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Existence and multiplicity results for a coupled system of Kirchhoff type equations

Dengfeng Lü and Jianhai Xiao

B

School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei 432000, P.R. China Received 4 September 2013, appeared 15 March 2014

Communicated by Johnny Henderson

Abstract. This paper deals with a coupled system of Kirchhoff type equations inR3. Under suitable assumptions on the potential functions V(x) andW(x), we obtain the existence and multiplicity of nontrivial solutions when the parameter λis sufficiently large. The method combines the Nehari manifold and the mountain-pass theorem.

Keywords:Kirchhoff type equation, ground state solution, mountain-pass theorem.

2010 Mathematics Subject Classification:35J50, 35J10, 35Q60.

1 Introduction

In this paper, we consider the coupled system of Kirchhoff type equations













a+b Z

R3|∇u|2dx

∆u+λV(x)u=

α+β|u|α2u|v|β inR3,

a+b Z

R3|∇v|2dx

∆v+λW(x)v=

α+β|u|α|v|β2v inR3, u(x)→0, v(x)→0, as|x| →,

(K)λ

wherea>0,b>0 are constants,λ>0 is a parameter,α>2,β>2 satisfyα+β<2 =6, and V(x),W(x)are nonnegative continuous potential functions onR3.

In recent years, many papers have extensively considered the scalar Kirchhoff equation





a+b Z

|∇u|2dx

∆u= f(x,u) inΩ,

u=0 on∂Ω,

(1.1)

whereΩ ⊂ R3is a smooth bounded domain, one can see [1,4,6,11,12,15] and the references therein. Problem (1.1) is related to the stationary analogue of the equation

utt

a+b Z

|∇u|2dx

∆u= f(x,u), (1.2)

BCorresponding author. Email: dengfeng1214@163.com (D. Lü), jhxmath@sina.cn (J. Xiao)

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which was proposed by Kirchhoff in [8] as an extension of the classical d’Alembert wave equa- tion for free vibrations of elastic strings. Kirchhoff’s model considers the changes in length of the string produced by transverse vibrations.

There are also many works on the existence and multiplicity results for the scalar case of (K)1









a+b Z

R3|∇u|2dx

∆u+V(x)u= f(u) inR3,

u∈ H1(R3), u>0 inR3,

(1.3)

where f is a subcritical function and satisfies certain conditions. We would mention the re- cent paper [14], by applying symmetric mountain-pass theorem, the author obtained the ex- istence results for nontrivial solutions and a sequence of high energy solutions for problem (1.3). Subsequently, Liu and He [9] proved the existence of infinitely many high energy so- lutions for (1.3) when f is a subcritical nonlinearity which does not need to satisfy the usual Ambrosetti–Rabinowitz conditions. Further related results can be seen in [7,10,13] and the references therein.

The purpose of this paper is to study the existence and multiplicity results for a coupled system of Kirchhoff type equations inR3. To the best of our knowledge, problem(K)λhas not been considered before, the main difficulties lie in the appearance of the non-local term and the lack of compactness due to the unboundedness of the domainR3. Motivated by the work mentioned above, we will get the existence and multiplicity results of nontrivial solutions for λlarge enough by exploiting the Nehari manifold method and the mountain-pass theorem.

Before stating our main results, we need to introduce some assumptions and notations:

(A1) V(x),W(x) ∈ C(R3,[0,+)) andΩ := int(V1(0)) = int(W1(0))is nonempty with smooth boundary andΩ= V1(0) =W1(0);

(A2) there existM1, M2 >0 such that

L({x∈R3 |V(x)≤ M1})<∞, L({x ∈R3 |W(x)≤ M2})<∞, whereLdenotes the Lebesgue measure inR3.

The hypothesis (A2)was first introduced by Bartsch and Wang [3] in the study of a non- linear Schrödinger equation. Let EV = {u ∈ H1(R3) : R

R3V(x)u2dx < +} and EW = {v∈ H1(R3):R

R3W(x)v2dx <+}with the normskuk2λ,V = R

R3(a|∇u|2+λV(x)u2)dxand kvk2λ,W =R

R3(a|∇v|2+λW(x)v2)dxrespectively. For any givenλ>0, we consider the Hilbert spaceE:=EV×EW endowed with the norm

k(u,v)kλ= q

kuk2λ,V+kvk2λ,W. The energy functional associated with(K)λis defined onEby

Iλ(u,v) = 1

2k(u,v)k2λ+ b

4 Υ2(u) +Υ2(v)2 α+β

Z

R3|u|α|v|βdx, whereΥ(w) = R

R3|∇w|2dx. In view of the assumptions(A1)and(A2), the energy functional Iλ(u,v)is well defined and belongs toC1(E,R). It is well known that the weak solutions of problem(K)λare the critical points of the energy functionalIλ(u,v).

The main results we get are the following:

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Theorem 1.1. Suppose that(A1)and(A2)hold. Then there isλ > 0such that for all λλ, the system(K)λhas a ground state solution.

Theorem 1.2. Suppose that(A1)and(A2)hold. Then for any given k∈N, there existsΛk >0such that for eachλΛk, the system(K)λpossesses at least k pairs of nontrivial solutions.

This paper is organized as follows. In Section 2, we will prove some important lemmas that will be used for the proofs of the main results. Section 3 is devoted to the proofs of Theorems 1.1and1.2.

2 Some preliminary lemmas

In this paper,C, C1, C2, . . . denote positive (possibly different) constants. →(respectively*) denotes strong (respectively weak) convergence. on(1)denoteson(1)→0 asn→∞. Brdenotes a ball centered at the origin with radiusr >0. For a given setK⊂ R3, we setKc =R3\K. We define the minimaxcλas

cλ = inf

(u,v)∈NλIλ(u,v), (2.1)

whereNλdenotes the Nehari manifold associated withIλgiven by Nλ =(u,v)∈E\ {(0, 0)}:hIλ0(u,v),(u,v)i=0 ,

andh·,·iis the duality product betweenEand its dual spaceE1. A ground state solution of (K)λ means a solution(u,v)of(K)λ withIλ(u,v) = cλ. Note thatNλcontains every nonzero solution of problem(K)λ. Hereafter, we suppose that(A1)and(A2)are satisfied.

Lemma 2.1. Let(u,v)∈ Nλ, then there existsσ>0which is independent ofλsuch thatk(u,v)kλσ.

Proof. First, by Young’s inequality, we get

|u|α|v|βα

α+β|u|α+β+ β

α+β|v|α+β,

then by the continuity of the Sobolev embeddingEV,→Ls(R3)andEW,→Ls(R3)for 2≤s≤6, we obtain

Z

R3|u|α|v|βdx≤ α α+β

Z

R3|u|α+βdx+ β α+β

Z

R3|v|α+βdx

≤C1kukαV+β+C2kvkαW+β ≤Ck(u,v)kα+β

λ , (2.2)

whereC>0 is independent ofλ. So, by (2.2), for any(u,v)∈ Nλ we have 0= hIλ0(u,v),(u,v)i=k(u,v)k2λ+b(Υ2(u) +Υ2(v))−2

Z

R3|u|α|v|βdx

≥ k(u,v)k2λ−2Ck(u,v)kαλ+β. Note thatα+β>2, thus there existsσ>0 such thatk(u,v)kλσ.

Lemma 2.2. Suppose that{(un,vn)}is a(PS)c-sequence forIλ(u,v). Then we have (i) {(un,vn)}is bounded in E;

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(ii) if c6=0, then c≥c0, for some c0>0is independent ofλ.

Proof. Let {(un,vn)} be a (PS)c sequence for Iλ(u,v), that is, Iλ(un,vn) = c+on(1) and Iλ0(un,vn) =on(1). Then we have that

c+on(1)−1

4on(k(un,vn)kλ) =Iλ(un,vn)−1

4hIλ0(un,vn),(un,vn)i

= 1

2k(un,vn)k2λ+ 1

2− 2 α+β

Z

R3|un|α|vn|βdx

1

2k(un,vn)k2λ, (2.3)

which implies that{(un,vn)}is bounded inE.

On the other hand, we have

on(k(un,vn)kλ) =hIλ0(un,vn),(un,vn)i

=k(un,vn)k2λ+b(Υ2(un) +Υ2(vn))−2 Z

R3|un|α|vn|βdx

≥ k(un,vn)k2λ−2Ck(un,vn)kαλ+β(by (2.2)), sinceα+β>2, there exists 0<σ1 <1 such that

hIλ0(un,vn),(un,vn)i ≥ 1

4k(un,vn)k2λ, fork(un,vn)kλ <σ1. (2.4) Now, ifc< σ221 and{(un,vn)}is a(PS)c-sequence ofIλ, then by (2.3)

nlimk(un,vn)k2λ ≤2c<σ12. Hence,k(un,vn)kλ <σ1fornlarge, then by (2.4)

1

4k(un,vn)k2λ ≤ hIλ0(un,vn),(un,vn)i= on(k(un,vn)kλ),

which impliesk(un,vn)kλ →0 asn→andc=0, it follows that (ii) holds forc0=σ12/2.

Lemma 2.3. Suppose that (A1)–(A2) hold and let C be fixed. Given ε > 0 there exist Λε = Λ(ε,C) > 0andρε = ρ(ε,C) > 0such that, if {(un,vn)}is a (PS)c-sequence ofIλ(u,v)with c≤ C,λΛε, then

lim sup

n Z

Bcρε

|un|α|vn|βdx≤ε. (2.5) Proof. Forρ>0, we set

A(ρ):={x ∈R3:|x| ≥ρ,V(x)≥ M1}, B(ρ):={x∈R3 :|x| ≥ρ,V(x)<M1}, then

Z

A(ρ)

|un|2dx≤ 1 λM1

Z

R3λV(x)u2ndx

1 λM1

Z

R3 a|∇un|2+λV(x)u2n dx

1 λM1

2c+on(k(un,vn)kλ)

1 λM1

2C+on(k(un,vn)kλ)

→0 asλ∞. (2.6)

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Using the Hölder inequality and (2.3), for 1<q<3 we obtain Z

B(ρ)

|un|2dx ≤

Z

R3|un|2qdx1q

· L(B(ρ))

q1 q

≤C4kunk2H1(R3)· L(B(ρ))

q1 q

≤C4·2C· L(B(ρ))q

1

q →0 asρ∞, (2.7)

where C4 = C4(q) is a positive constant. Setting θ = 3(α+β2)

2(α+β) and using the Gagliardo–

Nirenberg inequality, we obtain Z

Bρc

|un|α+βdx≤CZ

Bcρ

|∇un|2dx

(α+β)θ

2 ·

Z

Bcρ

|un|2dx

(α+β)(1θ) 2

≤C5k(un,vn)k(α+β)θ

λ ·

Z

A(ρ)

|un|2dx+

Z

B(ρ)

|un|2dx(α+β)(21θ)

≤C6Z

A(ρ)

|un|2dx+

Z

B(ρ)

|un|2dx(α+β)(21θ)

→0 asλ,ρ(by (2.6) and (2.7)). (2.8) Similarly,

Z

Bcρ

|vn|α+βdx ≤ε forλ,ρlarge. (2.9) At last, using the Hölder inequality, (2.8) and (2.9) we have that

lim sup

n Z

Bcρε

|un|α|vn|βdx≤lim sup

n

Z

Bcρε

|un|α+βdxα+αβZ

Bcρε

|vn|α+βdxα+ββ

ε.

This concludes the proof of Lemma2.3.

The following Brézis–Lieb type lemma is proved in [5, Lemma 4.2].

Lemma 2.4. Let{(un,vn)} ⊂E be a sequence such that(un,vn)*(u,v)weakly in E. Then we have Z

R3|un|α|vn|βdx−

Z

R3|un−u|α|vn−v|βdx=

Z

R3|u|α|v|βdx+on(1). Lemma 2.5. Letλ>0be fixed and{(un,vn)}is a(PS)c-sequence ofIλ. Then

(i) up to a subsequence(un,vn)*(u,v)in E with(u,v)being a weak solution of(K)λ; (ii) {(un−u,vn−v)}is a(PS)d-sequence forIλ with d=c− Iλ(u,v).

Proof. (i) Since{(un,vn)}is bounded in E(see Lemma 2.2(i)), then there is a subsequence of {(un,vn)}such that(un,vn) * (u,v)in Eas n → ∞. In order to see that(u,v) is a critical point of Iλ, we recall that (un,vn) * (u,v)in E, (un,vn) → (u,v)for almost everyx ∈ R3, (un,vn)→(u,v)inLsloc1 (R3)×Lsloc2 (R3), 2≤ s1,s2 <6. It is easy to see that for any(ϕ,ψ)∈ E, we have

hIλ0(u,v),(ϕ,ψ)i= lim

nhIλ0(un,vn),(ϕ,ψ)i=0.

Therefore(u,v)is a critical point ofIλ.

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(ii) Let(uen,ven) = (un−u,vn−v). Now we verify that

Iλ(uen,evn) =c− Iλ(u,v) asn→ (2.10) and

Iλ0(uen,ven)→0 asn→∞. (2.11) By the Brézis–Lieb lemma, we have that

k(uen,evn)k2λ =k(un,vn)k2λ− k(u,v)k2λ+on(1), Z

R3|∇uen|2dx2

=

Z

R3|∇un|2dx2

Z

R3|∇u|2dx2

+on(1), Z

R3|∇evn|2dx2

=

Z

R3|∇vn|2dx2

Z

R3|∇v|2dx2

+on(1). To show (2.10) we observe

Iλ(uen,ven) = 1 2

Z

R3(a|∇uen|2+λV(x)|uen|2)dx+ 1 2

Z

R3(a|∇evn|2+λW(x)|ven|2)dx + b

4 Z

R3|∇uen|2dx2

+

Z

R3|∇evn|2dx2

2 α+β

Z

R3|uen|α|evn|βdx

=Iλ(un,vn)− Iλ(u,v) +on(1)

+ 2

α+β Z

R3|un|α|vn|βdx−

Z

R3|u|α|v|βdx−

Z

R3|uen|αevn|βdx

. (2.12) From Lemma2.4, R

R3|un|α|vn|βdx−R

R3|u|α|v|βdx−R

R3|uen|α|evn|βdx → 0 as n → ∞. Thus from (2.12) we obtain (2.10).

In order to show (2.11), let(ϕ,ψ)∈ E. We note that

hIλ0(uen,evn),(ϕ,ψ)i=hIλ0(un,vn),(ϕ,ψ)i − hIλ0(u,v),(ϕ,ψ)i − α+β

Z

R3|uen|α2|ven|βuenϕdx

α+β

Z

R3|uen|α|ven|β2venψdx+ α+β

Z

R3|un|α2|vn|βunϕdx +

α+β Z

R3|un|α|vn|β2vnψdx− α+β

Z

R3|u|α2|v|βuϕdx

α+β

Z

R3|u|α|v|β2vψdx. (2.13)

SinceIλ0(un,vn)→0 andun →u, vn→vinLs(R3)(2≤ s<6), we have

nlim sup

kϕkλ,V1

Z

R3

|uen|α2|ven|βuen− |un|α2|vn|βun+|u|α2|v|βu

ϕdx=0, (2.14)

nlim sup

kψkλ,W1

Z

R3

|uen|α|ven|β2ven− |un|α|vn|β2vn+|u|α|v|β2v

ψdx=0. (2.15) Thus combining (2.13)–(2.15) we obtain that

nlimhIλ0(uen,ven),(ϕ,ψ)i= 0, ∀(ϕ,ψ)∈ E, which implies (2.12) and this completes the proof of Lemma2.5.

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3 Proof of the main results

We begin with the following lemma.

Lemma 3.1. Suppose that(A1)and(A2)hold. Then for any C0 >0, there existsΛ0>0such thatIλ satisfies the(PS)c-condition for all c≤C0andλΛ0.

Proof. Let c0 > 0 be given by Lemma2.2 (ii) and choose ε > 0 such that ε < c0(α+β)

α+β2. Then for given C0 > 0, we chooseΛε > 0 and ρε > 0 as in Lemma2.3. We claim thatΛ0 = Λε is just required in Lemma3.1. Let{(un,vn)} ⊂ Ebe a (PS)c-sequence of Iλ(u,v)with c ≤ C0 and λΛ0. By Lemma 2.5, we may suppose that (un,vn) * (u,v) weakly in E and then {(uen,ven)}={(un−u,vn−v)}is a(PS)d-sequence ofIλ withd= c− Iλ(u,v). We claim that d = 0. Arguing by contradiction, assume thatd 6= 0. Lemma 2.2(ii) implies thatd ≥ c0 > 0.

Since(uen,evn)is a(PS)d-sequence ofIλ, we have

Iλ(uen,ven) =d+on(1), Iλ0(uen,evn) =on(1). Then we get

d+on(1)−1

2on(k(un,vn)kλ) =Iλ(uen,ven)−1

2hIλ0(uen,ven),(uen,ven)i

=−b

4(Υ2(uen) +Υ2(ven)) +

1− 2 α+β

Z

R3|uen|α|ven|βdx

1− 2 α+β

Z

R3|uen|α|ven|βdx, (3.1) from which we deduce that

nlim Z

R3|uen|α|ven|βdx ≥d

1− 2 α+β

1

α+β

α+β−2c0. (3.2) On the other hand, Lemma2.3implies

lim sup

n Z

Bcρε

|uen|α|evn|βdx≤ε< c0(α+β) α+β−2.

This implies (uen,evn) * (u,v) in E with (u,v) 6= (0, 0), which is a contradiction. Therefore d=0 and it follows from (2.3) that

nlimk(uen,evn)k2λ ≤2d=0,

hence(uen,ven)→(0, 0)inE, that is,(un,vn)→ (u,v)inE. This completes the proof of Lemma 3.1.

The following lemma implies thatIλpossesses the mountain-pass geometry.

Lemma 3.2. The functionalIλsatisfies the following conditions.

(i) There existρ,η>0such thatIλ(u,v)≥ηfor allk(u,v)kλ =ρ.

(ii) There exists(u0,v0)∈ Bcρ(0)such thatIλ(u0,v0)<0.

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Proof. (i) By (2.2) we have that Iλ(u,v) = 1

2k(u,v)k2λ+ b

4 Υ2(u) +Υ2(v)2 α+β

Z

R3|u|α|v|βdx

1

2k(u,v)k2λ−Ck(u,v)kα+β

λ , (3.3)

sinceα+β>2, we can choose someη>0, ρ>0 such thatIλ(u,v)≥ηfork(u,v)kλ = ρ.

(ii) We note that for eachλ > 0, Iλ(0, 0) = 0. Furthermore, for (u,v) ∈ E\ {0, 0}, since α+β>4, we get that

Iλ(t(u,v)) = t2

2k(u,v)k2λ+ bt

4

4 (Υ2(u) +Υ2(v))− 2t

α+β

α+β Z

R3|u|α|v|βdx

→ − as t → +∞. Hence, we can choose t0 > 0 large enough such that kt0(u,v)kλ > ρ and Iλ(t0(u,v))<0. Let(u0,v0) =t0(u,v), then (ii) holds.

Now we give the proof of Theorem1.1.

Proof of Theorem1.1. By Lemma 3.2, the functional Iλ satisfies the mountain-pass geometry, then using a version of the mountain-pass theorem without (PS) condition, there exists a (PS)- sequence{(un,vn)} ⊂Esatisfying

Iλ(un,vn)→cλandIλ0(un,vn)→0.

Moreover, by Lemma2.2(i),{(un,vn)}is bounded inE. Then, up to a subsequence,(un,vn)* (u,v)weakly in Eand(un,vn) → (u,v)for almost everyx ∈ R3. By Lemma3.1, there exists λ > 0, such that(un,vn) → (u,v)inEforλλ. Furthermore, by Lemma2.5we have that Iλ0(u,v) =0. By Lemma2.1, we know that(u,v)6= (0, 0), then(u,v)∈ Nλ, and using Fatou’s lemma we get

Iλ(u,v) =Iλ(u,v)−1

4hIλ0(u,v),(u,v)i

= 1

4k(u,v)k2λ+ 1

2− 2 α+β

Z

R3|u|α|v|βdx

≤lim inf

n

1

4k(un,vn)k2λ+ 1

2− 2 α+β

Z

R3|un|α|vn|βdx)

=lim inf

n

Iλ(un,vn)−1

4hIλ0(un,vn),(un,vn)i

=cλ.

Hence,Iλ(u,v) ≤cλ. On the other hand, from the definition ofcλ, we havecλ ≤ Iλ(u,v). So, Iλ(u,v) =cλ, that is(u,v)is a ground state solution of problem(K)λ.

To prove Theorem1.2we need the following version of the symmetric mountain-pass the- orem [2].

Theorem 3.3. Let X be a real Banach space and W ⊂ X a finite dimensional subspace. Suppose that J ∈ C1(X,R)is an even functional satisfying J(0) =0and

(a) there exists a constantρ>0such that J|∂B

ρ(0) ≥0;

(b) there exists M0 >0such thatsupzWJ(z)< M0.

If J satisfies(PS)c for any 0 < c < M0, then J possesses at leastdimW pairs of nontrivial critical points.

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Proof of Theorem1.2. Obviously,Iλ(u,v)is an even functional. Givenk∈ N, we set W =span{(φ1,φ1), . . . ,(φk,φk)},

where φi is the eigenfunction corresponding to the i-th eigenvalue of (−∆,H01())and Ωis defined in assumption(A1), then dimW = k. Since all norms in a finite dimensional space are equivalent, for eachi=1, . . . ,k, we have that

t→+limIλ(t(φi,φi)) = lim

t→+ at2 Z

|∇φi|2dx+ bt

4

2 Z

|∇φi|2dx 2

2t

α+β

α+β Z

|φi|α+βdx

!

=−

uniformly in λ. SinceW has finite dimension we obtain Mk > 0, independent ofλ > 0, such that

sup

(u,v)∈W

Iλ(u,v)< Mk.

Moreover, similar to the proof of Lemma 3.2(i) we may obtainρ > 0, independent ofλ > 0, such that

Iλ(u,v)≥0 for k(u,v)kλ =ρ.

In view of Lemma 3.1, there exists Λk > 0 such that Iλ satisfies (PS)c for any c ≤ Mk and λΛk. Thus, for any fixedλΛk we may apply Theorem3.3to obtainkpairs of nontrivial solutions. Theorem1.2is proved.

Acknowledgements

The authors would like to thank the referee for carefully reading the manuscript and making valuable comments and suggestions. This research was supported by Educational Commission of Hubei Province of China (D20142702).

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