# A Class of Variable-Order Fractionalp·-Kirchhoff-Type Systems

Authors: Yong Wu; Zhenhua Qiao; Mohamed Karim Hamdani; Bingyu Kou; Libo Yang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractionalpx-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.


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## Body

## 1. Introduction

In this article, we discuss the following variable-order fractionalp·-Kirchhoff-type system:
(1)M1∬ℝ2N1px,yux−uypx,yx−yN+px,ysx,ydxdy−Δp·s·ux=fu,v+axinΩ,M2∬ℝ2N1px,yvx−vypx,yx−yN+px,ysx,ydxdy−Δp·s·vx=gu,v+bxinΩ,u=v=0inℝN\Ω,where Ω⊂ℝN is a bounded smooth domain with N>px,ysx,y for any x,y∈Ω¯×Ω¯. Here, the main operator −Δp·s· is the variable-order fractional p·-Laplacian given by
(2)−Δp·s·φx=P.V.∫ℝNφx−φypx,y−2φx−φyx−yN+px,ysx,ydy,x∈ℝN,along any φ∈C0∞ℝN, where P.V. denotes the Cauchy principle value.From now on, in order to simplify the notation, we denote(3)p−=minx,y∈ℝ2Npx,y,p+=maxx,y∈ℝ2Npx,y,s−=minx,y∈ℝ2Nsx,y,s+=maxx,y∈ℝ2Nsx,y.We will assume thatM1,M2:ℝ+⟶ℝ+ are continuous functions satisfying the conditionM: there exist m>0 and γ>1/p− such that
(4)M1t,M2t>mtγ−1,forallt>0.Note that the Kirchhoff functionsM1,M2 may be singular at t=0 for γ∈0,1.Moreover,H:ℝ2⟶ℝ is a C1-function verifying
(5)Z1:∂H∂uu,v=fu,vand∂H∂vu,v=gu,vforallu,v∈ℝ2;(Z2): there exists K>0 such that
(6)Hu,v=Hu+K,v+Kforallu,v∈ℝ2.Finally, we suppose thatAB:ax,bx∈LqxΩ,1/px+1/qx=1,1<qx<ps∗x, where ps∗x=Np¯x/N−s¯xp¯x, p¯x=px,x, s¯x=sx,x.PS:p·:ℝ2N⟶1,∞ is a continuous function fulfilling 0<s−<s+<1<p−≤p+, and p· and s· are symmetric, that is, px,y=py,x and sx,y=sy,x for any x,y∈ℝ2N.On the one hand, whens·≡1, the operators in (1) reduce to the integer order, i.e., p·-Laplacian Δp·. 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 [4–9], for example, Boulaaras and Allahem ([4]) studied the existence of positive solutions of a px-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 [11–13].Of course, whenpx≡porp=2 and s·≡s, the operators in (1) reduce to the classical non-local fractional p-Laplacian, i.e., −Δps (−Δs). 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. ([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:
(7)ρ∂2v∂t2−p0l+e2L∫0L∂v∂x2dx∂2v∂x2=0,where ρ,l,e,L,p0 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 functionalI will be introduced in “Abstract Framework.”Theorem 1.
LetΩ be a bounded smooth domain of ℝN, with N>px,ysx,y for any x,y∈Ω¯×Ω¯, where p· and s· verify PS. Assume that M, AB, and Z1−Z2 are satisfied. Then, problem (1) admits a weak solution if I is differentiable at u0,v0.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 functionalI 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 [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
(8)C+Ω¯=p∈CΩ¯:px>1forallx∈Ω¯.For anyp∈C+Ω¯, we define the variable exponent Lebesgue space as
(9)Lp·Ω=u:thefunctionu:Ω⟶ℝismeasurable,∫Ωuxpxdx<∞,the vector space endowed with the Luxemburgnorm(10)∥u∥p·=infλ>0:∫Ωuxλpxdx≤1.Then,Lp·Ω,·p· is a separable reflexive Banach space ( see [23], Theorem 2.5). Let q∈C+Ω¯ be the conjugate exponent of p, that is
(11)1px+1qx=1,forallx∈Ω¯.Then, we have the following Hölder inequality, whose proof can be found in [23], (Theorem 2.1).Lemma 2.
Assume thatu∈Lp·Ω and v∈Lq·Ω, then
(12)∣∫Ωuvdx∣≤1p−+1q−∥u∥p·∥v∥q·≤2∥u∥p·∥v∥q·.
The variable-order fractional Sobolev spaces with variable exponent is defined by(13)Ws·,p·Ω=u∈Lp¯·Ω:∬Ω×Ωux−uypx,yx−yN+px,ysx,ydxdy<∞,with the norm us,p·=up·+us·,p·, where
(14)us·,p·=infλ>0:∬Ω×Ωux−uypx,yλpx,yx−yN+px,ysx,ydxdy<1.
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
(15)X=u:ℝN⟶ℝ:uΩ∈Lp¯·Ω,∬Qux−uypx,yλpx,yx−yN+px,ysx,ydxdy<∞,forsomeλ>0,where Q≔ℝ2N\Ωc×Ωc. The space X is endowed with the norm
(16)∥u∥X=∥u∥p·+uX,where
(17)uX=infλ>0:∬Qux−uypx,yλpx,yx−yN+px,ysx,ydxdy<1.
We know that the norms∥·∥s,p· 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 Ws·,p·Ω×Ws·,p·Ω not sufficient for investigating the class of problems like (1).
For this, we set space as(18)X0=u∈X:u=0a.e.inℝN\Ω.
The spaceX0 is a separable reflexive Banach space, see [25], with respect to the norm
(19)∥u∥X0=infλ>0:∬Qux−uypx,yλpx,yx−yN+px,ysx,ydxdy=∬ℝ2Nux−uypx,yλpx,yx−yN+px,ysx,ydxdy<1,where last equality is a consequence of the fact that u=0 a.e. in ℝN\Ω.
In the following Lemma, we give a compact embedding result. For the proof, we refer the reader to [24].Lemma 3.
LetΩ⊂ℝn be a smooth bounded domain and s·∈0,1. Let px,y be continuous variable exponents with sx,ypx,y<Nforx,y∈Ω¯×Ω¯. Assume that q:Ω¯⟶1,∞ is a continuous function such that
(20)ps∗x>qx≥q−>1,forallx∈Ω¯.
Then, there exists a constantC=CN,s,p,q,Ω such that for every u∈X0, it holds that
(21)∥u∥qx≤C∥u∥X0.
The spaceX0 is continuously embedded in LqxΩ. Moreover, this embedding is compact.
We define the fractional modular functionρp·s·:X0⟶ℝ, by
(22)ρp·s·u=∬ℝ2Nux−uypx,yx−yN+px,ysx,ydxdy.
We also have the next result of ([24], Proposition 2.2).Lemma 4.
Assume thatu∈X0 and uj⊂X0, then
(23)∥u∥X0<1resp.=1,>1⇔ρp·s·u<1resp.=1,>1,∥u∥X0<1⇒∥u∥X0p+≤ρp·s·u≤∥u∥X0p−,∥u∥X0>1⇒∥u∥X0p−≤ρp·s·u≤∥u∥X0p+,limj⟶∞∥uj∥X0=0∞⇔limj⟶∞ρp·s·uj=0∞,limj⟶∞∥uj−u∥X0=0⇔limj⟶∞ρp·s·uj−u=0.
Finally, we define our workspaceS=X0×X0, which is endowed with the norm
(24)∥u,v∥S=∥u∥X0+∥v∥X0.
We say that a pair of functionsu,v∈S is the weak solution of problem (1), if for all ϕ,φ∈S one has
(25)M1δp·u∬ℝ2Nux−uypx,y−2vx−vyϕx−ϕyx−yN+px,ysx,ydxdy=∫Ωfu,v+axϕdx,M2δp·v∬ℝ2Nvx−vypx,y−2vx−vyφx−φyx−yN+px,ysx,ydxdy=∫Ωgu,v+bxφdx,where
(26)δp·u=∬ℝ2N1px,yux−uypx,yx−yN+px,ysx,ydxdy.
Let us consider the following functional associated to problem (1), defined by I:S⟶ℝ(27)Iu,v=M1~δp·u−M2~δp·v−∫ΩHu,vdx−∫Ωaxudx−∫Ωbxvdx,for all u,v∈S, where Mi~t=∫0tMiτdτ. Obviously, the continuity of M yields that I is well defined and of class C1 on S\0,0. Furthermore, for every u,v∈S\0,0, the derivative of I is given by
(28)I′u,v,ϕ,φ=M1δp·u∬ℝ2Nux−uypx,y−2ux−uyϕx−ϕyx−yN+px,ysx,ydxdy+M2δp·v∬ℝ2Nvx−vypx,y−2vx−vyφx−φyx−yN+px,ysx,ydxdy−∫Ωfu,v+axϕdx−∫Ωgu,v+bxφdx,for any ϕ,φ∈S. Therefore, the weak solution u,v∈S\0,0 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.
LetX be a Banach space and I:X⟶ℝ be a C1 function which is bounded from below. Then, for any ε>0, there exists ϖε∈X such that
(29)Iϖε≤infXI+εand∥I′ϖε∥X∗≤ε.
Throughout the paper, for simplicity, we useci,i∈ℕ to denote different nonnegative or positive constant.


## 3. The Main Result

Lemma 6.
Under the same assumptions of Theorem1, then I is coercive and bounded from below.Proof.
Firstly, we know that functionalI is well defined. Indeed, it is sufficient to prove that the functional T:S⟶ℝ, Tu,v=∫ΩHu,vdx, is well defined. Since H is continuous on 0,K×0,K and Hu,v=Hu+K,v+K for all u,v∈ℝ2, we get ∣Hu,v∣≤c1 for all u,v∈ℝ2. Thus,
(30)Tu,v≤∫ΩHu,vdx≤c1Ω,forallu,v∈ℝ2,i.e., T is well defined, where ∣Ω∣ is the Lebesgue measure of Ω. Next, we will prove that I is coercive and bounded from below. Let u,v∈S, and we have
(31)Iu,v=M1~δp·u−M2~δp·v−∫ΩHu,vdx−∫Ωaxudx−∫Ωbxvdx≥M1~δp·u−M2~δp·v−c1∣Ω∣−∫Ωaxudx−∫Ωbxvdx.
By the conditionAB and Lemma 2, we get
(32)Iu,v≥M1~δp·u−M2~δp·v−c1∣Ω∣−2∥ax∥qx∥u∥px−2∥bx∥qx∥v∥px.
It follows fromM and Lemmas 3 and 4 that
(33)Iu,v≥m∫01p+ρp·suτγ−1dτ+m∫01p+ρp·s·vτγ−1dτ−c3∥u∥X0−c4∥v∥X0−c2=mγp+γρp·s·uγ+ρp·s·vγ−c3∥u∥X0−c4∥v∥X0−c2≥mγp+γmin∥u∥X0γp−,∥u∥X0γp++min∥v∥X0γp−,∥v∥X0γp+−maxc3,c4∥u∥X0+∥v∥X0−c2.Since γp+>γp−>1, when ∥u,v∥S⟶+∞, at least one of ∥u∥X0 and ∥v∥X0 converges to infinity. So, I is coercive and bounded from below. The proof of Lemma 6 is complete.Proof of Theorem 1.
Obviously, sinceI∈C1S,ℝ is weakly lower semicontinuous and bounded from below, by means of the Ekeland variational principle, we have uj,vj⊂S such that
(34)Iuj,vj⟶infSIandI′uj,vj⟶0.
Furthermore, by the above expression, we getIuj,vj≤c5. Thus, it follows from (33) that
(35)c6≤Iuj,vj≤c5,which implies that the sequences uj and vj are bounded in X0. So, without the loss of generality, there exist subsequences uj and vj such that uj⇀u0 and vj⇀v0 in X0, and thus
(36)∫Ωaxujdx⟶∫Ωaxu0dxand∫Ωbxvjdx⟶∫Ωaxv0dx.
According to compact embedding theorem, which is Lemma3, we obtain
(37)ujx⟶u0xandvjx⟶v0x,a.e.,x∈Ω.
Again, by continuity ofH, we get
(38)Hujx,vjx⟶Hu0x,v0xa.e.x∈Ω.
And becauseH is bounded, we get the following convergence from the Lebesgue dominated convergence theorem:
(39)∫ΩHuj,vjdx⟶∫ΩHu0,v0dx.
By (34), we note that
(40)infSI=limIuj,vj=limM1~δp·uj−M2~δp·vj−∫ΩHuj,vjdx−∫Ωaxujdx−∫Ωbxvjdx.
In view of Fatou’s lemma, we have(41)δp·u0≤liminfδp·ujandδp·v0≤liminfδp·vj.
By the continuous monotone increasing property ofM1~ and M2~, we get
(42)M1~δp·u0≤limM1~δp·ujandM2~δp·v0≤limM2~δp·vj.
In conclusion,(43)infSI≥M1~δp·u0−M2~δp·v0−∫ΩHu0,v0dx−∫Ωaxu0dx−∫Ωbxv0dx=Iu0,v0,which implies Iu0,v0=infSI. Thus, u0,v0∈S is a weak solution of problem (1) if I is differentiable at u0,v0. The proof is complete.



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*Source: 5558074-2021-02-27.xml*