# Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

Authors: Yating Guo; Guoju Ye
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven byp·-fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.


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## Body

## 1. Introduction

In recent years, the fractional differential operators and equations have increasingly attracted much attention, since they are good at describing memory and heredity of some complex systems compared with the integer-order derivative [1, 2]. So far, the fractional differential operators have been applied in various research fields, such as optimization [3], fractional quantum mechanics [4], finance [5], image process [6], and biomedical engineering [7]. For more relevant references, we refer the readers to [8–10].The variable-order fractional derivative extends the study of constant order fractional derivative, which was first proposed by Samko and Ross [11] in 1993. In this concept, the order can change continuously as a function of either dependent or independent variables to better describe the change of memory property with time or space [12]. Later, Lorenzo and Hartley put the variable-order fractional operator to describe the diffusion process in [13], which may also describe the change in temperature [14]. From this, many applications of fractional variable-order spaces have been explored in considerable details [15–17]. The extensive applications urgently need systematic studies on the existence, uniqueness of solutions to these variable-order fractional differential equations. In [18], the infinitely many solutions to Kirchhoff-type variable-order fractional Laplacian equations have been discussed. Xiang [19] has introduced variable-order fractional Laplace −Δs· and explores some problems involving this operator. Moreover, Heydari solved the variable-order fractional nonlinear diffusion-wave equation in [20]. Considering that for some nonhomogeneous materials, the commonly used methods in Lebesgue and Sobolev spaces LpΩ and W1,pΩ are not sufficient; many scholars have begun to study the differential operator with variable exponent [21–23]. Similar to Lebesgue spaces with variable exponents, Kaufmann [24] introduced the fractional derivative involving variable exponents. In [25], Chen introduced a framework for image restoration using a variable exponents Laplacian. For more literature, see [26–32].On the other hand, the research on Kirchhoff-type problems has aroused great interest over recent years. Specifically, Kirchhoff built the model given by the equation(1)ρ∂2v∂t2−P0h+E2L∫0L∂v∂y2dy∂v∂y2=0,in [33] to extend the famous D’Alembert wave equation by further investigating the influence of the changes in the length variation during vibrations. Where v is displacement of a string, L is the length of the string, E is the Young modulus of the material, P0 is the initial tension, ρ is the mass density, and h is the area of cross-section. So far, many researchers have discussed the fractional Kirchhoff-type problems and wide applications. Pucci et al. [34] studied a Kirchhoff-type eigenvalue problem which has a critical nonlinearity and nonlocal fractional Laplace. Later, Molica Bisci et al. [35] centered their work on Kirchhoff nonlocal fractional equations and using three critical point theorem to obtain three solutions. For more results, see [36–38].To our knowledge, the results of the variable-order fractional Sobolev spaces with variable exponents and fractionalp·-Laplace equations with variable order are few. Motivated by these observations, we focus on the following variable-order fractional Laplacian equation with variable exponents:
(2)−Δp·s·vy+vyqy−2vy=gy,y∈Ω;vy=0,y∈∂Ω,and the Kirchhoff-type problem:
(3)a+b∫Ω∫Ω1py,zvy−vzpy,zy−zN+py,zsy,zdydz−Δp·s·vy+vyqy−2vy=gy,y∈Ω;vy=0,y∈ℝN\Ω.Where the nonlocal operator−Δp·s· is defined as
(4)−Δp·s·vy=P.V.∫Ωvy−vzpy,z−2vy−vzy−zN+sy,zpy,zdz,with
(5)s·∈CℝN×ℝN,0,1,p·∈CℝN×ℝN,1,∞,andqy∈CℝN,1,∞,and P.V. is a commonly used abbreviation in the principal value sense.The remainder of this paper is arranged as follows: in Section2, we review some basic knowledge. In Section 3, we research the existence and uniqueness of the weak solutions to equation (2). In Section 4, we investigate the weak solutions to Kirchhoff-type equation (3).


## 2. Preliminaries

In this section, we introduce the main tools and some theorems which will be used in this article.For notational convenience, we define(6)s−=miny,z∈ℝN×ℝNsy,z,s+=maxy,z∈ℝN×ℝNsy,z,p−=miny,z∈ℝN×ℝNpy,z,p+=maxy,z∈ℝN×ℝNpy,z,q−=miny∈Ω¯qy,q+=maxy∈Ω¯qy.Concerning the functions·, p·, and q· satisfied the followings:(A.1)s· is symmetric, i.e., sy,z=sz,y and continuous for all y,z∈ℝN×ℝN with 0<s−≤s+<1.(A.2)p· is symmetric, i.e., py,z=pz,y and continuous for all y,z∈ℝN×ℝN and ps<N for all y,z∈Ω¯×Ω¯.(A.3)p· and q· are both bounded away from 1 and ∞, i.e., there exist 1<q−<q+<+∞ and 1<p−<p+<+∞ so that q−≤qy≤q+ and p−≤py,z≤p+ for every y,z∈ℝN×ℝN.The Banach spaceLqyΩ is given by
(7)LqyΩ=v:functionv:Ω⟶ℝismeasurableand∃ξ>0:∫Ωvyξqydy<∞,with the norm
(8)vLqyΩ=infξ>0:∫Ωvyξqydy<1.It follows from [39] that LqyΩ,∙LqyΩ is a separable and reflexive Banach space.Consider the space(9)C+Ω¯=ψ∈CΩ¯:1<ψforally∈Ω¯.Letq′∈C+Ω¯ be the conjugate exponent of q, i.e.,
(10)1qy+1q′y=1,forally∈Ω¯.Then, we haveTheorem 1 (Holder’s inequality, [39]).
Suppose thatu∈Lq·Ω and v∈Lq′·Ω. Then,
(11)∫Ωuvdx≤1q−+1q′−uLq·ΩvLq′·Ω≤2uLq·ΩvLq′·Ω.In addition, the Sobolev fractional space with variable exponents and variable order for0<s·<1 is defined as follows:
(12)W=Wsy,z,qy,py,zΩ=v:Ω⟶ℝ:v∈LqyΩ:∫Ω∫Ωvy−vzpy,zξpy,zy−zN+sy,zpy,zdydz<∞,forsomeξ>0,and the variable exponent seminorm:
(13)vsy,z,py,zΩ=infξ>0:∫Ω∫Ωvy−vzpy,zξpy,zy−zN+sy,zpy,z<1.If we equipW with
(14)vW=vLqyΩ+vsy,z,py,zΩ,then W is a Banach space.LetE=ℝN×ℝN\Ωc×Ωc. It is proved that the norm ∙W is different from ∙s·,p· in [40] since Ω×Ω⊂E but Ω×Ω≠E. Then, let
(15)W0=v∈W:v=0a.e.inℝN\Ωbe the workspace in Sections 3 and 4. W0 is a Banach space which is separable and reflexive with the norm
(16)vW0=infξ>0:∫∫Evy−vzpy,zξpy,zy−zN+py,zsy,zdydz=∫∫ℝN×ℝNvy−vzpy,zξpy,zy−zN+py,zsy,zdydz<1,where the inequality is a result of v=0 a.e. in ℝN\Ω. Similar to the proof of the Theorem 2.4 in [40], there is a constant C>0 so that
(17)vP¯·≤CvW0,for all v∈W0, where p¯y=py,y. Then, ∙W and ∙W0 are topological equivalent in W0.Theorem 2 (Sobolev-type embedding theorem [41]).
SupposeΩ⊂ℝN (N≥2) is a smooth bounded domain and p·, s· satisfy the conditions (A1) and (A2), respectively. Assume α∈C+Ω¯, β∈C+Ω¯ such that αy≥py,y and βy<ps∗y≔Npy,y/N−sy,ypy,y for all y∈Ω¯. Then, there exists a constant J=JN,s,p,α,β,Ω>0 such that for every v∈W(18)vLβyΩ≤JvW.Furthermore, the embedding is compact.Remark 3.
According to Theorem 2.2 in [42], Theorem 2 holds true in W0.For classical Sobolev space theory such as constant order and constant exponential, see [43–45]. And for variable order, variable exponent cases, see [21, 41].


## 3. Equations in Variable-Order Fractional Laplacian Equations with Variable Exponents

In this section, we consider the existence and uniqueness of the solution to equation (2). Suppose g∈LkyΩ with 1<k−<ky<k+<+∞ for each y∈Ω¯, and define the following functional
(19)Iv=∫Ω∫Ωvy−vzpy,zy−zN+sy,zpy,zpy,zdydz+∫Ωvyqyqydy−∫Ωgyvydy,and the weak solution to equation (2) in space W0.Definition 4.
We say thatv is a weak solution to equation (2) if v∈W0 and
(20)∫Ω∫Ωvy−vzpy,z−2vy−vzwy−wzy−zN+sy,zpy,zdydz+∫Ωvqy−2vywydy=∫Ωgywydy,for any w∈W0.Now, we show that there is a unique minimizer ofI in W0 and the minimizer is also a unique weak solution to equation (2).Theorem 5.
Supposes·∈0,1 is variable order, and qy, py,z are continuous variable exponents with p−·s·>1. Let g∈LkyΩ, with 1<k−≤ky≤k+<+∞. If we have
(21)Npy,yN−sy,ypy,y>kyky−1>1for any y∈Ω¯, then there exists a unique minimizer of equation (19) in W0 which is also a unique weak solution to equation (2).Proof.
We prove the theorem by using the variational method. By simple verification, the functionalI in equation (19) is strictly convex and bounded below (due to the strict convexity of the function t⟶tpy,z for any y and z).
According to Remark3, W0 is compactly embedded in LβyΩ for βy<ps∗y and especially compactly embedded in Lky/ky−1Ω.
It follows from Theorem1 that
(22)Iv=∫Ω∫Ωvy−vzpy,zy−zN+sy,zpy,zpy,zdydz+∫Ωvyqyqydy−∫Ωgyvydy≥1p+vW0p−+∫Ωvyqyqydy−gLkyΩvLkyky−1Ω≥1p+vW0p−+∫Ωvyqyqydy−CvW0.
SupposevW0>1, it follows that
(23)IvvW0≥1p+vW0p−−1−C.
Choose a sequencevj so that vjW0⟶∞ (j⟶∞). Then, we have
(24)Ivj≥1p+vjW0p−−CvjW0⟶∞.
Thus,I is coercive. There is a unique minimizer of I.
Finally, let us verify that whenv is a minimum of equation (19), it is also a weak solution to equation (2). For w∈W0, we obtain
(25)0=ddtIv+twt=0=∫Ω∫Ωddtvy−vz+twy−wzpy,zpy,zy−zN+sy,zpy,zdydzt=0+∫Ωddtvy+twyqyqydyt=0−∫Ωddtgyvy+twydyt=0=∫Ω∫Ωvy−vzpy,z−2vy−vzwy−wzy−zN+sy,zpy,zdydz+∫Ωvyqy−2vywydy−∫Ωgywydy.
Sincev is a minimizer of equation (19), v is a weak solution to equation (2).
Therefore, the proof is completed.


## 4. The Kirchhoff-Type Problem Driven by ap·-Fractional Laplace Operator with Variable Order

In this section, we consider the existence and uniqueness of the solution to Kirchhoff-type problem (3) in the following.Supposeg∈LkyΩ with 1<k−≤ky≤k+<+∞ for each y∈Ω¯. And discuss the functional associated to equation (3), defined by Jv:W0⟶ℝ(26)Jv=a∫Ω∫Ω1py,zvy−vzpy,zy−zN+py,zsy,zdydz+b2∫Ω∫Ω1py,zvy−vzpy,zy−zN+py,zsy,zdydz2+∫Ωvyqyqydy−∫Ωgyvydy.Definition 6.
v is said to be a weak solution to equation (3) if v∈W0 and
(27)a+b∫Ω∫Ωvy−vzpy,zy−zN+py,zsy,zdydz∫Ω∫Ωvy−vzpy,z−2vy−vzwy−wzy−zN+sy,zpy,zdydz+∫Ωvqy−2vywydy=∫Ωgywydy,for any w∈W0.Theorem 7.
Supposea,b>0. Assume that s·∈0,1 is variable order, and qy, p· are continuous variable exponents with p−s·>1. Let g∈LkyΩ, with 1<k−≤ky≤k+<+∞ for any y∈Ω¯ so that
(28)Npy,yN−sy,ypy,y>kyky−1>1,then there is a unique minimizer of equation (26) in W0 which is also a unique weak solution to equation (3).Proof.
Let us prove thatJ is coercive. We have
(29)Jv=a∫Ω∫Ω1py,zvy−vzpy,zy−zN+py,zsy,zdydz+b2∫Ω∫Ω1py,zvy−vzpy,zy−zN+py,zsy,zdydz2+∫Ωvyqyqydy−∫Ωgyvydy≥ap+vW0p−+b2p+2vW02p−+∫Ωvyqyqydy−∫Ωgyvydy≥ap+vW0p−+b2p+2vW02p−+∫Ωvyqyqydy−gLkyΩvLkyky−1Ω≥ap+vW0p−+b2p+2vW02p−+∫Ωvyqyqydy−CvW0.
SupposevW0>1, we have
(30)JvvW0≥ap+vW0p−−1+b2p+2vW02p−−1+1vW0∫Ωvyqyqydy−C≥ap+vW0p−−1+b2p+2vW02p−−1−C.
Select a sequencevj so that vW0⟶∞ (j⟶∞). Then, we have
(31)Jvj≥ap+vW0p−+b2p+2vjW02p−−CvjW0⟶∞.
Thus,J is coercive. The functional J has a unique minimizer.
Next, we verify that whenv is a minimum of equation (26), it is a weak solution to equation (3). For v∈W0, we have
(32)0=ddtJv+twt=0=ddta∫Ω∫Ω1py,zvy−vz+twy−wzpy,zy−zN+py,zsy,zdydz+b2∫Ω∫Ω1py,zvy−vz+twy−wzpy,zy−zN+py,zsy,zdydz2+∫Ωvy+twyqyqydy−∫Ωgyvy+twydyt=0=a+b∫Ω∫Ωvy−vzpy,zy−zN+py,zsy,zdydz∫Ω∫Ωvy−vzpy,z−2vy−vzwy−wzy−zN+sy,zpy,zdydz+∫Ωvqy−2vywydy−∫Ωgywydy.Hence,v is a weak solution to the equation (3).Remark 8.
Whena=0,b>0, the equation (3) becomes a degenerate Kirchhoff-type equation and Theorem 7 still holds.



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*Source: 6686213-2021-01-06.xml*