n k } of {u n } which converges almost everywhere to u. It follows that {v n k } converges for almost every pair (x, y) to v(x, y) = (u(x)u(y))|x -y| -s . Applying the Fatou's lemma, we obtain

Ω Ω Φ x,y λ |u(x) -u(y)| |x -y| s dxdy |x -y| N lim inf k→∞ Ω Ω Φ x,y λ |u n k (x) -u n k (y)| |x -y| s dxdy |x -y| N < ∞,
for some λ > 0. It follows that u ∈ W s L Φ x,y (Ω). Thus, we find that u n → u in W s L Φ x,y (Ω) by taking the limit from m → ∞ in (5.10).

with α = 2||u|| C 1 (Ω) . Then for λ = 1 α , we have,

Ω Ω Φ x,y |u(x) -u(y)| α|x -y| s dxdy |x -y| N Ω Ω Φ x,y δ (x, y) |x -y| s dxdy |x -y| N = Ω Ω∩|x-y| 1 Φ x,y δ (x, y) |x -y| s dxdy |x -y| N + Ω Ω∩|x-y| 1 Φ x,y δ (x, y) |x -y| s dxdy |x -y| N = Ω Ω∩|x-y| 1 Φ x,y |x -y| |x -y| s dxdy |x -y| N + Ω Ω∩|x-y| 1 Φ x,y 1 |x -y| s dxdy |x -y| N sup (x,y)∈Ω×Ω Φ x,y (1) Ω Ω∩|x-y| 1 dxdy |x -y| N+s-1 + Ω Ω∩|x-y| 1 dxdy |x -y| N+s .
Note that the kernel |x -y| N+s-1 is summable with respect to y if |x -y| 1 since N + s -1 < N and, on the other hand, the kernel |x -y| N+s is summable when |x -y| 1 since N + s > N. Finally, the above two integrals are finite. Hence, u ∈ W s L Φ x,y (Ω).

Remark 24.

-As a trivial consequence of Theorem 5.

2.2, C ∞ 0 (Ω) ⊂ C 2 0 (Ω) ⊂ W s L Φ x,y (Ω). Then, W s L Φ x,y (Ω)
is non-empty.

-Note that we are proved that C ∞ 0 (Ω) ⊂ W s L Φ x,y (Ω) without using the local integrability condition (see Remark 23-(ii)) which is necessary in the proof of this result in the integer case when s ∈ N (see [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF]).

Lemma 5.2.1. Assume that (Φ 1 ) is satisfied. Then Φ x,y (ϕ x,y (t)) ϕ + Φ x,y (t) for all (x, y) ∈ Ω × Ω and all t 0.

Proof. We know that ϕ x,y (t) = sup {s : ϕ(x, y, s) t} for all (x, y) ∈ Ω × Ω and all t 0 and Φ x,y (t) = t 0 ϕ x,y (τ)dτ for all (x, y) ∈ Ω × Ω and all t 0.

On the other hand, we know that for all (x, y) ∈ Ω × Ω, ϕ x,y : R → R is an increasing homeomorphism, in particular, from R + into R + . It follows that for each (x, y) ∈ Ω×Ω, the function t → ϕ x,y (t)

has an inverse function denoted by t → ϕ -1 x,y (t). Thus, ϕ x,y (τ) t if and only if τ ϕ -1 x,y (t). Taking into account the above pieces of information, we deduce that ϕ x,y (t) = ϕ -1 x,y (t). Consequently, we x,y (τ)dτ for all (x, y) ∈ Ω × Ω and all t 0.

Next, since

Φ x,y (ϕ -1 x,y (τ)) = ϕ -1
x,y (τ) 0 ϕ x,y (θ )dθ for all (x, y) ∈ Ω × Ω and all t 0, taking ϕ x,y (θ ) = r, we obtain

Φ x,y (ϕ -1 x,y (τ)) = τ 0 r. d dr ϕ -1 x,y (r) = τ.ϕ -1 x,y (τ) -Φ x,y (τ), ∀x, y ∈ Ω × Ω, ∀τ 0.
The above relation implies

Φ x,y (τ) τ.ϕ -1
x,y (τ) for all (x, y) ∈ Ω × Ω and all τ 0.

Taking in the above inequality τ = ϕ x,y (t), we get Φ x,y (ϕ x,y (t)) t.ϕ x,y (t) for all (x, y) ∈ Ω × Ω and all t 0.

Combining the last inequality with (Φ 1 ), we deduce that Φ x,y (ϕ x,y (t)) ϕ + Φ x,y (t) for all (x, y) ∈ Ω × Ω and all t 0. Lemma 5.2.2. Assume that (Φ 1 ) is satisfied. Then the following inequalities hold true : Thus, we deduce Φ x,y (σt) σ ϕ -Φ x,y (t) for all t > 0 and σ > 1.

Φ x,y (σt) σ ϕ - Φ x,y (
By the similar techniques as those used in the proof of relation (5.11), we deduce that (5.12) -(5.14) hold true.

For any u ∈ W s L Φ x,y (Ω), we define the following modular function

Ψ (u) = Ω Ω Φ x,y |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx. Proposition 5.2.1. On W s L Φ x,y (Ω) the following norms ||u|| s,Φ x,y = ||u|| Φ x + [u] s,Φ x,y , ||u|| max = max ||u|| Φ x , [u] s,Φ x,y , ||u|| = inf λ > 0 : Ψ u λ 1 are equivalent. Proof. First,
Ψ (u) = Ω Ω Φ x,y ||u|| |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + Ω Φ x ||u|| |u(x)| ||u|| dx ||u|| ϕ + Ω Ω Φ x,y |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + ||u|| ϕ + Ω Φ x |u(x)| ||u|| dx = ||u|| ϕ + Ω Ω Φ x,y |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + Ω Φ x |u(x)| ||u|| dx ||u|| ϕ + .
Next, assume that ||u|| > 1. Let β ∈ (1, ||u||), by (5.11), we have

Ω Ω Φ x,y |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx β ϕ - Ω Ω Φ x,y |u(x) -u(y)| β |x -y| s dxdy |x -y| N + β ϕ - Ω Φ x |u(x)| β dx = β ϕ - Ω Ω Φ x,y |u(x) -u(y)| β |x -y| s dxdy |x -y| N + Ω Φ x |u(x)| β dx .
Since β < ||u||, we find

Ω Ω Φ x,y |u(x) -u(y)| β |x -y| s dxdy |x -y| N + Ω Φ x |u(x)| β dx > 1.
Thus, we have

Ω Ω Φ x,y |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx β ϕ -.
Letting β ||u||, we deduce that (5.18) holds true.

Next, we show that Ψ (u) ||u|| ϕ -for all u ∈ W s L Φ x,y (Ω) with ||u|| < 1. Using the definition of the Luxemburg norm and (5.14), we obtain

Ψ (u) ||u|| ϕ - Ω Ω Φ x,y |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + ||u|| ϕ - Ω Φ x |u(x)| ||u|| dx = ||u|| ϕ - Ω Ω Φ x,y |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + Ω Φ x |u(x)| ||u|| dx ||u|| ϕ -.
Chapitre 5. Fractional Musielak-Sobolev spaces

Let ξ ∈ (0, ||u||). From (5.12), it follows that

Ω Ω Φ x,y |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx ξ ϕ + Ω Ω Φ x,y |u(x) -u(y)| ξ |x -y| s dxdy |x -y| N + ξ ϕ + Ω Φ |u(x)| ξ dx = ξ ϕ + Ω Ω Φ x,y |u(x) -u(y)| ξ |x -y| s dxdy |x -y| N + Ω Φ |u(x)| ξ dx .
(5.20)

Defining v(x) = u(x) ξ for all x ∈ Ω. Then, ||v|| = ||u|| ξ > 1.
Using relation (5.18), we find

Ω Ω Φ x,y |v(x) -v(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|v(x)|) dx ||v|| ϕ -> 1.
(5.21)

Combining (5.20) and (5.21), we deduce that

Ω Ω Φ x,y |u(x) -u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx ξ ϕ -.
Letting ξ ||u|| in the above inequality, we obtain that relation (5.19) holds true.

Similar to Proposition 5.2.2, we obtain the following results.

Proposition 5.2.3. Assume that (Φ 1 ) is satisfied, Then, for any u ∈ W s L Φ x,y (Ω), the following assertions hold true :

[u] s,Φ x,y > 1 =⇒ [u] ϕ - s,Φ x,y φ (u) [u] ϕ + s,Φ x,y , (5.22)
[u] s,Φ x,y < 1 =⇒ [u] ϕ + s,Φ x,y φ (u) [u] ϕ - s,Φ x,y , (5.23)
where φ

(u) = Ω Ω Φ x,y |u(x) -u(y)| |x -y| s dxdy |x -y| N .

Some embedding results

This subsection is devoted to the embedding results of the new fractional Musielak-Sobolev spaces W s L Φ x,y (Ω). To this end, we follow the same approach used to obtain the embedding results in the fractional Orlicz-Sobolev space W s L Φ (Ω) established in chapter 2.

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 Φ x,y (Ω) is continuously embedded in W s 2 L Φ x,y (
< s 2 s 1 < 1. Assume that (Φ 1 )-(Φ 2 ) holds true. Then, W s 1 L Φ x,y (Ω) continuously embedded in W s 2 L Φ x,y (Ω). Proof. Let u ∈ W s 1 L Φ x,y (Ω) and λ > 0, If we define Ω 1 = (x, y) ⊂ Ω × Ω |D s 1 u| λ 1 and Ω 2 = Ω × Ω \ Ω 1 ,
where D s i u = u(x)u(y) |x -y| s i , i = 1, 2, and we set dµ = dxdy |x -y| N . Then

Ω Ω Φ x,y |D s 2 u| λ dµ = Ω Ω Φ x,y |D s 1 u| λ 1 |x -y| s 2 -s 1 dµ Ω Ω Φ x,y |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) = Ω 1 + Ω 2 Φ x,y |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) = I Ω 1 + I Ω 2 where p = 1 if |x -y| 1 or p = ϕ + if |x -y| < 1 for all (x, y) ∈ Ω × Ω. Notice that I Ω 1 = Ω 1 Φ x,y |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) sup (x,y)∈Ω×Ω Φ x,y (1)
Ω 1 dxdy |x -y| N+p(s 2 -s 1 ) := c 1 .
Since N > N + p(s 2s 1 ), it follows that the above integral is finite. On the other hand, we have

I Ω 2 Ω 2 Φ x,y |D s 1 u| λ dxdy |x -y| N+p(s 2 -s 1 ) d p(s 1 -s 2 ) Ω 2 Φ x,y |D s 1 u| λ dxdy |x -y| N d p(s 1 -s 2 ) Ω Ω Φ x,y |D s 1 u| λ dµ < ∞, where d = sup (x,y)∈Ω×Ω |x -y|. Hence, Ω Ω Φ x,y |D s 2 u| [u] s 1 ,Φ x,y dµ c 1 + d p(s 1 -s 2 ) .
This fact implies that

[u] s 2 ,Φ x,y c 1 + d p(s 1 -s 2 ) [u] s 1 ,Φ x,y . (5.24) Therefore, ||u|| s 2 ,Φ x,y c||u|| s 1 ,Φ x,y ,
where c = (1 + c 1 + d p(s 1 -s 2 ) ).

Chapitre 5. Fractional Musielak-Sobolev spaces

Given s ∈ (0, 1) and let Φ x as defined in (5.2). We denote by Φ -1

x the inverse function of Φ x which satisfies the following conditions :

1 0 Φ -1 x (τ) τ N+s N dτ < ∞ for all x ∈ Ω, (5.25)
∞ 1 Φ -1 x (τ) τ N+s N dτ = ∞ for all x ∈ Ω. (5.26)
Note that, if ϕ x,y (t) = |t| p(x,y)-1 , then (5.25) holds precisely when sp(x, y) < N for all (x, y) ∈ Ω × Ω.

If (5.26) is satisfied, we define the inverse Musielak conjugate function of Φ x as follows is compact for all B x ≺≺ Φ * x,s .

( Φ * x,s ) -1 (t) = t 0 Φ -1 x (τ) τ N+s N dτ. ( 5
The proof will be carried out in a several lemmas. The first one establishes an estimate for the Musielak conjugate function Φ * x,s defined in (5.27).

Lemma 5.2.3. Let s ∈ (0, 1), we assume that (5.25)-(5.26) holds true and let Φ * x,s be defined by (5.27). Then for all s ∈ (0, s), the following conclusions may be drawn.

[ Φ * x,s (t)]

N-s N is a Musielak function, in particular, Φ * x,s is a Musielak function.

2. For every ε > 0, there exists a constant K ε > 0 such that for every t,

[ Φ * x,s (t)] N-s N 1 2ε Φ * x,s (t) + K ε ε t.
(5.30)

The proof is similar to Lemma 2.3.1.

Lemma 5.2.4. Let Ω be an open subset of R N , and 0 < s < 1. Let f satisfies a Lipschitz-condition on R and f (0) = 0, then, (i) For every u ∈ W s,1 loc (Ω), if g(x) = f (|u(x)|), then g ∈ W s,1 loc (Ω).

(ii) For every u ∈ W s L Φ x,y (Ω), if g(x) = f (|u(x)|), then g ∈ W s L Φ x,y (Ω).

In particular, for every u ∈ W s,1 (Ω), if g(x) = f (|u(x)|), then g ∈ W s,1 (Ω).

To prove Lemma 5.2.4, we follow the same approach as in Lemma 2.3.2.

Lemma 5.2.5. Let Ω be a bounded open subset of R N and let 0 < s < s < 1. Assume that the condition (Φ 1 )-(Φ 2 ) are satisfied, then the space W s L Φ x,y (Ω) is continuously embedded in W s ,q (Ω) for all q ∈ [1, ϕ -].

Proof. By (5.4) there exist c > 0 such that , note that by definition of Φ x and (Φ 2 ), 0 < c < ∞.

|t| ϕ - c Φ x (t)
Then, for u ∈ W s L Φ x,y (Ω), we have (5.35)

From (5.30) and since L Φ x (Ω) → L 1 (Ω), then for ε = k 1 , we have We start with the following lemma. The main result of this subsection is given by the following theorem. To prove this theorem, we used Lemma 5.2.6 and we follow the same approach used to obtain the extend results in the fractional Orlicz-Sobolev space W s L Φ (Ω) established in chapter 2. For this, we need to assume that Φ x,y satisfies the following condition |t| p(x,y) MΦ x,y (t) for all (x, y) ∈ Ω × Ω, and all t 0, (Φ 4 )

k 1 I 1 1 2 Ω Φ * x,
where p ∈ C(Ω × Ω) is as given in (5.8)-(5.9) with sp + < N, and M is a positive constant.

In what follows, we will work with the modular norm . and we denote by (W s L Φ x,y (Ω)) * , ||.|| * the dual space of W s L Φ x,y (Ω), ||.|| .

Remark 26. By (Φ 4 ) we deduce that W s L Φ x,y (Ω) is continuously embedded in W s,p(x,y) (Ω), on the other hand, by Theorem 5.1.2, W s,p(x,y) (Ω) is continuously embedded in L r(x) (Ω) for any r ∈ C(Ω) with 1 < r -r(x) < p * s (x) for all x ∈ Ω. Thus, the space W s L Φ x,y (Ω) is continuously embedded in L r(x) for any r ∈ C(Ω) with 1 < r -r(x) < p * s (x) for all x ∈ Ω, that is, there exists a positive constant c 0 > 0 such that for any u ∈ W s L Φ x,y (Ω) u L r(x) (Ω) c 0 u .

Moreover, this embedding is compact.

Next, we suppose that f : Ω × R → R is a Carathéodory function such that

| f (x,t)| c 1 |t| q(x)-1 , ( C 1 )
c 2 |t| q(x) F(x,t) :

= t 0 f (x, τ)dτ, ( C 2 )
for all x ∈ Ω and all t ∈ R N , where c 1 and c 2 are two positive constants, and q ∈ C(Ω) with 1 < q -q(x) < p * s (x) for all x ∈ Ω. -f (x,t) = q(x)|t| q(x)-2 t, and F(x,t) = |t| q(x) , where q ∈ C(Ω) satisfies 2 q(x) < p * s (x) for all x ∈ Ω.

-f (x,t) = q(x)|t| q(x)-2 t +(q(x)-2) log(1+t 2 )|t| q(x)-4 t + t 1 + t 2 |t| q(x)-2 , and F(x,t) = |t| q(x) + log(1 + t 2 )|t| q(x)-2 , where q ∈ C(Ω) satisfies 4 q < p * s (x) for all x ∈ Ω.

-f (x,t) = q(x)|t| q(x)-2 t + (q(x) -1) sin(sint) × |t| q(x)-3 t cos(sint) cost|t| q(x)-1 , and F(x,t) = |t| q (x) + sin(sint)|t| q(x)-1 , where q ∈ C(Ω) satisfies 3 q(x) < p * s (x) for all x ∈ Ω. (C 2 ). If ϕ -> q + , then problem (P a ) has a nontrivial weak solution.

In order to prove the existence result, we consider the energy functional J : W s L Φ x,y (Ω) -→ R (5.42)

The functional J is well defined. Indeed, if u ∈ W s L Φ x,y (Ω), then, by Remark 26, we have u ∈ L q(x) (Ω) such that 1 < q(x) < p * s (x) for all x ∈ Ω. Hence, by the condition (C 1 ), we get

|F(x, u)| u 0 | f (x,t)|dt = c 1 |u| q(x) , consequently, Ω |F(x, u)|dx < ∞.
Now, we establish some basic properties of the functional J. for all u, v ∈ W s L Φ x,y (Ω).

Proof. First, it is easy to see that

I 1 (u), v = Ω Ω
a (x,y) (|D s u|) D s uD s uvdµ + Ω a x (|u|)uvdx, (5.43) for all u, v ∈ W s L Φ x,y (Ω).

Next, we prove that I 1 ∈ C 1 (W s L Φ x,y (Ω), R). Let {u n } ⊂ W s L Φ x,y (Ω) with u n -→ u strongly in for all u, v ∈ W s L Φ x,y (Ω).

W s L Φ x,y (
Proof. First, we observe that from the assumption (C 1 ) and Remark 26, I 2 is well-defined on W s L Φ x,y (Ω). By a standard argument, we have that I 2 is Gâteaux-differentiable on W s L Φ x,y (Ω) with the derivative is given by (5.46). Actually, let {u n } ⊂ W s L Φ x,y (Ω) be a sequence converges strongly to u ∈ W s L Φ x,y (Ω). Since 1 < q -q(x) < p * s (x) for all x ∈ Ω, then, W s L Φ x,y (Ω) is continuously embedded in L q(x) (Ω). It follows that {u n } converges strongly to u in L q(x) (Ω). So, there exists a subsequence of {u n }, still denoted by {u n }, and a function u ∈ L q(x) (Ω) such that {u n } converges to u almost everywhere in Ω, and |u n | |u| for all n ∈ N, and almost everywhere in Ω. Using the assumption (C 1 ), we have that for all measurable functions u : Ω -→ R, the operator defined by u -→ f (., u(.)) maps L q(x) (Ω) into L q(x) (Ω) where q ∈ C(Ω) such that 1 q(x) + 1 q(x) = 1 for all x ∈ Ω. Fix v ∈ W s L Φ x,y (Ω) with ||v|| 1. Using the Hölder's inequality and Remark 26, we obtain

I 2 (u n ) -I 2 (u), v = Ω ( f (x, u n (x)) -f (x, u(x))) v(x)dx ,
|| f (x, u n (x))f (x, u(x))|| q(x) ||v|| q(x) , c 0 || f (x, u n (x))f (x, u(x))|| q(x) ||v||, Thus, passing to the supremum for ||v|| 1, we get ||I 2 (u n ) -I 2 (u)|| * || f (x, u n (x))f (x, u(x))|| q(x) .

202

Chapitre 5. Fractional Musielak-Sobolev spaces By (C 1 ), we deduce that f (x, u n (x))f (x, u(x)) -→ n→+∞ 0, and | f (x, u n (x))f (x, u(x))| c 1 (|u(x)| q(x)-1 + |u(x)| q(x)-1 ) ∈ L q(x) (Ω)

for almost everywhere x ∈ Ω. Hence, in the light of the dominate convergence theorem, we deduce that || f (x, u n (x))f (x, u(x))|| q(x) → 0 as n → ∞. This proves that I 2 is continuous. Consequently, we find

J(u) lim inf n→∞ J(u n ).
Hence, J is weakly lower semi continuous. |u| q(x) dx ||u|| ϕ -c 1 c 0 ||u|| q + + ||u|| q -. Since ϕ -> q + q -, the above inequality implies that J(u) -→ ∞ as ||u|| → ∞, that is, J is coercive.

Proof of Theorem 5.3.1. By Lemma 5.3.3 and Lemma 5.3.4, we infer that J is weakly lower semi continuous and coercive on W s L Φ x,y (Ω). Then, Theorem 1.5.2 implies that there exists u ∈ W s L Φ x,y (Ω) a global minimizer of J which is a weak solution of problem (P a ). L q(x) t ϕ -||φ || ϕ -c 2 t q + ||φ || q + L q(x) .

Since p -> q + and ||φ || q + L q(x) > 0, we have I(t 0 φ ) < 0 for t 0 ∈ (0,t) sufficiently small. Hence, the critical point u of functional I satisfies I(u) I(t 0 φ ) < 0 = I(0), that is u = 0. Consequently, u ∈ W s L Φ x,y (Ω) is a nontrivial weak solution of problem (P a ).

Examples

We point certain examples of functions ϕ x,y and Φ x,y which illustrate the results of this section.

Example 5. As a first example, we can take ϕ x,y (t) = p(x, y)|t| p(x,y)-2 t and Φ x,y = |t| p(x,y) , for all t 0, where p ∈ C(Ω × Ω) satisfies 2 p(x, y) < N for all (x, y) ∈ Ω × Ω.

In this case the problem (P a ) reduces to the following fractional p(x, .)-Laplacian problem

(P 1 )            (-∆ p(x,.) ) s u + |u| p(x) u = f (x, u) in Ω u = 0 in R N \ Ω,
where p(x) = p(x, x) for all x ∈ Ω. Here, the operator (-∆ p(x,.) ) s is the fractional p(x, .)-Laplacian operator defined as follows (-∆ p(x,.) ) s u(x) = p.v.

Ω |u(x)u(y)| p(x,y)-2 (u(x)u(y)) |x -y| N+sp(x,y) dy for all x ∈ Ω.

It easy to see that Φ x,y is a Musielak function and satisfy conditions (Φ 1 ), (Φ 2 ) and (Φ 4 ). In this case we can take ϕ -= p -and ϕ + = p + . Then, we can extract the following result Corollary 5.3.1. Assume that f satisfies (C 1 ) and (C 2 ). If p -> q + . Then, problem (P 1 ) has a nontrivial weak solution u ∈ W s,p(x,y) (Ω).

Example 6. As a second example, we can take ϕ x,y (t) = ϕ 1 (x, y,t) = p(x, y) |t| p(x,y)-2 t log(1 + |t|) for all t 0, and thus, Thus, (Φ 1 ) holds true with ϕ -= p --1 and ϕ + = p + .

Next, Φ x,y satisfies condition (Φ 4 ) since Φ x,y (t) |t| p(x,y)-1 ∀(x, y) ∈ Ω × Ω, ∀t 0.

Hence, we derive an existence result for problem (P 2 ) which is given by the following corollary.

Corollary 5.3.2. Assume that f satisfies (C 1 ) and (C 2 ). If p --1 > q + . Then, problem (P 2 ) has a nontrivial weak solution.

Example 7. As a third example, we can take ϕ x,y (t) = ϕ 2 (x, y,t) = p(x, y) log(1 + α + |t|)|t| p(x,y)-2 t for all t 0 and so, By the above information and taking ϕ -= p -, we have (5.53)

Hence, {u n } is bounded in W s 1 0 L Φ x,y (Ω). Since W s 1 0 L Φ x,y (Ω) is a reflexive space, we may assume that u n converges weakly to u in W s 1 0 L Φ (Ω). Further, since the embedding of W s 1 0 L Φ x,y (Ω)

into L G (Ω) is compact, we obtain that u n -→ u in L G (Ω). Then, since I ∈ C 1 (W s 1 0 L Φ x,y (Ω), R),

  numbers R set of real numbers R + set of positive real numbers N positive integer greater than or equals to 1 R N Euclidean space of N-dimensional vectors Ω open bounded subset of R n ∂ Ω boudary of Ω Ω closure of Ω (i.e., Ω plus its boundary) |Ω| measure of the set |Ω| x (x 1 , ..., x N ) point in R N dx dx 1 , ..., dx N Lebesgue measure in Ω C(Ω) continuous functions from Ω to R ∇u gradient of u

a

  |u(x)u(y)| |x -y| s u(x)u(y) |x -y| s dy |x -y| N+s .

1. 2 . 1 .+ 1 p = 1 .

 2111 Lebesgue spaces Definition 1.2.1. Let Ω be an open subset of R N , for any p ∈ [1, ∞) we define the Lebesgue spaces L P (Ω) as follows :L p (Ω) = u : Ω → R N measurable : Ω |u(x)| p dx < ∞ .1.2. Fractional Sobolev spacesThis is a normed space thanks to the Minkowski inequality. The norm, which is denoted by .If p = ∞, we denote L ∞ (Ω) = u : Ω → R N measurable : |u(x)| c a.e in Ω for some c > 0 .It is easily verified that the functional defined byu ∞ = inf {c > 0 : |u(x)| c a.e in Ω} , is a norm on L ∞ (Ω).Theorem 1.2.1. (Holder's inequality) If 1 p ∞ and u ∈ L p (Ω), u ∈ L p (Ω) such that 1 p Then uv ∈ L 1 (Ω) and Ω |uv|dx u p v p . Theorem 1.2.2. (cf. [1]) Let Ω be an open subset in R N , -L p (Ω) is a Banach space for all p ∈ [1, ∞].

  W s,p (Ω) = u ∈ L p (Ω) : |u(x)u(y)| |x -y| N p +s ∈ L p (Ω × Ω) ,that is, an intermediary Banach space, endowed with its natural norm||u|| s,p = Ω |u| p dx + Ω Ω |u(x)u(y)| p |x -y| sp+N dxdy 1 p

Chapitre 2 .

 2 Fractional Orlicz-Sobolev spaces rem, there exist a subsequence D s u σ (n) and a function k in L A (Ω × Ω, dµ) such that |D s u σ (n) | |k(x, y)| for almost every pair (x, y), and we have D s u σ (n) -→ D s u for almost every pair (x, y), this implies by dominated convergence theorem that,

Lemma 2 . 3 . 3 .

 233 (see : [45, Proposition 2.9]) Let Ω be a bounded open subset of R N and let 0

Definition 2 . 4 . 1 .Remark 7 .

 2417 For any s ∈ (0, 1) and any N-function A, we say that an open set Ω ⊂ R N is an extension domains for W s L A (Ω) if there exists a positive constant c > 0 such that : for every function u ∈ W s L A (Ω) there exists u ∈ W s L A (R N ) with u(x) = u(x) for all x in Ω and || u|| W s L A (R N ) c||u|| W s L A (Ω) . In general, an arbitrary open set is not an extension domain for W s L A . The problem of characterizing the class of sets that are extension domains for W s L A is open. When s is an integer,

. 11 )

 11 By (3.10) and (3.11),we have :

3 . 13 )

 313 and (3.14), we get |F(x,t)| ε|t| α p + c 6 |t| r for each (x,t) ∈ Ω × R. So by Theorem 1.2.4, the embedding W s,p 0 (Ω) in L α p (Ω) and in L r (Ω) is Chapitre 3. On some nonlocal problems in fractional Sobolev spaces

3. 2 .

 2 Three solutions for a Kirchhoff type problem involving nonlocal fractional p-Laplacian 59 there exists a number δ > 0 with the property described in the conclusion of Theorem 1.5.7. Fix λ ∈ Λ and g ∈ A . Put

3. 3 .

 3 Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 63 endowed with the norm :

. 25 )

 25 Combining (3.23)-(3.24)-(3.25) with the Hölder's inequality, we have

3. 3 .

 3 Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 69 that is, lim n→∞ ||u n || = ||u||.

  So, let u, v ∈ W , with u = v and δ , β ∈ [0, 1] with δ + β = 1. By (3.27), we have φ : W → W * is strictly monotone, so by [164, Proposition 25.10], φ is strictly convex. Moreover, since M is nondecreasing, the function M is convex in R + , thus we have

3. 3 .

 3 Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 73 there exists a number δ > 0 with the property described in the conclusion of Theorem 1.5.7. Fix λ ∈ Λ and g ∈ A . Put

3. 4 .

 4 Ekeland's variational principle for a nonlocal p-Kirchhoff type eigenvalue problem 3.4 Ekeland's variational principle for a nonlocal p-Kirchhoff type eigenvalue problem

Theorem 3 . 4 . 1 .

 341 Assume M verify condition (M 1 ) and the functionals f and F satisfy conditions (F 1 ) and (F 2 ). Then there exist λ * and λ * , such that for any λ ∈ (0, λ * ) ∪ [λ * , ∞), problem (P s ) has a nontrivial weak solution.

3. 4 .

 4 Ekeland's variational principle for a nonlocal p-Kirchhoff type eigenvalue problem 79 On the other hand, by Theorem 1.2.6 we have u n -→ u strongly in W s,p (Ω), and since I 2 is of class C 1 . Then lim n→∞ I 2 (u n ) = I 2 (u).

Lemma 3 . 4 . 3 .

 343 Assume the hypotheses of Theorem 3.4.1 are fulfilled. Then there exist ρ, α > 0 and λ * > 0 such that for any λ ∈ (0, λ * ) : J λ (u) α > 0 for any u ∈ W s,p (Ω) with ||u|| = ρ.

Lemma 3 . 4 . 4 .

 344 Assume the hypotheses of Theorem 3.4.1 are fulfilled. Then there exists θ ∈ W s,p (Ω)

. 51 )

 51 By (3.50) and (3.51), we have :

  and thus u λ is a nontrivial weak solution of problem (P s ) for any λ ∈ [λ * , ∞). Therefore, problem (P s ) has a nontrivial weak solutions for all λ ∈ (0, λ * ) ∪ [λ * , ∞).On some nonlocal problems in fractionalOrlicz-Sobolev spaces4.1 IntroductionIn this chapter, great attention has been devoted to the study of nonlinear problems involving nonlocal operators in fractional Orlicz-Sobolev spaces W s L Φ (Ω). The study of variational problems where the function a satisfies a nonpolynomial growth conditions instead of having the usual p-structure arouses much interest with the development of applications to electrorheological fluids as an important class of non-Newtonian fluids (sometimes referred to as smart fluids).

4. 2 . 91 4. 2

 2912 Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces

Theorem 4 . 2 . 1 .

 421 Assume that conditions (C 1 )-(C 5 ) are fulfilled. If ϕ -> q + then the problem (P a ) has a nontrivial weak solution in W s 0 L Φ (Ω).

4. 2 . 0 f

 20 Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces 93 For u ∈ W s 0 L Φ (Ω), we defineJ(u) = Ω Ω Φ |u(x)u(y)| |x -y| s dxdy |x -y| N and H(u) = Ω F(x, u)dx I(u) = J(u) -H(u) (4.6)where F(x,t) = t (x,t)dτ. Obviously the energy functional I : W s 0 L Φ (Ω) -→ R associated with problem (P a ) is well defined.

94 Chapitre 4 .

 944 there exist a subsequence of {u n }, still denoted by {u n }, and a function u ∈ L G (Ω) such that {u n } converges to u almost everywhere in Ω and |u n | |u| for all n ∈ N and almost everywhere in Ω. By (C 1 ), we have for all measurable functions u : Ω -→ R, the operator defined by u -→ f (., u(.)) maps L G (Ω) into L G (Ω).Fix v ∈ W s 0 L Φ (Ω) with ||v|| 1. By using the Hölder inequality and the embedding of W s 0 L Φ (Ω) On some nonlocal problems in fractional Orlicz-Sobolev spaces into L G (Ω), we have

  dµ). So by dominated convergence theorem, there exist a subsequence {D s u n k } and a function h in L Φ (Ω × Ω, dµ) such that, a(|D s u n k |)D s u n k -→ a(|D s u|)D s u a.e. in ∈ Ω × Ω and by Lemma 1.1.1, we have |a(|D s u n k |)D s u n k |a(|h|)h| ∈ L Φ (Ω) a.e in Ω × Ω.

Lemma 4 . 2 . 3 .

 423 Let (C 1 ) be satisfied. Then the functional I ∈ C 1 (W s 0 L Φ (Ω), R) is weakly lower semicontinuous.

  Lemma 4.2.3 I is weakly lower semicontinuous on W s 0 L Φ (Ω), then by Theorem 1.5.2 functional I has a minimum point u 0 in W s 0 L Φ (Ω) and u 0 is a weakly solution of problem (P a ).

Theorem 4 . 3 . 1 . 4 . 3 .

 43143 Suppose that M satisfies (M 1 ), (M 2 ) and f satisfies (C 1 )-(C 6 ). Then problem (P a ) has a nontrivial solution u ∈ W s 0 L Φ (Ω) which is a critical point of mountain pass type for the functional :I(u) = M Ω Ω Φ |u(x)u(y)| |x -y| s dxdy |x -y| N -Ω F(x, u)dx.(4.12) Mountain pass type solution for a nonlacal fractional a-Kirchhoff type problem 101 Let us denote by I i : W s 0 L Φ (Ω) -→ R, i = 1, 2, the functionals I 1 (u) = M Ω Ω Φ |u(x)u(y)| |x -y| s dxdy |x -y| N and I 2 (u) = Ω F(x, u)dx.

Lemma 4 . 3 . 1 .

 431 Let Ω be a bounded open subset of R N , and let s ∈ (0, 1). Assume condition (1.2) is satisfied, then there exists a positive constant λ 1 such that,

Lemma 4 . 3 . 3 .

 433 The functional I 1 :W s 0 L Φ (Ω) -→ R is of class C 1 and I 1 (u), v = M(Ψ(u)) Ω Ω a |u(x)u(y)| |x -y| s u(x)u(y) |x -y| s v(x)v(y) |x -y| s dxdy |x -y| N ,for all u, v ∈ W s 0 L Φ (Ω).

4. 3 .

 3 Mountain pass type solution for a nonlacal fractional a-Kirchhoff type problem 103 Or, the continuity of M implies that

Lemma 4 . 3 . 4 .

 434 Let (C 1 )-(C 4 ) hold true. Let I be the functional defined in (4.12), and let {u n } be a sequence in W s 0 L Φ (Ω) such that

) that there exists c 3 >

 3 0 such that |I(u n )| c 3 and | I (u n ), u n | c 3 ||u n ||.By assumption (C 1 ) and (C 4 ), we have 0 < tϕ(t) ϕ + Φ(t) for all t > 0,(4.18)

4. 3 .

 3 Mountain pass type solution for a nonlacal fractional a-Kirchhoff type problem 105 and therefore that lim n→∞ I(u n ) = I(u).

Lemma 4 . 3 . 5 .

 435 Under the assumptions of Theorem 4.3.1, the geometric condition (G 1 ) of the mountain pass Theorem 1.5.5 hold for the functional I defined in (4.12).

4. 3 . 2 and thus setting r = α ε 2 ,Lemma 4 . 3 . 6 .any t 1 . 108 Chapitre 4 .

 32243611084 Mountain pass type solution for a nonlacal fractional a-Kirchhoff type problem Proposition 2.2.1, we get ||u|| -→ 0 ⇐⇒ Ψ(u) -→ 0, hence for ρ > 0 as given in (4.29), there exists a α = α(ρ) > 0 such that for all u with ||u|| = ρ, we obtain that (G 1 ) is satisfied. Under the assumptions of Theorem 4.3.1, the geometric condition (G 2 ) of the mountain pass Theorem 1.5.5 hold for the functional I defined in (4.12).Proof. First, by assumption (M 2 ), we get that From (C 5 ) it follows thatF(x, ξ ) r -β min {F(x, r), F(x, -r)} |ξ | β (4.31)for all |ξ | > r and a.e. x ∈ Ω. Thus by (4.31) andF(x, ξ ) max |ξ | r F(x, ξ ) for all |ξ | r, we obtain F(x, ξ ) r -β min {F(x, r), F(x, -r)} |ξ | βmax |ξ | r F(x, ξ )min {F(x,r), F(x, -r)} (4.32) for any ξ ∈ R and a.e. x ∈ Ω. By Theorem 2.2.3, we can fix u 0 ∈ C ∞ 0 (Ω) such that ||u 0 || = 1 and let t 1. Combining (4.30) with On some nonlocal problems in fractional Orlicz-Sobolev spaces (4.32), we haveI(tu 0 ) = M(Ψ(tu 0 )) -Ω F(x,tu 0 )dx M(||tu 0 || ϕ + ) -Ω F(x,tu 0 )dx M(1)t ϕ + θr -β |t| β Ω min {F(x, r), F(x, -r)} |u 0 (x)| β dx + Ω max |ξ | r F(x, ξ ) + min {F(x, r), F(x, -r)} dx.From assumptions (C 1 ) and (C 6 ), we get that 0< F(x, ξ ) c 1 (|r| + G(|r|)) for |ξ | r a.e. x ∈ Ω.Thus, 0 < min {F(x, r), F(x, -r)} < c 1 (|r| + G(|r|)), a.e. x ∈ Ω. Since β > ϕ + θ by assumption (C 5 ), passing to the limit as t → ∞, we obtain that I(tu 0 ) → -∞. Thus, the assertion follows by taking e = Tu 0 with T sufficiently large. 4.3.2.3 Proof of Theorem 4.3.1. It follows from Lemma 4.3.5 and Lemma 4.3.6 that the hypotheses of Theorem 1.5.5 are satisfied. So the Lemma 4.3.4 implies the existence of a nontrivial critical point of the functional I.

  3) holds. By Example 2 [65, p. 243] it follows that ϕ + = p + 1 and ϕ -= p. (4.38) Then (4.3) hold true.

4 . 3 .

 43 τ)] s/N = ∞ or, by (4.42), ∞ Φ -1 (1) dτ [log(1 + τ)] s/N τ sp/N = ∞. (4.45) Since log(1 + x) x, for all x > 0, Mountain pass type solution for a nonlacal fractional a-Kirchhoff type problem 111 we deduce that, 1

  dy.The problem (P a ) has been motivated by some works, which the authors have used different methods to get the existence of solutions. For examples in[START_REF] Mihäilescu | Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces[END_REF] Rȃdulescu et al, by means of Ekeland's variational principle and direct variational approach. They obtained the existence of nontrivial weak solutions for an eigenvalue problem in Orlicz-Sobolev spaces. In[START_REF] Azroul | Eigenvalue problems involving the fractional p(x)-Laplacian operator[END_REF] Azroul et al. have studied an eigenvalue problem in fractional Sobolev space with the variable exponent. Mainly, based on Ekeland's variational principle, the authors obtained the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin. Moreover, in[START_REF] Bonder | A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians[END_REF] Salort has studied the eigenvalues and minimizers of a fractional non-standard growth problem in fractional Orlicz-Sobolev spaces and it has established several properties on these quantities and their corresponding eigenfunctions.

Theorem 4 . 4 . 1 .

 441 Assume that conditions (C 1 ) and (C 2 ) are fulfilled. Then there exist λ * and λ * , such that for any λ ∈ (0, λ * ) ∪ [λ * , ∞), problem (P a ) has a nontrivial weak solutions.

(4. 50 ) 4 . 4 .

 5044 Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space 115

  |v n k |)v n ka(|v|)v)D s wdµ -→ 0.(4.53) Combining (4.52) -(4.53), we have

4. 4 .Lemma 4 . 4 . 2 .

 4442 Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space 117 The proof of Lemma 4.4.1 is completed. If f satisfies assumption (C 1 ) . Then the functional I 2 : W s L Φ (Ω) → R is of class C 1

Chapitre 4 .Lemma 4 . 4 . 3 .

 4443 On some nonlocal problems in fractional Orlicz-Sobolev spaces Combining Lemma 4.4.2 and Lemma 4.4.1 we infer that Proposition 4.4.1 holds true. The functional I 1 is weakly lower semi continuous.

n→∞ I 1

 1 (u n ), that is, the map u -→ Ω Ω Φ |u(x)u(y)| |x -y| s dxdy |x -y| N + Ω Φ (|u(x)|) dxis weakly lower semi continuous. Proposition 4.4.2. The functional J λ is weakly lower semi continuous.

4. 4 .

 4 Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space 119 Therefore, J λ is weakly lower semi continuous and Proposition 4.4.2 is verified. Lemma 4.4.4. Assume that the sequence {u n } converges weakly to u in W s L Φ (Ω) and lim sup n→∞ I 1 (u n ), u nu 0. (4.55)

Lemma 4 . 4 . 5 .

 445 Assume the hypotheses of Theorem 4.4.1 are fulfilled. Then there exist ρ, α > 0 and λ * > 0 such that for any λ ∈ (0, λ * ), J λ (u) α > 0 for any u ∈ W s L Φ (Ω) with ||u|| = ρ.

4. 4 .Lemma 4 . 4 . 6 .

 4446 Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space 121 The proof of Lemma 4.4.5 is completed. Assume the hypotheses of Theorem 4.4.1 are fulfilled. Then there exists θ ∈ W s L Φ (Ω)

Lemma 4 . 4 . 7 .

 447 Assume the hypotheses of Theorem 4.4.1 are fulfilled. Then for any λ > 0 the functional J λ is coercive.

4. 4 .

 4 Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space 123 inequality implies

  are two Carathéodory functions, λ and µ are tow real parameters and (-∆) s a(.) is a nonlocal integro-differential operator of elliptic type defined as follows (-∆) s a(.) u(x) = 2 lim ε 0 R N \B ε (x) a |u(x)u(y)| |x -y| s u(x)u(y) |x -y| s dy |x -y| N+s .for all x ∈ R N .

  Chung in[START_REF] Chung | Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities[END_REF], studied the problem (P) with Au = -M Ω φ (|∇u|)dx div(a(|∇u|∇u)). That is it 4.5. Multiple solutions for a nonlocal Kirchhoff problem in Fractional Orlicz-Sobolev spaces 125 studied the following problem in Orlicz-Soblev spaces :

Definition 4 . 5 . 1 .

 451 Let 0 < s < s < 1, if N > s ϕ -, we denote by A the class of all Carathéodory functions f :

Now, let {u

  n } ⊂ W s 0 L Φ (Ω) with u n u weakly in W s 0 L Φ (Ω), then by the continuity and monotonicity of the function t → M(t), we get lim inf n→∞ I(u n ) = lim inf n→∞ M(Ψ(u n ))

. 68 )Lemma 4 . 5 . 2 .

 68452 from (4.66) and (4.68) we obtain a contradiction. This shows that {u n } converges strongly to u and the functional Ψ belongs to the class W W s 0 L Φ (Ω) . Assume that the sequence {u n } converges weakly to u in W s 0 L Φ (Ω) and lim sup n→∞ Ω Ω a(|D s u n |)D s u n (D s u n -D s u) dµ 0. (4.69)

4. 5 .

 5 Multiple solutions for a nonlocal Kirchhoff problem in Fractional Orlicz-Sobolev spaces 129

132 Chapitre 4 .

 1324 is compact. Then for some positive constant c 4 , one has for all u ∈ W s 0 L Φ (Ω) with |u| η 1 On some nonlocal problems in fractional Orlicz-Sobolev spaces and ||u|| < 1,

4. 5 .

 5 Multiple solutions for a nonlocal Kirchhoff problem in Fractional Orlicz-Sobolev spaces 133

3 144Chapitre 4 .

 34 On some nonlocal problems in fractional Orlicz-Sobolev spaces large enough such that J(u) 0 for every u ∈ Y k and ||u|| d k .(4.92)

Lemma 4 . 7 . 3 . 1 →∞

 4731 The following relations hold true lim ||u||

( 4 . 102 )Lemma 4 . 7 . 4 .

 4102474 holds true. The proof of Lemma 4.7.3 is completed. There exists u ∈ W s 1 0 L Φ 1 (Ω) \ {0} such that J(u) I(u) = λ 1 .

4. 7 .

 7 Eigenvalue problem associated with nonhomogeneous integro-differential operators 157 this implies that T λ (v λ )

4. 8 .

 8 The Nehari manifold approach for Fractional a(.)-Laplacian problem 161 Theorem 4.8.1.

(4. 120 )Lemma 4 . 8 . 1 .

 120481 Hence, N λ contains every nontrivial weak solution of problem (P a ). Moreover, we have the fallowing result. Under the assumptions of Theorem 4.8.2. Then the functional J λ is coercive and bounded below in N λ .

Remark 19 .

 19 As a consequence of Lemma 4.8.2, for any λ ∈ (0, λ * ) we have N λ = N + λ ∪ N - λ , and we define σ - λ := inf u∈N - λ J λ (u) and σ + λ := inf u∈N + λ J λ (u).

Lemma 4 . 8 . 3 .

 483 Under the assumptions of Theorem 4.8.2. If 0 < λ < λ * , then σ + λ < 0.

Chapitre 5 .

 5 Fractional Musielak-Sobolev spaces find Φ x,y (t) =

  Ω) when s 2 s 1 , as next result points out. Proposition 5.2.4. Let Ω be an open subset of R N and let 0

. 27 )Theorem 5 . 2 . 4 .

 27524 Theorem 5.2.3. Let Ω be a bounded open subset of R N with C 0,1 -regularity and bounded boundary. If (5.25) and (5.26) hold, then W s L Φ x,y (Ω) → L Φ * Let Ω be a bounded open subset of R N and C 0,1 -regularity with bounded boundary. If (5.25) and (5.26) hold, then the embedding W s L Φ x,y (Ω) → L B x (Ω), (5.29)

Ω 1 k 1 = k 1 I 1 +

 1111 |u(x)| ϕ - dx Ω∩{|u| 1} |u(x)| ϕ - dx + Ω∩{|u|>1} |u(x)| ϕ - dx |Ω| + c Ω Φ x (|u(x)|)dx.Hence,||u|| ϕ -c 1 ||u|| Φ x ,(5.32)where c 1 = |Ω| + c. On the other hand, similar to Proposition 5.2.4 we have[u] s ,ϕ -c 2 [u] s,Φ x,y .(5.33)Then, combining (5.32) with (5.33), we obtain Chapitre 5. Fractional Musielak-Sobolev spaces||u|| s ,ϕ -c[u] s,Φ x,y .This completes the proof.Proof of Theorem 5.2.3. Let 0 < s < s < 1, σ (t) = [ Φ * x,s (t)] N-s Nand u ∈ W s L Φ x,y (Ω), we suppose for the moment that u is bounded on Ω and not equal to zero in L Φ x (Ω), then Ω infinity to zero as λ increases from zero to infinity and according, assumes the value unity for some positive value k of λ , thusΩ Φ * x,s |u(x)| k dx = 1, k = ||u|| Φ * x,s . (5.34) Let f (x) = σ u(x) k . Using Lemma 5.2.1 u ∈ W s ,1 (Ω), and σ is Lipschitz function (see Lemma 5.2.3), so from Lemma 5.2.4 we have f ∈ W s ,1 (Ω), and since N > s , then by Theorem 1.2.|| f || L 1 + [ f ] s ,1 , || f || L 1 + [ f ] s ,1 k 1 I 2 .

Proof.,

  Let σ (t) = [ Φ x (t)]N-s N where 0 < s < s < 1 and u ∈ W s 0 L Φ x,y (Ω), then similarly to Lemma 5.2.4, for f (x) = σ u(x) k with k = ||u|| Φ x , we have f ∈ W s ,1 0 (Ω), and since N > s , then by Theroem 1.2.6, one has, this implies that,1 k 5 [ f ] s ,1 = k 5 Ω Ω | f (x)f (y)| |x -y| N+s dxdy ||u|| Φ x γ[u] s,Φ x,ywhere γ = k 5 c.Remark 25.a)-As a trivial consequence of Theorem 5.2.5, for a bounded open subset Ω of R N , there exists a positive constant λ 1 such that,Ω Φ x (u(x)) dx λ 1 Ω Ω Φ x,y |u(x)u(y)| |x -y| s dxdy |x -y| N ,(5.40)for all u ∈ W s 0 L Φ x,y (Ω).b)-From Theorem 5.2.5, we deduce that [.] s,Φ x,y is a norm on W s 0 L Φ x,y (Ω) which is equivalent to the norm ||u|| s,Φ x,y .5.2.2 Extending a W s L Φ x,y (Ω) function to the whole of R NGiven an open bounded domain Ω ∈ R N , For any s ∈ (0, 1) and any Musielak function Φ x,y , we say that an open set Ω is an extension domains for W s L Φ x,y (Ω) if for every function u ∈ W s L Φ x,y (Ω)there exists u ∈ W s L Φ x,y (R N ) with u(x) = u(x) for all x in Ω.Hence, our aim in this subsection is to show that any open bounded set Ω of class C 0,1 with bounded boundary is an extension domain for W s L Φ x,y . For this, we assume that Φ x,y ∈ B f and lim t→∞

Lemma 5 . 2 . 6 .

 526 Assume that (Φ 1 )-(Φ 3 ) holds. Then for all u ∈ W s L Φ x,y (Ω)Ω Φ x,y (|u(x)|)dx < ∞ ∀y ∈ Ω.Proof. First, seminary to (5.31), we haveΦ x,y (t) |t| ϕ + ∀t > 1. Then, Ω Φ x,y (|u(x)|)dx Ω Φ x,y (1)dx + Ω |u(x)| ϕ + dx sup (x,y)∈Ω×Ω Φ x,y (1)|Ω| + u ϕ + ϕ + .On the other hand, by (Φ 3 ) we can used Theorem 5.2.4 and we have u ϕ + c u s,Φ x,y .

5. 3 .Theorem 5 . 2 . 6 .

 3526 Application 1 : Existence of solutions for a nonlocal type problem in fractional Musielak-Sobolev spaces 197 Let Ω be an open subset of R N with C 0,1 -regularity and bounded boundary. Then, W s L Φ x,y (Ω) is continuously embedded in W s L Φ x,y (R N ), namely for any u ∈ W s L Φ x,y (Ω) there exists u ∈ W s L Φ x,y (R N ).

5. 3

 3 Application 1 : Existence of solutions for a nonlocal type problem in fractional Musielak-Sobolev spaces The aim of this section is to prove the existence in fractional Musielak-Sobolev spaces of the following nonlocal problem(P a ) s a (x,.) u + a x (|u|)u = f (x, u) in Ω, u = 0 in R N \ Ω,where Ω is an open bounded subset in R N , N 1, with Lipschitz boundary ∂ Ω, 0 < s < 1, f :Ω × R -→ R is a Carathéodory function,and (-∆) s a (x,y) is the nonlocal integro-differential operator of elliptic type defined as follows :(-∆) s a (x,.) u(x) = 2 lim ε 0 R N \B ε (x) a (x,y) |u(x)u(y)| |x -y| s u(x)u(y) |x -y| s dy |x -y| N+s .

Example 4 .

 4 We point out certain examples of function f which satisfies the hypotheses (C 1 ) and (C 2 ).

Definition 5 . 3 . 1 .Theorem 5 . 3 . 1 .

 531531 u ∈ W s L Φ x,y (Ω) is called a weak solution of problem (P a ) if, Ω Ω a (x,y) (|D s u|)D s uD s vdµ + Ω a x (|u|)uvdx -Ω f (x, u)vdx = 0,(5.41)for all v ∈ W s L Φ x,y (Ω).Now, we are ready to state our existence result. Assume that (Φ 1 ), (Φ 2 ) and (Φ 4 ) hold true, and the function f satisfying (C 1 ) and

5. 3 .

 3 Application 1 : Existence of solutions for a nonlocal type problem in fractional Musielak-Sobolev spaces 199 associated to (P a ), and it is defined by :J(u) = Ω Ω Φ x,y |u(x)u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx -Ω F(x, u)dx.

Proposition 5 . 3 . 1 .

 531 Assume that (Φ 1 ) and (C 1 ) are verified. Then J ∈ C 1 W s L Φ x,y (Ω), R . Moreover, its Gâteaux derivative is given byJ (u), v = Ω Ω a (x,y) (|D s u|)D s uD s vdµ + Ω a x (|u|)uvdx -Ω f (x, u)vdx for all u, v ∈ W s L Φ x,y (Ω).Note that the weak solutions of (P a ) coincide with the critical points of J. To prove Proposition 5.3.1, we define the following functionalsI i : W s L Φ x,y (Ω) -→ R, i = 1, 2, by I 1 (u) = Ω Ω Φ x,y |u(x)u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dx,andI 2 (u) = Ω F(x, u)dx.Lemma 5.3.1. Suppose that (Φ 1 ) hold true. Then, the functional I 1 ∈ C 1 (W s L Φ x,y (Ω), R), and I 1 (u), v = Ω Ω a (x,y) (|D s u|) D s uD s vdµ + Ω a x (|u|)uvdx.

5 . 3 . 1 | I 1 Lemma 5 . 3 . 2 .

 5311532 Ω), then u n -→ u in L Φ x (Ω), and v n := D s u n -→ v := D s u in L Φ x,y (Ω × Ω, dµ). So, by dominated convergence theorem, there exist a subsequence {u n k } (resp. in {v n k }), and a function h1 (resp. h 2 ) in L Φ x (Ω) (resp. L Φ x,y (Ω × Ω, dµ)) such that a x (|u n k |)u n k -→ a x (|u|)u a.e. x ∈ Ω, a (x,y) (|v n k |)v n k -→ a (x,y) (|v|)v a.e. x, y ∈ Ω × Ω,and by Lemma 5.2.1, we have| a x (|u n k |)u n k | | a x (|h 1 |)h 1 | ∈ L Φ x (Ω) a.e in Ω, |a (x,y) (|v n k |)v n k | |a (x,y) (|h 2 |)h 2 | ∈ L Φ x,y (Ω × Ω, dµ) a.e in Ω × Ω.Hence, for w ∈ W s L Φ x,y (Ω), we have D s w ∈ L Φ x,y (Ω × Ω, dµ), and by the Hölder's inequality, we obtainΩ Ω ( a x (|u n k |)u n ka x (|u|)u)wdx 2 || a x (|u n k |)u n ka x (|u|)u|| Φ x ||w|| Φ x 2 || a x (|u n k |)u n ka x (|u|)u|| Φ x ||w|| ,andΩ Ω (a (x,y) (|v n k |)v n ka (x,y) (|v|)v)D s wdµ 2 a (x,y) (|v n k |)v n ka (x,y) (|v|)v s,Φ x,y [w] s,Φ x,y 2 a (x,y) (|v n k |)v n ka (x,y) (|v|)v s,Φ x,y ||w|| .Then, by the dominated convergence theorem, we obtain that sup||w|| 1 Ω Ω ( a x (|u n k |)u n ka x (|u|)u)wdx -→ 0, (5.44) Application 1 : Existence of solutions for a nonlocal type problem in fractional Musielak-Sobolev spaces 201 and sup ||w|| 1 Ω Ω (a (x,y) (|v n k |)v n ka (x,y) (|v|)v)D s wdµ -→ 0. (5.45) Combining (5.44) -(5.45), we deduce||I 1 (u n ) -I 1 (u)|| * = sup ||v|| (u n ) -I 1 (u), v | -→ 0.The proof of Lemma 5.3.1 is completed. If f satisfies assumption (C 1 ). Then, I 2 ∈ C 1 (W s L Φ x,y (Ω), R) with its Gâteaux derivative is given byI 2 (u), v = Ω f (x, u(x))v(x)dx(5.46)

Proof of Proposition 5 . 3 . 1 .

 531 Combining Lemma 5.3.2 and Lemma 5.3.1, we infer that Proposition 5.3.1 holds true.

Lemma 5 . 3 . 3 .

 533 Under the same assumptions of Theorem 5.3.1, the functional J is weakly lower semi continuous.Proof. First, we show that the map :u -→ Ω Ω Φ x,y |u(x)u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dxis lower semi-continuous for the weak topology of W s L Φ x,y (Ω). Indeed, similar to Lemma 5.3.1, we obtain I 1 ∈ C 1 (W s L Φ x,y (Ω), R), andI 1 (u), v = Ω Ω a (x,y) (|D s u|) D s uD s v dxdy |x -y| N + Ω a x (|u|)uvdxfor all u, v ∈ W s L Φ x,y (Ω). On the other hand, since Φ x,y is a convex function, it folows that I 1 is also convex.Now, let {un } ⊂ W s L Φ x,y (Ω) with u n u weakly in W s L Φ x,y(Ω), then by convexity of I 1 we haveI 1 (u n ) -I 1 (u) I 1 (u), u nu .Hence, we obtain I 1 (u) lim inf n→∞ I 1 (u n ), that is, the mapu -→ Ω Ω Φ x,y |u(x)u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u(x)|) dxis weakly lower semi continuous, that is,I 1 (u) lim inf n→∞ I 1 (u n ).

5. 3 .

 3 Application 1 : Existence of solutions for a nonlocal type problem in fractional Musielak-Sobolev spaces 203 Now, using condition (C 1 ) and similarly to Lemma 5.3.2, we have lim n→∞ Ω F(x, u n )dx = Ω F(x, u)dx.

Lemma 5 . 3 . 4 .

 534 Assume that the hypotheses of Theorem 5.3.1 are fulfilled. Then, the functional J is coercive.Proof. For any u ∈ W s L Φ x,y (Ω) with ||u|| > 1, From Propositions 5.2.2 and 5.1.1, Remark 26 and(C 1 ), it follows that J(u) = Ω Ω Φ x,y |u(x)u(y)| |x -y| s dxdy |x -y| N + Ω Φ x (|u|)dx -Ω F(x, u)dx ||u|| ϕ -c 1 Ω

  Next, we need to verify that u is nontrivial. Letx 0 ∈ Ω 0 ⊂⊂ Ω, 0 < R < 1 satisfy B 2R (x 0 ) ⊂ Ω 0 ,where B 2R (x 0 ) is the ball of radius 2R with center at the pointx 0 in R N . Let φ ∈ C ∞ 0 (B 2R (x 0 )) satisfies 0 φ 1 and φ ≡ 1 in B R (x 0 ). Theorem 5.2.2 implies that ||φ || < ∞. Then for 0 < t < 1,by (C 2 ), we haveI(tφ ) = Ω Ω Φ x,y |tφ (x)tφ (y)| |x -y| s dxdy |x -y| N + Ω Φ x (|tφ |)dx -Ω F(x,tφ )dx ||tφ || ϕ -c 2 Ω |tφ | q(x) dx ||tφ || ϕ -c 2 ||tφ || q +

Φϕ 1 )

 1 x,y (t) = p(x, y) |t| p(x,y) log(1 + |t|)+ |t| 0 τ p(x,y) (1 + τ)(log(1 + τ))2 dτ, 5.3. Application 1 : Existence of solutions for a nonlocal type problem in fractional Musielak-Sobolev spaces 205with p ∈ C(Ω × Ω) satisfies 2 p(x, y) < N for all (x, y) ∈ Ω × Ω.Then, in this case problem (P a ) becomes(P 2 ) s u + p(x)|u| p(x)-2 u log(1 + |u|) = f (x, u) in Ω u = 0 in R N \ Ω, with (-∆ ϕ 1 ) s u(x) = p.v. Ω p(x,y)|D s u| p(x,y)-2 D s u log(1 + |D s u|)|x -y| N+s dy for all x ∈ Ω. It easy to see that Φ x,y is a Musielak function and satisfy condition (Φ 2 ). Moreover, for each (x, y) ∈ Ω × Ω fixed, by Example 3 on p 243 in [65], we have p(x, y) -1 tϕ x,y (t) Φ x,y (t) p(x, y) ∀(x, y) ∈ Ω × Ω, ∀t 0.

ΦChapitre 5 .ϕ 2 )

 52 x,y (t) = log(1 + |t|)|t| p(x,y) p ∈ C(Ω × Ω) satisfies 2 p(x, y) < N for all (x, y) ∈ Ω × Ω.206 Fractional Musielak-Sobolev spacesThen we consider the following fractional p(x, .)-problem(P 2 ) s u + p(x) log(1 + α + |u|)|u| p(x)-2 u = f (x, u) in Ω u = 0 in R N \ Ω,where(-∆ ϕ 2 ) s u(x) = p.v. Ω p(x,y) log(1 + α + |D s u|).|D s u| p(x,y)-2 D s u |x -y| N+s dy for all x ∈ Ω. It easy to see that Φ x,y is a Musielak function and satisfy condition (Φ 2 ). Next, we remark that for each (x, y) ∈ Ω × Ω fixed, we have p(x, y) tϕ x,y (t) Φ x,y (t) for all t 0.

5. 4

 4 Application 2 : Bi-nonlocal type problem in fractional Musielak-Sobolev spaceIn this section, we investigate the existence of weak solution in the new fractional Musielak-Sobolev space of the following nonlocal problem(P a ) ) s 1 a (x,.) u + (-∆) s 2 a (x,.) u = f (x, u) in Ω, u = 0 in R N \ Ω,whereΩ is an open bounded subset in R N , N 1, with Lipschitz boundary ∂ Ω, 0 < s 2 s 1 < 1, f : Ω × R -→ R is aCarathéodory function satisfies some suitable conditions. Moreover, for anyi = 1, 2, (-∆) s i a (x,.) is the nonlocal integro-differential operator of elliptic type defined as follows(-∆) s i a (x,.) u(x) = 2 lim ε 0 R N \B ε (x) a (x,y) |u(x)u(y)| |x -y| s i u(x)u(y) |x -y| s i dy |x -y| N+s i , for all x ∈ R N . ||u n || ϕ ∓c 4 .

using (H 1 ) 1 .

 11 -(H 3 ), we get lim n→∞ I(u n ) = I(u) and lim n→∞ I (u n ) = I (u) in (W s 1 0 L Φ x,y (Ω)) * . Moreover, from(5.49), we haveJ (u n ) → 0 in (W s 1 0 L Φ x,y (Ω)) * . Hence, Ψ (u n ) -→ I (u) in (W s 1 0 L Φ x,y (Ω)) * . (5.54)Now, since Ψ is convex, then we haveΨ(u n ) Ψ(u) + Ψ (u n ), u nu .Therefore, using (5.54), we may deduce thatlim sup n→∞ Ψ(u n ) Ψ(u).It further follows from the convexity of Ψ that it is weakly lower semicontinuous and hencelim inf n→∞ Ψ(u n ) Ψ(u),From Theorem 5.2.2, we can fix u 0 ∈ C ∞ 0 (Ω) such that ||u 0 || s 1 = 1 and let t 1. By (5.63), we haveJ(tu 0 ) = Ψ s 1 (tu 0 ) + Ψ s 2 (tu 0 ) -Ω F(x,tu 0 )dx ||tu 0 || ϕ + s 1 + ||tu 0 || ϕ + s 2 + ||tu 0 || ϕ - s 2 -Ω F(x,tu 0 )dx ||tu 0 || ϕ + s 1 + c ϕ + ||tu 0 || ϕ + s 1 + c ϕ -||tu 0 || ϕ - s 1 -Ω F(x,tu 0 )dx (1 + c ϕ + )t ϕ + + c ϕ + t ϕ -r -θ |t| θ Ω min {F(x, r), F(x, -r)} |u 0 (x)| θ dx + Ω max |ξ | r F(x, ξ ) + min {F(x, r), F(x, -r)} dx.From assumptions (H 1 ) and (H 5 ), we get 0< F(x, ξ ) c 1 (|r| + G(|r|)) for |ξ | r a.e. x ∈ Ω. Thus, 0 < min {F(x, r), F(x, -r)} < c 1 (|r| + G(|r|)), a.e. x ∈ Ω. Since θ > ϕ + ϕ -by assumption (H 4 ),passing to the limit as t → ∞, we obtain that J(tu 0 ) → -∞. Thus, the assertion (G 2 ) follows by taking e = Tu 0 with T sufficiently large. 5.4.2 Proof of Theorem 5.4.It follows from Lemma 5.4.2 and Lemma 5.4.3 that the hypotheses of Theorem 1.5.5 are satisfied. So Lemma 5.4.1 implies the existence of a nontrivial critical point of the functional J which is a weak solutions to our problem (P a ).

  Then A satisfies the global ∆ 2 -condition (see[START_REF] Mihäilescu | Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces[END_REF] Proposition 2.3]).Lemma 1.1.1. (see.[START_REF] Bonder | Fractional order Orlicz-Soblev spaces[END_REF] Lemma 2.9]). Let A be an N-function which satisfies the global ∆ 2 -

	1.1. Orlicz Spaces					3
	condition. Then we have,
	A(a(t)) cA(t) for all t 0	(1.3)
	s>0	sa(s) A(s)	p + := sup s>0	sa(s) A(s)	< +∞.	(1.2)

Proposition 1.1.1. Assume that 1 < p -:= inf

  Chapitre 1. Preliminaries Theorem 1.1.1. Let Ω be an open subset of R N . 1) L A (Ω) is separable if and only if (A, Ω) is ∆-regular,2) L A (Ω) is reflixive if and only if (A, Ω) and (A, Ω) are ∆-regular, Using the Young's inequality, it is possible to prove a Hölder type inequality, that is,

	3) L A (Ω) is uniformly convex space if (A, Ω) is ∆-regular and A( √ t) is convex.
	Ω	uvdx	2||u|| A ||v|| A	for all u ∈ L A (Ω) and all v ∈ L A (Ω).	(1.6)
	Proposition 1.1.2. (see. [114, Proposition 2.1]) Let A be an N-function, assume condition (1.2) is
	satisfied. Then the following relations holds true
	||u|| p -

A Ω A(|u|)dx ||u|| p + A , ∀u ∈ L A (Ω) with ||u|| A > 1,

  Chapitre 2. Fractional Orlicz-Sobolev spaces namely a fractional Sobolev space constructed from an Orlicz space at the place of L p (Ω). As we know, the Orlicz spaces represent a generalization of classical Lebesgue spaces in which the role usually played by the convex function t p /p is assumed by a more general convex function A(t),

So, in this chapter, we propose what we believe is the natural fractional version of these spaces, 16

  ||u|| max |x -y| s for all u ∈ W s L A (Ω).

	and
	2A	|u(x) -u(y)| 2||u|| max |x -y| s		A	|u(x) -u(y)|
						(2.14)
	On the other hand, by relation (2.5), we have
			A(2t) 2A(t) for all t > 0.
	Thus, we deduce that
		2A	|u(x)| 2||u|| max	A	|u(x)| ||u|| max	for all u ∈ W s L A (Ω),

s,A ||u|| and ||u|| A ||u||, this implies that ||u|| s,A 2||u|| for all u ∈ W s L A (Ω).

  when s 2 s 1 , as next result points out. Let Ω be an open subset of R N and let 0 < s 2 s 1 < 1. Assume condition(1.2)

	Proposition 2.3.1.

  n } is bounded and by Lemma 2.3.2 it is belongs to W s L A (Ω). Moreover ||u n || A * k 4 ||u n || s,A k 4 ||u|| s,A . || A * = k, then k k 4 ||u|| s,A . By Fatou's Lemma we get,

	Let lim n→∞	||u n Ω	A *	|u(x)| k	dx lim n→∞ Ω	A *	|u n (x)| ||u n || A *	dx < 1,
	so u ∈ L A Proof of Theorem 2.3.2. By Lemma 2.3.3, we have,
				W s L A (Ω) → W s ,1 (Ω) → L 1 (Ω).
	The latter embedding being compact by Theorem 1.2.6. A bounded subset S of W s L A (Ω) is also a
	bounded subset of L A * (Ω) and precompact in L 1 (Ω), hence by Theorem 1.1.2 it is precompact in
	L B (Ω).
	By combining Lemma 2.3.3 and Theorem 1.2.6, we obtain the following results.
	Corollary 2.3.1. Let Ω be a bounded open subset of R N with C 0,1 -regularity and bounded boundary.
	Let 0 < s < s < 1 and let A be an N-function satisfies the condition (1.2), we define
				p * s

* (Ω) and ||u|| A * k.

  Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 67 as well as a.e. in R N

  3.3. Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators 71 Proof of Theorem 3.3.1. We want to apply theorem 1.5.7, taking X = W , Ψ and J are as before, by Lemma 3.3.4 J is C 1 -functional with compact derivative. Moreover by Lemma 3.3.6, Ψ is a sequentially weakly lower continuous and C 1 -functional belonging to W W , also by Lemma 3.3.8, the operator Ψ admits a continuous inverse on W * .

  .33) On the other hand, by (F 3 ), for all ε > 0, there exists η 2 > 0 such that

|F(x,t)| ε|t| α p (3.34) for all x ∈ R N and |t| > η 2 , and since f ∈ A , for r ∈ (p, α p), we can choose c 8 > 0 such that |F(x,t)| c 8 |t| r (3.35) for each x ∈ R N and |t| η 2 . By Theorem 3.3.3, we have J(u)

  .49) 3.4. Ekeland's variational principle for a nonlocal p-Kirchhoff type eigenvalue problem

		83
	Relations (3.48) and (3.49) imply
	lim n→∞	I 1

  . Since u n converges weakly to u in W s L Φ (Ω), then {||u n ||} is a bounded sequence of real numbers, this, implies that {||u n || Φ } and {[u n ] s,Φ } are bounded. Then by Proposition 2.2.1 and Proposition 1.1.2, we deduce that {I 1 (u n )} is bounded. So for a subsequence, we deduce that,

  4.5. Multiple solutions for a nonlocal Kirchhoff problem in Fractional Orlicz-Sobolev spaces 127 By a standard argument to Lemma 3.2.1 and Lemma 4.3.3, we have J

  Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spacesWe assume by contradiction that {u n } does not converge to u in W s 0 L Φ (Ω). Hence, there exists a subsequence of {u n }, still denoted by {u n } and there exists ε 0 > 0 such that

			u n -u 2	ε 0 2	,
	by Proposition 2.2.1, we have
		Ψ	u n -u 2	max ε p -0 , ε p + 0	.
	On the other hand, by the conditions (4.3) and (4.4), we can apply [96, Lemma 2.1] in order to
	obtain
	1
	2
						.65)
	On the other hand, since	u n + u 2	converges weakly to u in W s 0 L Φ (Ω), we have
			lim inf n→∞	Ψ	u n + u 2	Ψ(u).	(4.66)

  .71) 4.5. Multiple solutions for a nonlocal Kirchhoff problem in Fractional Orlicz-Sobolev spaces 131 By (4.70) and (4.71), we have :

  Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces By a standard argument to Lemma 4.2.1 and Lemma 4.2.2, we can use (H 3

y| s dxdy |x -y| N .

138

  4.7. Eigenvalue problem associated with nonhomogeneous integro-differential operators 151 By a standard argument to Lemma 4.2.1 and Lemma 4.3.3, we have J

  1 0 L Φ 1 (Ω) with ||u|| 1 > 1,by Proposition 2.2.1 and relations (4.103), (4.104),

	we deduce

  Thus, λ is an eigenvalue of problem (P a ) if and only if there exists u λ ∈W s 1 0 L Φ 1 (Ω) \ {0} critical point of T λ .For that, by Theorem 1.5.2 it is enough to prove that T λ is coercive and weakly lower semi continuous. Indeed, for each u ∈ W s 1 0 L Φ 1 (Ω) with ||u|| 1 > 1, relations (4.103) and (4.104) imply

  for any u ∈ W s L Φ 1 (Ω) with ||u|| > 1, by Proposition 2.2.1, Proposition 1.1.2 and Corollary Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces 4.8.1, we have

  Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

	This implies that
	Ω	β 3 (x)a 3 (|u|)|u| 2 dx	ϕ + 1 -ϕ -2 ϕ -2 3 -ϕ -	Ω Ω	a 1 (|D s u|)|D s u| 2 dµ +	Ω	a 1 (|u|)|u| 2 dx .	(4.126)
	So, from (4.120), (4.123) and (4.126), we have
									.125)
	combining (4.124) with (4.125), we obtain,
	ϕ + 1 -ϕ -2	Ω Ω	a 1 (|D s u|)|D s u| 2 dµ +

Ω a 1 (|u|)|u| 2 dx + ϕ - 2ϕ -

  Hence, we have u +

		n	u + 0
	weakly in W s L Φ 1 (Ω). By Theorem 4.8.2, we get
	u + n -→ u + 0 strongly in L Φ 2 (Ω). β 2	(4.129)
	Now, we need to show that u + n → u + 0 strongly in W

s L Φ 1 (Ω). Indeed we suppose that u + n u + 0 in W s L Φ 1 (Ω). By lemma 4.4.3, Ψ is weakly lower semicontinuous, then we have

  Orlicz spaces W s L Φ x,y (Ω), namely the fractional Orlicz Sobolev spaces W s L Φ (Ω) (resp. fractional Sobolev spaces with variable exponents W s,p(x,y) (Ω)) constructed from a Musielak spaces L Φ

x (Ω) at place the of Orlicz spaces L Φ (Ω) (resp. Lebesgue spaces with variable exponent L p(x) (Ω)).

As we know, Musielak spaces represent a generalization of Orlicz spaces (resp. Lebesgue spaces with variable exponent) in which the role usually played by the N-function Φ (resp. (x,t) → |t| p(x) )

  .3) Theorem 5.1.1. ([START_REF] Benkirane | An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces Bull[END_REF]). Let Ω be an open subset of R N which has a finite volume. Let A x (t), B x (t) : R + × Ω -→ R + be two Musielak functions such that B x ≺≺ A x . Then any bounded subset S of L A x (Ω) which is precompact in L 1 (Ω), is also precompact in L B x (Ω).

	Now, due to the non-locality of the operator (-∆) s a (x,.) , we introduce the new fractional Musielak-
	Sobolev space as follows

  Proof. Since ϕ -tϕ x,y (t) Φ x,y (t) for all t > 0, it follows that for letting σ > 1, we have log(Φ x,y (σt))log(Φ x,y (t)) =

			t	σt	ϕ x,y (τ) Φ x,y (τ)	dτ	t	σt	ϕ -τ	dτ = log(σ ϕ
			t) for all t > 0 and any σ > 1,	(5.11)
	Φ x,y (σt) σ ϕ +	Φ x,y (t) for all t > 0 and any σ ∈ (0, 1),	(5.12)
	Φ x,y (σt) σ ϕ +	Φ x,y (t) for all t > 0 and any σ > 1,	(5.13)

Φ x,y (t) σ ϕ -Φ x,y t σ for all t > 0 and any σ ∈ (0, 1).

(5.14) -).

  by the definition of ||.|| s,Φ x,y and ||.|| max we find 2||u|| max ||u|| s,Φ x,y ||u|| max for all u ∈ W s L Φ x,y (Ω). ||.|| s,Φ x,y and ||.|| max are equivalent. [u] s,Φ x,y ||u|| and ||u|| Φ x ||u||. Then ||u|| s,Φ x,y 2||u|| for all u ∈ W s L Φ x,y (Ω). ||u|| max |x -y| s for all u ∈ W s L Φ x,y (Ω). Using the definition of the Luxemburg norm and the relation (5.13), we get

	So, (5.16)
	On the other hand, by (5.11), we have
					Φ x,y (2t) 2Φ x,y (t) for all t > 0.
	Thus, we deduce that
		2 Φ x		|u(x)| 2||u|| max		Φ x	|u(x)| ||u|| max	for all u ∈ W s L Φ x,y (Ω),
	and
	2Φ x,y	|u(x) -u(y)| 2||u|| max |x -y| s		Φ x,y	|u(x) -u(y)|
	Hence, we get
	Ω Ω	Φ x,y	|u(x) -u(y)| 2||u|| max |x -y| s	dxdy |x -y| N +	Ω	Φ x	|u(x)| 2||u|| max	dx
		1 2 Ω Ω	Φ x,y	|u(x) -u(y)| ||u|| max |x -y| s	dxdy |x -y| N +	Ω	Φ x	|u(x)| ||u|| max	dx
		1 2		Ω Ω	Φ x,y	|u(x) -u(y)| [u] s,Φ x,y |x -y| s	dxdy |x -y| N +	Ω	Φ x	|u(x)| ||u|| Φ x	dx
		1.
	Then, combining this fact with (5.15), we have
										(5.15)
	||u|| 2||u|| max 2||u|| s,Φ x,y . From (5.15)-(5.17), we deduce that Proposition 5.2.1 hold true. That is, Next, we remark that	(5.17)
	Ω Ω Proposition 5.2.2. Assume that (Φ 1 ) is satisfied. Then, for any u ∈ W s L Φ x,y (Ω), the following rela-Φ x,y |u(x) -u(y)| ||u|||x -y| s dxdy |x -y| N + Ω Φ x |u(x)| ||u|| dx 1.
	tions hold true :
	It follows that
		Ω Ω	Φ x,y	|u(x) -u(y)| ||u|||x -y| s	dxdy |x -y| N 1 and	Ω	Φ x	|u(x)| ||u||	dx 1.

||u|| > 1 =⇒ ||u|| ϕ - Ψ (u) ||u|| ϕ + , (5.18) ||u|| < 1 =⇒ ||u|| ϕ + Ψ (u) ||u|| ϕ -. (

5

.19) Proof. First, we show that if ||u|| > 1, then Ψ (u) ||u|| ϕ + . Indeed, let u ∈ W s L Φ x,y (Ω) such that 5.2. Some qualitative properties of fractional Musielak-Sobolev spaces 187 ||u|| > 1.

  Some qualitative properties of fractional On the other hand, as σ is a Lipschitz function, then there exists c 1 > 0 such thatk 1 I 2 c 1 k Ω Ω |u(x)u(y)| |x -y| N+s dxdy.We posek 3 = c 1 k 1 c 2 .Combining (5.36)-(5.38), we obtain ||u|| Φ x + k 3 [u] s,Φ x,y . 4 ||u|| s,Φ x,y , where k 4 = max {2k ε , 2k 3 }. Now, for u ∈ W s L Φ x,y (Ω) arbitrary, we define Then, {u n } is bounded and by Lemma 5.2.4, u n ∈ W s L Φ x,y (Ω). Moreover ||u n || Φ * x,s k 4 ||u n || s,Φ x,y k 4 ||u|| s,Φ x,y . = k, then k k 4 ||u|| s,Φ x,y . Using the Fatou's Lemma, we get,

	5.2. Next, by Lemma 5.2.1, we have
				Ω Ω	|u(x) -u(y)| |x -y| N+s dxdy c 2 [u] s,Φ x,y ,	(5.37)
	thus,
						k 1 I 2		k 1 c 1 k	c 2 [u] s,Φ x,y .	(5.38)
				1		1 2	+	k ε k	||u|| Φ x +	k 3 k	[u] s,Φ x,y ,
	this implies that,
	k 2 k ε Hence, we obtain,
				k = ||u|| Φ * x,s
						
				u n (x) =	 	u(x)		if |u(x)| n,
						 
	Let lim n→∞	||u n || Φ * x,s
		Ω	Φ * x,s	|u(x)| k		dx lim n→∞ Ω	Φ x,s	|u
			s	|u(x)| k		dx +	k ε k Ω	|u(x)|dx	1 2	+	k ε k	||u|| Φ x ,	(5.36)

k n sgn u(x) if |u(x)| > n. n (x)| ||u n || Φ * x,s dx < 1, Consequently, u ∈ L Φ * x,s

(

Ω) and ||u|| Φ * x,s k k 4 ||u|| s,Φ x,y .

  .4. Application 2 : Bi-nonlocal type problem in fractional Musielak-Sobolev space 207 Now, since ϕ x,y (t) ϕ -. log(1 + α).t p(x,y)-1 for all (x, y) ∈ Ω × Ω and any t 0 Hence Φ x,y (t) ϕ - ϕ + . log(1 + α).t p(x,y) for all (x, y) ∈ Ω × Ω and any t 0. The above relation assures that relation (Φ 4 ) is verified. Corollary 5.3.3. Assume that f satisfies (C 1 ) and (C 2 ). If p -> q + . Then, problem (P 3 ) has a nontrivial weak solution. Note that The problems (P 1 ), (P 2 ) and (P 3 ) and the results obtained in Corollaries 5.3.1, 5.3.2 and 5.3.3 are all new.

	Thus, we remark that	t.ϕ x,y (t) Φ x,y (t)	is continuous on Ω × Ω × [0, ∞). Moreover,
			1 < p -lim t→0	t.ϕ x,y (t) Φ x,y (t)	p + + 1 < ∞,
	and
			1 < p -lim t→∞	t.ϕ x,y (t) Φ x,y (t)	p + + 1 < ∞.
	It follows that
				ϕ

1 < p -t.ϕ x,y (t) Φ x,y (t) for all (x, y) ∈ Ω × Ω and all t 0. On the other hand, some simple computations imply lim t→∞ t.ϕ x,y (t) Φ x,y (t) = p(x, y) for all (x, y) ∈ Ω × Ω, and lim t→0 t.ϕ x,y (t)

Φ x,y (t) = p(x, y) + 1 for all (x, y) ∈ Ω × Ω, + < ∞.

We conclude that relation (Φ 1 ) is satisfied.

5

By means of Ekeland's variational principle and direct variational approach, we investigate the existence of nontrivial weak solution for the above problem.

Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces J λ (u k ) = 0 and J λ (u k ) -→ ∞ as k → ∞. This completes the proof.

Ω β 3 (x)a 3 (|u|)|u| 2 dx > 0.

Acknowledgments

My first thanks go to my co-supervisor Professor Abdelmoujib BENKIRANE for everything that brought me during all years of my university studies, in license, in master, in M2 project, as well as throughout my thesis. During all this time, he didn't only teach me mathematics, but also showed me how to build my mathematical intuition, how to do mathematics correctly, how i can trust in myself. In addition to his mathematical qualities, we always find the human qualities that made these years enjoyable and enriching. Sometimes when I'm not motivated, a discussion with him makes me feel confident again and ready to tackle exciting questions with a smile. He was always available, despite a busy professional and social life, he has never refused my requests. A big thank-you.

Acknowledgments

Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces and by the continuity and monotonicity of the functions t → M i (t) for any i = 1, 2, we get

Thus, the functional J is sequentially weakly lower semicontinuous. Thus, we find lim inf n→∞ J(u n ) J(u).

(4.108)

Since W s 1 0 L Φ 1 (Ω) is compactly embedded in L q(x) (Ω). Then, u n converges strongly in L q(x) (Ω) and by Proposition 1.3.1, we have

Relations (4.108) and (4.109) imply that if u = 0, then

Thus, in order to conclude that the Lemma holds true it is enough to show that u cannot be trivial.

Assume by contradiction the contrary. Then, u n converges weakly to 0 in W s 1 0 L Φ 1 (Ω) and strongly in L q(x) (Ω). So by Proposition 1.3.1, we have

Letting ε ∈ (0, λ 1 ) be fixed, by relation (4.107) we deduce that for n large enough, we have

that is,

Passing to the limit in the above inequalities and taking into account that relation (4.110) holds true, or, by condition (M 1 ), we have

then, the above inequality imply

That fact combined with the Proposition 2.2.1 implies that actually u n converges strongly to 0 in

Form this information and relation (4.102), we get :

and this is a contradiction. Thus u = 0. The proof of Lemma 4.7.4 is completed.

By Lemma 4.7.4, we conclude that there exists

J(w) I(w) .

Then, for any v ∈ W s 1 0 L Φ 1 (Ω), we have

A simple computation yields

Relation (4.111) combined with the fact that J(u) = λ 1 I(u) and I(u) = 0 implies that λ 1 is an eigenvalue of problem (P a ). Thus Step 2 is verified.

• Step 3. We show that any λ ∈ (λ 1 , ∞) is an eigenvalue of problem (P a ).

Chapitre 4. On some nonlocal problems in fractional Orlicz-Sobolev spaces

On the other hand, by (4.128), we have

. Then

which is a contradiction with the Lemma 4.8.3. Hence

Consequently, u + 0 is a minimizer of J λ on N + λ .

Theorem 4.8.4. Under the assumptions of Theorem 4.8.2. Then for all λ ∈ (0, λ ), there exists a

Proof. J λ is bounded below on N λ , in particular is bounded below on N - λ . Then there exists a minimizing sequence {u

τ)dτ for all (x, y) ∈ Ω × Ω, and all t 0.

Then, Φ x,y is a Musielak function (see [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF]), that is

-For every t 0, Φ(., .,t) :

defined by :

Then, Φ x is also a Musielak function.

Remark 20. For all (x, y) ∈ Ω × Ω, Φ x,y and Φ x are two convex and increasing functions from

) is a nonlocal integro-differential operator of elliptic type which can be seen as a generalization of the fractional p(x, .)-Laplacian operator (-∆ p(x,.) ) s (when a (x,y) (t) = |t| p(x,y)-2 , see for instance [START_REF] Azroul | Three solutions for fractional p(x, .)-Laplacian Dirichlet problems with weight[END_REF][START_REF] Azroul | On a class of fractional p(x)-Kirchhoff type problems[END_REF][START_REF] Azroul | Eigenvalue problems involving the fractional p(x)-Laplacian operator[END_REF][START_REF] Kaufmann | Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians[END_REF]), and of the fractional p-Laplacian operator (-∆) s p in the constant exponent case (when p(x, y) = p = constant, i.e. a (x,y) (t) = |t| p-2 ). Moreover, this operator 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 [START_REF] Azroul | Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces[END_REF][START_REF] Bonder | Fractional order Orlicz-Soblev spaces[END_REF]). On the other hand, we remark that is the fractional version of the well-known a(.)-Laplacian operator div a(x, |∇u(x)|)∇u(x) which is associated with the Chapitre 5. Fractional Musielak-Sobolev spaces

Throughout this section, we assume that there exist two positive constants ϕ + and ϕ -such that 1 < ϕ -tϕ x,y (t) Φ x,y (t) ϕ + < +∞ for all (x, y) ∈ Ω × Ω and all t 0. (Φ 1 )

This relation implies that

ϕ + < +∞, for all x ∈ Ω and all t 0.

(5.4)

It follows that Φ x,y and Φ x satisfy the global ∆ 2 -condition, written Φ x,y ∈ ∆ 2 and Φ x ∈ ∆ 2 , (see [START_REF] Mihäilescu | Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces[END_REF]), that is,

and

where K 1 and K 2 are two positive constants. The inequality (5.6) implies that

(see [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF]).

An important role in manipulating the Musielak spaces is played by the modular of the space ). Assume that the condition (5.4) is satisfied. Then, for all u ∈ L Φ x (Ω), the following relations hold true

for some (resp for each) a > 0 and t 0 (depending on a). Now, to show the reflexivity and separability of the W s L Φ x,y (Ω), we define the operator T :

Clearly, T is an isometry. Since L Φ x (Ω) and L Φ x,y (Ω × Ω, dµ) are separable and uniformly convex (reflexive) spaces (see [START_REF] Mihäilescu | Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces[END_REF][START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF]), then W s L Φ x,y (Ω) is also a reflexive, separable and uniformly convex space.

Remark 23. In the integer case, that is, when s ∈ N (see for instance [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF]), to prove the completeness of Musielak space we need to the two following conditions i)inf

ii)-local integrability condition :

Note that in our functional framework we are proved that W s L Φ x,y (Ω) is Banach space without using the above two conditions.

Definition 5.2.1. We say that Φ x,y satisfies the fractional boundedness condition, written

Theorem 5.2.2. Let Ω be an open subset of R N , and 0 < s < 1. Assume that Φ x,y ∈ B f . Then,

On the other hand, we notice that

Accordingly, we get

• If s ϕ -< N, then W s L Φ x,y (Ω) → L q (Ω), for all q ∈ [1, ϕ * s ] and the embedding W s L Φ x,y (Ω) → L q (Ω) is compact for all q ∈ [1, ϕ * s ).

• If s ϕ -= N, then W s L Φ x,y (Ω) → L q (Ω), for all q ∈ [1, ∞] and the embedding W s L Φ x,y (Ω) → L q (Ω) is compact for all q ∈ [1, ∞).

Next, we introduce a closed linear subspace of W s L Φ x,y (Ω) as follows

Then, we have the following generalized Poincaré type inequality.

Theorem 5.2.5. Let Ω be a bounded open subset of R N with C 0,1 -regularity and bounded boundary, let s ∈ (0, 1). Then there exists a positive constant γ such that ||u|| Φ x γ[u] s,Φ x,y for all u ∈ W s 0 L Φ x,y (Ω).

Chapitre 5. Fractional Musielak-Sobolev spaces

We start by considering the function α : R -→ R such that g : R -→ R defined by

which is an increasing homeomorphism from R onto itself. We assume that f :

Carathéodory function satisfies the following condition

where c 1 is a nonnegative constant. Furthermore, if we set

with g(t) = sup {s : g(s) t}, then we obtain complementary Musielak function which define their corresponding Musielak spaces L G and L G .

Moreover, we assume that there exist two positive constants g + and g -such that

Also, we assume the following assumptions :

There exist θ > ϕ + and r > 0 such that for all |t| r and a.e. x ∈ Ω

where F(x,t) := continuously embedded in W s 2 0 L Φ x,y (Ω). Thus a solution for a problem of type (P a ) will be sought in W s 1 0 L Φ x,y (Ω).

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

Next, we give the definition of weak solutions for problem (P a ).

)

Now, we are ready to state our existence result.

Theorem 5.4.1. Suppose that (H 1 )-(H 5 ) hold true. Then problem (P a ) has a nontrivial weak solution u ∈ W s 1 0 L Φ x,y (Ω) which is a critical point of mountain pass type for the energy functional

Let us denote by Ψ, I :

Remark 28. We note that the functional J :

Ω), then by Proposition 5.2.4, we have u ∈ W s 2 0 L Φ x,y (Ω), thus, Ψ(u) < ∞. Moreover, by condition (H 3 ), we have that u ∈ L G (Ω) and thus u ∈ L 1 (Ω). Hence, by the condition (H 1 ),

It follows that

Chapitre 5. Fractional Musielak-Sobolev spaces By a standard argument to Lemmas 5.3.2 and 5.3.1 we have J ∈ C 1 W s 1 0 L Φ x,y (Ω), R and its Gâteaux derivative is given by

where ., . denotes the usual duality between W s 1 0 L Φ x,y (Ω), . s 1 and its dual space W s 1 0 L Φ x,y (Ω) * , . s 1 , * .

Next, we show an important lemma, namely that if the functional J defined in (5.48) satisfies the conclusion of Theorem 1.5.5, then it has a critical point.

Lemma 5.4.1. Let (H 1 )-(H 3 ) hold true and let {u n } be a sequence in W s 1 0 L Φ x,y (Ω) such that

(5.49)

Then there exists u ∈ W s 1 0 L Φ x,y (Ω) such that ) and (H 1 )-(H 3 ), we have 0 < tϕ x,y (t) ϕ + Φ x,y (t) for all t > 0, (5.50) 0 < tg(t) g + G(t) for all t > 0, (

and

Consequently, by the uniqueness of the limit, we deduce that, J (u) = 0. The convexity of Ψ implies that Ψ is monotone and hence

By (5.54), we have

Ω) and t ∈ R + , then, we get

for all h ∈ W s 1 0 L Φ x,y (Ω). Letting t → 0 and using the fact that h is arbitrary in W s 1 0 L Φ x,y (Ω), we find that

It follows that u is a critical point of J.

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 1.5.5 will hold. Proof. For all u ∈ W s 1 0 L Φ x,y (Ω) \ {0}, the functional I is satisfied :

( Using the condition (H 5 ), we have that there exists ε ∈ (0, 1) and t 0 > 0 such that

for all |t| t 0 and all x ∈ Ω.

We pose Ω 0 := {x ∈ Ω : |u(x)| t 0 }, then we have

(5.56) By (5.40), we have Since W s 1 0 L Φ x,y (Ω) → L G (Ω) and W s 1 0 L Φ x,y (Ω) → L 1 (Ω), we obtain Ω 0 F(x, u(x))dx c 1 c 2 (||u|| s 1 + ||u|| g - s 1 + ||u|| g + s 1 ).

(5.58)

Then, for ||u|| s 1 1, we find Hence for ρ > 0 as given in (5.61), there exists a α = α(ρ) > 0 such that for all u with ||u|| s 1 = ρ, we have

We therefore obtain

Thus, if we set r = α ε 2 , we obtain that (G 1 ) is satisfied.