# A New Proof of Existence of Positive Weak Solutions for Sublinear Kirchhoff Elliptic Systems with Multiple Parameters

Authors: Salah Mahmoud Boulaaras; Rafik Guefaifia; Bahri Cherif; Sultan Alodhaibi
Journal: Complexity
 (2020)

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## Abstract

This paper deals with the study of the existence of weak positive solutions for sublinear Kirchhoff elliptic systems with zero Dirichlet boundary condition in bounded domainΩ⊂ℝN by using the subsuper solutions method.


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## Body

## 1. Introduction

In this paper, we consider the following system of differential equations:(1)−A∫Ω∇u2dxΔu=λ1ua+μ1vbinΩ,−B∫Ω∇u2dxΔv=λ2uc+μ2vdinΩ,u=v=0on∂Ω,where Ω⊂ℝN N≥3 is a bounded smooth domain with C2 boundary ∂Ω, A, B: ℝ+⟶ℝ+ are continuous functions, and λ1,λ2,μ1, and μ2 are positive parameters, where a+c<1 and b+d<1. The peculiarity of this type of problem, and by far the most important, is that it is not local. This is based on the presence of the operator(2)−A∫Ω∇u2dxΔurespectively−B∫Ω∇u2dxΔu,which contains an integral on all the fields and implies that the equation is not a specific identity. It is clear that these problems contribute to the transition from academia to application. Indeed, very popular for its physical motivations, problem (1) is none other than a stationary version of the following model which regulates the behavior of elastic whose ends are fixed and which is subjected to nonlinear vibrations:(3)utt−M∫Ω∇u2dxΔu=hx,u,inΩ×0,T,u=0,in∂Ω×0,T,ux,0=u0x,utx,0=u1x,where T is a positive constant and u0 and u1 are given functions. In such problems, u expresses the displacement, hx,u the extreme force, Mr=a1r+b1, b1 the initial tension, and a1 relates to the intrinsic properties of the wire material (such as Young’s modulus). For more details, see [1], as well as their references. Basically, this is a generalization to larger dimensions of the model originally proposed in one dimension by Kirchhoff [2] in (1883):(4)∂2u∂t2−ρ0+ρ1∫0L∂u∂x2dx∂u∂x=0,where ρ0 is the initial tension, ρ1 represents Young’s modulus of the material of the wire, and L its length. The latter is known to be an extension of the equation of D’Alembert waves. Indeed, Kirchhoff took into account the changes caused by transverse oscillations along the length of the wire. With their implications in other disciplines, and given the breadth of their fields of application, nonlocal problems will be used to model several physical phenomena, and they also intervene in biological systems or describe a process dependent on its average, such as particle density population. Moreover, With this significant impact strengthening the field of applications, this type of problem has caught the interest of mathematicians and a lot of work on the existence of solutions has emerged, particularly after the coup de force provided by the famous Lions article [3], where the latter has adopted an approach based on functional analysis. Nevertheless, in most of these articles, the benefit method is purely topological. It is only in the last decades that this approach has been removed from variational methods when Alves and his colleagues [4] obtained for the first time the results of their existence through these methods. Since then, a very fruitful development has given rise to many works based on this advantageous axis (see [1, 3, 5]).In recent years, problems relating to Kirchhoff operators have been studied in several papers (we refer to [6]), where the authors used different methods to obtain solutions (1) in the case of single equation (see [6]). The concept of weak sub- and supersolutions was first formulated by Hess and Deuel in [7, 8] to obtain existence results for weak solutions of semilinear elliptic Dirichlet problems and was subsequently continued by several authors (see, e.g., [9–18]).In our recent paper [19], we have discussed the existence of weak positive solution for the following Kirchhoff elliptic systems:(5)−A∫Ω∇u2dxΔu=λ′fv+μ1′huinΩ,−B∫Ω∇u2dxΔv=λ2′gu+μ2′τvinΩ,u=v=0on∂Ω.Motivated by the ideas of [20], which the authors considered a system (1) in the case At=Bt=1, more precisely, under suitable conditions on f, g, we will prove that the problem which is defined in (1) admits a positive solution In current paper, motivated by previous works in ([19, 20]), we discuss the existence of weak positive solution for sublinear Kirchhoff elliptic systems in bounded domains by using subsupersolutions method combined with comparison principle (see Lemma 2.1 in [4]).The outline of the paper is as follows. In the second section, we give some assumptions and definitions related to problem (1). In Section 3, we prove our main result.


## 2. Assumptions and Definitions

Let us assume the following assumption.H
1 Assume that A,B:ℝ+⟶ℝ+ are two continuous and increasing functions and there exists ai,bi>0,i=1,2, such that(6)a1≤At≤a2,b1≤Bt≤b2,for allt∈ℝ+.H
2 Suppose that a,d≥0, b, c>0, a+c<1 and b+d<1.Now, in order to discuss our main result of problem (1), we need the following two definitions.Definition 1.
Letu,v∈H01Ω×H01Ω; u,v is said to be a weak solution of (1) if it satisfies(7)Au2∫Ω∇u∇ϕdx=λ1∫Ωuaϕdx+μ1vbϕdxinΩ,Bv2∫Ω∇v∇ψdx=λ2∫Ωucψdx+μ2∫ΩvdψdxinΩ,for all ϕ,ψ∈H01Ω×H01Ω.Definition 2.
A pair of nonnegative functionsu¯,v¯,u¯,v¯ in H01Ω×H01Ω is called a weak subsolution and supersolution of (1) if they satisfy u¯,v¯,u¯,v¯=0,0 on ∂Ω:(8)Au¯2∫Ω∇u¯∇ϕdx≤λ1∫Ωu¯aϕdx+μ1∫Ωv¯bϕdxinΩ,Bv¯2∫Ω∇v¯∇ψdx≤λ2∫Ωu¯cψdx+μ2∫Ωv¯dψdxinΩ,Au¯2∫Ω∇u¯∇ϕdx≥λ1∫Ωu¯aϕdx+μ1∫Ωv¯bϕdxinΩ,Bv¯2∫Ω∇v¯∇ψdx≥λ2∫Ωu¯cψdx+μ2∫Ωv¯dψdxinΩ,for all ϕ,ψ∈H01Ω×H01Ω.Lemma 1 [4].
Assume thatM:ℝ+⟶ℝ+ is a continuous and nonincreasing function satisfying(9)Ms>m0,for alls≥s0,where m0 is a positive constant and assume that u,v are two nonnegative functions such that(10)−Mu2Δu≥−Mv2ΔvinΩ,u=v=0on∂Ω,and then u≥v a.e. in Ω.


## 3. Main Result

In this section, we shall state and prove the main result of this paper.Theorem 1.
Suppose thatH1-H2 hold and M is a nonincreasing function satisfying (9). Then, problem (1) has a large positive weak solution for each positive parameters λ1,λ2,μ1, and μ2.Proof of Theorem 1.
Letσ be the first eigenvalue of −Δ with Dirichlet boundary conditions and ϕ1 the corresponding eigenfunction with ϕ1=1 satisfying ϕ1>0 in Ω and ∇ϕ1>0 on ∂Ω.
Sincebc<1−a1−d, we can take k such that(11)c1−d<k<b1−a.
We shall verify thatu¯,v¯=εϕ12,εkϕ12 is a subsolution of problem (1), where ε>0 is small and specified later.
A simple calculation:(12)Au¯2∫Ω∇u¯⋅∇ϕdx=2εAu¯2∫Ωϕ1∇ϕ1⋅∇ϕdx=2εAu¯2×∫Ω∇ϕ1∇ϕ1⋅ϕdx−∫Ω∇ϕ12ϕdx=2εAu¯2∫Ωσϕ12−∇ϕ12ϕdx≤2a2ε∫Ωσϕ12−∇ϕ12ϕdx.
Similarly,(13)Bv¯2∫Ω∇v¯⋅∇ψdx=2εkBv¯2∫Ωσϕ12−∇ϕ12ϕdx≤2b2εk∫Ωσϕ12−∇ϕ12ϕdx.
Letη>0, μ>0 be such that(14)σϕ12−∇ϕ12≤0,x∈Ω¯η,and μ≤ϕ1≤1 on Ω\Ω¯η where Ω¯η=x∈Ω:dx,∂Ω≤η.
We have from (14) that(15)A∫Ω¯η∇u¯2dx∫Ω¯η∇u¯⋅∇ϕdx≤0≤λ1∫Ω¯ηu¯aϕdx+μ1∫Ω¯ηv¯bϕdx,(16)B∫Ω¯η∇v¯2dx∫Ω¯η∇v¯⋅∇ψdx≤0≤λ2∫Ω¯ηu¯cψdx+μ2∫Ω¯ηv¯dψdx.
On the other hand, inΩ\Ω¯η, let(17)r1=1−ac,r2=1−a1−a−c,s1=1−db,s2=1−d1−d−b.
Note that(18)1r1+1r2=1,1s1+1s2=1.
We have from (11) that(19)1−ar1−kbr2≥1−a−kb>0,k1−ds2−cs1≥k1−d−c>0.
Thus, we chooseε>0 such that(20)2a2ε1−a/r1−kb/r2σϕ12≤λ11/r1μ11/r2μ2+aδ,x∈Ω\Ω¯η,2b2εk1−d/s2−c/s1σϕ12≤λ21/s1μ21/s2μ2+γd,x∈Ω\Ω¯η,where δ=2/1−a, γ=2/1−d. Furthermore,(21)aδr1=2a1−a−c≥2a,γds2=2d1−d−b≥2d,2s1=21−db>2c1−a≥2c,2r2=21−ac>2b1−d≥2b.
These relations and Young inequality show that(22)2a2ε∫Ω\Ω¯ησϕ12−∇ϕ12ϕdx≤2a2ε∫Ω\Ω¯ησϕ12⋅ϕdx≤∫Ω\Ω¯ηλ11/r1εa/r1μaδμ11/r2εkb/r2μ2ϕdx≤∫Ω\Ω¯ηλ11/r1εa/r1μaδr1r1+μ11/r2εkb/r2μ2r2r2ϕdx≤∫Ω\Ω¯ηλ11/r1εa/r1μaδr1+μ11/r2εkb/r2μ2r2ϕdx=∫Ω\Ω¯ηλ1εaμaδr1+μ1εkbμ2r2ϕdx≤∫Ω\Ω¯ηλ1εaϕ12a+μ1εkbϕ12bϕdx=∫Ω\Ω¯ηλ1u¯a+μ1v¯bϕdx,(23)2b2εk∫Ω\Ω¯ησϕ12−∇ϕ12ψdx≤2b2εk∫Ω\Ω¯ησϕ12⋅ψdx≤∫Ω\Ω¯ηλ21/s1εc/s1μ2μ21/s2εkd/s2μγdψdx≤∫Ω\Ω¯ηλ21/s1εc/s1μ2s1s1+μ21/s2εkd/s2μγds2s2ψdx≤∫Ω\Ω¯ηλ21/s1εc/s1μ2s1+μ21/s2εkd/s2μγds2ψdx=∫Ω\Ω¯ηλ2εcμ2s1+μ2εkdμγds2ψdx≤∫Ω\Ω¯ηλ2εcμ2c+μ2εkdμ2dψdx≤∫Ω\Ω¯ηλ2εcϕ12c+μ2εkdϕ12dψdx=∫Ω\Ω¯ηλ2u¯c+μ2v¯dψdx.
Hence, from (15)–(23), it follows that(24)A∫Ω∇u¯2dx∫Ωη∇u¯∇ϕdx+∫Ω\Ω¯η∇u¯∇ϕdx=A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≤λ1u¯a+μ1v¯bϕdx,(25)B∫Ω∇v¯2dx∫Ωη∇v¯∇ψdx+∫Ω\Ω¯η∇v¯∇ψdx=B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≤λ2u¯c+μ2v¯dψdx.
Then, by (24) and (25), u¯,v¯ is a subsolution of (1).
Next, we shall construct a supersolution of problem (1). Let ω be the solution of the following problem:(26)−Δe=1inΩ,e=0on∂Ω.
Let(27)u¯=C1e,v¯=C2e,where e is given by (26) and C1,C2>0 are large positive real numbers to be chosen later. We shall verify that u¯,v¯ is a supersolution of problem (1). Let ϕ∈H01Ω with ϕ≥0 in Ω. Then, we obtain from (26) and the condition H1 that(28)A∫Ω∇u¯2dx∫Ω∇u¯⋅∇ϕdx=A∫Ω∇u¯2dxC1∫Ω∇e⋅∇ϕdx=A∫Ω∇u¯2dxC1∫Ωϕdx≥a1C1∫ΩϕdxB∫Ω∇v¯2dx∫Ω∇v¯⋅∇ψdx=B∫Ω∇v¯2dxC2∫Ω∇e⋅∇ψdx=B∫Ω∇v¯2dxC2∫Ωψdx≥b1C2∫Ωψdx.
Letl=e∞. Since a<1, d<1, these imply that there exist positive large constants α=a1C1,β=b1C2 such that(29)α≥λ1αla+μ1βlb,β≥λ2αlc+μ2βld.
Thus,(30)a1C1∫Ωϕdx≥∫Ωλ1u¯a+μ1v¯bϕdx,(31)b1C2∫Ωψdx≥∫Ωλ2u¯c+μ2v¯dϕdx.
From (26) and (30), we can deduce that the couple u¯,v¯ is a subsolution of problem (1) with u¯≤u¯ and v¯≤v¯ for C1, C2 large.
In order to obtain a weak solution of problem (1) we shall use the arguments by Azouz and Bensedik [19]. For this purpose, we define a sequence un,vn⊂H01Ω×H01Ω as follows: u0≔u¯,v0=v¯ and un,vn is the unique solution of the system(32)−A∫Ω∇un2dxΔun=λ1un−1a+μ1vn−1binΩ,−B∫Ω∇vn2dxΔvn=λ2un−1c+μ2vn−1dinΩ,un=vn=0on∂Ω.
Problem (32) is A,B−linear in the sense that if un−1,vn−1∈H01Ω×H01Ω is given, the right hand sides of (32) are independent of un,vn.
SetAt=tAt2, Bt=tBt2. Then, since Aℝ=ℝ, Bℝ=ℝ, fun−1=un−1a, hvn−1=vn−1b,gun−1=un−1c, and τvn−1=vn−1d∈L2Ω.
We deduce from a result in [4] that system (32) has a unique solution un,vn∈H01Ω×H01Ω.
By using (32) and the fact that u0,v0 is a supersolution of (1), we have(33)−A∫Ω∇u02dxΔu0≥λ1u0a+μ1v0b=−A∫Ω∇u12dxΔu1,−B∫Ω∇v02dxΔv0≥λ2u0c+μ2v0d=−B∫Ω∇v1dxΔv1,and by Lemma 1, u0≥u1 and v0≥v1. Also, since u0≥u¯, v0≥v¯ and the monotonicity of f, h, g, and τ one has(34)−A∫Ω∇u12dxΔu1=λ1u0a+μ1v0b≥λ1u¯a+μ1v¯b≥−A∫Ω∇u¯2dxΔu¯,−B∫Ω∇v12dxΔv1=λ2u0c+μ2v0d≥λ2u¯c+μ2v¯d≥−B∫Ω∇v¯2dxΔv¯,from which, rding to Lemma 1, u1≥u¯, v1≥v¯ for u2,v2 we write(35)−A∫Ω∇u12dxΔu1=λ1u0a+μ1v0b≥λ1u1a+μ1v1b=−A∫Ω∇u22dxΔu2,−B∫Ω∇v1dxΔv1=λ2u0c+μ2v0d≥λ2u1c+μ2v1d=−B∫Ω∇v22dxΔv2,and then u1≥u2, v1≥v2. Similarly, u2≥u¯ and v2≥v¯ because(36)−A∫Ω∇u22dxΔu2=λ1u0a+μ1v0b≥λ1u¯a+μ1v¯b≥−A∫Ω∇u¯2dxΔu¯,−B∫Ω∇v22dxΔv2=λ2u1c+μ2v1d≥λ2u¯c+μ2v¯d≥−B∫Ω∇v¯2dxΔv¯.
Repeating this argument, we get a bounded monotone sequenceun,vn⊂H01Ω×H01Ω satisfying(37)u¯=u0≥u1≥u2≥⋯≥un≥⋯≥u¯>0,(38)v¯=v0≥v1≥v2≥⋯≥vn≥⋯≥v¯>0.
Using the continuity of the functionsf,h, g, and t and the definition of the sequences un,vn,there exist constants Ci>0, i=1,…,4 independent of n such that(39)fvn−1≤C1,hun−1≤C2,gun−1≤C3,(40)τun−1≤C4,for alln.
From (39), multiplying the first equation of (32) by un and integrating using the Holder inequality and Sobolev embedding, we can show that(41)a1∫Ω∇un2dx≤A∫Ω∇un2dx∫Ω∇un2dx=λ1fvn−1undx+μ1∫Ωhun−1undx≤λ1∫Ωfvn−1undx+μ1∫Ωhun−1undx≤C1λ1∫Ωundx+C2μ1∫Ωundx≤C5unH01Ω,or(42)unH01Ω≤C5,∀n,where C5>0 is a constant independent of n. Similarly, there exists C6>0 independent of n such that(43)vnH01Ω≤C6,∀n.
From (42) and (43), we infer that un,vn has a subsequence which weakly converges in H01Ω,ℝ2 to a limit u,v with the properties u≥u¯>0 and v≥v¯>0. Being monotone and also using a standard regularity argument, un,vn converges itself to u,v. Now, letting n⟶+∞ in (32), we deduce that u,v is a positive solution of system (1). The proof of theorem is now completed.


## 4. Conclusion

In this work, we study the existence of weak positive solutions for a sublinear Kirchhoff elliptic systems in bounded domains by using the subsuper solutions method (SSM) combined with comparison principle which have been widely applied in many work (see for example [4, 19, 21–25]).Validity of the comparison principle and of the SSM for local and nonlocal problems as the stationary Kirchhoff Equation was an important subject in the last few years (see, for example, [26] and [23]. Moreover, the two conditions that M is nonincreasing and H is increasing turn out to be necessary and sufficient, at least for the validity of the comparison principle. It is worth to notice that in [4], Alves and Correa developed a new SSM for problem (1) to deal with the increasing M case. The result is obtained by using a kind of Minty–Browder theorem for a suitable pseudomonotone operator, but instead of constructing a subsolution, the authors assumed the existence of a whole family of functions which satisfy a stronger condition than just being subsolutions; the inconvenience is that these stronger conditions restrict the possible right hand sides in (1). Another SSM for nonlocal problems is obtained in [4] for a problem involving a nonlocal term with a Lebesgue norm, instead of the Sobolev norm appearing in (1). In our next study, we will try to apply an alternative approach using the variational principle which has been presented in [27–29].



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