Embedding theorems for variable exponent fractional Sobolev spaces and an application

: In this paper, we mainly discuss the embedding theory of variable exponent fractional Sobolev space W s ( · ) , p ( · ) ( Ω ), and apply this theory to study the s ( x )- p ( x )-Laplacian equation: where x ∈ Ω ⊂ R n , ( − ∆ ) s ( · ) p ( · ) is s ( x )- p ( x )-Laplacian operator with 0 < s ( x ) < 1 < p ( x ) < ∞ and p ( x ) s ( x ) < n , the nonlinear term f : Ω × R → R is a Carath´eodory function, V : R n → R is a potential function and g : R n → R is a perturbation term.


Introduction
Variable exponent Lebesgue spaces were first studied by Orlicz in 1931 (see [33]). Since the 1990s, variable exponent Lebesgue spaces and variable exponent Sobolev spaces have been used in a variety of fields, the most important of which is the mathematical modeling of electrorheological fluids. In 1997, the variable exponent Lebesgue spaces were applied to the study of image processing: In image reconstruction, the variable exponent interpolation technique can be used to obtain a smoother image. For the theory and applications of variable exponent Lebesgue spaces and variable exponent Sobolev spaces, see [10,12,15,21,28] and the references therein.
As a part of the theory of variable exponent function spaces, variable exponent fractional Sobolev spacea are also developing vigorously. In [27], Kaufmann et al gave a class of variable exponent fractional Sobolev spaces: W s,q(x),p(x,y) (Ω) := u ∈ L q(x) (Ω) : Ω Ω |u(x) − u(y)| p(x,y) λ p(x,y) |x − y| n+sp(x,y) dxdy < ∞ for some λ > 0 , (1.1) where s ∈ (0, 1), Ω ⊂ R n is a bounded domain with Lipschitz boundary, q :Ω → (1, ∞) and p : Ω ×Ω → (1, ∞) are two continuous functions bounded away from 1 and ∞. Assume further that p is symmetric, i.e. p(x, y) = p(y, x). Afterwards some scholars did further research on theory and applications of this kind of spaces (see [3, 5-7, 13, 25, 32] and the references therein). In [31], we considered the case that the index s is a function s(x), p(x, y) is p(x)+p(y) 2 , q(x) is p(x), established the so called variable exponent fractional Sobolev spaces W s(·),p(·) (Ω) and gave some basic properties and an application. In this paper, we will further study basic properties of this kind of spaces, for example: Embedding.
Embedding is always a classical topic in functional analysis, partial differential equations and other fields. The first task of this paper is to give embedding theorems for W s(·),p(·) (Ω). Related to embedding theorems, we refer to [14,18,24,35] and the references therein.
In [4], Azroul et al studied the existence of nontrivial weak solutions for fractional p(x, y)-Kirchhoff type problems. In [3], the existence of eigenvalues of fractional p(x, y)-Laplacian is studied by means of Ekeland variational principle. These problems are considered under the condition that the exponent s is constant.
In [34], Xiang et al used the mountain pass theorem and Ekeland variational principle to study the elliptic problems of Laplacian with variable exponent s and constant pc under appropriate assumptions: It is proved that there are at least two different solutions to the above problems. Furthermore, the existence of infinite many solutions for the limit problems is obtained.
In [11], Cheng et al further studied the existence of weak solutions for nonlinear elliptic equations where the exponents s and p are of variable forms, i.e.
As we know that when people studied nonlinear problems of fractional Laplace operators with variable exponents, they mainly focus on the case that the exponent s is constant and p is variable. For the cases that the exponent s is variable and p is constant or both the exponents s and p are variables, there are still few results.
Under the quantum mechanics background, in [29,30] Laskin expanded the Feynman way integrals from the kind of Braun quantum mechanics way to the kind of Lévy quantum mechanics way, proposed the nonlinear fractional Schrödinger equation. Subsequently, results on the fractional Schrödinger equation gradually appeared |x − y| n+2s dy and f satisfies some conditions, which are stated in details in [17,22].
As a direct application of embedding theorems for W s(·),p(·) (Ω), the second task of this paper is to study the existence of multiple solutions for Dirichlet boundary value problem of the s(x)-p(x)-Laplacian equations in W s(·),p(·) (Ω): |x − y| n+ s(x)p(x)+s(y)p(y) 2 dy, x ∈ Ω.
When p(x) = 2 and s(x) = s(constant), Eq (1.5) becomes a fractional Laplacian equation This can be seen as fractional form of the following classic stationary Schrödinger equation Therefore, we think it is meaningful to study problem (1.5), and further, it is very necessary to study the application of s(x)-p(x)-Laplace equation in W s(·),p(·) (Ω).

Preliminaries
First we provide some basic concepts and related notations. Suppose that Ω be a Lebesgue measurable subset of R n with positive measure. Let For a Lebesgue measurable function u : Ω → R, define The space W s(·),∞ (Ω) is defined as the set of functions When the exponent s is constant, it is the space W s,∞ (Ω) mentioned in [1,26]. The norm can be defined as where the Hölder semi-norm is defined by where Ω ∞ = {x ∈ Ω : p(x) = ∞}. The variable exponent Lebesgue space L p(·) (Ω) is defined by We define a norm, so called Luxembourg norm, for this space by The variable exponent fractional Sobolev space W s(·),p(·) (Ω) is defined by be the corresponding variable exponent Gagliardo semi-norm. The norm is equipped as It is easy to verify that under this norm this space is a Banach space. For the sake of convenience, we give some notations. For the variable exponent p : In view of ρ p(·) and L p(·) (Ω), we can define modularρ p(·,·) and variable exponent Lebesgue spaces L p(·,·) on Ω × Ω. The conclusions on L p(·) (Ω) can be moved to L p(·,·) (Ω × Ω). Here we give another modular and norm in W s(·),p(·) (Ω). In this case, we only consider the case of p + < ∞. Modular is defined as:ρ According to this modular, we define the norm as: |||u||| W s(·),p(·) (Ω) = inf λ > 0 :ρ s(·),p(·),Ω ( u λ ) < 1 .
The following conclusions are what we will use later.
Proof. Let u ∈ C ∞ 0 (Ω) with suppu ⊂ Ω, we already know u ∈ L p(·) (Ω). Now we prove: Now we estimate I 1 and for x ∈ B r (0), y ∈ B 2r (0), 0 < θ < 1. So dz is finite and further I 1 is also finite. Next |u(x)| and constant C depends on M, r, p − and p + . Since n + s − p − > n, we have R n \B 2 (0) 1 |z| n+s − p − dz is finite and further I 2 is also finite. Based on the discussion above, we arrive at the conclusion.
The embedding theorem given in [11] (the space involved is X k(·),α(·) ), the exponent α(·) is restricted by the exponent p 1 (·) in the space L p 1 (·) under the condition: α(z, s) < p 1 (z) for (z, s) ∈Ω ×Ω, but the conclusion of our theorem does not require such a requirement. In addition, in the statement of the embedding theorem in this paper, the case that the variable exponent p and q are equal to 1 is considered, which is not mentioned in references [8,11].
In order to prove this embedding theorem, we will use embedding theorem for constant exponent fractional Sobolev space. In order to make the proof more clear, we list this theorem here. Let Ω ⊂ R n be an extension domain for W s,p (Ω). Then there exists a positive constant C = C(n, p, s, Ω) such that for any u ∈ W s,p (Ω), we have u L q (Ω) ≤ C u W s,p (Ω) for any q ∈ [p, p * ]. i.e. the space W s,p (Ω) is continuously embedded in L q (Ω) for any q ∈ [p, p * ].
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 * ]. Moreover, this embedding is compact for q ∈ [1, p * ).
With these preparations, we will now prove the Theorem 3.2.
Proof. Since p, s, q are continuous onΩ and Ω is bounded, there exists a positive constant ξ such that for every x ∈Ω.
In view of the continuity of p and (3.1), we can find a constant ε = ε(n, p, q, s, Ω) and a fnite family of disjoint Lipschitz sets O i such that for every x, y, z ∈ O i . We can choose constant p i and t i , with p i = inf for each x ∈ O i . By Theoremn 3.3, there exists a constant C = C(n, ε, t i , p i , O i ), such that Now, we prove the following inequalities. (a) There exists a constant c 1 such that (b) There exists a constant c 2 such that If the above three inequalities hold, a conclusion can be drawn by combining (3.3) and Theorem 3.1 as the following: First prove (a). We have .
According to Theorem 2.1, we have In this way, (a) is proved.
Next prove (b). Set Now the proof is finished.

An application
For problem (1.5), we make the following assumptions.
Proof. By (F1) and (F3), we can get for all x ∈ Ω and t ∈ R.
[2] Let X be Banach space. I is a functional on X. We say that I satisfies PS condition in X, if any PS sequence {u n } n ⊂ X, i.e. {I(u n )} n is bounded and I (u n ) → 0 as n → ∞, admits a strongly convergent subsequence in X. (Ω). Then there exists C > 0 such that | I (u n ), u n | ≤ C u n W s(·),p(·) 0 (Ω) and |I(u n )| ≤ C. Thus by (F2), Proposition 2.2 and Theorem 3.2, we get (Ω) .
By Hölder inequality, hence B ψ is continuous. By (F1) and (F3), there exists a constant C > 0 such that for all x ∈ Ω and t ∈ R. By Hölder inequality, (4.10) The fact that I satisfies PS condition in W s(·),p(·)
Proof of Theorem 4.1. By Lemma 4.5, Lemma 4.6 and Lemma 4.8, I has mountain pass structure. By Mountain Pass Theorem, there exists a critical value C 1 ≥ α 0 > 0 and a corresponding critical point u 1 ∈ W s(·),p(·) (Ω) such that I(u 1 ) = C 1 , where α 0 is the one in Lemma 4.5.
Since I is lower semi-continuous, by Ekeland variational principle and Lemma 4.5, there exists a sequence {u n } ⊂ B ρ 0 such that C 2 ≤ I(u n ) ≤ C 2 + 1 n and I(v) ≥ I(u n ) − 1 n v − u n W s(·),p(·) (Ω) for all v ∈ B ρ 0 . Then we can infer that {u n } is a PS sequence. By Lemma 4.5 and Lemma 4.8, there exists a critical point u 2 ∈ B ρ 0 such that I(u 2 ) = C 2 < 0 and u 1 u 2 0.

Conclusions
We obtain embedding theorems for variable exponent fractional Sobolev space W s(·),p(·) (Ω): In the case that Ω is a bounded open set, if s 2 (x) ≥ s 1 (x), space W s 2 (·),p(·) (Ω) can be continuously embedded into W s 1 (·),p(·) (Ω). In the case that Ω is a Lipschitz bounded domain, if s(x)p(x) < n, for continuous function q with 1 < q(x) < p * (x), W s(·),p(·) (Ω) can not only be continuously embedded, but also be compactly embedded into L q(·) (Ω). As an application of the embedding theorems, we obtain that the problem (1.5) of s(x)-p(x)-Laplacian equations has at least two nontrivial weak solutions when the nonlinear function f satisfies conditions (F1)-(F3), the potential function V satisfies condition (V), the exponen p, q, s satisfies condition (PQS) and g satisfies condition (G).