A Class of Variable-Order Fractional p ð · Þ -Kirchhoff-Type Systems

This paper is concerned with an elliptic system of Kirchho ﬀ type, driven by the variable-order fractional p ð x Þ -operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our ﬁ rst attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.

From now on, in order to simplify the notation, we denote We will assume that M 1 , M 2 : ℝ + ⟶ ℝ + are continuous functions satisfying the condition ðMÞ: there exist m > 0 and γ > 1/p − such that Note that the Kirchhoff functions M 1 , M 2 may be singular at t = 0 for γ ∈ ð0, 1Þ.
On the one hand, when sð·Þ ≡ 1, the operators in (1) reduce to the integer order, i.e., pð·Þ-Laplacian Δ pð·Þ . This kind of variable exponent problem has a wide range of real applications, such as electrorheological fluids (see [1]), elastic mechanics ( [2]), and image restoration ( [3]). For this kind of operator combined with Kirchhoff function system problem, we recall [4][5][6][7][8][9], for example, Boulaaras and Allahem ( [4]) studied the existence of positive solutions of a pðxÞ -Kirchhoff system by using subsuper solutions concepts. A very interesting question arose is that whether there are other ways to solve this class of problems. And also, can we consider the nonlocal variable-order case? We know that the fractional variable-order derivatives proposed by Lorenzo and Hartley in [10] appeared in different nonlinear diffusion processes. Subsequently, many results of the variable-order problem have appeared in the literature [11][12][13].
Of course, when pðxÞ ≡ pðorp = 2Þ and sð·Þ ≡ s, the operators in (1) reduce to the classical non-local fractional p -Laplacian, i.e., ð−ΔÞ s p (ð−ΔÞ s ). Similarly, for the Kirchhofftype system cases, the papers [14][15][16][17][18][19][20] introduce a lot of related work in recent years, where many authors studied the existence and multiplicity of solutions by applying variational methods. For instance, based on the three critical points theorem, Azroul et al. ( [15]) discussed an elliptic system with the homogeneous Dirichlet boundary conditions, and they obtained the existence of three weak solutions.
On the other hand, it is worth mentioning that Kirchhoff in 1883 (see [21]) presented a stationary version of differential equation, the so-called Kirchhoff equation: where ρ, l, e, L, p 0 are positive constants which represent the corresponding physical meanings. It is a generalization of D'Alembert equation. It is very interesting to combine this model with various operators due to its nonlocal nature. Inspired by the above works, we consider a new fractional Kirchhoff-type system (1). As far as we know, this is the first attempt on variable-order fractional situations to study a binonlocal problem with variable exponent. In order to overcome the difficulty, we use the direct variational method and Ekeland variational principle to deal with it. Our result is new to the variable-order fractional system with variable exponent. Now, we give the main result of this paper; our energy functional I will be introduced in "Abstract Framework." where pð·Þ and sð·Þ verify ðPSÞ. Assume that ðMÞ, ðABÞ, and The paper is organized as follows. In "Abstract Framework," we state some interesting properties of variable exponent Lebesgue spaces and variable-order fractional Sobolev spaces with variable exponent. In "The Main Result," we prove the functional I is bounded from below and give the proof of Theorem 1.

Abstract Framework
In this section, first of all, we recall some basic properties about the variable exponent Lebesgue spaces in [22] and variable-order fractional Sobolev spaces. Secondly, we give some necessary lemmas that will be used in this paper. Finally, we introduce the definition of weak solutions for problem (1) and build the corresponding energy functional. Consider the set For any p ∈ C + ð ΩÞ, we define the variable exponent Lebesgue space as the vector space endowed with the Luxemburgnorm Then, ðL pð·Þ ðΩÞ, k·k pð·Þ Þ is a separable reflexive Banach space ( see [23], Theorem 2.5). Let q ∈ C + ð ΩÞ be the conjugate exponent of p, that is Then, we have the following Hölder inequality, whose proof can be found in [23], (Theorem 2.1). Journal of Function Spaces Lemma 2. Assume that u ∈ L pð·Þ ðΩÞ and v ∈ L qð·Þ ðΩÞ, then The variable-order fractional Sobolev spaces with variable exponent is defined by with the norm kuk s,pð·Þ = kuk pð·Þ + ½u sð·Þ,pð·Þ , where For a more detailed introduction of this space, we can refer to [24]. For the reader's convenience, we now list some of the results in reference [24] which will be used in our paper. We define the new variable-order fractional Sobolev spaces with variable exponent where Q ≔ ℝ 2N \ ðΩ c × Ω c Þ. The space X is endowed with the norm where We know that the norms ∥·∥ s,pð·Þ and ∥·∥ X are not the same due to the fact that Ω × Ω ⊂ Q and Ω × Ω ≠ Q. This makes the variable-order fractional Sobolev space with variable exponent W sð·Þ,pð·Þ ðΩÞ × W sð·Þ,pð·Þ ðΩÞ not sufficient for investigating the class of problems like (1).
For this, we set space as The space X 0 is a separable reflexive Banach space, see [25], with respect to the norm where last equality is a consequence of the fact that u = 0 a.e. in ℝ N \ Ω.
In the following Lemma, we give a compact embedding result. For the proof, we refer the reader to [24].

Lemma 4.
Assume that u ∈ X 0 and fu j g ⊂ X 0 , then Finally, we define our workspace S = X 0 × X 0 , which is endowed with the norm

Journal of Function Spaces
We say that a pair of functions ðu, vÞ ∈ S is the weak solution of problem (1), if for all ðϕ, φÞ ∈ S one has where Let us consider the following functional associated to problem (1), defined by I : for all ðu, vÞ ∈ S, where f M i ðtÞ = Ð t 0 M i ðτÞdτ: Obviously, the continuity of M yields that I is well defined and of class C 1 on S \ f0, 0g. Furthermore, for every ðu, vÞ ∈ S \ f0, 0g, the derivative of I is given by for any ðϕ, φÞ ∈ S. Therefore, the weak solution ðu, vÞ ∈ S \ f0, 0g of problem (1) is a nontrivial critical point of I . Now, we recall the following well-known Ekeland variational principle found in [7], which will be used to prove our conclusion, that is Theorem 1.

Theorem 5.
Let X be a Banach space and I : X ⟶ ℝ be a C 1 function which is bounded from below. Then, for any ε > 0, there exists ϖ ε ∈ X such that I ϖ ε ð Þ ≤ inf X I + ε and∥I ′ ϖ ε ð Þ∥ X * ≤ ε: ð29Þ Throughout the paper, for simplicity, we use fc i , i ∈ ℕg to denote different nonnegative or positive constant.

The Main Result
Lemma 6. Under the same assumptions of Theorem 1, then I is coercive and bounded from below.