# Existence of Positive Weak Solutions for Quasi-Linear Kirchhoff Elliptic Systems via Sub-Supersolutions Concept

Authors: Amor Menaceur; Salah Mahmoud Boulaaras; Rafik Guefaifia; Asma Alharbi
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

By using the sub- and supersolutions concept (Schmitt, 2007), we prove in this paper the existence of positive solutions of quasi-linear Kirchhoff elliptic systems in bounded smooth domains. This work is an extension of the recent work of Boulaaras et al., 2020.


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## Body

## 1. Introduction

The scope of nonlinear partial differential equations is quite wide. One of the main advances in the development of nonlinear PDEs has been the study of wave propagation, then comes the equations related to chemical and biological phenomena, and later, the equations related to solid mechanics, fluid dynamics, acoustics, nonlinear optics, plasma physics, quantum field theory, and engineering.Studying these equations is a daunting task because there are no general methods for solving them. Each problem requires an appropriate approach depending on the type of linearity ([1–10]).Thep-Laplacian operator is a model of quasi-linear elliptic operators which makes it possible to model physical phenomena such as the flow of non-Newtonian aids, reaction flow systems, nonlinear elasticity, the extraction of petroleum, astronomy, through porous media, and glaciology. Several authors in this field obtained many results of existence (see, for example, [1, 3, 5, 11, 12]).In this work, we consider the following quasi-linear elliptic system:(1)−A∫Ω∇u2dxΔu=λuαvγ,inΩ,−B∫Ω∇v2dxΔv=λuδvβ,inΩ,v=u=0,on∂Ω,where Ω⊂ℝNN≥3 is a bounded domain and its boundary ∂Ω. Also, A and B are two continuous functions on ℝ+, and the parameters α, β, δ, and γ satisfy the following conditions:(2)0≤α<1,0≤β<1,δ,γ>0,θ=1−α1−β−γδ>0for eachλ>0.Within previous studies [13–15], some nonlocal elliptical problems of the Kirchhoff type of the following model were extensively studied:(3)M∫Ω∇u2dx△u=hx,u,inΩ,u=0,in∂Ω,where Ω is a bounded open domain of ℝn with a smooth boundary ∂Ω and hx,u the right hand side is defined for some exceptional functions similar to those in [13–16]. In addition, M is a defined and continuous function on ℝ+ with values in ℝ+∗. In recent years, various Kirchhoff or px-Kirchhoff-type problems have been widely studied by many authors due to their theoretical and practical importance. Such problems are often referred to as nonlocal due to the presence of a full term on Ω or in ℝn. It is well known that this problem is analogous to the stationary problem of a model introduced by Kirchhoff [17].(4)ρ∂2u∂t2−P0h+E2L∫0L∂u∂x2dx∂2u∂x2=0.More specifically, Kirchhoff proposed this model as an extension of the wave equation of the Alembert classic by considering the effects of variations in the length of the strings during vibration. The parameters of the above equation have the following meanings:E is Young’s modulus of the material, ρ is the mass density, L is the length of the chain, h is the section area, and P0 is the initial tension.In recent work in [18], we have discussed the existence of the weak positive solution for the following Kirchhoff elliptic systems:(5)−A∇uL2Ω△u=λ1uα+μ1′vβ,inΩ,−B∇uL2Ω△v=λ2′uc+μ2′vd,inΩ,u=v=0,on∂Ω,where λ1,μ1′,λ2′, and μ2′ are positive parameters, α+c<1, and β+d<1.Motivated by the recent work in [13, 14, 18, 19] and by using the sub- and supersolution method which is defined in [20], existence of positive solutions of quasi-linear Kirchhoff elliptic systems is shown in bounded smooth domains.The paper outline is as follows: some assumptions and definitions related to problem (1) are given in Section 2. Finally, our main result is given in Section 3.


## 2. Preliminaries and Assumptions

We assume the following hypothesis:H1: we assume that M:ℝ+⟶ℝ+ is a nonincreasing and continuous function which satisfies(6)limt⟶0+Mt=m0,wherem0>0, and there exists ai,bi>0,i=1,2 such that(7)a1≤At≤a2,b1≤Bt≤b2for all t∈ℝ+.H2: and(8)α,β∈CΩ¯,αx≥α0>0,βx≥β0>0for allx∈Ω.H3:f, g, h, and τ are C1 on 0,+∞ and increasing functions, where(9)limt⟶+∞ft=+∞,limt⟶+∞gt=+∞,limt⟶+∞ht=+∞=limt⟶+∞τt=+∞.H4:∃γ>0 such that(10)limt⟶+∞htfkgtγt=0,for allk>0,limt⟶+∞τktγtγ−1=0,for allk>0.Lemma 1 (see [14]).
Under assumption (H1),we suppose further that function Ht:=tMt2 is increasing on ℝ.
We assume thatu and v are couple nonnegative functions, where(11)−M∫Ω∇u2dxΔu≥−M∫Ω∇v2dxΔv,inΩ,u=v=0,on∂Ω,and then u≥va.e. in Ω.Lemma 2 (see [1]).
IfM verifies the conditions of Lemma 1, then for each f∈L2Ω, there exists a unique solution u∈H01Ω to the M-linear problem:(12)−M∫Ω∇u2dxΔu=fxinΩandu=0in∂Ω.Lemma 3 (see [1]).
Letw solve Δw=ginΩ. If g∈CΩ, then w∈C1,αΩ for any α∈0,1, so particularly, w is continuous in Ω.Definition 1.
Letu,v∈H01Ω∩L∞Ω×H01Ω∩L∞Ω, and u,v is said a weak solution of (1) if it satisfies(13)A∫Ω∇u2dx∫Ω∇u∇ϕdx=λ∫Ωuαvγϕdx,inΩ,B∫Ω∇v2dx∫Ω∇v∇ψdx=λ∫Ωuδvβψdx,inΩ,for all ϕ,ψ∈H01Ω×H01Ω.Definition 2.
We call the following nonnegative functionsu¯,v¯, respectively; u¯,v¯ in H01Ω∩L∞Ω×H01Ω∩L∞Ω are a weak subsolution (respectively, upersolution) of (1) if they verify u¯,v¯ and u¯,v¯=0,0 in ∂Ω:(14)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≤λ∫Ωu¯αv¯γϕdxinΩ,B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≤λ∫Ωu¯δv¯βψdxinΩ,A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≥λ∫Ωu¯αv¯γϕdxinΩ,B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≥λ∫Ωu¯δv¯βψdxinΩ,for all ϕ,ψ∈H01Ω×H01Ω.
Before proving our main result, we need to prove the existence of weak supersolution and subsolution in the following section.


## 3. Weak Existence Results

### 3.1. Existence of Weak Supersolution

The existence of a positive weak supersolution for system (1) is established such that each component belongs to C0,ρΩ¯, for ρ∈0,1.Lemma 4.
Suppose thatH1 holds, 0≤α,β<1,δ,γ>0, and θ=1−α1−β−γδ>0. Then, system (1) possesses a positive weak supersolution(15)u¯,v¯∈L20,T,C0,ρ1Ω¯×L20,T,C0,ρ2Ω¯,for ρi∈0,1,i=1,2 and λ>0.Proof.
Letei∈C0,ρiΩ¯, for i=1,2, ρi>0, be the solution of the following problem:(16)−△ei=1,inΩ,ei=0,on∂Ω.
Then, by the strong maximum principle, we getei>0 in Ω, i=1,2.
We define(17)u¯,v¯=C1e1,C2e2,where C1 and C2 are positive constants which we will fix them later.
Letϕ,ψ⊂H01Ω×H01Ω, with ϕ,ψ≥0.
Then, we obtain(18)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx=A∫Ω∇u¯2dxC1∫Ω∇e1∇ϕdx=A∫Ω∇u¯2dxC1∫Ωϕdx≥a1C1∫Ωϕdx,and similarly,(19)B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx=B∫Ω∇v¯2dxC2∫Ωψdx≥b1C2∫Ωψdx.
If(20)l=e1∞,L=e2∞,0≤α<1,0≤β<1,λ>0,θ>0,and H1 holds, it is easy to prove that there exist positive constants C1 and C2 such that(21)a1C11−α=λC2γlαLγ,b1C21−β=λC1γlδLβ.
Thus, from (21), we obtain for all x∈Ω(22)λu¯αv¯γ≤λC1αC2γlαLγ≤a1C1,λu¯δv¯β≤λC1δC2βlδLβ≤b1C2.
Therefore, by using (18), (19), and (22), we conclude that(23)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≥λ∫Ωu¯αv¯γϕdx,inΩ,B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≥λ∫Ωu¯δv¯βψdx,inΩ.
Hence,u¯,v¯∈C0,ρ1Ω¯×C0,ρ2Ω¯ is a positive weak supersolution of system (1).


### 3.2. Existence of Weak Subsolution

Existence of a positive weak subsolution for system (1) is proved such that each component belongs to C0Ω¯.Lemma 5.
We assume thatH1 holds:(24)0≤α,β<1,δ,γ>0,θ=1−α1−β−γδ>0.Therefore, system (1) possesses a positive weak subsolution u¯,v¯∈C0Ω¯×C0Ω¯, for all λ>0.Proof.
We assume thatλ1 is the first eigenvalue of −Δ with Dirichlet condition with ϕ1 which is its corresponding eigenfunction and ϕ1belongs to C0,ρ1Ω¯×C0,μ1Ω¯, ϕ1>0 in Ω and ∇ϕ1≥σ1 on ∂Ω, for some positive constants σ1,μ1, and ρ1.
We define(25)u¯,v¯=cϕ12,ckϕ12which belongs to C0Ω¯∩C1Ω¯×C0Ω¯∩C1Ω¯, with c>0 to be fixed later, and(26)δ1−β<k<1−αγbecause θ>0, 1−α>0, and 1−β>0. Then, for all ϕ,ψ∈H01Ω×H01Ω, with ϕ, ψ≥0, we have(27)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx=2cA∫Ω∇u¯2dx∫Ωϕ1∇ϕ1∇ϕ,=2cA∫Ω∇u¯2dx∫Ωλ1ϕ12−∇ϕ12ϕdx.
Similarly,(28)B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx=2ckB∫Ω∇v¯2dx∫Ωλ1ϕ12−∇ϕ12ψdx.
Sinceϕ1=0 and ∇ϕ1≥σ1 on ∂Ω, there exists η>0 such that, for every x∈Ωη=x∈Ω:dx,∂Ω≤η, we have(29)λ1ϕ12−∇ϕ12≤0,λ1ψ12−∇ψ12≤0.
Then, for eachλ>0, we get(30)A∫Ω∇u¯2dx∫Ωη∇u¯∇ϕdx≤0≤λ∫Ωηu¯αu¯γϕdx,for all ϕ∈H01Ω, ϕ≥0, and(31)B∫Ω∇v¯2dx∫Ωη∇v¯∇ψdx≤0≤λ∫Ωηu¯δu¯βψdx,for all ψ∈H01Ω and ψ≥0.
Now, asϕ1>0 in Ω and ϕ1 is continuous, then there exists μ>0 such that ϕ1x≥μ>0 for all x∈Ω\Ω¯η. Therefore, from (26), we obtain a0>0 such that the following inequalities hold:(32)2b2λ1ck1−β−δϕ12−2βx≤λμ2δ≤λϕ12δx,∀x∈ΩΩ¯η,(33)2a2λ1c1−α−kγϕ12−2αx≤λμ2γ≤λϕ12γx,∀x∈ΩΩ¯η,for each c∈0,a0.
Then,(34)2cA∫Ω∇u¯2dxλ1ϕ12−∇ϕ12ϕ≤2a2cλ1ϕ12=2a2λ1c1−α−kγϕ12−2αckγcαϕ12α.
By (33), we have(35)2cA∫Ω∇u¯2dxλ1ϕ12−∇ϕ12≤λϕ12γckγcαϕ12α=λu¯αv¯γ.
And similarly, from (32), we have(36)2ckB∫Ω∇v¯2dxλ1ϕ12−∇ϕ12≤λu¯δv¯βin Ω/Ω¯η and each c∈0,a0.
Therefore,(37)A∫Ω∇u¯2dx∫Ω/Ω¯η∇u¯∇ϕdx≤λ∫Ω/Ω¯ηu¯αv¯γϕdx,(38)B∫Ω∇v¯2dx∫Ω/Ω¯η∇v¯∇ψdx≤λ∫Ω/Ω¯ηu¯δv¯βψdx.
Hence, from (30), (31), (37), and (38), it follows that(39)A∫Ω∇u¯2dx∫Ωη∇u¯∇ϕdx+∫Ω/Ω¯η∇u¯∇ϕdx=A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≤∫Ωu¯αv¯γϕdx,(40)B∫Ω∇v¯2dx∫Ωη∇v¯∇ψdx+∫Ω/Ω¯η∇v¯∇ψdx=B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≤λ∫Ωu¯δu¯βψdx.
Then, by (39) and (40), u¯,v¯ is a positive weak subsolution of system (1), for each c∈0,a0.


## 3.1. Existence of Weak Supersolution

The existence of a positive weak supersolution for system (1) is established such that each component belongs to C0,ρΩ¯, for ρ∈0,1.Lemma 4.
Suppose thatH1 holds, 0≤α,β<1,δ,γ>0, and θ=1−α1−β−γδ>0. Then, system (1) possesses a positive weak supersolution(15)u¯,v¯∈L20,T,C0,ρ1Ω¯×L20,T,C0,ρ2Ω¯,for ρi∈0,1,i=1,2 and λ>0.Proof.
Letei∈C0,ρiΩ¯, for i=1,2, ρi>0, be the solution of the following problem:(16)−△ei=1,inΩ,ei=0,on∂Ω.
Then, by the strong maximum principle, we getei>0 in Ω, i=1,2.
We define(17)u¯,v¯=C1e1,C2e2,where C1 and C2 are positive constants which we will fix them later.
Letϕ,ψ⊂H01Ω×H01Ω, with ϕ,ψ≥0.
Then, we obtain(18)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx=A∫Ω∇u¯2dxC1∫Ω∇e1∇ϕdx=A∫Ω∇u¯2dxC1∫Ωϕdx≥a1C1∫Ωϕdx,and similarly,(19)B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx=B∫Ω∇v¯2dxC2∫Ωψdx≥b1C2∫Ωψdx.
If(20)l=e1∞,L=e2∞,0≤α<1,0≤β<1,λ>0,θ>0,and H1 holds, it is easy to prove that there exist positive constants C1 and C2 such that(21)a1C11−α=λC2γlαLγ,b1C21−β=λC1γlδLβ.
Thus, from (21), we obtain for all x∈Ω(22)λu¯αv¯γ≤λC1αC2γlαLγ≤a1C1,λu¯δv¯β≤λC1δC2βlδLβ≤b1C2.
Therefore, by using (18), (19), and (22), we conclude that(23)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≥λ∫Ωu¯αv¯γϕdx,inΩ,B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≥λ∫Ωu¯δv¯βψdx,inΩ.
Hence,u¯,v¯∈C0,ρ1Ω¯×C0,ρ2Ω¯ is a positive weak supersolution of system (1).


## 3.2. Existence of Weak Subsolution

Existence of a positive weak subsolution for system (1) is proved such that each component belongs to C0Ω¯.Lemma 5.
We assume thatH1 holds:(24)0≤α,β<1,δ,γ>0,θ=1−α1−β−γδ>0.Therefore, system (1) possesses a positive weak subsolution u¯,v¯∈C0Ω¯×C0Ω¯, for all λ>0.Proof.
We assume thatλ1 is the first eigenvalue of −Δ with Dirichlet condition with ϕ1 which is its corresponding eigenfunction and ϕ1belongs to C0,ρ1Ω¯×C0,μ1Ω¯, ϕ1>0 in Ω and ∇ϕ1≥σ1 on ∂Ω, for some positive constants σ1,μ1, and ρ1.
We define(25)u¯,v¯=cϕ12,ckϕ12which belongs to C0Ω¯∩C1Ω¯×C0Ω¯∩C1Ω¯, with c>0 to be fixed later, and(26)δ1−β<k<1−αγbecause θ>0, 1−α>0, and 1−β>0. Then, for all ϕ,ψ∈H01Ω×H01Ω, with ϕ, ψ≥0, we have(27)A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx=2cA∫Ω∇u¯2dx∫Ωϕ1∇ϕ1∇ϕ,=2cA∫Ω∇u¯2dx∫Ωλ1ϕ12−∇ϕ12ϕdx.
Similarly,(28)B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx=2ckB∫Ω∇v¯2dx∫Ωλ1ϕ12−∇ϕ12ψdx.
Sinceϕ1=0 and ∇ϕ1≥σ1 on ∂Ω, there exists η>0 such that, for every x∈Ωη=x∈Ω:dx,∂Ω≤η, we have(29)λ1ϕ12−∇ϕ12≤0,λ1ψ12−∇ψ12≤0.
Then, for eachλ>0, we get(30)A∫Ω∇u¯2dx∫Ωη∇u¯∇ϕdx≤0≤λ∫Ωηu¯αu¯γϕdx,for all ϕ∈H01Ω, ϕ≥0, and(31)B∫Ω∇v¯2dx∫Ωη∇v¯∇ψdx≤0≤λ∫Ωηu¯δu¯βψdx,for all ψ∈H01Ω and ψ≥0.
Now, asϕ1>0 in Ω and ϕ1 is continuous, then there exists μ>0 such that ϕ1x≥μ>0 for all x∈Ω\Ω¯η. Therefore, from (26), we obtain a0>0 such that the following inequalities hold:(32)2b2λ1ck1−β−δϕ12−2βx≤λμ2δ≤λϕ12δx,∀x∈ΩΩ¯η,(33)2a2λ1c1−α−kγϕ12−2αx≤λμ2γ≤λϕ12γx,∀x∈ΩΩ¯η,for each c∈0,a0.
Then,(34)2cA∫Ω∇u¯2dxλ1ϕ12−∇ϕ12ϕ≤2a2cλ1ϕ12=2a2λ1c1−α−kγϕ12−2αckγcαϕ12α.
By (33), we have(35)2cA∫Ω∇u¯2dxλ1ϕ12−∇ϕ12≤λϕ12γckγcαϕ12α=λu¯αv¯γ.
And similarly, from (32), we have(36)2ckB∫Ω∇v¯2dxλ1ϕ12−∇ϕ12≤λu¯δv¯βin Ω/Ω¯η and each c∈0,a0.
Therefore,(37)A∫Ω∇u¯2dx∫Ω/Ω¯η∇u¯∇ϕdx≤λ∫Ω/Ω¯ηu¯αv¯γϕdx,(38)B∫Ω∇v¯2dx∫Ω/Ω¯η∇v¯∇ψdx≤λ∫Ω/Ω¯ηu¯δv¯βψdx.
Hence, from (30), (31), (37), and (38), it follows that(39)A∫Ω∇u¯2dx∫Ωη∇u¯∇ϕdx+∫Ω/Ω¯η∇u¯∇ϕdx=A∫Ω∇u¯2dx∫Ω∇u¯∇ϕdx≤∫Ωu¯αv¯γϕdx,(40)B∫Ω∇v¯2dx∫Ωη∇v¯∇ψdx+∫Ω/Ω¯η∇v¯∇ψdx=B∫Ω∇v¯2dx∫Ω∇v¯∇ψdx≤λ∫Ωu¯δu¯βψdx.
Then, by (39) and (40), u¯,v¯ is a positive weak subsolution of system (1), for each c∈0,a0.


## 4. Main Result

In this section, we give the result of the existence of the positive weak solution to quasi-linear elliptic system (1) by using the sub- and supersolution method which has been already used for some classical elliptic equations by known authors (see [1, 4, 11, 19, 21]).Theorem 1.
Suppose thatH1 holds, 0≤α,β<1,δ,γ>0, and θ=1−α1−β−γδ>0 as well as under the results of Lemma 4 and 5. Then, system (1) possesses a weak solution u,v∈H01Ω×H01Ω, where each component is positive and belongs to C0,ρΩ¯∩C1,μΩ for some ρ∈0,1, μ>0, and each λ>0.Proof 3.
In order to obtain a weak solution of problem (1), we shall use the arguments by Azzouz and Bensedik [13]. 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(41)−A∫Ω∇un2dx△un=λun−1αvn−1γ,inΩ,−B∫Ω∇vn2dx△vn=λun−1δvn−1β,inΩ,un=vn=0,on∂Ω.
Problem (41) is A,B−linear in the sense that if(42)un−1,vn−1∈H01Ω×H01Ωis given, the right-hand sides of (41) are independent of un,vn.
Set(43)At=tAt2,Bt=tBt2.
Then, since(44)Aℝ=ℝ,Bℝ=ℝ,fun−1,vn−1=un−1αvn−1γ∈L2Ω,gun−1,vn−1=un−1δvn−1β∈L2Ω.
According to the result in [1], we can deduce that system (41) admits a unique solution(45)un,vn∈H01Ω×H01Ω.
By using (41) and the fact that u0,v0 is a supersolution of (1), we have(46)−A∫Ω∇u02dx△u0≥λu0αv0γ=−A∫Ω∇u12dx△u1,−B∫Ω∇v02dx△v0≥λu0δv0β=−B∫Ω∇v12dx△v1.
Also, by using Lemma1, u0≥u1 and v0≥v1, and since u0≥u¯, v0≥v¯, and the monotonicity of fu,v=uαvγ and gu,v=uδvβ, one has(47)−A∫Ω∇u12dxΔu1=λu0αv0γ≥λu¯αv¯γ≥−A∫Ω∇u¯2dxΔu¯,−B∫Ω∇v12dxΔv1=λu0δv0β≥λu¯δv¯β≥−B∫Ω∇v¯2dxΔv¯,from which, according to Lemma 1, u1≥u¯ and v1≥v¯. For u2,v2, we write(48)−A∫Ω∇u12dxΔu1=λu0αv0γ≥λu1αv1γ=−A∫Ω∇u22dxΔu2,−B∫Ω∇v1dxΔv1=λu0δv0β≥λu1δv1β=−B∫Ω∇v22dxΔv2,and then u1≥u2 and v1≥v2. Similarly, u2≥u¯ and v2≥v¯ because(49)−A∫Ω∇u22dxΔu2=λu1αv1γ≥λu¯1αv¯1γ≥−A∫Ω∇u¯2dxΔu¯,−B∫Ω∇v22dxΔv2=λu1δv1β≥λu¯1δv¯1β≥−B∫Ω∇v¯2dxΔv¯.
Repeating this argument, we get a bounded monotone sequenceun,vn⊂H01Ω×H01Ω satisfying(50)u¯=u0≥u1≥u2≥…≥un≥…≥u¯>0,(51)v¯=v0≥v1≥v2≥…≥vn≥…≥v¯>0.
Using the continuity of the functionsf and g and the definition of the sequence un,vn, there exist constants Ci>0, i=1,…4, independent of n such that(52)fun−1,vn−1≤C1,gun−1,vn−1≤C2,for alln.
From (52), we multiply the first equation of (41) by un; in addition, by using the Holder inequality combined with Sobolev embedding, we have(53)a1∫Ω∇un2dx≤A∫Ω∇un2dx∫Ω∇un2dx=λ∫Ωfun−1,vn−1undx≤λ∫Ωfun−1,vn−1undx≤C1λ∫Ωun21/2dx≤C3unH01ΩorunH01Ω≤C3,∀n,where C3>0 is a constant independent of n. Similarly, there exists C2>0 independent of n such that(54)vnH01Ω≤C4,∀n.
From (53) and (54), we deduce that the couple un,vn converges weakly in H01Ω,ℝ2 to the couple u,vwithu≥u¯>0 and v≥v>0.
By using a standard regularity argument,un,vn converges to u,v. Thus, when n⟶+∞ in (41), we can see that u,v is a positive solution of system (1).
The proof is completed.


## 5. Conclusion

As a conclusion of this contribution, we have proved the existence of positive solutions of quasi-linear Kirchhoff elliptic systems in bounded smooth domains by using the sub- and super-solution method [20], which is an extension of our recent works of Boulaaras et al. in [18]. In the next work, some other methods such as variational and Galerkin methods (see, for example, [15]) will be used for this problem, and some numerical examples will also be given [9, 22].



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