Research ArticleANewProof of Existence of PositiveWeak Solutions for SublinearKirchhoff Elliptic Systems with Multiple Parameters

Salah Mahmoud Boulaaras 12 Rafik Guefaifia 3 Bahri Cherif 1

and Sultan Alodhaibi 1

1Department of Mathematics College of Sciences and Arts in Al-Rass Qassim University Buraidah Saudi Arabia2Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO) University of Oran 1 Ahmed Benbella Algeria3Department of Mathematics Faculty of Exact Sciences University of Tebessa Tebessa 12002 Algeria

Correspondence should be addressed to Salah Mahmoud Boulaaras sboularasquedusa

Received 8 November 2019 Accepted 19 December 2019 Published 25 January 2020

Guest Editor Sundarapandian Vaidyanathan

Copyright copy 2020 SalahMahmoud Boulaaras et alis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper deals with the study of the existence of weak positive solutions for sublinear Kirchhoff elliptic systems with zeroDirichlet boundary condition in bounded domain Ω sub RN by using the subsuper solutions method

1 Introduction

In this paper we consider the following system of differentialequations

minus A 1113946Ω

|nablau|2dx1113874 1113875Δu λ1ua + μ1vb inΩ

minus B 1113946Ω

|nablau|2dx1113874 1113875Δv λ2uc + μ2vd inΩ

u v 0 on zΩ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

where Ω sub RN (Nge 3) is a bounded smooth domain withC2 boundary zΩ A B R+⟶ R+ are continuous func-tions and λ1 λ2 μ1 and μ2 are positive parameters wherea + clt 1 and b + dlt 1 e peculiarity of this type ofproblem and by far the most important is that it is not localis is based on the presence of the operator

minus A 1113946Ω

|nablau|2dx1113874 1113875Δu respectively minus B 1113946

Ω|nablau|

2dx1113874 1113875Δu1113874 1113875

(2)

which contains an integral on all the fields and impliesthat the equation is not a specific identity It is clear thatthese problems contribute to the transition from aca-demia to application Indeed very popular for its physical

motivations problem (1) is none other than a stationaryversion of the following model which regulates the be-havior of elastic whose ends are fixed and which issubjected to nonlinear vibrations

utt minus M 1113946Ω

|nablau|2dx1113874 1113875Δu h(x u) inΩ times(0 T)

u 0 in zΩ times(0 T)

u(x 0) u0(x) ut(x 0) u1(x)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)

where T is a positive constant and u0 and u1 are givenfunctions In such problems u expresses the displacementh(x u) the extreme force M(r) a1r + b1 b1 the initialtension and a1 relates to the intrinsic properties of the wirematerial (such as Youngrsquos modulus) For more details see[1] as well as their references Basically this is a general-ization to larger dimensions of the model originally pro-posed in one dimension by Kirchhoff [2] in (1883)

z2u

zt2minus ρ0 + ρ1 1113946

L

0

zu

zx

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

2dx1113888 1113889

zu

zx 0 (4)

where ρ0 is the initial tension ρ1 represents Youngrsquosmodulus of the material of the wire and L its length e

HindawiComplexityVolume 2020 Article ID 1924085 6 pageshttpsdoiorg10115520201924085

latter is known to be an extension of the equation ofDrsquoAlembert waves Indeed Kirchhoff took into account thechanges caused by transverse oscillations along the length ofthe wire With their implications in other disciplines andgiven the breadth of their fields of application nonlocalproblems will be used to model several physical phenomenaand they also intervene in biological systems or describe aprocess dependent on its average such as particle densitypopulation Moreover With this significant impactstrengthening the field of applications this type of problemhas caught the interest of mathematicians and a lot of work onthe existence of solutions has emerged particularly after thecoup de force provided by the famous Lions article [3] wherethe latter has adopted an approach based on functionalanalysis Nevertheless in most of these articles the benefitmethod is purely topological It is only in the last decades thatthis approach has been removed from variational methodswhen Alves and his colleagues [4] obtained for the first timethe results of their existence through these methods Sincethen a very fruitful development has given rise tomany worksbased on this advantageous axis (see [1 3 5])

In recent years problems relating to Kirchhoff oper-ators have been studied in several papers (we refer to [6])where the authors used different methods to obtain so-lutions (1) in the case of single equation (see [6]) econcept of weak sub- and supersolutions was first for-mulated by Hess and Deuel in [7 8] to obtain existenceresults for weak solutions of semilinear elliptic Dirichletproblems and was subsequently continued by severalauthors (see eg [9ndash18])

In our recent paper [19] we have discussed the existenceof weak positive solution for the following Kirchhoff ellipticsystems

minus A 1113946Ω

|nablau|2dx1113874 1113875Δu λprimef(v) + μ1primeh(u)inΩ

minus B 1113946Ω

|nablau|2dx1113874 1113875Δv λ2primeg(u) + μ2primeτ(v)inΩ

u v 0 on zΩ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)

Motivated by the ideas of [20] which the authorsconsidered a system (1) in the case A(t) B(t) 1 moreprecisely under suitable conditions on f g we will provethat the problem which is defined in (1) admits a positivesolution In current paper motivated by previous works in([19 20]) we discuss the existence of weak positive solutionfor sublinear Kirchhoff elliptic systems in bounded domainsby using subsupersolutions method combined with com-parison principle (see Lemma 21 in [4])

e outline of the paper is as follows In the secondsection we give some assumptions and definitions related toproblem (1) In Section 3 we prove our main result

2 Assumptions and Definitions

Let us assume the following assumption(H1) Assume that A B R+⟶ R+ are two continuous

and increasing functions and there exists ai bi gt 0 i 1 2

such that

a1 leA(t)le a2

b1 leB(t)le b2

for all t isin R+

(6)

(H2) Suppose that a dge 0 b cgt 0 a + clt 1 andb + dlt 1

Now in order to discuss our main result of problem (1)we need the following two definitions

Definition 1 Let (u v) isin (H10(Ω) times H1

0(Ω)) (u v) is said tobe a weak solution of (1) if it satisfies

A u2

1113872 11138731113946Ωnablaunablaϕdx λ11113946

Ωu

aϕdx + μ1vbϕdx inΩ

B v2

1113872 11138731113946Ωnablavnablaψdx λ21113946

Ωu

cψdx + μ21113946Ω

vdψdx inΩ

(7)

for all (ϕψ) isin (H10(Ω) times H1

0(Ω))

Definition 2 A pair of nonnegative functions (u v) (u v) in(H1

0(Ω) times H10(Ω)) is called a weak subsolution and super-

solution of (1) if they satisfy (u v) (u v) (0 0) on zΩ

A u2

1113872 11138731113946Ωnablaunablaϕdxle λ11113946

Ωuaϕdx + μ11113946

Ωvbϕdx inΩ

B v2

1113872 11138731113946Ωnabla vnablaψdxle λ21113946

Ωucψ dx + μ21113946

Ωvdψdx in Ω

A u2

1113872 11138731113946Ωnablaunablaϕdxge λ11113946

Ωu

aϕdx + μ11113946Ω

vbϕdx inΩ

B v2

1113872 11138731113946Ωnablavnablaψdxge λ21113946

Ωu

cψdx + μ21113946Ω

vdψdx inΩ

(8)

for all (ϕψ) isin (H10(Ω) times H1

0(Ω))

Lemma 1 [4] Assume that M R+⟶ R+ is a continuousand nonincreasing function satisfying

M(s)gtm0 for all sge s0 (9)

where m0 is a positive constant and assume that u v are twononnegative functions such that

minus M u21113872 1113873Δuge minus M v21113872 1113873Δv inΩ

u v 0 on zΩ

⎧⎨

⎩ (10)

and then uge v ae in Ω

3 Main Result

In this section we shall state and prove the main result ofthis paper

Theorem 1 Suppose that (H1)-(H2) hold and M is anonincreasing function satisfying (9) 8en problem (1) has alarge positive weak solution for each positive parametersλ1 λ2 μ1 and μ2

2 Complexity

Proof of 8eorem 1 Let σ be the first eigenvalue of minus Δ withDirichlet boundary conditions and ϕ1 the correspondingeigenfunction with ϕ1 1 satisfying ϕ1 gt 0 in Ω and|nablaϕ1|gt 0 on zΩ

Since bclt (1 minus a)(1 minus d) we can take k such thatc

1 minus dlt klt

b

1 minus a (11)

We shall verify that (u v) (εϕ21 εkϕ21) is a subsolution

of problem (1) where εgt 0 is small and specified laterA simple calculation

A u

21113872 11138731113946Ωnabla u middotnablaϕdx 2εA u

21113872 11138731113946Ωϕ1nablaϕ1 middot nablaϕdx

2εA u

21113872 1113873 times 1113946Ωnablaϕ1nabla ϕ1 middot ϕ( 1113857dx1113882

minus 1113946Ωnablaϕ1

111386811138681113868111386811138681113868111386811138682ϕdx1113883

2εA u

21113872 11138731113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

le 2a2ε1113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

(12)

Similarly

B v

21113872 11138731113946Ωnabla v middotnablaψdx 2εk

B v

21113872 11138731113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

le 2b2εk1113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

(13)

Let ηgt 0 μgt 0 be such that

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682 le 0

x isin Ωη(14)

and μleϕ1 le 1 on ΩΩη where Ωη x isin Ω d(x zΩ)le η1113864 1113865

We have from (14) that

A 1113946Ωηnabla u

1113868111386811138681113868111386811138682dx1113888 11138891113946

Ωηnabla u middotnablaϕdxle 0le λ11113946

Ωηuaϕdx + μ11113946

Ωηvbϕdx

(15)

B 1113946Ωηnabla v

1113868111386811138681113868111386811138682dx1113888 11138891113946

Ωηnabla v middotnablaψdxle 0le λ21113946

Ωηucψdx + μ21113946

Ωηvdψdx

(16)

On the other hand in ΩΩη let

r1 1 minus a

c

r2 1 minus a

1 minus a minus c

s1 1 minus d

b

s2 1 minus d

1 minus d minus b

(17)

Note that1r1

+1r2

1

1s1

+1s2

1

(18)

We have from (11) that

1 minusa

r1minus

kb

r2ge 1 minus a minus kbgt 0

k 1 minusd

s21113888 1113889 minus

c

s1ge k(1 minus d) minus cgt 0

(19)

us we choose εgt 0 such that

2a2ε1minus ar1( )minus kbr2( )σϕ21 le λ

1r11 μ1r21 μ2+aδ

x isin ΩΩη

2b2εk 1minus ds2( )( )minus cs1( )σϕ21 le λ

1s12 μ1s22 μ2+c d

x isin ΩΩη

(20)

where δ 2(1 minus a) c 2(1 minus d) Furthermore

aδr1 2a

1 minus a minus cge 2a

cds2 2 d

1 minus d minus bge 2 d

2s1 21 minus d

b1113888 1113889gt 2

c

1 minus a1113874 1113875ge 2c

2r2 21 minus a

c1113874 1113875gt 2

b

1 minus d1113888 1113889ge 2b

(21)

ese relations and Young inequality show that

2a2ε1113946ΩΩη

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdxle 2a2ε1113946ΩΩη

σϕ21 middot ϕdx

le1113946ΩΩη

λ1r11 εar1μaδ1113872 1113873 μ1r21 εkbr2μ21113872 1113873ϕdx

le1113946ΩΩη

λ1r11 εar1μaδ1113872 1113873r1

r1+

μ1r21 εkbr2μ21113872 1113873r2

r2

⎡⎢⎢⎣ ⎤⎥⎥⎦ϕdx

le1113946ΩΩη

λ1r11 εar1μaδ1113872 1113873

r1+ μ1r21 εkbr2μ21113872 1113873

r21113876 1113877ϕdx

1113946ΩΩη

λ1εaμaδr1 + μ1ε

kbμ2r21113872 1113873ϕdx

le1113946ΩΩη

λ1εaϕ2a

1 + μ1εkbϕ2b

11113872 1113873ϕdx

1113946ΩΩη

λ1ua

+ μ1vb

1113888 1113889ϕ dx

(22)

Complexity 3

2b2εk1113946ΩΩη

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ψ dxle 2b2εk1113946ΩΩη

σϕ21 middot ψdx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873 μ1s22 εk ds2μc d1113872 1113873ψdx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873s1

s1+

μ1s22 εk ds2μc d1113872 1113873s2

s2

⎡⎢⎢⎣ ⎤⎥⎥⎦ψ dx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873s1

+ μ1s22 εk ds2μc d1113872 1113873

s21113876 1113877ψdx

1113946ΩΩη

λ2εcμ2s1 + μ2ε

k dμc ds21113872 1113873ψdx

le1113946ΩΩη

λ2εcμ2c

+ μ2εk dμ2 d

1113872 1113873ψdx

le1113946ΩΩη

λ2εcϕ2c

1 + μ2εk dϕ2d

11113872 1113873ψdx

1113946ΩΩη

λ2uc

+ μ2vd

1113888 1113889ψdx

(23)

Hence from (15)ndash(23) it follows that

A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875 1113946

Ωηnabla unablaϕdx + 1113946

ΩΩηnabla unablaϕdx1113890 1113891

A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnabla unablaϕdxle λ1u

a+ μ1v

b1113872 1113873ϕdx

(24)

B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875 1113946

Ωηnabla vnablaψdx + 1113946

ΩΩηnabla vnablaψdx1113890 1113891

B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnabla vnablaψdxle λ2u

c+ μ2v

d1113872 1113873ψ dx

(25)

en 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

minus Δe 1 inΩ

e 0 on zΩ1113896 (26)

Let

u C1e

v C2e(27)

where e is given by (26) and C1 C2 gt 0 are large positivereal numbers to be chosen later We shall verify that (u v)

is a supersolution of problem (1) Let ϕ isin H10(Ω) with ϕge 0

in Ω en we obtain from (26) and the condition (H1)

that

A 1113946Ω

|nablau|2dx1113874 11138751113946

Ωnablau middot nablaϕdx A 1113946

Ω|nablau|

2dx1113874 1113875C1 1113946

Ω

nablae middot nablaϕdx

A 1113946Ω

|nablau|2dx1113874 1113875C11113946

Ωϕdx

ge a1C11113946Ωϕdx

B 1113946Ω

|nablav|2dx1113874 11138751113946

Ωnablav middot nablaψdx B 1113946

Ω|nablav|

2dx1113874 1113875C21113946Ωnablae middot nablaψdx

B 1113946Ω

|nablav|2dx1113874 1113875C21113946

Ωψdx

ge b1C21113946Ωψdx

(28)

Let l einfin Since alt 1 dlt 1 these imply that thereexist positive large constants α a1C1 β b1C2 such that

αge λ1(αl)a

+ μ1(βl)b

βge λ2(αl)c

+ μ2(βl)d

(29)

us

a1C11113946Ωϕdxge1113946

Ωλ1u

a+ μ1v

b1113872 1113873ϕdx (30)

b1C21113946Ωψ dxge1113946

Ωλ2u

c+ μ2v

d1113872 1113873ϕdx (31)

From (26) and (30) we can deduce that the couple (u v)

is a subsolution of problem (1) with u le u and v le v for C1

C2 largeIn order to obtain a weak solution of problem (1) we shall

use the arguments by Azouz and Bensedik [19] For thispurpose we define a sequence (un vn)1113864 1113865 sub (H1

0(Ω) times

H10(Ω)) as follows u0 ≔ u v0 v and (un vn) is the unique

solution of the system

minus A 1113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δun λ1ua

nminus 1 + μ1vbnminus 1 inΩ

minus B 1113946Ωnablavn

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δvn λ2uc

nminus 1 + μ2vdnminus 1 inΩ

un vn 0 on zΩ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Problem (32) is (A B)minus linear in the sense that if(unminus 1 vnminus 1) isin (H1

0(Ω) times H10(Ω)) is given the right hand

sides of (32) are independent of un vn

Set A(t) tA(t2) B(t) tB(t2) en since A(R) R

B(R) R f(unminus 1) uanminus 1 h(vnminus 1) vb

nminus 1 g(unminus 1) ucnminus 1

and τ(vnminus 1) vdnminus 1 isin L2(Ω)

We deduce from a result in [4] that system (32) has aunique solution (un vn) isin (H1

0(Ω) times H10(Ω))

By using (32) and the fact that (u0 v0) is a supersolutionof (1) we have

minus A 1113946Ωnablau0

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu0 ge λ1ua

0 + μ1vb0 minus A 1113946

Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1

minus B 1113946Ωnablav0

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv0 ge λ2uc

0 + μ2vd0 minus B 1113946

Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(33)

4 Complexity

and by Lemma 1 u0 ge u1 and v0 ge v1 Also since u0 ge uv0 ge v and the monotonicity of f h g and τ one has

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(34)

from which rding to Lemma 1 u1 ge u v1 ge v for u2 v2 wewrite

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a1 + μ1v

b1

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2

minus B 1113946Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c1 + μ2v

d1

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2

(35)

and then u1 ge u2 v1 ge v2 Similarly u2 ge u and v2 ge v because

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2 λ2u

c1 + μ2v

d1 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(36)

Repeating this argument we get a bounded monotonesequence (un vn)1113864 1113865 sub (H1

0(Ω) times H10(Ω)) satisfying

u u0 ge u1 ge u2 ge middot middot middot ge un ge middot middot middot ge u gt 0 (37)

v v0 ge v1 ge v2 ge middot middot middot ge vn ge middot middot middot ge v gt 0 (38)

Using the continuity of the functions f h g and t andthe definition of the sequences un1113864 1113865 vn1113864 1113865there exist con-stants Ci gt 0 i 1 4 independent of n such that

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC1

h unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 leC2

g unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC3

(39)

τ unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC4 for all n (40)

From (39) multiplying the first equation of (32) by un

and integrating using the Holder inequality and Sobolevembedding we can show that

a11113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dxleA 1113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx

λ1f vnminus 1( 1113857undx + μ11113946Ω

h unminus 1( 1113857undx

le λ11113946Ω

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx + μ11113946

Ωh unminus 1( 1113857

11138681113868111386811138681113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx

leC1λ11113946Ω

un

11138681113868111386811138681113868111386811138681113868dx + C2μ11113946

Ωun

11138681113868111386811138681113868111386811138681113868dx

leC5 un

H1

0(Ω)

(41)

or

un

H1

0(Ω)leC5 foralln (42)

where C5 gt 0 is a constant independent of n Similarly thereexists C6 gt 0 independent of n such that

vn

H1

0(Ω)leC6 foralln (43)

From (42) and (43) we infer that (un vn)1113864 1113865 has a sub-sequence which weakly converges in H1

0(ΩR2) to a limit(u v) with the properties uge ugt 0 and vge v gt 0 Beingmonotone and also using a standard regularity argument(un vn)1113864 1113865 converges itself to (u v) Now letting n⟶ +infinin (32) we deduce that (u v) is a positive solution of system(1) e proof of theorem is now completed

4 Conclusion

In this work we study the existence of weak positive so-lutions for a sublinear Kirchhoff elliptic systems in boundeddomains by using the subsuper solutions method (SSM)combined with comparison principle which have beenwidely applied in many work (see for example[4 19 21ndash25])Validity of the comparison principle and ofthe SSM for local and nonlocal problems as the stationaryKirchhoff Equation was an important subject in the last fewyears (see for example [26] and [23] Moreover the twoconditions that M is nonincreasing and H is increasing turnout to be necessary and sufficient at least for the validity ofthe comparison principle It is worth to notice that in [4]Alves and Correa developed a new SSM for problem (1) todeal with the increasing M case e result is obtained byusing a kind of MintyndashBrowder theorem for a suitablepseudomonotone operator but instead of constructing asubsolution the authors assumed the existence of a wholefamily of functions which satisfy a stronger condition thanjust being subsolutions the inconvenience is that thesestronger conditions restrict the possible right hand sides in(1) Another SSM for nonlocal problems is obtained in [4] fora problem involving a nonlocal term with a Lebesgue norminstead of the Sobolev norm appearing in (1) In our nextstudy we will try to apply an alternative approach using thevariational principle which has been presented in [27ndash29]

Complexity 5

Data Availability

e data used to support the findings of the study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

All authors contributed equally to this article ey have allread and approved the final manuscript

References

[1] P DrsquoAncona and S Spagnolo ldquoGlobal solvability for thedegenerate Kirchhoff equation with real analytic datardquoInventiones Mathematicae vol 108 pp 247ndash262 1992

[2] H Medekhel S Boulaaras K Zennir and A AllahemldquoExistence of positive solutions and its asymptotic behavior of(p(x) q(x))-Laplacian parabolic systemrdquo Symmetry vol 11no 3 p 332 2019

[3] J L Lions ldquoOn some questions in boundary value problems ofmathematical physicsrdquo Contemporary Developments inContinuum Mechanics and Partial Differential EquationsProceedings of the International Symposium on ContinuumMechanics and Partial Differential Equations vol 30pp 284ndash346 1978

[4] C O Alves and F J S A Correa ldquoOn existence of solutionsfor a class of problem involving a nonlinear operatorrdquoCommunications on Applied Nonlinear Analysis vol 8pp 43ndash56 2001

[5] Y Li F Li and J Shi ldquoExistence of positive solutions toKirchhoff type problems with zero massrdquo Journal of Math-ematical Analysis and Applications vol 410 no 1 pp 361ndash374 2014

[6] B Ricceri ldquoOn an elliptic Kirchhoff-type problem dependingon two parametersrdquo Journal of Global Optimization vol 46no 4 pp 543ndash549 2010

[7] J Deuel and P Hess ldquoInequations variationelless elliptiquesnon coercivesrdquo C R Acad Sci Paris vol 279 pp 719ndash7221974

[8] P Hess ldquoOn the solvability of nonlinear elliptic boundaryvalue problemsrdquo Indiana University Mathematics Journalvol 25 no 5 pp 461ndash466 1976

[9] S Carl V K Le and D Motreanu ldquoe sub-supersolutionmethod and extremal solutions for quasilinear hemivaria-tional inequalitiesrdquo Differential and Integral Equationsvol 17 pp 165ndash178 2004

[10] E N Dancer and G Sweers ldquoOn the existence of a maximalweak solution for a semilinear elliptic equationrdquo Differentialand Integral Equations vol 2 pp 533ndash540 1989

[11] T Kura ldquoe weak supersolution-subsolution method forsecond order quasilinear elliptic equationsrdquo HiroshimaMathematical Journal vol 19 p 136 1989

[12] B Mairi R Guefaifia S Boulaaras and T Bouali ldquoExistenceof positive solutions for a new class of nonlocal p(x)-Kirchhoffelliptic systems via sub-super solutions conceptrdquo AppliedScience APPS vol 20 pp 117ndash128 2018

[13] S Boulaaras R Guefaifia and K Zennir ldquoExistence ofpositive solutions for nonlocal p(x)-Kirchhoff elliptic

systemsrdquo Advances in Pure and Applied Mathematics vol 10no 1 pp 1867ndash1152 2019

[14] V K Le and K Schmitt ldquoSome general concepts of sub- andsupersolutions for nonlinear elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 28 pp 87ndash103 2006

[15] N Mezouar and S Boulaaras ldquoGlobal existence and decay ofsolutions for a class of viscoelastic Kirchhoff equationrdquoBulletin of the Malaysian Mathematical Sciences Societyvol 43 no 1 pp 725ndash755 2018

[16] V K Le ldquoSubsolution-supersolution method in variationalinequalitiesrdquo Nonlinear Analysis vol 45 pp 775ndash800 2001

[17] V K Le ldquoSubsolution-supersolutions and the existence ofextremal solutions in noncoercive variational inequalitiesrdquoJournal of Inequalities in Pure and Applied Mathematicsvol 2 p 116 2001

[18] S Boulaaras D Ouchenane and F Mesloub ldquoGeneral decayfor a viscoelastic problem with not necessarily decreasingkernelrdquo Applicable Analysis vol 98 no 9 pp 1677ndash16932019

[19] N Azouz and A Bensedik ldquoExistence result for an ellipticequation of Kirchhoff-type with changing sign datardquo Funk-cialaj Ekvacioj vol 55 no 1 pp 55ndash66 2012

[20] D D Hai and R Shivaji ldquoAn existence result on positivesolutions for a class of p-Laplacian systemsrdquo NonlinearAnalysis 8eory Methods amp Applications vol 56 no 7pp 1007ndash1010 2004

[21] S Boulaaras and R Guefaifia ldquoExistence of positive weaksolutions for a class of Kirrchoff elliptic systems with multipleparametersrdquo Mathematical Methods in the Applied Sciencesvol 41 no 13 pp 5203ndash5210 2018

[22] S Boulaaras R Guefaifia and S Kabli ldquoAn asymptotic be-havior of positive solutions for a new class of elliptic systemsinvolving of $$left(pleft(xright) qleft(xright) right) $$ px q x -Laplacian systemsrdquo Boletın de la Sociedad MatematicaMexicana vol 25 no 1 pp 145ndash162 2019

[23] F J S A Correa and G M Figueiredo ldquoOn an ellipticequation of pminus Kirchhoff type via variational methodsrdquoBulletin of the Australian Mathematical Society vol 74 no 2pp 263ndash277 2006

[24] R Guefaifia and S Boulaaras ldquoExistence of positive solutionfor a class of (p(x) q(x))-Laplacian systemsrdquo Rendiconti delCircolo Matematico di Palermo Series 2 vol 67 pp 93ndash1032018

[25] S Boulaaras and A Allahem ldquoExistence of positive solutionsof nonlocal p(x)-Kirchhoff evolutionary systems via sub-supersolutions conceptrdquo Symmetry vol 11 no 2 p 253 2019

[26] M Chipot and B Lovat ldquoSome remarks on non local ellipticand parabolic problemsrdquo Nonlinear Analysis 8eoryMethods amp Applications vol 30 no 7 pp 4619ndash4627 1997

[27] S Boulaaras ldquoSome existence results for elliptic Kirchhoffequation with changing sign data and a logarithmic non-linearityrdquo Journal of Intelligent and Fuzzy Systems vol 37no 6 pp 1ndash10 2019

[28] S Boulaaras ldquoA well-posedness and exponential decay ofsolutions for a coupled lame system with viscoelastic term andlogarithmic source termsrdquoApplicable Analysis pp 1ndash19 2019

[29] J-H He ldquoVariational principle for the generalized KdV-burgers equation with fractal derivatives for shallow waterwavesrdquo Journal of Applied and Computational Mechanicsvol 6 no 4 httpjacmscuacirarticle_14813html 2020

6 Complexity

Proof of 8eorem 1 Let σ be the first eigenvalue of minus Δ withDirichlet boundary conditions and ϕ1 the correspondingeigenfunction with ϕ1 1 satisfying ϕ1 gt 0 in Ω and|nablaϕ1|gt 0 on zΩ

Since bclt (1 minus a)(1 minus d) we can take k such thatc

1 minus dlt klt

b

1 minus a (11)

We shall verify that (u v) (εϕ21 εkϕ21) is a subsolution

of problem (1) where εgt 0 is small and specified laterA simple calculation

A u

21113872 11138731113946Ωnabla u middotnablaϕdx 2εA u

21113872 11138731113946Ωϕ1nablaϕ1 middot nablaϕdx

2εA u

21113872 1113873 times 1113946Ωnablaϕ1nabla ϕ1 middot ϕ( 1113857dx1113882

minus 1113946Ωnablaϕ1

111386811138681113868111386811138681113868111386811138682ϕdx1113883

2εA u

21113872 11138731113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

le 2a2ε1113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

(12)

Similarly

B v

21113872 11138731113946Ωnabla v middotnablaψdx 2εk

B v

21113872 11138731113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

le 2b2εk1113946Ω

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdx

(13)

Let ηgt 0 μgt 0 be such that

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682 le 0

x isin Ωη(14)

and μleϕ1 le 1 on ΩΩη where Ωη x isin Ω d(x zΩ)le η1113864 1113865

We have from (14) that

A 1113946Ωηnabla u

1113868111386811138681113868111386811138682dx1113888 11138891113946

Ωηnabla u middotnablaϕdxle 0le λ11113946

Ωηuaϕdx + μ11113946

Ωηvbϕdx

(15)

B 1113946Ωηnabla v

1113868111386811138681113868111386811138682dx1113888 11138891113946

Ωηnabla v middotnablaψdxle 0le λ21113946

Ωηucψdx + μ21113946

Ωηvdψdx

(16)

On the other hand in ΩΩη let

r1 1 minus a

c

r2 1 minus a

1 minus a minus c

s1 1 minus d

b

s2 1 minus d

1 minus d minus b

(17)

Note that1r1

+1r2

1

1s1

+1s2

1

(18)

We have from (11) that

1 minusa

r1minus

kb

r2ge 1 minus a minus kbgt 0

k 1 minusd

s21113888 1113889 minus

c

s1ge k(1 minus d) minus cgt 0

(19)

us we choose εgt 0 such that

2a2ε1minus ar1( )minus kbr2( )σϕ21 le λ

1r11 μ1r21 μ2+aδ

x isin ΩΩη

2b2εk 1minus ds2( )( )minus cs1( )σϕ21 le λ

1s12 μ1s22 μ2+c d

x isin ΩΩη

(20)

where δ 2(1 minus a) c 2(1 minus d) Furthermore

aδr1 2a

1 minus a minus cge 2a

cds2 2 d

1 minus d minus bge 2 d

2s1 21 minus d

b1113888 1113889gt 2

c

1 minus a1113874 1113875ge 2c

2r2 21 minus a

c1113874 1113875gt 2

b

1 minus d1113888 1113889ge 2b

(21)

ese relations and Young inequality show that

2a2ε1113946ΩΩη

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ϕdxle 2a2ε1113946ΩΩη

σϕ21 middot ϕdx

le1113946ΩΩη

λ1r11 εar1μaδ1113872 1113873 μ1r21 εkbr2μ21113872 1113873ϕdx

le1113946ΩΩη

λ1r11 εar1μaδ1113872 1113873r1

r1+

μ1r21 εkbr2μ21113872 1113873r2

r2

⎡⎢⎢⎣ ⎤⎥⎥⎦ϕdx

le1113946ΩΩη

λ1r11 εar1μaδ1113872 1113873

r1+ μ1r21 εkbr2μ21113872 1113873

r21113876 1113877ϕdx

1113946ΩΩη

λ1εaμaδr1 + μ1ε

kbμ2r21113872 1113873ϕdx

le1113946ΩΩη

λ1εaϕ2a

1 + μ1εkbϕ2b

11113872 1113873ϕdx

1113946ΩΩη

λ1ua

+ μ1vb

1113888 1113889ϕ dx

(22)

Complexity 3

2b2εk1113946ΩΩη

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ψ dxle 2b2εk1113946ΩΩη

σϕ21 middot ψdx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873 μ1s22 εk ds2μc d1113872 1113873ψdx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873s1

s1+

μ1s22 εk ds2μc d1113872 1113873s2

s2

⎡⎢⎢⎣ ⎤⎥⎥⎦ψ dx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873s1

+ μ1s22 εk ds2μc d1113872 1113873

s21113876 1113877ψdx

1113946ΩΩη

λ2εcμ2s1 + μ2ε

k dμc ds21113872 1113873ψdx

le1113946ΩΩη

λ2εcμ2c

+ μ2εk dμ2 d

1113872 1113873ψdx

le1113946ΩΩη

λ2εcϕ2c

1 + μ2εk dϕ2d

11113872 1113873ψdx

1113946ΩΩη

λ2uc

+ μ2vd

1113888 1113889ψdx

(23)

Hence from (15)ndash(23) it follows that

A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875 1113946

Ωηnabla unablaϕdx + 1113946

ΩΩηnabla unablaϕdx1113890 1113891

A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnabla unablaϕdxle λ1u

a+ μ1v

b1113872 1113873ϕdx

(24)

B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875 1113946

Ωηnabla vnablaψdx + 1113946

ΩΩηnabla vnablaψdx1113890 1113891

B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnabla vnablaψdxle λ2u

c+ μ2v

d1113872 1113873ψ dx

(25)

en 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

minus Δe 1 inΩ

e 0 on zΩ1113896 (26)

Let

u C1e

v C2e(27)

where e is given by (26) and C1 C2 gt 0 are large positivereal numbers to be chosen later We shall verify that (u v)

is a supersolution of problem (1) Let ϕ isin H10(Ω) with ϕge 0

in Ω en we obtain from (26) and the condition (H1)

that

A 1113946Ω

|nablau|2dx1113874 11138751113946

Ωnablau middot nablaϕdx A 1113946

Ω|nablau|

2dx1113874 1113875C1 1113946

Ω

nablae middot nablaϕdx

A 1113946Ω

|nablau|2dx1113874 1113875C11113946

Ωϕdx

ge a1C11113946Ωϕdx

B 1113946Ω

|nablav|2dx1113874 11138751113946

Ωnablav middot nablaψdx B 1113946

Ω|nablav|

2dx1113874 1113875C21113946Ωnablae middot nablaψdx

B 1113946Ω

|nablav|2dx1113874 1113875C21113946

Ωψdx

ge b1C21113946Ωψdx

(28)

Let l einfin Since alt 1 dlt 1 these imply that thereexist positive large constants α a1C1 β b1C2 such that

αge λ1(αl)a

+ μ1(βl)b

βge λ2(αl)c

+ μ2(βl)d

(29)

us

a1C11113946Ωϕdxge1113946

Ωλ1u

a+ μ1v

b1113872 1113873ϕdx (30)

b1C21113946Ωψ dxge1113946

Ωλ2u

c+ μ2v

d1113872 1113873ϕdx (31)

From (26) and (30) we can deduce that the couple (u v)

is a subsolution of problem (1) with u le u and v le v for C1

C2 largeIn order to obtain a weak solution of problem (1) we shall

use the arguments by Azouz and Bensedik [19] For thispurpose we define a sequence (un vn)1113864 1113865 sub (H1

0(Ω) times

H10(Ω)) as follows u0 ≔ u v0 v and (un vn) is the unique

solution of the system

minus A 1113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δun λ1ua

nminus 1 + μ1vbnminus 1 inΩ

minus B 1113946Ωnablavn

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δvn λ2uc

nminus 1 + μ2vdnminus 1 inΩ

un vn 0 on zΩ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Problem (32) is (A B)minus linear in the sense that if(unminus 1 vnminus 1) isin (H1

0(Ω) times H10(Ω)) is given the right hand

sides of (32) are independent of un vn

Set A(t) tA(t2) B(t) tB(t2) en since A(R) R

B(R) R f(unminus 1) uanminus 1 h(vnminus 1) vb

nminus 1 g(unminus 1) ucnminus 1

and τ(vnminus 1) vdnminus 1 isin L2(Ω)

We deduce from a result in [4] that system (32) has aunique solution (un vn) isin (H1

0(Ω) times H10(Ω))

By using (32) and the fact that (u0 v0) is a supersolutionof (1) we have

minus A 1113946Ωnablau0

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu0 ge λ1ua

0 + μ1vb0 minus A 1113946

Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1

minus B 1113946Ωnablav0

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv0 ge λ2uc

0 + μ2vd0 minus B 1113946

Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(33)

4 Complexity

and by Lemma 1 u0 ge u1 and v0 ge v1 Also since u0 ge uv0 ge v and the monotonicity of f h g and τ one has

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(34)

from which rding to Lemma 1 u1 ge u v1 ge v for u2 v2 wewrite

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a1 + μ1v

b1

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2

minus B 1113946Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c1 + μ2v

d1

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2

(35)

and then u1 ge u2 v1 ge v2 Similarly u2 ge u and v2 ge v because

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2 λ2u

c1 + μ2v

d1 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(36)

Repeating this argument we get a bounded monotonesequence (un vn)1113864 1113865 sub (H1

0(Ω) times H10(Ω)) satisfying

u u0 ge u1 ge u2 ge middot middot middot ge un ge middot middot middot ge u gt 0 (37)

v v0 ge v1 ge v2 ge middot middot middot ge vn ge middot middot middot ge v gt 0 (38)

Using the continuity of the functions f h g and t andthe definition of the sequences un1113864 1113865 vn1113864 1113865there exist con-stants Ci gt 0 i 1 4 independent of n such that

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC1

h unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 leC2

g unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC3

(39)

τ unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC4 for all n (40)

From (39) multiplying the first equation of (32) by un

and integrating using the Holder inequality and Sobolevembedding we can show that

a11113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dxleA 1113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx

λ1f vnminus 1( 1113857undx + μ11113946Ω

h unminus 1( 1113857undx

le λ11113946Ω

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx + μ11113946

Ωh unminus 1( 1113857

11138681113868111386811138681113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx

leC1λ11113946Ω

un

11138681113868111386811138681113868111386811138681113868dx + C2μ11113946

Ωun

11138681113868111386811138681113868111386811138681113868dx

leC5 un

H1

0(Ω)

(41)

or

un

H1

0(Ω)leC5 foralln (42)

where C5 gt 0 is a constant independent of n Similarly thereexists C6 gt 0 independent of n such that

vn

H1

0(Ω)leC6 foralln (43)

From (42) and (43) we infer that (un vn)1113864 1113865 has a sub-sequence which weakly converges in H1

0(ΩR2) to a limit(u v) with the properties uge ugt 0 and vge v gt 0 Beingmonotone and also using a standard regularity argument(un vn)1113864 1113865 converges itself to (u v) Now letting n⟶ +infinin (32) we deduce that (u v) is a positive solution of system(1) e proof of theorem is now completed

4 Conclusion

In this work we study the existence of weak positive so-lutions for a sublinear Kirchhoff elliptic systems in boundeddomains by using the subsuper solutions method (SSM)combined with comparison principle which have beenwidely applied in many work (see for example[4 19 21ndash25])Validity of the comparison principle and ofthe SSM for local and nonlocal problems as the stationaryKirchhoff Equation was an important subject in the last fewyears (see for example [26] and [23] Moreover the twoconditions that M is nonincreasing and H is increasing turnout to be necessary and sufficient at least for the validity ofthe comparison principle It is worth to notice that in [4]Alves and Correa developed a new SSM for problem (1) todeal with the increasing M case e result is obtained byusing a kind of MintyndashBrowder theorem for a suitablepseudomonotone operator but instead of constructing asubsolution the authors assumed the existence of a wholefamily of functions which satisfy a stronger condition thanjust being subsolutions the inconvenience is that thesestronger conditions restrict the possible right hand sides in(1) Another SSM for nonlocal problems is obtained in [4] fora problem involving a nonlocal term with a Lebesgue norminstead of the Sobolev norm appearing in (1) In our nextstudy we will try to apply an alternative approach using thevariational principle which has been presented in [27ndash29]

Complexity 5

Data Availability

e data used to support the findings of the study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

All authors contributed equally to this article ey have allread and approved the final manuscript

References

[1] P DrsquoAncona and S Spagnolo ldquoGlobal solvability for thedegenerate Kirchhoff equation with real analytic datardquoInventiones Mathematicae vol 108 pp 247ndash262 1992

[2] H Medekhel S Boulaaras K Zennir and A AllahemldquoExistence of positive solutions and its asymptotic behavior of(p(x) q(x))-Laplacian parabolic systemrdquo Symmetry vol 11no 3 p 332 2019

[3] J L Lions ldquoOn some questions in boundary value problems ofmathematical physicsrdquo Contemporary Developments inContinuum Mechanics and Partial Differential EquationsProceedings of the International Symposium on ContinuumMechanics and Partial Differential Equations vol 30pp 284ndash346 1978

[4] C O Alves and F J S A Correa ldquoOn existence of solutionsfor a class of problem involving a nonlinear operatorrdquoCommunications on Applied Nonlinear Analysis vol 8pp 43ndash56 2001

[5] Y Li F Li and J Shi ldquoExistence of positive solutions toKirchhoff type problems with zero massrdquo Journal of Math-ematical Analysis and Applications vol 410 no 1 pp 361ndash374 2014

[6] B Ricceri ldquoOn an elliptic Kirchhoff-type problem dependingon two parametersrdquo Journal of Global Optimization vol 46no 4 pp 543ndash549 2010

[7] J Deuel and P Hess ldquoInequations variationelless elliptiquesnon coercivesrdquo C R Acad Sci Paris vol 279 pp 719ndash7221974

[8] P Hess ldquoOn the solvability of nonlinear elliptic boundaryvalue problemsrdquo Indiana University Mathematics Journalvol 25 no 5 pp 461ndash466 1976

[9] S Carl V K Le and D Motreanu ldquoe sub-supersolutionmethod and extremal solutions for quasilinear hemivaria-tional inequalitiesrdquo Differential and Integral Equationsvol 17 pp 165ndash178 2004

[10] E N Dancer and G Sweers ldquoOn the existence of a maximalweak solution for a semilinear elliptic equationrdquo Differentialand Integral Equations vol 2 pp 533ndash540 1989

[11] T Kura ldquoe weak supersolution-subsolution method forsecond order quasilinear elliptic equationsrdquo HiroshimaMathematical Journal vol 19 p 136 1989

[12] B Mairi R Guefaifia S Boulaaras and T Bouali ldquoExistenceof positive solutions for a new class of nonlocal p(x)-Kirchhoffelliptic systems via sub-super solutions conceptrdquo AppliedScience APPS vol 20 pp 117ndash128 2018

[13] S Boulaaras R Guefaifia and K Zennir ldquoExistence ofpositive solutions for nonlocal p(x)-Kirchhoff elliptic

systemsrdquo Advances in Pure and Applied Mathematics vol 10no 1 pp 1867ndash1152 2019

[14] V K Le and K Schmitt ldquoSome general concepts of sub- andsupersolutions for nonlinear elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 28 pp 87ndash103 2006

[15] N Mezouar and S Boulaaras ldquoGlobal existence and decay ofsolutions for a class of viscoelastic Kirchhoff equationrdquoBulletin of the Malaysian Mathematical Sciences Societyvol 43 no 1 pp 725ndash755 2018

[16] V K Le ldquoSubsolution-supersolution method in variationalinequalitiesrdquo Nonlinear Analysis vol 45 pp 775ndash800 2001

[17] V K Le ldquoSubsolution-supersolutions and the existence ofextremal solutions in noncoercive variational inequalitiesrdquoJournal of Inequalities in Pure and Applied Mathematicsvol 2 p 116 2001

[18] S Boulaaras D Ouchenane and F Mesloub ldquoGeneral decayfor a viscoelastic problem with not necessarily decreasingkernelrdquo Applicable Analysis vol 98 no 9 pp 1677ndash16932019

[19] N Azouz and A Bensedik ldquoExistence result for an ellipticequation of Kirchhoff-type with changing sign datardquo Funk-cialaj Ekvacioj vol 55 no 1 pp 55ndash66 2012

[20] D D Hai and R Shivaji ldquoAn existence result on positivesolutions for a class of p-Laplacian systemsrdquo NonlinearAnalysis 8eory Methods amp Applications vol 56 no 7pp 1007ndash1010 2004

[21] S Boulaaras and R Guefaifia ldquoExistence of positive weaksolutions for a class of Kirrchoff elliptic systems with multipleparametersrdquo Mathematical Methods in the Applied Sciencesvol 41 no 13 pp 5203ndash5210 2018

[22] S Boulaaras R Guefaifia and S Kabli ldquoAn asymptotic be-havior of positive solutions for a new class of elliptic systemsinvolving of $$left(pleft(xright) qleft(xright) right) $$ px q x -Laplacian systemsrdquo Boletın de la Sociedad MatematicaMexicana vol 25 no 1 pp 145ndash162 2019

[23] F J S A Correa and G M Figueiredo ldquoOn an ellipticequation of pminus Kirchhoff type via variational methodsrdquoBulletin of the Australian Mathematical Society vol 74 no 2pp 263ndash277 2006

[24] R Guefaifia and S Boulaaras ldquoExistence of positive solutionfor a class of (p(x) q(x))-Laplacian systemsrdquo Rendiconti delCircolo Matematico di Palermo Series 2 vol 67 pp 93ndash1032018

[25] S Boulaaras and A Allahem ldquoExistence of positive solutionsof nonlocal p(x)-Kirchhoff evolutionary systems via sub-supersolutions conceptrdquo Symmetry vol 11 no 2 p 253 2019

[26] M Chipot and B Lovat ldquoSome remarks on non local ellipticand parabolic problemsrdquo Nonlinear Analysis 8eoryMethods amp Applications vol 30 no 7 pp 4619ndash4627 1997

[27] S Boulaaras ldquoSome existence results for elliptic Kirchhoffequation with changing sign data and a logarithmic non-linearityrdquo Journal of Intelligent and Fuzzy Systems vol 37no 6 pp 1ndash10 2019

[28] S Boulaaras ldquoA well-posedness and exponential decay ofsolutions for a coupled lame system with viscoelastic term andlogarithmic source termsrdquoApplicable Analysis pp 1ndash19 2019

[29] J-H He ldquoVariational principle for the generalized KdV-burgers equation with fractal derivatives for shallow waterwavesrdquo Journal of Applied and Computational Mechanicsvol 6 no 4 httpjacmscuacirarticle_14813html 2020

6 Complexity

2b2εk1113946ΩΩη

σϕ21 minus nablaϕ11113868111386811138681113868

11138681113868111386811138682

1113872 1113873ψ dxle 2b2εk1113946ΩΩη

σϕ21 middot ψdx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873 μ1s22 εk ds2μc d1113872 1113873ψdx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873s1

s1+

μ1s22 εk ds2μc d1113872 1113873s2

s2

⎡⎢⎢⎣ ⎤⎥⎥⎦ψ dx

le1113946ΩΩη

λ1s12 εcs1μ21113872 1113873s1

+ μ1s22 εk ds2μc d1113872 1113873

s21113876 1113877ψdx

1113946ΩΩη

λ2εcμ2s1 + μ2ε

k dμc ds21113872 1113873ψdx

le1113946ΩΩη

λ2εcμ2c

+ μ2εk dμ2 d

1113872 1113873ψdx

le1113946ΩΩη

λ2εcϕ2c

1 + μ2εk dϕ2d

11113872 1113873ψdx

1113946ΩΩη

λ2uc

+ μ2vd

1113888 1113889ψdx

(23)

Hence from (15)ndash(23) it follows that

A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875 1113946

Ωηnabla unablaϕdx + 1113946

ΩΩηnabla unablaϕdx1113890 1113891

A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnabla unablaϕdxle λ1u

a+ μ1v

b1113872 1113873ϕdx

(24)

B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875 1113946

Ωηnabla vnablaψdx + 1113946

ΩΩηnabla vnablaψdx1113890 1113891

B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnabla vnablaψdxle λ2u

c+ μ2v

d1113872 1113873ψ dx

(25)

en 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

minus Δe 1 inΩ

e 0 on zΩ1113896 (26)

Let

u C1e

v C2e(27)

where e is given by (26) and C1 C2 gt 0 are large positivereal numbers to be chosen later We shall verify that (u v)

is a supersolution of problem (1) Let ϕ isin H10(Ω) with ϕge 0

in Ω en we obtain from (26) and the condition (H1)

that

A 1113946Ω

|nablau|2dx1113874 11138751113946

Ωnablau middot nablaϕdx A 1113946

Ω|nablau|

2dx1113874 1113875C1 1113946

Ω

nablae middot nablaϕdx

A 1113946Ω

|nablau|2dx1113874 1113875C11113946

Ωϕdx

ge a1C11113946Ωϕdx

B 1113946Ω

|nablav|2dx1113874 11138751113946

Ωnablav middot nablaψdx B 1113946

Ω|nablav|

2dx1113874 1113875C21113946Ωnablae middot nablaψdx

B 1113946Ω

|nablav|2dx1113874 1113875C21113946

Ωψdx

ge b1C21113946Ωψdx

(28)

Let l einfin Since alt 1 dlt 1 these imply that thereexist positive large constants α a1C1 β b1C2 such that

αge λ1(αl)a

+ μ1(βl)b

βge λ2(αl)c

+ μ2(βl)d

(29)

us

a1C11113946Ωϕdxge1113946

Ωλ1u

a+ μ1v

b1113872 1113873ϕdx (30)

b1C21113946Ωψ dxge1113946

Ωλ2u

c+ μ2v

d1113872 1113873ϕdx (31)

From (26) and (30) we can deduce that the couple (u v)

is a subsolution of problem (1) with u le u and v le v for C1

C2 largeIn order to obtain a weak solution of problem (1) we shall

use the arguments by Azouz and Bensedik [19] For thispurpose we define a sequence (un vn)1113864 1113865 sub (H1

0(Ω) times

H10(Ω)) as follows u0 ≔ u v0 v and (un vn) is the unique

solution of the system

minus A 1113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δun λ1ua

nminus 1 + μ1vbnminus 1 inΩ

minus B 1113946Ωnablavn

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δvn λ2uc

nminus 1 + μ2vdnminus 1 inΩ

un vn 0 on zΩ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(32)

Problem (32) is (A B)minus linear in the sense that if(unminus 1 vnminus 1) isin (H1

0(Ω) times H10(Ω)) is given the right hand

sides of (32) are independent of un vn

Set A(t) tA(t2) B(t) tB(t2) en since A(R) R

B(R) R f(unminus 1) uanminus 1 h(vnminus 1) vb

nminus 1 g(unminus 1) ucnminus 1

and τ(vnminus 1) vdnminus 1 isin L2(Ω)

We deduce from a result in [4] that system (32) has aunique solution (un vn) isin (H1

0(Ω) times H10(Ω))

By using (32) and the fact that (u0 v0) is a supersolutionof (1) we have

minus A 1113946Ωnablau0

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu0 ge λ1ua

0 + μ1vb0 minus A 1113946

Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1

minus B 1113946Ωnablav0

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv0 ge λ2uc

0 + μ2vd0 minus B 1113946

Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(33)

4 Complexity

and by Lemma 1 u0 ge u1 and v0 ge v1 Also since u0 ge uv0 ge v and the monotonicity of f h g and τ one has

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(34)

from which rding to Lemma 1 u1 ge u v1 ge v for u2 v2 wewrite

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a1 + μ1v

b1

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2

minus B 1113946Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c1 + μ2v

d1

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2

(35)

and then u1 ge u2 v1 ge v2 Similarly u2 ge u and v2 ge v because

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2 λ2u

c1 + μ2v

d1 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(36)

Repeating this argument we get a bounded monotonesequence (un vn)1113864 1113865 sub (H1

0(Ω) times H10(Ω)) satisfying

u u0 ge u1 ge u2 ge middot middot middot ge un ge middot middot middot ge u gt 0 (37)

v v0 ge v1 ge v2 ge middot middot middot ge vn ge middot middot middot ge v gt 0 (38)

Using the continuity of the functions f h g and t andthe definition of the sequences un1113864 1113865 vn1113864 1113865there exist con-stants Ci gt 0 i 1 4 independent of n such that

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC1

h unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 leC2

g unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC3

(39)

τ unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC4 for all n (40)

From (39) multiplying the first equation of (32) by un

and integrating using the Holder inequality and Sobolevembedding we can show that

a11113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dxleA 1113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx

λ1f vnminus 1( 1113857undx + μ11113946Ω

h unminus 1( 1113857undx

le λ11113946Ω

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx + μ11113946

Ωh unminus 1( 1113857

11138681113868111386811138681113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx

leC1λ11113946Ω

un

11138681113868111386811138681113868111386811138681113868dx + C2μ11113946

Ωun

11138681113868111386811138681113868111386811138681113868dx

leC5 un

H1

0(Ω)

(41)

or

un

H1

0(Ω)leC5 foralln (42)

where C5 gt 0 is a constant independent of n Similarly thereexists C6 gt 0 independent of n such that

vn

H1

0(Ω)leC6 foralln (43)

From (42) and (43) we infer that (un vn)1113864 1113865 has a sub-sequence which weakly converges in H1

0(ΩR2) to a limit(u v) with the properties uge ugt 0 and vge v gt 0 Beingmonotone and also using a standard regularity argument(un vn)1113864 1113865 converges itself to (u v) Now letting n⟶ +infinin (32) we deduce that (u v) is a positive solution of system(1) e proof of theorem is now completed

4 Conclusion

In this work we study the existence of weak positive so-lutions for a sublinear Kirchhoff elliptic systems in boundeddomains by using the subsuper solutions method (SSM)combined with comparison principle which have beenwidely applied in many work (see for example[4 19 21ndash25])Validity of the comparison principle and ofthe SSM for local and nonlocal problems as the stationaryKirchhoff Equation was an important subject in the last fewyears (see for example [26] and [23] Moreover the twoconditions that M is nonincreasing and H is increasing turnout to be necessary and sufficient at least for the validity ofthe comparison principle It is worth to notice that in [4]Alves and Correa developed a new SSM for problem (1) todeal with the increasing M case e result is obtained byusing a kind of MintyndashBrowder theorem for a suitablepseudomonotone operator but instead of constructing asubsolution the authors assumed the existence of a wholefamily of functions which satisfy a stronger condition thanjust being subsolutions the inconvenience is that thesestronger conditions restrict the possible right hand sides in(1) Another SSM for nonlocal problems is obtained in [4] fora problem involving a nonlocal term with a Lebesgue norminstead of the Sobolev norm appearing in (1) In our nextstudy we will try to apply an alternative approach using thevariational principle which has been presented in [27ndash29]

Complexity 5

Data Availability

e data used to support the findings of the study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

All authors contributed equally to this article ey have allread and approved the final manuscript

References

[1] P DrsquoAncona and S Spagnolo ldquoGlobal solvability for thedegenerate Kirchhoff equation with real analytic datardquoInventiones Mathematicae vol 108 pp 247ndash262 1992

[2] H Medekhel S Boulaaras K Zennir and A AllahemldquoExistence of positive solutions and its asymptotic behavior of(p(x) q(x))-Laplacian parabolic systemrdquo Symmetry vol 11no 3 p 332 2019

[3] J L Lions ldquoOn some questions in boundary value problems ofmathematical physicsrdquo Contemporary Developments inContinuum Mechanics and Partial Differential EquationsProceedings of the International Symposium on ContinuumMechanics and Partial Differential Equations vol 30pp 284ndash346 1978

[4] C O Alves and F J S A Correa ldquoOn existence of solutionsfor a class of problem involving a nonlinear operatorrdquoCommunications on Applied Nonlinear Analysis vol 8pp 43ndash56 2001

[5] Y Li F Li and J Shi ldquoExistence of positive solutions toKirchhoff type problems with zero massrdquo Journal of Math-ematical Analysis and Applications vol 410 no 1 pp 361ndash374 2014

[6] B Ricceri ldquoOn an elliptic Kirchhoff-type problem dependingon two parametersrdquo Journal of Global Optimization vol 46no 4 pp 543ndash549 2010

[7] J Deuel and P Hess ldquoInequations variationelless elliptiquesnon coercivesrdquo C R Acad Sci Paris vol 279 pp 719ndash7221974

[8] P Hess ldquoOn the solvability of nonlinear elliptic boundaryvalue problemsrdquo Indiana University Mathematics Journalvol 25 no 5 pp 461ndash466 1976

[9] S Carl V K Le and D Motreanu ldquoe sub-supersolutionmethod and extremal solutions for quasilinear hemivaria-tional inequalitiesrdquo Differential and Integral Equationsvol 17 pp 165ndash178 2004

[10] E N Dancer and G Sweers ldquoOn the existence of a maximalweak solution for a semilinear elliptic equationrdquo Differentialand Integral Equations vol 2 pp 533ndash540 1989

[11] T Kura ldquoe weak supersolution-subsolution method forsecond order quasilinear elliptic equationsrdquo HiroshimaMathematical Journal vol 19 p 136 1989

[12] B Mairi R Guefaifia S Boulaaras and T Bouali ldquoExistenceof positive solutions for a new class of nonlocal p(x)-Kirchhoffelliptic systems via sub-super solutions conceptrdquo AppliedScience APPS vol 20 pp 117ndash128 2018

[13] S Boulaaras R Guefaifia and K Zennir ldquoExistence ofpositive solutions for nonlocal p(x)-Kirchhoff elliptic

systemsrdquo Advances in Pure and Applied Mathematics vol 10no 1 pp 1867ndash1152 2019

[14] V K Le and K Schmitt ldquoSome general concepts of sub- andsupersolutions for nonlinear elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 28 pp 87ndash103 2006

[15] N Mezouar and S Boulaaras ldquoGlobal existence and decay ofsolutions for a class of viscoelastic Kirchhoff equationrdquoBulletin of the Malaysian Mathematical Sciences Societyvol 43 no 1 pp 725ndash755 2018

[16] V K Le ldquoSubsolution-supersolution method in variationalinequalitiesrdquo Nonlinear Analysis vol 45 pp 775ndash800 2001

[17] V K Le ldquoSubsolution-supersolutions and the existence ofextremal solutions in noncoercive variational inequalitiesrdquoJournal of Inequalities in Pure and Applied Mathematicsvol 2 p 116 2001

[18] S Boulaaras D Ouchenane and F Mesloub ldquoGeneral decayfor a viscoelastic problem with not necessarily decreasingkernelrdquo Applicable Analysis vol 98 no 9 pp 1677ndash16932019

[19] N Azouz and A Bensedik ldquoExistence result for an ellipticequation of Kirchhoff-type with changing sign datardquo Funk-cialaj Ekvacioj vol 55 no 1 pp 55ndash66 2012

[20] D D Hai and R Shivaji ldquoAn existence result on positivesolutions for a class of p-Laplacian systemsrdquo NonlinearAnalysis 8eory Methods amp Applications vol 56 no 7pp 1007ndash1010 2004

[21] S Boulaaras and R Guefaifia ldquoExistence of positive weaksolutions for a class of Kirrchoff elliptic systems with multipleparametersrdquo Mathematical Methods in the Applied Sciencesvol 41 no 13 pp 5203ndash5210 2018

[22] S Boulaaras R Guefaifia and S Kabli ldquoAn asymptotic be-havior of positive solutions for a new class of elliptic systemsinvolving of $$left(pleft(xright) qleft(xright) right) $$ px q x -Laplacian systemsrdquo Boletın de la Sociedad MatematicaMexicana vol 25 no 1 pp 145ndash162 2019

[23] F J S A Correa and G M Figueiredo ldquoOn an ellipticequation of pminus Kirchhoff type via variational methodsrdquoBulletin of the Australian Mathematical Society vol 74 no 2pp 263ndash277 2006

[24] R Guefaifia and S Boulaaras ldquoExistence of positive solutionfor a class of (p(x) q(x))-Laplacian systemsrdquo Rendiconti delCircolo Matematico di Palermo Series 2 vol 67 pp 93ndash1032018

[25] S Boulaaras and A Allahem ldquoExistence of positive solutionsof nonlocal p(x)-Kirchhoff evolutionary systems via sub-supersolutions conceptrdquo Symmetry vol 11 no 2 p 253 2019

[26] M Chipot and B Lovat ldquoSome remarks on non local ellipticand parabolic problemsrdquo Nonlinear Analysis 8eoryMethods amp Applications vol 30 no 7 pp 4619ndash4627 1997

[27] S Boulaaras ldquoSome existence results for elliptic Kirchhoffequation with changing sign data and a logarithmic non-linearityrdquo Journal of Intelligent and Fuzzy Systems vol 37no 6 pp 1ndash10 2019

[28] S Boulaaras ldquoA well-posedness and exponential decay ofsolutions for a coupled lame system with viscoelastic term andlogarithmic source termsrdquoApplicable Analysis pp 1ndash19 2019

[29] J-H He ldquoVariational principle for the generalized KdV-burgers equation with fractal derivatives for shallow waterwavesrdquo Journal of Applied and Computational Mechanicsvol 6 no 4 httpjacmscuacirarticle_14813html 2020

6 Complexity

and by Lemma 1 u0 ge u1 and v0 ge v1 Also since u0 ge uv0 ge v and the monotonicity of f h g and τ one has

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(34)

from which rding to Lemma 1 u1 ge u v1 ge v for u2 v2 wewrite

minus A 1113946Ωnablau1

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu1 λ1u

a0 + μ1v

b0 ge λ1u

a1 + μ1v

b1

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2

minus B 1113946Ωnablav1

11138681113868111386811138681113868111386811138681113868dx1113874 1113875Δv1 λ2u

c0 + μ2v

d0 ge λ2u

c1 + μ2v

d1

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2

(35)

and then u1 ge u2 v1 ge v2 Similarly u2 ge u and v2 ge v because

minus A 1113946Ωnablau2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δu2 λ1u

a0 + μ1v

b0 ge λ1u

a+ μ1v

b ge

minus A 1113946Ωnabla u

1113868111386811138681113868111386811138682dx1113874 1113875Δ u

minus B 1113946Ωnablav2

111386811138681113868111386811138681113868111386811138682dx1113874 1113875Δv2 λ2u

c1 + μ2v

d1 ge λ2u

c+ μ2v

d ge

minus B 1113946Ωnabla v

1113868111386811138681113868111386811138682dx1113874 1113875Δ v

(36)

Repeating this argument we get a bounded monotonesequence (un vn)1113864 1113865 sub (H1

0(Ω) times H10(Ω)) satisfying

u u0 ge u1 ge u2 ge middot middot middot ge un ge middot middot middot ge u gt 0 (37)

v v0 ge v1 ge v2 ge middot middot middot ge vn ge middot middot middot ge v gt 0 (38)

Using the continuity of the functions f h g and t andthe definition of the sequences un1113864 1113865 vn1113864 1113865there exist con-stants Ci gt 0 i 1 4 independent of n such that

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC1

h unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 leC2

g unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC3

(39)

τ unminus 1( 11138571113868111386811138681113868

1113868111386811138681113868leC4 for all n (40)

From (39) multiplying the first equation of (32) by un

and integrating using the Holder inequality and Sobolevembedding we can show that

a11113946Ωnablaun

111386811138681113868111386811138681113868111386811138682dxleA 1113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx1113874 11138751113946

Ωnablaun

111386811138681113868111386811138681113868111386811138682dx

λ1f vnminus 1( 1113857undx + μ11113946Ω

h unminus 1( 1113857undx

le λ11113946Ω

f vnminus 1( 11138571113868111386811138681113868

1113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx + μ11113946

Ωh unminus 1( 1113857

11138681113868111386811138681113868111386811138681113868 un

11138681113868111386811138681113868111386811138681113868dx

leC1λ11113946Ω

un

11138681113868111386811138681113868111386811138681113868dx + C2μ11113946

Ωun

11138681113868111386811138681113868111386811138681113868dx

leC5 un

H1

0(Ω)

(41)

or

un

H1

0(Ω)leC5 foralln (42)

where C5 gt 0 is a constant independent of n Similarly thereexists C6 gt 0 independent of n such that

vn

H1

0(Ω)leC6 foralln (43)

From (42) and (43) we infer that (un vn)1113864 1113865 has a sub-sequence which weakly converges in H1

0(ΩR2) to a limit(u v) with the properties uge ugt 0 and vge v gt 0 Beingmonotone and also using a standard regularity argument(un vn)1113864 1113865 converges itself to (u v) Now letting n⟶ +infinin (32) we deduce that (u v) is a positive solution of system(1) e proof of theorem is now completed

4 Conclusion

In this work we study the existence of weak positive so-lutions for a sublinear Kirchhoff elliptic systems in boundeddomains by using the subsuper solutions method (SSM)combined with comparison principle which have beenwidely applied in many work (see for example[4 19 21ndash25])Validity of the comparison principle and ofthe SSM for local and nonlocal problems as the stationaryKirchhoff Equation was an important subject in the last fewyears (see for example [26] and [23] Moreover the twoconditions that M is nonincreasing and H is increasing turnout to be necessary and sufficient at least for the validity ofthe comparison principle It is worth to notice that in [4]Alves and Correa developed a new SSM for problem (1) todeal with the increasing M case e result is obtained byusing a kind of MintyndashBrowder theorem for a suitablepseudomonotone operator but instead of constructing asubsolution the authors assumed the existence of a wholefamily of functions which satisfy a stronger condition thanjust being subsolutions the inconvenience is that thesestronger conditions restrict the possible right hand sides in(1) Another SSM for nonlocal problems is obtained in [4] fora problem involving a nonlocal term with a Lebesgue norminstead of the Sobolev norm appearing in (1) In our nextstudy we will try to apply an alternative approach using thevariational principle which has been presented in [27ndash29]

Complexity 5

Data Availability

e data used to support the findings of the study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

All authors contributed equally to this article ey have allread and approved the final manuscript

References

[1] P DrsquoAncona and S Spagnolo ldquoGlobal solvability for thedegenerate Kirchhoff equation with real analytic datardquoInventiones Mathematicae vol 108 pp 247ndash262 1992

[2] H Medekhel S Boulaaras K Zennir and A AllahemldquoExistence of positive solutions and its asymptotic behavior of(p(x) q(x))-Laplacian parabolic systemrdquo Symmetry vol 11no 3 p 332 2019

[3] J L Lions ldquoOn some questions in boundary value problems ofmathematical physicsrdquo Contemporary Developments inContinuum Mechanics and Partial Differential EquationsProceedings of the International Symposium on ContinuumMechanics and Partial Differential Equations vol 30pp 284ndash346 1978

[4] C O Alves and F J S A Correa ldquoOn existence of solutionsfor a class of problem involving a nonlinear operatorrdquoCommunications on Applied Nonlinear Analysis vol 8pp 43ndash56 2001

[5] Y Li F Li and J Shi ldquoExistence of positive solutions toKirchhoff type problems with zero massrdquo Journal of Math-ematical Analysis and Applications vol 410 no 1 pp 361ndash374 2014

[6] B Ricceri ldquoOn an elliptic Kirchhoff-type problem dependingon two parametersrdquo Journal of Global Optimization vol 46no 4 pp 543ndash549 2010

[7] J Deuel and P Hess ldquoInequations variationelless elliptiquesnon coercivesrdquo C R Acad Sci Paris vol 279 pp 719ndash7221974

[8] P Hess ldquoOn the solvability of nonlinear elliptic boundaryvalue problemsrdquo Indiana University Mathematics Journalvol 25 no 5 pp 461ndash466 1976

[9] S Carl V K Le and D Motreanu ldquoe sub-supersolutionmethod and extremal solutions for quasilinear hemivaria-tional inequalitiesrdquo Differential and Integral Equationsvol 17 pp 165ndash178 2004

[10] E N Dancer and G Sweers ldquoOn the existence of a maximalweak solution for a semilinear elliptic equationrdquo Differentialand Integral Equations vol 2 pp 533ndash540 1989

[11] T Kura ldquoe weak supersolution-subsolution method forsecond order quasilinear elliptic equationsrdquo HiroshimaMathematical Journal vol 19 p 136 1989

[12] B Mairi R Guefaifia S Boulaaras and T Bouali ldquoExistenceof positive solutions for a new class of nonlocal p(x)-Kirchhoffelliptic systems via sub-super solutions conceptrdquo AppliedScience APPS vol 20 pp 117ndash128 2018

[13] S Boulaaras R Guefaifia and K Zennir ldquoExistence ofpositive solutions for nonlocal p(x)-Kirchhoff elliptic

systemsrdquo Advances in Pure and Applied Mathematics vol 10no 1 pp 1867ndash1152 2019

[14] V K Le and K Schmitt ldquoSome general concepts of sub- andsupersolutions for nonlinear elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 28 pp 87ndash103 2006

[15] N Mezouar and S Boulaaras ldquoGlobal existence and decay ofsolutions for a class of viscoelastic Kirchhoff equationrdquoBulletin of the Malaysian Mathematical Sciences Societyvol 43 no 1 pp 725ndash755 2018

[16] V K Le ldquoSubsolution-supersolution method in variationalinequalitiesrdquo Nonlinear Analysis vol 45 pp 775ndash800 2001

[17] V K Le ldquoSubsolution-supersolutions and the existence ofextremal solutions in noncoercive variational inequalitiesrdquoJournal of Inequalities in Pure and Applied Mathematicsvol 2 p 116 2001

[18] S Boulaaras D Ouchenane and F Mesloub ldquoGeneral decayfor a viscoelastic problem with not necessarily decreasingkernelrdquo Applicable Analysis vol 98 no 9 pp 1677ndash16932019

[19] N Azouz and A Bensedik ldquoExistence result for an ellipticequation of Kirchhoff-type with changing sign datardquo Funk-cialaj Ekvacioj vol 55 no 1 pp 55ndash66 2012

[20] D D Hai and R Shivaji ldquoAn existence result on positivesolutions for a class of p-Laplacian systemsrdquo NonlinearAnalysis 8eory Methods amp Applications vol 56 no 7pp 1007ndash1010 2004

[21] S Boulaaras and R Guefaifia ldquoExistence of positive weaksolutions for a class of Kirrchoff elliptic systems with multipleparametersrdquo Mathematical Methods in the Applied Sciencesvol 41 no 13 pp 5203ndash5210 2018

[22] S Boulaaras R Guefaifia and S Kabli ldquoAn asymptotic be-havior of positive solutions for a new class of elliptic systemsinvolving of $$left(pleft(xright) qleft(xright) right) $$ px q x -Laplacian systemsrdquo Boletın de la Sociedad MatematicaMexicana vol 25 no 1 pp 145ndash162 2019

[23] F J S A Correa and G M Figueiredo ldquoOn an ellipticequation of pminus Kirchhoff type via variational methodsrdquoBulletin of the Australian Mathematical Society vol 74 no 2pp 263ndash277 2006

[24] R Guefaifia and S Boulaaras ldquoExistence of positive solutionfor a class of (p(x) q(x))-Laplacian systemsrdquo Rendiconti delCircolo Matematico di Palermo Series 2 vol 67 pp 93ndash1032018

[25] S Boulaaras and A Allahem ldquoExistence of positive solutionsof nonlocal p(x)-Kirchhoff evolutionary systems via sub-supersolutions conceptrdquo Symmetry vol 11 no 2 p 253 2019

[26] M Chipot and B Lovat ldquoSome remarks on non local ellipticand parabolic problemsrdquo Nonlinear Analysis 8eoryMethods amp Applications vol 30 no 7 pp 4619ndash4627 1997

[27] S Boulaaras ldquoSome existence results for elliptic Kirchhoffequation with changing sign data and a logarithmic non-linearityrdquo Journal of Intelligent and Fuzzy Systems vol 37no 6 pp 1ndash10 2019

[28] S Boulaaras ldquoA well-posedness and exponential decay ofsolutions for a coupled lame system with viscoelastic term andlogarithmic source termsrdquoApplicable Analysis pp 1ndash19 2019

[29] J-H He ldquoVariational principle for the generalized KdV-burgers equation with fractal derivatives for shallow waterwavesrdquo Journal of Applied and Computational Mechanicsvol 6 no 4 httpjacmscuacirarticle_14813html 2020

6 Complexity

Data Availability

e data used to support the findings of the study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

All authors contributed equally to this article ey have allread and approved the final manuscript

References

[1] P DrsquoAncona and S Spagnolo ldquoGlobal solvability for thedegenerate Kirchhoff equation with real analytic datardquoInventiones Mathematicae vol 108 pp 247ndash262 1992

[2] H Medekhel S Boulaaras K Zennir and A AllahemldquoExistence of positive solutions and its asymptotic behavior of(p(x) q(x))-Laplacian parabolic systemrdquo Symmetry vol 11no 3 p 332 2019

[3] J L Lions ldquoOn some questions in boundary value problems ofmathematical physicsrdquo Contemporary Developments inContinuum Mechanics and Partial Differential EquationsProceedings of the International Symposium on ContinuumMechanics and Partial Differential Equations vol 30pp 284ndash346 1978

[4] C O Alves and F J S A Correa ldquoOn existence of solutionsfor a class of problem involving a nonlinear operatorrdquoCommunications on Applied Nonlinear Analysis vol 8pp 43ndash56 2001

[5] Y Li F Li and J Shi ldquoExistence of positive solutions toKirchhoff type problems with zero massrdquo Journal of Math-ematical Analysis and Applications vol 410 no 1 pp 361ndash374 2014

[6] B Ricceri ldquoOn an elliptic Kirchhoff-type problem dependingon two parametersrdquo Journal of Global Optimization vol 46no 4 pp 543ndash549 2010

[7] J Deuel and P Hess ldquoInequations variationelless elliptiquesnon coercivesrdquo C R Acad Sci Paris vol 279 pp 719ndash7221974

[8] P Hess ldquoOn the solvability of nonlinear elliptic boundaryvalue problemsrdquo Indiana University Mathematics Journalvol 25 no 5 pp 461ndash466 1976

[9] S Carl V K Le and D Motreanu ldquoe sub-supersolutionmethod and extremal solutions for quasilinear hemivaria-tional inequalitiesrdquo Differential and Integral Equationsvol 17 pp 165ndash178 2004

[10] E N Dancer and G Sweers ldquoOn the existence of a maximalweak solution for a semilinear elliptic equationrdquo Differentialand Integral Equations vol 2 pp 533ndash540 1989

[11] T Kura ldquoe weak supersolution-subsolution method forsecond order quasilinear elliptic equationsrdquo HiroshimaMathematical Journal vol 19 p 136 1989

[12] B Mairi R Guefaifia S Boulaaras and T Bouali ldquoExistenceof positive solutions for a new class of nonlocal p(x)-Kirchhoffelliptic systems via sub-super solutions conceptrdquo AppliedScience APPS vol 20 pp 117ndash128 2018

[13] S Boulaaras R Guefaifia and K Zennir ldquoExistence ofpositive solutions for nonlocal p(x)-Kirchhoff elliptic

systemsrdquo Advances in Pure and Applied Mathematics vol 10no 1 pp 1867ndash1152 2019

[14] V K Le and K Schmitt ldquoSome general concepts of sub- andsupersolutions for nonlinear elliptic problemsrdquo TopologicalMethods in Nonlinear Analysis vol 28 pp 87ndash103 2006

[15] N Mezouar and S Boulaaras ldquoGlobal existence and decay ofsolutions for a class of viscoelastic Kirchhoff equationrdquoBulletin of the Malaysian Mathematical Sciences Societyvol 43 no 1 pp 725ndash755 2018

[16] V K Le ldquoSubsolution-supersolution method in variationalinequalitiesrdquo Nonlinear Analysis vol 45 pp 775ndash800 2001

[17] V K Le ldquoSubsolution-supersolutions and the existence ofextremal solutions in noncoercive variational inequalitiesrdquoJournal of Inequalities in Pure and Applied Mathematicsvol 2 p 116 2001

[18] S Boulaaras D Ouchenane and F Mesloub ldquoGeneral decayfor a viscoelastic problem with not necessarily decreasingkernelrdquo Applicable Analysis vol 98 no 9 pp 1677ndash16932019

[19] N Azouz and A Bensedik ldquoExistence result for an ellipticequation of Kirchhoff-type with changing sign datardquo Funk-cialaj Ekvacioj vol 55 no 1 pp 55ndash66 2012

[20] D D Hai and R Shivaji ldquoAn existence result on positivesolutions for a class of p-Laplacian systemsrdquo NonlinearAnalysis 8eory Methods amp Applications vol 56 no 7pp 1007ndash1010 2004

[21] S Boulaaras and R Guefaifia ldquoExistence of positive weaksolutions for a class of Kirrchoff elliptic systems with multipleparametersrdquo Mathematical Methods in the Applied Sciencesvol 41 no 13 pp 5203ndash5210 2018

[22] S Boulaaras R Guefaifia and S Kabli ldquoAn asymptotic be-havior of positive solutions for a new class of elliptic systemsinvolving of $$left(pleft(xright) qleft(xright) right) $$ px q x -Laplacian systemsrdquo Boletın de la Sociedad MatematicaMexicana vol 25 no 1 pp 145ndash162 2019

[23] F J S A Correa and G M Figueiredo ldquoOn an ellipticequation of pminus Kirchhoff type via variational methodsrdquoBulletin of the Australian Mathematical Society vol 74 no 2pp 263ndash277 2006

[24] R Guefaifia and S Boulaaras ldquoExistence of positive solutionfor a class of (p(x) q(x))-Laplacian systemsrdquo Rendiconti delCircolo Matematico di Palermo Series 2 vol 67 pp 93ndash1032018

[25] S Boulaaras and A Allahem ldquoExistence of positive solutionsof nonlocal p(x)-Kirchhoff evolutionary systems via sub-supersolutions conceptrdquo Symmetry vol 11 no 2 p 253 2019

[26] M Chipot and B Lovat ldquoSome remarks on non local ellipticand parabolic problemsrdquo Nonlinear Analysis 8eoryMethods amp Applications vol 30 no 7 pp 4619ndash4627 1997

[27] S Boulaaras ldquoSome existence results for elliptic Kirchhoffequation with changing sign data and a logarithmic non-linearityrdquo Journal of Intelligent and Fuzzy Systems vol 37no 6 pp 1ndash10 2019

[28] S Boulaaras ldquoA well-posedness and exponential decay ofsolutions for a coupled lame system with viscoelastic term andlogarithmic source termsrdquoApplicable Analysis pp 1ndash19 2019

[29] J-H He ldquoVariational principle for the generalized KdV-burgers equation with fractal derivatives for shallow waterwavesrdquo Journal of Applied and Computational Mechanicsvol 6 no 4 httpjacmscuacirarticle_14813html 2020

6 Complexity