(School of Mathematics and Statistics, Henan University of Science and Technology,Luoyang 471023, China)

Abstract: In this paper, we study a coupled fourth-order system of Kirchhoff type.Under appropriate hypotheses of Vi(x) for i=1,2, f and g, we obtained two main existence theorems of weak solutions for the problem by variational methods. Some recent results are extended.

Keywords: A coupled fourth-order system of Kirchhoff type; Positive solutions; Lack of compactness; Variational methods

§1. Introduction

We study a coupled fourth-order system of Kirchhoff type as follows

whereai>0,bi>0,λi>0 are constants for i=1,2,k>0 is a parameter,Vi(x)∈C(RN×R,R)fori=1,2,f ∈C(RN×R,R) andg ∈C(RN×R,R). In order to prove the existence and multiplicity of solutions for system (1.1), we make the following assumptions forVi(x) (i=1,2),fandg.

(V1)whered1is a constant. This is a constantK1>0 such that the setis nonempty andwheremeasis the Lebesgue measure in RN.

(V2)whered2is a constant. There exists a constantK2>0 such that the setis nonempty andwheremeasis the Lebesgue measure in RN.

(A1)and|g(x,v)|≤C(1+|v|p?1), where 2?=2N/(N ?4) ifN>4 and 2?=∞ifN ≤4.

(A2) For?x∈RN,|f(x,u)|=o(|u|) as|u|→0, and|g(x,v)|=o(|v|) as|v|→0.

(A3) This isμ1>4 such thatμ1F(x,u)≤uf(x,u) for?(x,u)∈RN×R+and there existsμ2>4 such thatμ2G(x,v)≤vg(x,v) for?(x,v)∈RN×R+.

(A4)

(A5) There existsμ1>4 such thatis monotonically increasing on (0,+∞) and there existsμ2>4 such thatis monotonically increasing on (0,+∞).

(A6)f(x,?u)=?f(x,u)for?(x,u)∈RN×R+andg(x,?v)=?g(x,v)for?(x,v)∈RN×R+.

The biharmonic equation has been widely concerned by mathematicians and physicists.There are some interesting results can be found in [4,6,7,10,14,15]. In [14], using variational methods, Yin and Wu obtained the existence and multiplicity results for the following fourth order elliptic equations

In [15], Zhang, Tang and Zhang studied equations (1.2) in asymptotically linear case and obtained the existence of ground state solutions. Ifa1=a2=1,b1=b2=0,λ1=λ2=1, andk=0, problem (1.1) is related to problem (1.2), which is used to describe the static deflection of an elastic plate in a fluid and the traveling wave in suspension bridge. In [7], by variational methods, Liu, Chen and Wu obtained the existence and multiplicity results for the following problem

In [6], Liang, Zhang and Luo considered a class of singularly perturbed fourth order equations with critical nonlinearity and proved the existence and multiplicity of solutions. Ifk=0 andλ1=λ2=1, problem (1.1) is related to the following fourth order elliptic equations of Kirchhoff type

In recent years, a lot of scholars were concerned about the problem (1.4). Some meaningful works can be seen in [9,11–13,16]. In [11], Song and Chen proved the multiplicity of solutions for equations (1.4) by using symmetric mountain pass theorem, whereN ≤7,a,b>0,V(x)∈C(RN,R)andf ∈C(RN×R,R)satisfies the Ambrosetti-Rabinowitz condition. If ? is a bounded domain in RNandV(x)≡0, problem (1.4) can reduce to the following Kirchhoff type problem

The system (1.5) is related to the stationary analogue of the Kirchhoff type problem

Since it is considered as a good approximation for describing nonlinear vibrations of beams or plates (see [1,2]), problem (1.6) was concerned by some physics and engineering scholars. In [9],Ru and An considered the following a class of biharmonic equations of Kirchhoff type

where ? is a bounded domain inandThey obtained existence and multiplicity of solutions by variational methods.

Compared with problem (1.3), the nonlocal terms and couplings in problem (1.1) which will make it more difficult to overcome the lack of compactness and prove the (PS) condition. As we mentioned above, all results in the literatures are concerned the single equation. A natural question is can we extend the results to the system of fourth-order.

Inspired by the works of these scholars, we aim to study the existence and multiplicity of positive solutions for problem (1.1). Our main results are as follows.

Theorem 1.1.If(V1)?(V2)and(A1)?(A4)hold, there existsΛ?large enough such that problem (1.1) has at least one positive solution for all λ1,λ2>Λ?. In addition, if(A6)is satisfied, then problem (1.1) has infinitely many positive solutions.

Theorem 1.2.If(V1)?(V2),(A1)?(A2)and(A4)?(A5)are satisfied, then there existsΛ?large enough such that problem (1.1) has at least one positive solution for all λ1,λ2>Λ?. In addition, if(A6)is satisfied, then problem (1.1) has infinitely many positive solutions.

§2. Preliminaries

andOnH, for any (u1,v1),(u2,v2)∈H, we define an inner product as follows

and then the corresponding norm isLetX ?H. OnX, for all(u1,v1),(u2,v2)∈X, define its inner product be

and the corresponding norm beLet

with inner product and the norm

DefineE=E1×E2with inner product

and the corresponding norm isLetf ∈Lp(?) for 1≤p≤∞. Setf+=max{f,0}andf?=min{f,0}, thenf=f++f?. Denoteas theLp-norm off.LetLp(?)×Lp(?) be the Cartesian product [5] of the twoLp(?) spaces, andfor any (f,g)∈Lp(?)×Lp(?).

Remark 2.1.Let the energy functional of equations (1.1) be

Obviously, under the assumptions(V1),(V2)and(A1)?(A6), we obtain I ∈C1(E,R). Moreover,for any(?,ψ)∈E, we have

Lemma 2.1.Under the assumptions(V1)and(V2), we obtain thatiscontinuous for any s∈[2,2?)if λ=min{λ1,λ2}≥1. Then, there exists a constant αs>0, whichis independent of λ, such thatfor each(u,v)∈E.

Proof.For any (u,v)∈E, we have

whereSdenotes the best Sobolev constant for the imbedding ofD2,2(RN) inL2?(RN),

andβ=max{β1,β2}. Thus, these two norms are equivalent. Next, we showis equivalent to the standard product norm onH. For any (u,v)∈H, we have

In fact,

Obviously, we haveThen, the embeddingis continuous for anys∈[2,2?) andλ=min{λ1,λ2}≥1. Thus there exists a constantαs>0, which is independent ofλ, such thatfor each (u,v)∈E.

Let’s first state two lemmas, and we’ll use them to prove our conclusions.

Lemma 2.2.[8, Theorem 2.2] Let E be a real Banach space and I ∈C1(E,R). Moreover, I satisfies(PS)condition. Then I has a critical value η ≥α if I(0)=0and

(I1)there exists constants ρ,α>0such that I?Bρ(0)≥α,

(I2)there exists ansuch that I(e)≤0.

Lemma 2.3.[8, Theorem 9.12] Let E be a infinite dimension Banach space. Let I(0)=0,I ∈C1(E,R)be even and satisfy(PS)condition. Suppose thatwhere V is finitedimensional. If I satisfies the following conditions:

(I3)there are constants ρ,α>0such thatand

(I4)for any finite dimensional subspacethere exists a positive constantsuch that

Then I has an unbounded sequence of critical values.

§3. Proof of main results

In order to prove our main theorems, the following lemmas are given.

Lemma 3.1.[3] Let??RN be an open set. Set the functional f ∈C(?×R,R)such that|f(x,u)|≤C3(|u|r+|u|s)for some C3>0and1≤r0and un(x)→u(x)a.e. in x∈?, where s≤p<∞, r ≤q<∞, q>1. Letbesuch that χ(t)=1for t≤1, χ(t)=0for t≥2,Rn is a positive constant sequence withRn →∞as n→∞. Set the space Lp(?)∩Lq(?)with the norm

and the space Lp(?)+Lq(?)with the norm

Then, there is a sequencein Lp(?)∩Lq(?), and

Lemma 3.2.Ifin E, then we can take a subsequencesuch thatin E. And if {(un,vn)} is a(PS)c sequence, i.e.,

then,

Proof.Letbe such thatχ(t)=1 fort≤1,χ(t)=0 fort≥2, Rnis a positive constant sequence with Rn →∞asn→∞.IfinE, we may choose a subsequencesuch that, asn→∞,inE. In fact,from the definition of the cut-off functionχin Lemma 3.1,then?ε>0, there is an enough large integernsuch thatFor any (u,v)∈E,then there exists a constantρ1=ρ(ε)>0 such that

and

whereThen,

Thus, asn→∞, we haveby Lebesgue dominated converge theorem.UsinginE, we obtain

Using lower semi-continuity of the norm andwe get

and

ForinE, passing to a subsequence, we may suppose thatun →uinfor allt∈[2,2?) andun(x)→u(x) a.e. in RN. If (A1) and (A2) are satisfied, then for anyε1>0, there is a constantC(ε1)>0 such that

and

Setr=2,s=p, andq=2 in Lemma 3.1, then we get

Similarly, we obtain

Obviously,

Sinceis bounded andonH2(RN), theninLq(RN).Similarly,inLq(RN). By, we have

Thus,

From (3.1)-(3.4) and (3.6)-(3.8), we haveNext, we will show

for any (?,ψ)∈E. Therefore, for any (?,ψ)∈E, we get

Based on the previous discussions, we have

And, asn→∞,

Thus, we getasn→∞.

Lemma 3.3.Let(A1)?(A3)be satisfied. Thenforsomeand all(x,u)with |u| sufficiently large.

Proof.Suppose that (A1)?(A3) are satisfied, then there areCi>0,i=1,2 such thatfor all (x,u)∈RN×R+. Obviously, we haveforp∈(2,2?). For a fixedif|u|≥1, then|f(x,u)|≤(C1+C2)|u|p?1. Set R≥1 large enough such thatwhenever|u|≥R. Then,for|u|sufficiently large,Therefore, we get

Remark 3.1.Similar to Lemma 3.3, if(A1)?(A3)hold, thenH?(x,v)for someand all(x,v)with |v| sufficiently large.

Lemma 3.4.Suppose(V1)?(V2)and(A1)?(A3)are satisfied. Then there existsΛ?>0such that, for any c∈R, I satisfies(PS)c condition for all λ>Λ?.

Proof.Let{(un,vn)}be a (PS)csequence, then, by (A3), fornsufficiently large, we get

whereμ=min{μ1,μ2}. It means that{(un,vn)}is bounded inE. Passing to subsequence, we can assume thatAccording to Lemma 3.2, there existssuch that(u,v)in E. SetthenBy (V1) and (V2), we get

Letwhereτhas been defined in Lemma 3.3, then 2

whereaq>0. By Lemma 3.2, we obtain

Sincenext, we showIn fact

By (A3), we haveThus,

By (3.10) and (3.11), we get

Therefore, by Lemma 3.3 and H¨older’s inequality, we obtain

Similarly, by Remark 3.1, we get

Then, by (3.9),

On the other hand,

and

Hence,

Then,

Thus, there is Λ?large enough such that

for anyλ1>Λ?andλ2>Λ?. Then (ωn,τn)→0 inE, for allλ1,λ2>Λ?. Sincethus, we get (un,vn)→(u,v) inE.

Proof of Theorem 1.1.Similar to the way to get (3.5), we obtain that there is a constantC(ε2)>0 such that

for anyε2>0. By (3.5), (3.12) and Sobolev embedding theorem, forand smallρ>0, we get

whereandC(ε)=max{C(ε1),C(ε2)}. Thus,

By Lemma 2.1, we knowSinceL2(RN)×L2(RN) is a separable Hilbert space,Ehas a countable orthogonal basis{ej}. LetE=Ek ⊕Zk,whereEk=span{e1,e2,···,ek}andZk=(Ek)⊥. Obviously,Ekis a finite dimension space andI?Bρ(0)∩Zk ≥α>0. Ifis any subspace ofE, then there is a positive integermsuch thatBecause all norms are equivalent in a finite dimension space, there exists a constantη>0 such thatfor any (u,v)∈Em, whereμ=min{μ1,μ2}. According to (3.5) and (A3)?(A4), there isC(ε1)>0 such that

Similarly, by (3.12) and (A3)?(A4), there existsC(ε2)>0 such that

Thus,

for each (u,v)∈Em. Sinceμ>4, there exists a large enough R>0 such thatI<0 onTherefore, we can find ane∈Esuch thatsuch thatI(e)<0. By (2.1), we haveI(0,0)=0. By Lemma 2.2 and Lemma 3.4, we obtain thatIhas a critical valueζ ≥α. In a word, the problem (1.1) has a nontrivial solution (u,v) inE. Since (u,v) solves

by maximum principle, we getu>0 andv>0. We will give the proof of multiplicity of solutions for problem (1.1). According to (A1), (A2) and (A6),F(x,u) andG(x,v) both are even. From(2.1), we can obtainIis even. Thus, by Lemma 2.3, we get problem (1.1) has infinitely many positive solutions.

Proof of Theorem 1.2.Similar to the proof in [7], we obtain that (A5) implies (A3). Thus we omit the proof. According to Theorem 1.1, Theorem 1.2 holds.

§4. Conclusions

In this article, we are concerned with the existence of solutions for problem (1.1). By using variational methods, we get two existence theorems of weak solutions for problem (1.1). First, if(V1), (V2) and (A1)?(A4) hold, we can obtain there exists Λ?large enough such that problem(1.1) has at least one positive solution for allλ1,λ2>Λ?. Moreover, if (A6) hold, then problem(1.1) has infinitely many positive solutions. Second, if (V1)?(V2), (A1)?(A2) and (A4)?(A5)hold, we can get there exists Λ?large enough such that problem (1.1) has at least one positive solution for allλ1,λ2>Λ?. In addition, if (A6) hold, then problem (1.1) has infinitely many positive solutions.