Nonlocal fractional p(·)-Kirchhoff systems with variable-order: Two and three solutions

In this article, we consider the following nonlocal fractional Kirchhoff-type elliptic systems  −M1 (∫ RN×RN |η(x)−η(y)| p(x,y)|x−y|N+p(x,y)s(x,y) dxdy + ∫ Ω |η|p(x) p(x) dx ) ( ∆ s(·) p(·)η − |η|p(x)η ) = λFη(x, η, ξ) + μGη(x, η, ξ), x ∈ Ω, −M2 (∫ RN×RN |ξ(x)−ξ(y)| p(x,y)|x−y|N+p(x,y)s(x,y) dxdy + ∫ Ω |ξ|p(x) p(x) dx ) ( ∆ s(·) p(·)ξ − |ξ|p(x)ξ ) = λFξ(x, η, ξ) + μGξ(x, η, ξ), x ∈ Ω, η = ξ = 0, x ∈ RN\Ω, where M1(t),M2(t) are the models of Kirchhoff coefficient, Ω is a bounded smooth domain in RN , (−∆)s(·) p(·) is a fractional Laplace operator, λ, μ are two real parameters, F,G are continuous differentiable functions, whose partial derivatives are Fη, Fξ,Gη,Gξ. With the help of direct variational methods, we study the existence of solutions for nonlocal fractional p(·)-Kirchhoff systems with variable-order, and obtain at least two and three weak solutions based on Bonanno’s and Ricceri’s critical points theorem. The outstanding feature is the case that the Palais-Smale condition is not requested. The major difficulties and innovations are nonlocal Kirchhoff functions with the presence of the Laplace operator involving two variable parameters.

Throughout this paper, s(·), p(·) ∈ C + (D) are two continuous functions that the following assumptions are satisfied.
Kirchhoff in [9] introduced the following Kirchhoff equation From then on, the existence, multiplicity, uniqueness, and regularity of solutions for various Kirchhoff-type equations have been studied extensively, such as, see [10][11][12][13][14] for further details. The continuous Kirchhoff terms M i (t) : R + 0 → R + , (i = 1, 2) are strictly increasing functions, which the following conditions are satisfied.
(M): There exist m i = m i (ι) > 0 and M i = M i (ι) > 0, (i = 1, 2) for any ι > 0 such that and put In recent years, a multitude of scholars has devoted themselves to the study of Kirchhoff-type systems. When M 1 (t) = 1 and M 2 (t) = 1, Chen et al. in [15] consider the nontrivial solutions for the following elliptic systems.
by utilizing Nehari manifold method and Fibering maps, they studied the existence of weak solutions for this kind of problem (1.4). Moreover, it has been applied in the local case s = 1 in [16].
In the famous literature [17], the three critical points theorem was established by Ricceri. Starting from this paper, Marano and Motreanu in [18] extended the result of Ricceri to non-differentiable functionals. Subsequently, Fan and Deng in [19] studied the version of Ricceri's result including variables exponents. Ricceri's result in [20] has been successfully applied to Sobolev spaces W 1,p 0 (Ω), and then at least three solutions are obtained. Furthermore, Bonanno in [21] established the existence of two intervals of positive real parameters λ for which the functional Φ − λJ has three critical points, and applied the result to obtain two critical points.
By using three critical points theorem, Azroul et al. [22] discussed the fractional p-Laplace systems with bounded domain thus, the existence and multiplicity of solutions were obtained by Azroul et al. In addition, there are many scholars who have used different methods to study the existence of elliptic systems on bounded and unbounded regions, for instance, see [23][24][25] for details.
With respect to the fractional p(·)-Laplace operators, Azroul et al. [26] dealt with the class of Kirchhoff type elliptic systems in nonlocal fractional Sobolev spaces with variable exponents and Based on the three critical points theorem introduced by Ricceri and on the theory of fractional Sobolev spaces with variable exponents, the existence of weak solutions for a nonlocal fractional elliptic system of (p(x, ·), q(x, ·))-Kirchhoff type with homogeneous Dirichlet boundary conditions was obtained. By using Ekeland's variational principle and dual fountain theorem, Bu et al. in [27] obtained some new existence and multiplicity of negative energy solutions for the fractional p(·)-Laplace operators with constant order without the Ambrosetti-Rabinowitz condition.
Previous studies have shown that the fractional p(·)-Laplace operators with variable-order are much more complex and difficult than p-Laplace operators. The investigation of these problems has captured the attention of a host of scholars. For example, Wu et al. in [28] considered the fractional Kirchhoff systems with a bounded set Ω in R N , as follows: by applying Ekeland variational principle, they obtained the existence of a solution for this class of problem.
When µ = 0, problem (1.1) reduces to the following fractional Kirchhoff-type elliptic systems Motivated by the above cited works, we take into account the nonlocal fractional Kirchhoff-type elliptic systems with variable-order. Our aims are to establish the existence of at least three solutions for problem (1.1) by utilizing Ricceri's result in [29] and obtain the existence of at least two solutions for problem (1.8) with the help of the multiple critical points theorem in [37]. The primary consideration of the paper is an extension of the results found in the literatures and our results are new to the Kirchhofftype systems in some ways.
For simplicity, C j ( j = 1, 2, ..., N) are used in various places to denote distinct constants, i = 1, 2, and we denote F : Ω × R 2 → R is a C 1 -function, whose partial derivatives are F η , F ξ , which satisfy the following conditions.
Define the corresponding functional I : X 0 → R associated with Kirchhoff systems (1.1), by The functions Φ, Ψ, J : X 0 → R are well defined, and we define their Gâteaux derivatives at (η, ξ) ∈ X 0 , by for all (υ, φ) ∈ X 0 . Hence, (η, ξ) ∈ X 0 is a (weak) solution of Kirchhoff systems (1.1) if and only if (η, ξ) is a critical point of the functional I, that is Definition 2. For s + p + < N, and denote by A: there exists a kind of functions F : for any ϑ(x) ∈ [1, p * s (x)). Now, let us show our results in this article. Theorem 1.1. For s(·), p(·) ∈ C + (D) with s + p + < N and F ∈ A, assume that (S), (P), (M), (F1) and (F2) are satisfied. There exist three constants a, c 1 , (1.20) Then, for any there exists a positive real number ρ such that the system (1.8) has at least two weak solutions w j = (η j , ξ j ) ∈ X 0 ( j = 1, 2) whose norms w j in X 0 are less than some positive constant ρ.
3) whose norms w j in X 0 are less than some positive constant ρ.
Remark 1.1. Existence results for the Kirchhoff-type elliptic systems with both boundary value problems and variational problems were obtained according to using critical points theorem by Ricceri and Bonanno, respectively, where the condition of Palais-Smale is not satisfied.
To deal with these difficulties, we suppose that M 1 (t), M 2 (t) are strictly increasing functions, and then prove that the function Φ is a homeomorphism.
The remaining of this article is organized as follows: Some fundamental results about the fractional Lebesgue spaces and Sobolve spaces are given in Section 2. In Section 3, in order to use critical point theory, we prove some technical lemmas. Theorem 1.1 and Theorem 1.2 are proved in Section 4. Finally, we make a conclusion in Section 5.

Variable exponents Lebesgue spaces
To study Laplacian problems with variable exponents, we need to recall a slice of preliminary theories on generalized Lebesgue spaces L ϑ(x) (Ω) and give some necessary lemmas and propositions.

Fractional Sobolev spaces with variable-order
From now on, we briefly review a slice of essential lemmas and propositions about the Sobolev spaces, which will be used later. The readers are invited to consult [33][34][35] and the references therein.
The fractional Sobolev spaces W s(·),p(·) (Ω) is defined as and it can be endowed with the norm then, the spaces (W, · W ) is a separable and reflexive Banach space, see [2,5] for a more detailed.
We define the new fractional Sobolev spaces W concerning variable exponent and variable-order for some χ > 0.

Some technical lemmas
In this section, in order to use critical point theory for Kirchhoff systems (1.1), we need the following crucial lemmas, which will play an important role in the proof of our results.
Since M 1 and M 2 is monotone, we have Hence, the operator Φ is bounded.

Proof of the main results
In this subsection, we firstly prove Theorem 1.1 by applying Theorem 2.2.
Proof of Theorem 1.1. Let X = X 0 , Ψ and Φ are given as (1.11) and (1.12), respectively. Note that Ψ is a compact derivative from Lemma 3.1, Lemma 3.2 ensures that Φ is a weakly lower semicontinuous and bounded operator in X 0 , and Φ admits a continuous inverse operator Φ : X * 0 → X 0 .
In what follows, we prove Theorem 1.2 Theorem by using Theorem 2.3.

Conclusions
In this article, we consider a kind of p(·)-fractional Kirchhoff systems. Under some reasonable assumptions, we obtain two and three solutions based on Bonanno's multiple critical points theorems and Ricceri's three critical points theorem, where the condition of Palais-Smale is not requested. Several recent results of literatures are extended and improved.