1 Introduction

In recent years, elliptic problems involving the fractional p(x, .) Laplacian operator have been subject of intensive investigation. In order to solve these problems, Kaufmann et al. [17] introduced in 2017 recent results about the new fractional Sobolev space with variable exponents \(W^{s,q(x), p(x,y)}\) defined as :

$$\begin{aligned} W^{s,q(.), p(.,.)}(\Omega ) =\Big \{ u \in L^{q(.)}(\Omega ) : \frac{|u(x)- u(y)|}{|x-y|^{s+\frac{d}{p(.,.)}} } \in L^{p(.,.)}(\Omega \times \Omega ) \Big \}, \end{aligned}$$

where \(s \in ( 0, 1),\) \(\Omega \subset {\mathbb {R}}^{N}\) is a bounded domain with Lipschitz boundary, \(q : \overline{\Omega } \rightarrow (1,\infty )\) and \(p :\overline{\Omega }\times \overline{\Omega } \rightarrow (1,\infty )\) are two continuous functions. Furthermore, they established a continuous and compact embedding theorem of these spaces into Lebesgue spaces. In addition, by applying a direct method of calculus of variations for the following energy fractional

$$\begin{aligned} \theta (u) = \int _{\Omega }\int _{\Omega }\frac{|u(x)- u(y)|^{p(x,y)}}{ {p(x,y)} |x-y|^{ N + sp(x,y) } }dxdy + \int _{\Omega }|u(x)|^{q(x)}dx, \end{aligned}$$
(1)

the authors proved existence and uniqueness of a solution for a nonlocal problem involving p(x, .) Laplacian operator. Later, many scholars further studied the theory and applications of this type of spaces, see for instance [7, 8, 10, 11, 15] and the references therein.

The notion of capacity plays an important role in the study of classical and nonlinear potential theory. Indeed, capacities are characterized by the property of giving the measure of small sets more precisely than the usual Lebesgue measure and they are absolutely necessary to understand point-wise behavior of functions in a Sobolev space. For the theory and applications of capacity see for instance [1,2,3,4, 14, 18, 23] and the references therein. Recently, Baalal and Berghout studied in [9] two kinds of capacities : Sobolev capacity \(C^{s}_{q(.),p(.,.)}\) and relative capacity connected with fractional Sobolev spaces with variable exponents, they proved that both capacities are Choquet capacities and all Borel sets are capacitable. In this context, fractional capacities for fixed exponent spaces were studied by many scholars such as [25,26,27]. Based on the fact that \(C^{s}_{q(.),p(.,.)}\) is an outer capacity, we deal on the convergence of a sequence in \(C^{s}_{q(.),p(.,.)}\)-capacity and we give sufficient conditions on a subset E of \({\mathbb {R}}^{N}\) ensuring that \(C^{s}_{q(.),p(.,.)}(E) = 0.\)

The Sobolev space with zero boundary is significant to specify or compare boundary values of Sobolev functions. In the classical Euclidean case with the Lebesgue measure, the first order Sobolev space with zero boundary values is defined as a completion of compacty supported smooth functions under the Sobolev norm [20]. Afterwards, Harjulehto et al. studied the definition of the variable exponent Sobolev space with zero boundary values involving a method introduced in [22] for metric measure. Later, this notion has been extended to the Musielak–Orlicz–Sobolev space and the weighted variable exponent Sobolev space on metric measure spaces (see [6, 24]). In a recent work [5], we have also studied the anisotropic Sobolev space with zero boundary values and showed some of its properties using results related to anisotropic Sobolev \(\vec {p}\)-capacity.

Our goal in this work is to study the fractional Sobolev space with variable exponents and zero boundary values using the concept of fractional capacities. More precisely, our results on the \(C^{s}_{q(.),p(.,.)}\)-capacity allow us to prove that the Sobolev space with variable exponent and zero boundary values \(H^{s,q(.),p(.,.)}(\Omega )\) is a Banach reflexive space and it coincides with \(H^{s,q(.),p(.,.)}( \Omega \backslash E)\) for all set E satisfying \(C^{s}_{q(.),p(.,.)}(E)=0.\)

For this purpose, the present paper is organized as follows. In Sect. 2, we briefly recall some preliminary knowledge on fractional Sobolev spaces with variable exponents and some properties of capacities. In Sect. 3, we mainly give furthers results related to the fractional \(C^{s}_{q(.),p(.,.)}\) capacity, we define also the fractional Sobolev space with variable exponents and zero boundary values and prove some of its properties. As an application of our results, we prove in Sect. 4 that the Dirichlet energy has a minimizer in the fractional Sobolev space with variable exponents and zero boundary values.

2 Some preliminary results

2.1 Fractional Sobolev spaces with variable exponents

We begin this section by remembering the definition of the variable exponent fractional Sobolev space as defined in [17]. Let \(\Omega\) be an open in \({\mathbb {R}}^{N} (N \in {\mathbb {N}})\). We set \(s \in (0,1)\) and consider two mesurables variable exponents functions , that is \(q: \Omega \rightarrow [0,+\infty )\) and \(p : \Omega \times \Omega \rightarrow [0,+\infty ) .\) We denote by \({\mathcal {P}}(\Omega )\) the set of variable exponents \(q: \Omega \rightarrow [0,+\infty )\) and by \({\mathcal {P}}(\Omega \times \Omega )\) the set of variable exponents \(p : \Omega \times \Omega \rightarrow [0,+\infty ).\) We set

$$\begin{aligned} p^{-}&= ess \inf p(x,y)_{(x,y) \in \Omega \times \Omega }, p^{+}= ess \sup p(x,y)_{(x,y) \in \Omega \times \Omega }\\ q^{-}&= ess \inf q(x)\displaystyle _{x \in \Omega }, q^{+}= ess \sup q(x)_{x \in \Omega }. \end{aligned}$$

In the following, we assume that

$$\begin{aligned}&1<p^{-}\le p(x,y)\le p^{+}<\infty \end{aligned}$$
(2)
$$\begin{aligned}&1<q^{-}\le q(x)\le q^{+}<\infty \end{aligned}$$
(3)

Notice that by proposition 4.1.7 in [12], we can extend q and p to all of \({\mathbb {R}}^{N}\) and \({\mathbb {R}}^{N}\times {\mathbb {R}}^{N}\) respectively. The variable exponent Lebesgue \(L^{q(.)}\) is defined as :

$$\begin{aligned} L^{q(.)}(\Omega )= \Bigg \{ u \in L^{0}(\Omega ) : \int _{\Omega }|u(x)|^{q(x)}dx < \infty \Bigg \}. \end{aligned}$$

We recall the definition of the fractional Sobolev space with variable exponents as follows:

$$\begin{aligned} W^{s,q(.), p(.,.)}(\Omega ) =\Bigg \{ u \in L^{q(.)}(\Omega ) : \frac{|u(x)- u(y)|}{|x-y|^{s+\frac{N}{p(.,.)}} } \in L^{p(.,.)}(\Omega \times \Omega ) \Bigg \}. \end{aligned}$$
(4)

The definition of the spaces \(L^{q(.)}({\mathbb {R}}^{N})\) and \(W^{s,q(.), p(.,.)}({\mathbb {R}}^{N})\) are analogous to \(L^{q(.)}(\Omega )\) and \(W^{s,q(.), p(.,.)}(\Omega );\) one just changes every occurrence of \(\Omega\) by \({\mathbb {R}}^{N}.\) We refer to [8, 11, 17] for more details. We denote by \(\varphi _{q(.),p(.,.)}^{s, \Omega }\) the modular on \(W^{s,q(.), p(.,.)}(\Omega )\) defined by :

$$\begin{aligned} \varphi _{q(.),p(.,.)}^{s,\Omega }(u) = \int _{\Omega }|u(x)|^{q(x)}dx + \int _{\Omega }\int _{\Omega }\frac{|u(x)- u(y)|^{p(x,y)}}{|x-y|^{ N + sp(x,y) } }dxdy. \end{aligned}$$
(5)

When \(\Omega = {\mathbb {R}}^{N},\) we denote it by \(\varphi _{q(.),p(.,.)}^{s}.\)

$$\begin{aligned}{}[u]^{s,p(.,.)}(\Omega )=\inf \Bigg \{ \lambda > 0 : \int _{\Omega } \int _{\Omega } \frac{|u(x)- u(y)|}{ \lambda ^{p(x,y)} |x-y|^{ N + sp(x,y) } }dxdy\le 1 \Bigg \} \end{aligned}$$

is the corresponding variable exponent Gagliardo semi-norm. By the same arguments as in [13] for the constant case, it easy to see that \(W^{s,q(.), p(.,.)}(\Omega )\) is a Banach space with the norm:

$$\begin{aligned} ||u||_{s,q(.), p(.,.)}= ||u ||_{L^{q(\Omega )}} + [u]^{s,p(.,.)}(\Omega ). \end{aligned}$$

We denote by \(C^{\infty }_{0}(\Omega )\) the set of infinitely differentiable functions with compact support on \(\Omega .\)

Let \(W_{0}^{s,q(.), p(.,.)}(\Omega )\) be the closure in \(W^{s,q(.), p(.,.)}(\Omega )\) of the set \(C^{\infty }_{0}(\Omega ).\) The modular \(\varphi _{q(.),p(.,.)}^{s, \Omega }\) induces a norm denoted by

$$\begin{aligned} ||u||_{\varphi _{q(.),p(.,.)}^{s,\Omega }} = \inf \Bigg \{ \lambda >0 : \varphi _{q(.),p(.,.)}^{s,\Omega } \Bigg (\frac{1}{\lambda }u\Bigg )\le 1\Bigg \} . \end{aligned}$$

which is equivalent to the norm \(||u||_{s,q(.), p(.,.)}.\)

Then \(u\in W^{s,q(.), p(.,.)}(\Omega )\) if and only if \(\varphi _{q(.),p(.,.)}^{ s,\Omega }(u) < \infty\).

Proposition 2.1

[9] Let \(s\in (0,1),\) \(q\in {\mathcal {P}}(\Omega )\) and \(p\in {\mathcal {P}}(\Omega \times \Omega ).\) The space \(W^{s,q(.), p(.,.)}(\Omega )\) is separable, reflexive and uniformly convex.

Remark 2.2

Recall that a normed space E has the Banach Saks property if: \(\frac{1}{n}\sum \nolimits ^{n}_{i=1}u_{i}\) converges to u strongly in E,  when every \(u_{i}\) weakly converges to u. By [16], every uniformly convex space has the Banach Saks property.

By application of the previous Proposition, we have the following result.

Corollary 2.3

[9] Let \(s\in (0,1),\) \(q\in {\mathcal {P}}(\Omega )\) and \(p\in {\mathcal {P}}(\Omega \times \Omega ).\) The space \(W^{s,q(.), p(.,.)}(\Omega )\) has the Banach Saks property.

Proposition 2.4

[9] Let \(s \in (0,1),\) \(q\in {\mathcal {P}}(\Omega )\) and \(p\in {\mathcal {P}}(\Omega \times \Omega ).\) Let \(u_{n}, u \in W^{s,q(.), p(.,.)}(\Omega )\) such that \(n\in {\mathbb {N}}.\) Then the following statements are equivalents :

  1. 1.

    \(\lim \nolimits _{n\rightarrow \infty } ||u_{n}-u||_{\varphi _{q(.),p(.,.)}^{s,\Omega } }=0 ;\)

  2. 2.

    \(\lim \nolimits _{n\rightarrow \infty } \varphi _{ q(.),p(.,.)}^{s,\Omega }(u_{n}-u)=0 ;\)

  3. 3.

    \((u_{n})_{n}\) converges to u in \(\Omega\) in measure and

    $$\begin{aligned} \lim \limits _{n\rightarrow \infty } \varphi _{q(.),p(.,.)}^{s, \Omega }(u_{n}) = \varphi _{q(.),p(.,.)}^{s,\Omega }(u). \end{aligned}$$

2.2 Capacity

Definition 2.1

Let E be a topological space and T be the class of Borel sets in E, and let C : \(T \rightarrow [0,+\infty ]\) be a function.

  1. (1)

    The function C is called a capacity if the following axioms are satisfied:

    1. (i)

      \(C(\emptyset )=0\).

    2. (ii)

      \(X\subset Y\Rightarrow C(X)\le C(Y)\) for all X and Y in T (monotonicity), 

    3. (iii)

      For all sequence \((X_{n})\subset T\)

      $$\begin{aligned} \displaystyle C\left( \bigcup \limits _{n}X_{n}\right) \le \sum \limits _{n}C(X_{n}). (countable\, subadditivity) \end{aligned}$$
  2. (2)

    The capacity C is called an outer capacity if, for all \(X \in T,\)

    $$\begin{aligned} \displaystyle C(X)=\displaystyle \inf \{C(O):O\supset X, O\quad \text{ is } \text{ open } \}. \end{aligned}$$
  3. (3)

    The capacity C is called an interior capacity if, for all \(X \in T,\)

    $$\begin{aligned} C(X)=\displaystyle \sup \{C(K):K\subset X, K\quad \text{ is } \text{ compact }\}. \end{aligned}$$
  4. (4)

    A property, that holds true except perhaps on a set of capacity zero is said to be true C-quasi everywhere \(\left( abbreviated \quad C-q.e \right) .\)

Definition 2.2

Let f be a real-valued function finite C-q.e and \((f_{n})\) be a sequence of real-valued function finite C-q.e. (1) We say that \((f_{n})\) converges to f in C-capacity if

$$\begin{aligned} \forall \varepsilon>0, \displaystyle \lim _{n\rightarrow +\infty }C\left( \left\{ x:|f_{n}(x)-f(x)|>\varepsilon \right\} \right) =0. \end{aligned}$$

(2) We say that \((f_{n})\) converges to f C-quasi-uniformly \(\left( abbreviated \quad C -q.u \right)\) if \(\forall \varepsilon >0, \exists X\in T: C(X)<\varepsilon\) and \((f_{n})\) converges to f uniformly on \(X^{c}.\)

Now, we recall the definition of capacity in the fractional Sobolev space \(W^{s,q(.), p(.,.)}({\mathbb {R}}^{N})\) given in [9].

Definition 2.3

[9] Let \(0<s<1,\) \(q\in {\mathcal {P}}({\mathbb {R}}^{N})\) and \(p \in {\mathcal {P}}({\mathbb {R}}^{N}\times {\mathbb {R}}^{N}).\) For a set \(E\subset {\mathbb {R}}^{N},\) we consider

$$\begin{aligned} B_{q(.), p(.,.)}^{s}(E)=\Big \{ u \in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N}): u\ge 1\quad a.e \quad on \quad a \quad neighbourhood \quad of \quad E \Big \}. \end{aligned}$$

The fractional Sobolev (s, q(.) , p(., .))-capacity of the set E is defined by

$$\begin{aligned} C^{s}_{q(.), p(.,.)}(E) = \displaystyle \inf _{ u \in B_{q(.), p(., .)}^{s}(E)} \Big \{ \varphi _{q(.),p(.,.)}^{s}(u) \Big \}, \end{aligned}$$

If \(B_{q(.), p(.,.)}^{s}(E) = \phi ,\) we set   \(C^{s}_{q(.), p(.,.)}(E)=\infty\). Functions belonging to \(B_{q(.), p(.,.)}^{s}(E)\) are called admissible functions for E.

Theorem 2.5

[9] The \(C^{s}_{q(.), p(.,.)}\)-capacity is a Choquet capacity.

3 Main results

In this section, we present our main results involving convergence in \(C^{s}_{q(.),p(.,.)}\)-capacity. We define the fractional Sobolev space with variable exponent and zero boundary values and we prove that it is a Banach reflexive space.

3.1 Convergence in \(C^{s}_{q(.), p(.,.)}\)-capacity

Proposition 3.1

Let \(E \subset {\mathbb {R}}^{N},\) then

  1. (i)

    we have :

    $$\begin{aligned} \mu (E) \le \quad C^{s}_{q(.), p(.,.)}(E), \end{aligned}$$

    where \(\mu\) is the Lebesgue’s measure of E.

  2. (ii)

    If there exists \(u \in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) such that \(u=+\infty\) on an open set containing E then \(C^{s}_{q(.),p(.,.)}(E)=0.\)

Proof

  1. (i)

    If \(C^{s}_{q(.), p(.,.)}(E)= \infty ,\) there is nothing to prove. We may assume that \(C^{s}_{q(.), p(.,.)}(E)< \infty .\) Let \(\varepsilon > 0\) and take \(u \in B_{q(.), p(.,.)}^{s}(E)\) such that

    $$\begin{aligned} \varphi _{q(.),p(.,.)}^{s}(u)\le C^{s}_{q(.), p(.,.)}(E)+ \varepsilon . \end{aligned}$$
    (6)

    There is an open \(O \supset E\) such that \(u\ge 1\) in O.

    $$\begin{aligned} \mu (E) \le \int _{O} \mid u \mid d\mu \le \varphi _{q(.),p(.,.)}^{s}(u). \end{aligned}$$
    (7)

    By using (6) and (7) and letting \(\varepsilon \rightarrow 0,\) we obtain the result.

  2. (ii)

    Let \(u \in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) such that \(u=+\infty\) on an open set O containing E,  then \(u\ge \alpha\) for all \(\alpha > 0,\) we have

    $$\begin{aligned} C^{s}_{q(.), p(.,.)}(E)\le \varphi _{q(.),p(.,.)}^{s,\Omega }\Bigg (\frac{1}{\alpha }u\Bigg )\le \max \Bigg ( \Bigg (\frac{1}{\alpha }\Bigg )^{q^{+}},\Bigg (\frac{1}{\alpha }\Bigg )^{p^{+}}\Bigg ) \quad \varphi _{q(.),p(.,.)}^{s}(u). \end{aligned}$$

    Since \(p^{+}<\infty ,\) \(q^{+}<\infty\) and by letting \(\alpha \rightarrow +\infty ,\) we obtain :

    $$\begin{aligned} C^{s}_{q(.),p(.,.)}(E)=0 . \end{aligned}$$

\(\square\)

Theorem 3.2

Let u and \((u_{n})_{n}\) be in \(W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) and consider the following propositions:

  1. (i)

    \(u_{n}\rightarrow u\) strongly in \(W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\).

  2. (ii)

    \(u_{n}\rightarrow u\) in \(C^{s}_{q(.), p(.,.)}\)-capacity .

  3. (iii)

    There is a subsequence \((u_{n_{j}})\) such that \(u_{n_{j}}\rightarrow u\) in \(C^{s}_{q(.), p(.,.)}\)-q.u.

  4. (iv)

    \(u_{n_{j}} \rightarrow u\) in \(C^{s}_{q(.), p(.,.)}-\) q.e.

Then we have

$$\begin{aligned} (i)\Rightarrow (ii)\Rightarrow (iii)\Rightarrow (iv). \end{aligned}$$

Proof

  • We show that    \((i)\Rightarrow (ii).\)

By Proposition 3.1, we have u and \(u_{n}\) are finite \(C^{s}_{q(.), p(.,.)}\)-q.e, for all n. Let \(\varepsilon > 0,\) then

$$\begin{aligned} C^{s}_{q(.), p(.,.)} \Big ( \Big \{ x:\vert u_{n}-u \vert (x) > \varepsilon \Big \}\Big ) \le \max \Big ( (\frac{1}{\varepsilon })^{q^{+}},(\frac{1}{\varepsilon })^{p^{+}}\Big ) \varphi _{q(.),p(.,.)}^{s}(u_{n}-u). \end{aligned}$$

Consequently, by Proposition 2.4, we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\varphi _{q(.),p(.,.)}^{s}(u_{n}-u)=0, \end{aligned}$$

and we have,

$$\begin{aligned} \lim _{n\rightarrow \infty } C^{s}_{q(.), p(.,.)} \Big ( \Big \{ x:\vert u_{n}-u \vert (x) > \varepsilon \Big \}\Big )=0. \end{aligned}$$

  • We show that    \((ii) \Rightarrow (iii).\)

Let \(\varepsilon >0,\) then there exists \(u_{n_{j}}\) such that

$$\begin{aligned} C^{s}_{q(.), p(.,.)}\Big (\Big \{x:\vert u_{n_{j}}-u\vert (x)>2^{-j} \Big \}\Big ) \le \varepsilon \cdot 2^{-j}. \end{aligned}$$

We put

$$\begin{aligned} E_{j}=\Big \{x:\vert u_{n_{j}}-u\vert (x) >2^{-j}\Big \}, \end{aligned}$$

and

$$\begin{aligned} G_{m}=\bigcup \limits _{j\ge m}E_{j}. \end{aligned}$$

Then we have

$$\begin{aligned} C^{s}_{q(.), p(.,.)}(G_{m})\le \sum \limits _{j\ge m}\varepsilon \cdot 2^{-j}<\varepsilon . \end{aligned}$$

On the other hand,

$$\begin{aligned} \forall x \in (G_{m})^{c}, \forall j \ge m, \quad \vert u_{n_{j}}-u\vert (x) \le 2^{-j}, \end{aligned}$$

thus

$$\begin{aligned} u_{n_{j}}\rightarrow u \quad C^{s}_{q(.), p(.,.)} -q.u. \end{aligned}$$

  • We show that \((iii)\Rightarrow (iv).\)

We have

$$\begin{aligned} \forall j \in {\mathbb {N}}, \exists X_{j}: C^{s}_{q(.), p(.)}(X_{j}) \le \frac{1}{j}, \end{aligned}$$

and

$$\begin{aligned} u_{n_{j}} \quad \text{ converges } \text{ uniformly } \text{ to } \quad u \text{ on } \quad (X_{j})^{C}. \end{aligned}$$

We put \(X=\bigcap \nolimits _{j}X_{j}\), then \(C^{s}_{q(.), p(.)}(X)=0\) and

$$\begin{aligned} u_{n_{j}}\rightarrow u \,\, \text{ on } \,\, X^{C}. \end{aligned}$$

\(\square\)

As an immediate consequence of Theorem 3.2 and Proposition 3.1, we have the following result.

Corollary 3.3

If \((u_{n})_{n}\) is a sequence which converges to u in \(W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\), then there exists a subsequence of \((u_{n})_{n}\) which converges to u, \(\mu\)-a.e.

Definition 3.1

A function \(u : {\mathbb {R}}^{N} \rightarrow [-\infty , +\infty ]\) is called a \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous function in \({\mathbb {R}}^{N}\) if for every \(\varepsilon > 0\) there is a set X such that \(C^{s}_{q(.), p(.,.)}(X) < \varepsilon\) and the restriction of u to \({\mathbb {R}}^{N},\) denoted by \(u_{{\mathbb {R}}^{N} \backslash X}\) is continuous.

Theorem 3.4

The \(C^{s}_{q(.), p(.,.)}\)-capacity satisfies the following properties :

  1. (1)

    If O is an open set of \({\mathbb {R}}^{N}\) and \(E \subset {\mathbb {R}}^{N}\) is such that \(\mu (E)=0,\) then

    $$\begin{aligned} C^{s}_{q(.), p(.,.)}(O)= C^{s}_{q(.), p(.,.)}(O-E). \end{aligned}$$
  2. (2)

    Let u and v be \(C^{s}_{q(.), p(.)}\) - quasicontinuous functions in \({\mathbb {R}}^{N},\) we have (i) if \(u=v\), almost everywhere in an open \(O \subset {\mathbb {R}}^{N},\) then

    $$\begin{aligned} u=v \quad C^{s}_{q(.), p(.,.)} \text{-quasi } \text{ everywhere } \text{ in } \quad O, \end{aligned}$$

    (ii) If \(u\le v\), almost everywhere in an open \(O\subset {\mathbb {R}}^{N},\) then

    $$\begin{aligned} u\le v \quad C^{s}_{q(.), p(.,.)}\text{-quasi } \text{ everywhere } \text{ in } \quad O. \end{aligned}$$

Proof

  1. (1)

    It is obvious that \(C^{s}_{q(.), p(.,.)}(O) \ge C^{s}_{q(.), p(.,.)}(O-E).\) Let \(u \in B_{q(.), p(.,.)}^{s}(O-E)\) thus \(u \ge 1\) in an open set containing \(O-E .\) Let the function f define as : \(f(x) = {\left\{ \begin{array}{ll} u(x), &{} \text { if }\quad x \in {\mathbb {R}}^{N} - E \\ 1, &{} \text { if }\quad x \in E. \end{array}\right. }\) We have \(f \in B_{q(.), p(.,.)}^{s}(O)\) and \(\varphi _{q(.),p(.,.)}^{s}(f)=\varphi _{q(.),p(.,.)}^{s}(u),\) thus

    $$\begin{aligned} C^{s}_{q(.), p(.,.)}(O) \le \varphi _{q(.),p(.,.)}^{s}(u), \end{aligned}$$

    by passing to \(\inf ,\) we obtain

    $$\begin{aligned} C^{s}_{q(.), p(.,.)}(O) \le C^{s}_{q(.), p(.,.)}(O-E). \end{aligned}$$
  2. (2)

    Since \(C^{s}_{q(.), p(.,.)}\) is an outer capacity, we get the results by [21].

\(\square\)

Now, we prove the following results which will be useful below.

Proposition 3.5

Let \(\Omega\) an open of \({\mathbb {R}}^{N}.\) If \(\left( u_{n}\right) _{n}\), \(u \in W^{s,q(.),p(.,.)}(\Omega )\) are such that \(u_{n}\rightharpoonup u\) weakly in \(W^{s,q(.),p(.,.)}(\Omega )\) then

$$\begin{aligned} \liminf ( u_{n})\le u\le \limsup ( u_{n}) \quad C^{s}_{q(.), p(.,.)}\text{-q.e }. \end{aligned}$$

Proof

Since \(W^{s,q(.),p(.,.)}(\Omega )\) is a reflexive space and \(u_{n} \rightharpoonup u\) weakly in \(W^{s,q(.),p(.,.)}(\Omega ).\) Then by applying Banach–Saks theorem, there is a subsequence denoted again by \((u_{n})\) such that the sequence \((h_{n})\) defined by \(h_{n}=\frac{1}{n}\sum \nolimits ^{n}_{i=1}u_{i}\) converges to u strongly in \(W^{s,q(.),p(.,.)}(\Omega )\). Moreover, by Theorem 3.2, there is a subsequence of \((h_{n})\) denoted again by \((h_{n})\), such that

$$\begin{aligned} \lim \limits _{n\rightarrow +\infty } h_{n}= u \quad C^{s}_{q(.), p(.,.)}\text{-q.e }. \end{aligned}$$

On the other hand,

$$\begin{aligned} \liminf u_{n} \le \lim \limits _{n\rightarrow +\infty } h_{n}. \end{aligned}$$

Therefore,

$$\begin{aligned} \liminf (u_{n})\le u C^{s}_{q(.), p(.,.)}\text{-q.e }. \end{aligned}$$

For the second inequality, it suffices to replace \(u_{n}\) by \((-u_{n})\) in the first inequality. \(\square\)

In the following, we assume that \(\Omega\) is an open set of \({\mathbb {R}}^{N}.\)

3.2 Fractional Sobolev spaces with variable exponents and zero boundary values

Definition 3.2

We say that a function u belongs to the fractional Sobolev space with variable exponents and zero boundary values, and we denote \(u \in H^{s,q(.),p(.,.)}(\Omega ),\) if there is a \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous function \({\tilde{u}} \in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N}),\) called canonical representative, such that \({\tilde{u}} = u\) almost everywhere in \(\Omega\) and \({\tilde{u}}=0\) quasi everywhere in \({\mathbb {R}}^{N} \backslash \Omega .\)

The set \(H^{s,q(.),p(.,.)}(\Omega )\) is endowed with the norm

$$\begin{aligned} || u ||_{H^{s, q(.), p(.,.)}(\Omega )}= || {\tilde{u}} ||_{W^{s, q(.), p(.,.)}({\mathbb {R}}^{N})}. \end{aligned}$$

Remark 3.6

Since \(C^{s}_{q(.), p(.,.)}(E) = 0\) implies that \(\mu (E)=0\) for every \(E\subset {\mathbb {R}}^{N},\) observe that the norm does not depend on the choice of the quasicontinuous representative .

Theorem 3.7

The fractional Sobolev space with variable exponent and zero boundary values \(H^{s,q(.),p(.,.)}(\Omega )\) is a Banach space.

Proof

Let \((u_{n})_{n}\) be a Cauchy sequence in \(H^{s,q(.,.),p(.,.)}(\Omega ),\) for every n there is a \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous function \(\tilde{u_{n}} \in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) such that \({\tilde{u}}_{n}= u_{n}\) almost everywhere in \(\Omega\) and \(\tilde{u_{n}}=0, C^{s}_{q(.), p(.,.)}\)-quasi everywhere in \({\mathbb {R}}^{N}\backslash \Omega .\) Since \(W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) is a Banach space, there is a function u such that \(\tilde{u_{n}}\rightarrow u\) in \(W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) as \(n \rightarrow +\infty .\) By applying Theorem 3.2, we deduce that u is \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous and by Proposition  3.5 we have \(u = 0\)   \(C^{s}_{q(.), p(.,.)}\)-q.e in   \({\mathbb {R}}^{N} \backslash \Omega .\) Consequently, \(u \in H^{s,q(.),p(.,.)}(\Omega )\) and we conclude that the space \(H^{s,q(.,.),p(.,.)}(\Omega )\) is complete. \(\square\)

Corollary 3.8

The space \(H^{s,q(.),p(.,.)}(\Omega )\) is reflexive.

Proof

Since \(W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) is a reflexive Banach space, by applying Theorem 3.7, we deduce that the space \(H^{s,q(.),p(.)}(\Omega )\) is closed, and therefore \(H^{s,q(.),p(.,.)}(\Omega )\) is reflexive. \(\square\)

Corollary 3.9

We have \(W_{0}^{s,q(.),p(.,.)}(\Omega )\subset H^{s,q(.),p(.,.)}(\Omega ) \subset W^{s,q(.),p(.,.)}(\Omega ).\)

Proof

Since \(C^{\infty }_{0}(\Omega ) \subset H^{s,q(.),p(.,.)}(\Omega )\) and by applying Theorem 3.7, we obtain the first inclusion. The second inclusion follows directly from the definition of the space \(H^{s,q(.,.),p(.)}(\Omega ).\) \(\square\)

Proposition 3.10

Let \(f\in H^{s,q(.),p(.,.)}(\Omega )\) and \(g \in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N}).\) If f is bounded and g is \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous, then \(fg \in H^{s,q(.),p(.,.)}(\Omega ).\)

Proof

Let \({\tilde{f}}\in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) be a \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous representative function of f. We have \({\tilde{f}}g\) is \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous in \({\mathbb {R}}^{N}.\) Let \(D= \Big \{x\in {\mathbb {R}}^{N} \backslash \Omega : {\tilde{f}}g\ne 0 \Big \}\), then \(D= G \cup H\) where

$$\begin{aligned} G = \Big \{ x\in {\mathbb {R}}^{N}\backslash \Omega : {\tilde{f}}\ne 0 \Big \} \quad \bigcap \quad \Big \{ x\in {\mathbb {R}}^{N}\backslash \Omega : g\ne 0 \quad and \quad g\ne \infty \Big \} \end{aligned}$$

and

$$\begin{aligned} H = \Big \{ x\in {\mathbb {R}}^{N} \backslash \Omega : g=\infty \Big \}. \end{aligned}$$

It is obvious that \(C^{s}_{q(.), p(.,.)} (G)= 0\) and by Proposition 3.1 we have \(C^{s}_{q(.), p(.,.)} (H)= 0\), thus

$$\begin{aligned} C^{s}_{q(.), p(.,.)} (D)=0 . \end{aligned}$$

Therefore,

$$\begin{aligned} {\tilde{f}}g =0 \, \, C^{s}_{q(.), p(.,.)} \text{-quasi } \text{ everywhere } \text{ in } \, \, {\mathbb {R}}^{N} \backslash \Omega . \end{aligned}$$

Since \({\tilde{f}}g=fg\) a.e in \(\Omega ,\) we get

$$\begin{aligned} fg \in H^{s,q(.),p(.,.)}(\Omega ). \end{aligned}$$

\(\square\)

Theorem 3.11

Let \(E\subset \Omega\) be such that \(C^{s}_{q(.), p(.,.)}(E)=0\), we have

$$\begin{aligned} H^{s,q(.),p(.,.)}(\Omega )= H^{s,q(.),p(.,.)}( \Omega \backslash E). \end{aligned}$$

Proof

It is obvious that \(H^{s,q(.),p(.,.)} (\Omega \backslash E) \subset H^{s,q(.),p(.,.)}(\Omega ).\) Let \(u \in H^{s,q(.),p(.,.)}(\Omega ),\) then there is a \(C^{s}_{q(.), p(.,.)}\)-quasicontinuous function \({\tilde{u}}\in W^{s,q(.),p(.,.)}({\mathbb {R}}^{N})\) such that \({\tilde{u}}= u\) a.e in \(\Omega\) and \({\tilde{u}}=0\) \(C^{s}_{q(.), p(.,.)}\)-quasi everywhere in \({\mathbb {R}}^{N} \backslash \Omega .\) Since \(C^{s}_{q(.), p(.,.)}(E)=0,\) we have \({\tilde{u}} = 0\) \(C^{s}_{q(.), p(.,.)}\)-quasi everywhere in \({\mathbb {R}}^{N} \backslash (\Omega \backslash E ).\) Thus, \(u \in H^{s,q(.),p(.,.)}(\Omega \backslash E).\) \(\square\)

Remark 3.12

If \(C^{s}_{q(.), p(.,.)}(\partial \Omega )=0,\) then \(H^{s,q(.),p(.,.)}({\mathring{\Omega }})= H^{s,q(.),p(.,.)}(\overline{\Omega }).\)

4 Application

4.1 The Dirichlet energy integral minimizers

Definition 4.1

Let \(g \in W^{s,q(.),p(.,.)}(\Omega ).\) For all \(u \in H^{s,q(.),p(.,.)}(\Omega ),\) we define the energy operator \(I^{s,p(.,.)}_{\Omega ,g}(u)\) corresponding to the boundary value function g by

$$\begin{aligned} I^{s,p(.,.)}_{\Omega ,g}(u)= & {} \int _{\Omega }|u(x)|^{q(x)}dx + \int _{\Omega }\int _{\Omega }\frac{|u(x)- u(y)|^{p(x,y)}}{|x-y|^{ N + sp(x,y) } }dxdy \\&+ \int _{\Omega }\int _{\Omega }\frac{|g(x)- g(y)|^{p(x,y)}}{|x-y|^{ N + sp(x,y) } }dxdy. \end{aligned}$$

Theorem 4.1

Let \(H^{s,q(.),p(.,.)}(\Omega )\) be the fractional Sobolev space with variable exponents and zero boundary values. Then there exists a function \(u \in H^{s,q(.),p(.,.)}(\Omega )\) such that

$$\begin{aligned} I^{s,p(.,.)}_{\Omega ,g}(u) = \inf _{v\in H^{s,q(.),p(.,.)}(\Omega ) } I^{s,p(.,.)}_{\Omega ,g}(v). \end{aligned}$$

Proof

It follows from Theorem 3.7 and Corollary 3.8 that \(H^{s,q(.),p(.,.)}(\Omega )\) is a reflexive Banach space.

Since the function \(x\rightarrow x^{p}\) is convex for every fixed \(1<p<\infty\), we deduce that \(I^{s,p(.,.)}_{\Omega ,g}\) is convex. Moreover, \(I^{s,p(.,.)}_{\Omega ,g}\) is lower semi-continuous. Next, we show that \(I^{s,p(.,.)}_{\Omega ,g}\) is coercive . Indeed, we can observe that for any \(u \in H^{s,q(.),p(.,.)}(\Omega )\) with \(|| u ||_{H^{s, q(.), p(.,.)}(\Omega )} > 1,\)

$$\begin{aligned} I^{s,p(.,.)}_{\Omega ,g}(u) > ||u||^{\min \{p_{-}, q_{-}\}}_{ H^{s, q(.), p(.,.)}(\Omega )}. \end{aligned}$$

This implies the coercivity of \(I^{s,p(.,.)}_{\Omega ,g},\) hence we conclude by applying the following Lemma \(\square\)

Lemma 4.2

[19] Let H be a reflexive Banach space. If \(I : H \rightarrow {\mathbb {R}}\) is a convex, lower semi-continuous and coercive operator, then there is an element in H that minimizes I.