ON A NEW FRACTIONAL SOBOLEV SPACE AND APPLICATIONS TO NONLOCAL VARIATIONAL PROBLEMS WITH VARIABLE EXPONENT

The content of this paper is at the interplay between function spaces Lp(x) and Wk,p(x) with variable exponents and fractional Sobolev spaces W s,p. We are concerned with some qualitative properties of the fractional Sobolev space W s,q(x),p(x,y), where q and p are variable exponents and s ∈ (0, 1). We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous p(x)-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

1. Introduction and preliminary results. Fractional Sobolev spaces and the corresponding nonlocal equations have major applications to various nonlinear problems, including phase transitions, thin obstacle problem, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes and flame propagation, ultra-relativistic limits of quantum mechanics, multiple scattering, minimal surfaces, material science, water waves, etc. We refer to Di Nezza, Palatucci and Valdinoci [9] for a comprehensive introduction to the study of nonlocal problems. After the seminal papers by Caffarelli et al. [6,7,8], a large amount of papers were written on problems involving the fractional diffusion operator (−∆) s (0 < s < 1). The cited results turn out to be very fruitful in order to recover an elliptic PDE approach in a nonlocal framework, and they have recently been used very often, see [1,2,14,17,20,21]. We also refer to the recent monographs [15,5,10] for a thorough variational approach of nonlocal problems.
A natural question is to see what results can be recovered when the standard Laplace operator is replaced by the fractional Laplacian. On the other hand, for some nonhomogeneous materials (such as electrorheological fluids, sometimes referred to as "smart fluids"), the standard approach based on Lebesgue and Sobolev spaces L p and W 1,p , is not adequate. This leads to the study of variable exponent Lebesgue and Sobolev spaces, L p(x) and W 1,p(x) , where p is a real-valued function. Variable exponent Lebesgue spaces appeared in the literature in 1931 in the paper by Orlicz [16]. Zhikov [24] started a new direction of investigation, which created the relationship between spaces with variable exponent and variational integrals with nonstandard growth conditions. We also point out the important contributions of Marcellini [13], who studied minimization problems with (p, q)-growth, namely The case corresponding to the variable exponent corresponds to F (x, t) = t p(x) , where p : Ω → (1, ∞) is a bounded function. We refer to [18,19] for the abstract treatment of function spaces with variable exponent.
It is therefore a natural question to see what results can be "recovered" when the p(x)-Laplace operator is replaced by the fractional p(x)-Laplacian. As far as we know, the only result about the fractional Sobolev spaces with variable exponent and the fractional p(x)-Laplacian is obtained in [12]. In particular, the authors generalize p(x)-Laplace operator to the fractional case. They also introduce a suitable functional space to study an equation in which a fractional variable exponent operator is present.
Let Ω be a smooth open set in R N . For any real s > 0 and for any functions q(x) and p(x, y), we want to define the fractional Sobolev space with variable exponent. We start by fixing 0 < s < 1 and q : Ω → R and p : Ω × Ω → R be two continuous functions. We assume that p is symmetric and p((x, y) − (z, z)) = p(x, y), ∀(x, y), (z, z) ∈ Ω × Ω, (P ) and We define the fractional Sobolev space with variable exponents via the Gagliardo approach as follows: (L q(x) (Ω) and |.| q(x) will be introduced in the next section), then E becomes a Banach space. Let E 0 denote the closure of C ∞ 0 (Ω) in E. Then E 0 is a Banach space with the norm u = [u] s,p(x,y) .
In [12], the authors prove the following basic theorem.
Theorem 1.1. Let Ω ⊂ R N be a smooth bounded domain and s ∈ (0, 1). Let q(x), p(x, y) be continuous variable exponents with sp(x, y) < N for (x, y) ∈ Ω × Ω and q(x) > p(x, x) for x ∈ Ω. Let (P ) and (Q) be satisfied. Assume that r : Ω → (1, ∞) is a continuous function such that for x ∈ Ω. Then there exists a constant C = C(N, s, p, q, r, Ω) such that for every Thus, the space W s,q(x),p(x,y) (Ω) is continuously embedded in L r(x) (Ω) for any r ∈ (1, p * ). Moreover, this embedding is compact.
Remark 1.2. The above theorem remains true if we replace E by E 0 .
As an application of Theorem 1.1 in [12], the authors study the existence of solutions to some nonlocal problems. Let us consider the operator L given by In the constant exponent case it is know as the fractional p-Laplacian. On the other hand, we remark that L is a fractional version of the well known p(x)-Laplacian.
The main purpose of this paper is to present some further basic results both on the function spaces E 0 , E and the fractional operator L. Then, we study the existence of solutions to some nonlocal problems. This paper is organized as follows. In Sect. 2, we give some definitions and fundamental properties of the spaces L p(x) (Ω) and W m,p(x) (Ω). In Sect. 3, we study the reflexivity, separability, density of E 0 and E. Moreover, we prove some basic properties of operator L. Finally, in Section 4, using a direct variational method, we give an application of our abstract results.
2. Terminology and abstract setting. In this section, we recall some necessary properties of variable exponent spaces. We refer to [11,19] and the references therein.
Consider the set For all p ∈ C + (Ω) we define For any p ∈ C + (Ω), we define the variable exponent Lebesgue space This vector space is a Banach space if it is endowed with the Luxemburg norm, which is defined by

ANOUAR BAHROUNI AND VICENŢ IU D. RȂDULESCU
We point out that if p(x) ≡ p ∈ [1, ∞) then the optimal choice in the above and v ∈ L q(x) (Ω) then the following Hölder-type inequality holds: Moreover, if p j ∈ C + (Ω) (j = 1, 2, . . . , k) and 1 An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L p(x) (Ω) space, which is the mapping ρ : Proposition 2.2. If u, u n ∈ L p(x) (Ω) and n ∈ N, then the following statements are equivalent each other: (2) lim n→+∞ ρ(u n − u) = 0.
From Theorems 1.6, 1.8 and 1.10 of [11], we obtain the following proposition: ) is a reflexive uniformly convex and separable space.
If k is a positive integer number and p ∈ C + (Ω), we define the variable exponent Sobolev space by On W k,p(x) (Ω) we consider the following norm Then W k,p(x) (Ω) is a reflexive and separable Banach space. Let W k,p(x) 0 (Ω) denote the closure of C ∞ 0 (Ω) in W k,p(x) (Ω).
Then v ∈ L p(x,y) (Ω × Ω). If > 0, by Theorem 2.3, there exists with is sufficiently small. This proves our claim (6). Moreover, for a.e. z ∈ B 1 , there exists a positive constant c such that for any > 0. Hence, using (6), (7) and the dominated convergence theorem, we infer that as → 0. From the above assertion, we obtain (4). Using this fact in combination with (3), we conclude our proof.

4.
Properties of the fractional operator L. In this section, we give some basic properties of the operator L. Let (P ) and (Q) be satisfied. In the sequel, we denote by K(x, y) = |x − y| −N −sp(x,y) . Consider the following functionals: and L 1 : E 0 → E * 0 such that for all u, ϕ ∈ E 0 Similarly, we consider L 2 : E → E * such that Lemma 4.1. The functional I 1 is well defined on E 0 . Moreover, I 1 ∈ C 1 (E 0 , R) with the derivative given by Proof. The proof is standard, see [15,19].
Recall the Simon inequalities [22], which imply the strict monotonicity of L 1 : By Theorem 1.1, we obtain This along with Fatou's lemma yield On the other hand, we have Now using Young's inequality, there exists a positive constant c such that As a consequence of (11), (12) and (13), we get Now from (10), (14) and the Brezis-Lieb lemma [4], our result is proved.
(iii) By (i), L 1 is an injection. In view of Proposition 2.1, we obtain Therefore, L 1 is coercive. Thus, in light of Minty-Browder theorem (see [23]), L 1 is a surjection. Hence L 1 has an inverse mapping L −1 1 : In view of the coercivity of L 1 , (u n ) is bounded in E 0 . We may assume that u n u 0 in E 0 . It follows that Using the fact that L 1 is of type (S + ), we conclude that u n → u 0 in E 0 . This concludes the proof. 5. Application to nonlocal fractional problems with variable exponent. In this section, we work under the hypotheses of Theorem 1.1. We investigate the existence of solutions of the following problem where λ > 0, 1 < r(x) < p − . We say that u ∈ E 0 is a weak solution of problem (15) Theorem 5.1. For every λ > 0, problem (15) admits at least one nontrivial weak solution.
Define the functional J λ : E 0 → R by By Lemma 4.1, J λ ∈ C 1 (E 0 , R) and Proof of Theorem 5.1. We first show that J λ satisfies the Palais-Smale condition. Let (u n ) ∈ E 0 be a (P S)−sequence of J λ . We claim that (u n ) is bounded in E 0 . Arguing by contradiction, we suppose that (u n ) is unbounded in E 0 . Without loss of generality, we can assume that u n > 1 for all n ≥ 1. There exists a positive constant c such that In the last inequality, we used Proposition 2.1 and Theorem 1.1. In view of r + < p − , it follows that (u n ) is bounded in E 0 . Thus, up to a subsequence and using Theorem 1.1, we can assume that u n u in E 0 , u n → u in L r(x) (Ω) and u n → u in L q(x) (Ω).
We show in what follows that u n → u in E 0 .
Since (u n ) is a (P S)−sequence and using the above assertions, we get J λ (u n ) − J λ (u n ), u n − u = L 1 (u n ) − L 1 (u), u n − u = 0. Now since L 1 is an operator of type (S + ), we conclude that u n → u in E 0 , which shows that the Palais-Smale condition is satisfied. Next, we show that J λ is coercive. Indeed, as we have observed, for any λ > 0 and u ∈ E 0 with u > 1, we have This implies the coercivity of J λ . We deduce that J λ ∈ C 1 (E 0 , R) is bounded from below, coercive and satisfies the Palais-Smale condition. These facts in combination with Ekeland's variational principle show that there exists u λ ∈ E a global minimum of J λ . It remains to prove that u λ = 0. Fix φ ∈ E 0 , φ = 0 and φ ≥ 0 in Ω. Then, for each t ∈ (0, 1), we have J λ (tφ) = I 1 (tφ) Taking into account r − < p + and r − < q + , for t small enough, we infer that J λ (tφ) < 0.

ANOUAR BAHROUNI AND VICENŢ IU D. RȂDULESCU
This completes the proof of our theorem.