On the variable exponential fractional Sobolev space W s ( · ) , p ( · )

: In this paper a new kind of variable exponential fractional Sobolev spaces is introduced. For this kind of spaces, some basic properties, such as separability, reﬂexivity, strict convexity and denseness, are established. At last as an application the existence of solutions for so called s ( · )- p ( · )-Laplacian equations is discussed.


Introduction
Let Ω be an open subset of R n . For any s ∈ (0, 1) and p ∈ [1, ∞), we can define the fractional Sobolev space W s,p (Ω). The study on fractional Sobolev space is a classical topic in functional analysis and harmonic analysis and the theory of fractional Sobolev space has been widely applied in different fields, such as optimization, phase transition, anomalous diffusion, material science, non-uniform elliptic problems, gradient potential theory etc (see [10]). In 1991, Kováčik and Rákosník studied variable exponential spaces L p(x) and W 1,p(x) (see [17]). Since then, some scholars have successively studied the theories and applications of these spaces (see [7, 9, 11-14, 20, 21] and their related references). With the vigorous development of variable index space theory, in recent years, some scholars have focused their research on the variable exponent fractional Sobolev spaces. In 2017, Kaufmann, Rossi and Vidal extended the constant exponent p of fractional Sobolev space to variable exponent p(x, y) type (not only that, but also another variable exponent q(x)), and studied the compact embedding of these spaces and obtained the existence and uniqueness of solutions for non-local problems of p(x)-Laplacian equations by means of these spaces (see [16]).
Due to the need of research, some scholars limited the exponent of the variable exponent fractional Sobolev spaces above to L¯p (x) (Ω), i.e. replace q(x) in the definition withp(x), wherep(x) = p(x, x), see for example [2].
In [3], the author adjusted the definition of (1.1) to give another form of variable exponent fractional Sobolev space: where the integral is extended from Ω × Ω to Q = R 2n \ (CΩ × CΩ) with CΩ = R n \ Ω. The authors used the closed subspace X 0 := {u ∈ X : u = 0 a.e. in R n \ Ω} to study the p(x)-Kirchhoff Dirichlet boundary problem λ p(x,y) |x−y| n+sp(x,y) dy for all x ∈ R n . Some basic properties, such as reflexivity, completeness, separability, uniform convexity, were also obtained.
In [23], the authors considered such variable order fractional Sobolev space H s(·,·) 0 (Ω), where for any function u ∈ L 2 (Ω) satisfying u = 0 in R n \ Ω, They studied an elliptic equation involving variable exponent driven by the fractional Laplace operator with variable order derivative.
In this paper we want to define a new kind of variable exponent fractional Sobolev spaces similar to the ones in [3] and [23], but with the variable order s(x) and the variable exponent p(x)+p(y)

2
. Some basic properties of this kind of spaces are discussed and an application on so called s(·)-p(·)-Laplacian equations is given.
Indeed, first we have We estimate I 1 and I 2 respectively. As 1 |z| n+s − dz is convergent. Then we get On the other hand, In this article, without confusion, [u] W s(·),p(·) (Ω) , ρ p(·),Ω (u) and ϕ s(·),p(·),Ω (u) can be abbreviated as [u], ρ(u) and ϕ(u) respectively. Definition 2.6. Let u k , u ∈ W s(·),p(·) (Ω). We say that u k is ϕ-convergent to u if there exists λ > 0 such that ϕ(λ(u k − u)) → 0 as k → ∞ and we denote this convergence by u k ϕ − → u. We say that u k is Definition 2.7. Let X be a normed linear space. If every chord of the unit sphere of X has its midpoint below the surface of the unit sphere, then X is called strictly convex.
. Then by Fatou Lemma and the definition of ).
Therefore, we can find λ sufficiently close to [u] such that ϕ( u λ ) < 1. But by the definition of [ · ], we must have ϕ( u λ ) ≥ 1. From this contradiction we see that equality holds.
, we come to the conclusion. 3. By 1. and 2., it is immediate.
Proof. The equivalence between statements 1 and 2 can be obtained from Theorem 2.69 in [7] and Proposition 2.5. Now we prove the equivalence between statements 2 and 3.
It is immediate by Proposition 2.5. Proof. We prove in following two cases.
Proof. Here we use the following equivalent definition of strictly convex spaces. Let (X, · ) be normed linear space, (X, · ) is called strictly convex if for every u ∈ X, v ∈ X, u 0, v 0, the equality u + v = u + v implies u = λv, where λ is positive.
x ∈ Ω, f (x, t) is monotonically decreasing with respect to t.
Definition 3.1. We say that u ∈ W s(·),p(·) 0 (Ω) is a weak solution of problem (3.1) if for all v ∈ W s(·),p(·)  We know that the critical point of I is the weak solution of the problem (3.1), so we only examine the critical point of I. Before proving Theorem 3.1, we give two theorems to be used. Theorem 3.2 can be inferred from Theorems 1.3 and 1.7 in [24], and Theorem 3.3 is derived from [7]. Theorem 3.2. Let X be a real reflexive Banach space. If the real functional I : X → R is coercive, strictly convex, and has bounded Gâteaux differential in X, then I has a unique minimum point, which is of course also a critical point. Theorem 3.3. Let p(·), q(·) ∈ P(Ω) and suppose |Ω \ Ω q(·) ∞ | < ∞. Then L p(·) ⊂ L q(·) if and only if q(x) ≤ p(x) a.e.. Furthermore Our task is to verify that I is coercive, strictly convex, and has bounded Gâteaux differential in W s(·),p(·) 0 (Ω), so that the only minimum point of I is the critical point, which is the weak solution of the problem (3.1).
Proof of Theorem 3.1.
On the other hand, for any h ∈ W s(·),p(·)
According to Lebesgue Dominated Convergence Theorem, we know that It is immediate that I is linear, so now we verify that I is a bounded functional of h ∈ W s(·),p(·) 0 (Ω). By Hölder inequality, |x − y| n+ s(x)p(x)+s(y)p(y) If we assume u > 1, then we have So as u → +∞, I(u) → +∞.

Conclusions
We define a class of variable exponent fractional Sobolev spaces W s(·),p(·) (Ω), which is a subspace of L p(·) (Ω), and has variable order s(x) and variable exponent p(x)+p(y) 2 . W s(·),p(·) (Ω) is a Banach space under the given norm. We give some basic properties, such as the closed unit ball is equivalent in the sense of [ · ] and ϕ, and that the [ · ]-convergent and ϕ-convergent are equivalent, norm convergent is equivalent to the ρ-convergent and the ϕ-convergent. If the exponent p(x) satisfies certain conditions, we obtain that W s(·),p(·) (Ω) is reflexive, separable, strictly convex and the set of all bounded measurable functions is dense in W s(·),p(·) (Ω). As an application, we obtain the existence and uniqueness of weak solutions in W s(·),p(·) 0 (Ω) for Dirichlet boundary value problems of s(x) − p(x)-Laplacian equations.

Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11771107).

Conflict of interest
All authors declare no conflicts of interest in this paper.