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Yong Wu, Zhenhua Qiao, Mohamed Karim Hamdani, Bingyu Kou, Libo Yang, "A Class of Variable-Order Fractional -Kirchhoff-Type Systems", Journal of Function Spaces, vol. 2021, Article ID 5558074, 6 pages, 2021. https://doi.org/10.1155/2021/5558074
A Class of Variable-Order Fractional -Kirchhoff-Type Systems
This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional -operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first 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.
In this article, we discuss the following variable-order fractional -Kirchhoff-type system: where is a bounded smooth domain with for any . Here, the main operator is the variable-order fractional -Laplacian given by along any , where P.V. denotes the Cauchy principle value.
From now on, in order to simplify the notation, we denote
We will assume that are continuous functions satisfying the condition
there exist and such that
Note that the Kirchhoff functions may be singular at for .
Moreover, is a -function verifying
(Z2): there exists such that
Finally, we suppose that
, where , ,
is a continuous function fulfilling , and and are symmetric, that is, and for any .
On the one hand, when , the operators in (1) reduce to the integer order, i.e., -Laplacian . This kind of variable exponent problem has a wide range of real applications, such as electrorheological fluids (see ), elastic mechanics (), and image restoration (). For this kind of operator combined with Kirchhoff function system problem, we recall [4–9], for example, Boulaaras and Allahem () studied the existence of positive solutions of a -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  appeared in different nonlinear diffusion processes. Subsequently, many results of the variable-order problem have appeared in the literature [11–13].
Of course, when and , the operators in (1) reduce to the classical non-local fractional -Laplacian, i.e., (). Similarly, for the Kirchhoff-type system cases, the papers [14–20] 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. () 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 ) presented a stationary version of differential equation, the so-called Kirchhoff equation: where 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 bi-nonlocal 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 will be introduced in “Abstract Framework.”
Theorem 1. Let be a bounded smooth domain of , with for any , where and verify . Assume that , , and are satisfied. Then, problem (1) admits a weak solution if is differentiable at .
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 is bounded from below and give the proof of Theorem 1.
2. Abstract Framework
In this section, first of all, we recall some basic properties about the variable exponent Lebesgue spaces in  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 we define the variable exponent Lebesgue space as the vector space endowed with the
Then, is a separable reflexive Banach space ( see , Theorem 2.5). Let be the conjugate exponent of , that is
Then, we have the following Hölder inequality, whose proof can be found in , (Theorem 2.1).
Lemma 2. Assume that and , then
The variable-order fractional Sobolev spaces with variable exponent is defined by
with the norm , where
For a more detailed introduction of this space, we can refer to . For the reader’s convenience, we now list some of the results in reference  which will be used in our paper. We define the new variable-order fractional Sobolev spaces with variable exponent
where . The space is endowed with the norm
We know that the norms and are not the same due to the fact that and . This makes the variable-order fractional Sobolev space with variable exponent not sufficient for investigating the class of problems like (1).
For this, we set space as The space is a separable reflexive Banach space, see , with respect to the norm where last equality is a consequence of the fact that a.e. in .
In the following Lemma, we give a compact embedding result. For the proof, we refer the reader to .
Lemma 3. Let be a smooth bounded domain and . Let be continuous variable exponents with for. Assume that is a continuous function such that
Then, there exists a constant such that for every , it holds that
The space is continuously embedded in . Moreover, this embedding is compact.
We define the fractional modular function , by We also have the next result of (, Proposition 2.2).
Lemma 4. Assume that and , then
Finally, we define our workspace , which is endowed with the norm
We say that a pair of functions is the weak solution of problem (1), if for all one has
Let us consider the following functional associated to problem (1), defined by for all , where Obviously, the continuity of yields that is well defined and of class on . Furthermore, for every the derivative of is given by
for any . Therefore, the weak solution of problem (1) is a nontrivial critical point of .
Now, we recall the following well-known Ekeland variational principle found in , which will be used to prove our conclusion, that is Theorem 1.
Theorem 5. Let be a Banach space and be a function which is bounded from below. Then, for any , there exists such that Throughout the paper, for simplicity, we use to denote different nonnegative or positive constant.
3. The Main Result
Lemma 6. Under the same assumptions of Theorem 1, then is coercive and bounded from below.
Proof. Firstly, we know that functional is well defined. Indeed, it is sufficient to prove that the functional , , is well defined. Since is continuous on and for all , we get for all . Thus, i.e., is well defined, where is the Lebesgue measure of . Next, we will prove that is coercive and bounded from below. Let , and we have By the condition and Lemma 2, we get It follows from and Lemmas 3 and 4 that Since , when , at least one of and converges to infinity. So, is coercive and bounded from below. The proof of Lemma 6 is complete.
Proof of Theorem 1. Obviously, since is weakly lower semicontinuous and bounded from below, by means of the Ekeland variational principle, we have such that Furthermore, by the above expression, we get . Thus, it follows from (33) that which implies that the sequences and are bounded in . So, without the loss of generality, there exist subsequences and such that and in , and thus According to compact embedding theorem, which is Lemma 3, we obtain Again, by continuity of , we get And because is bounded, we get the following convergence from the Lebesgue dominated convergence theorem: By (34), we note that In view of Fatou’s lemma, we have By the continuous monotone increasing property of and , we get In conclusion, which implies . Thus, is a weak solution of problem (1) if is differentiable at . The proof is complete.
We do not involve any data in our work.
Conflicts of Interest
The authors declare that they have no competing interests.
The third author Hamdani is supported by the Tunisian Military Research Center for the scientific and technological laboratory LR19DN01, and he would like to express his deepest gratitude to the Military School of Aeronautical Specialities, Sfax (ESA), for providing an excellent atmosphere for work. The fourth author is supported by the Fundamental Research Funds for Youth Development of The Army Engineering University of PLA (Grant No. KYJBJQZL2003). The fifth author is supported by the Natural Science Foundation of Huaiyin Institute of Technology (Grant/Award Number: 20HGZ002).
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