Existence of Solutions for an Indefinite KirchhoffEquation with New Sublinear Nonlinearities

CHEN Yusong(陳玉松),SHEN Zihui(申子慧),ZHU Xincai(朱新才)

(1.Department of Basic Education,Shangqiu Institute of Technology,Shangqiu 476000,China;2.School of Mathematics and Statistics,Xinyang Normal University,Xinyang 464000,China)

Abstract: In this paper,we study an indefinite Kirchhoffequation.Under the new sublinear growth condition and some suitable conditions,the existence of nontrivial solutions is obtained by using variational mathods.

Key words: Sublinear;(PS)-condition;Variational method

1.Introduction and Main Results

In this paper,we are concerned with the existence of solutions to the following nonlinear Kirchhoffequation:

wherea,b>0,f(x,u)∈C(R3×R,R).

In the past two decades,the Kirchhoffequations have been studied extensively due to its strong physical background.We refer the readers to [1-12,14-15,7-22]and the references therein.Recently,considerable attention has been paid to the solutions of sublinear Kirchhofftype equations.Specially,YE and TANG[20]proved the existence of infinitely many solutions for the equation similar to (K) withf(x,t)=γξ(x)|t|γ?2t,whereγ ∈(1,2) andξ(x)∈.Then,DUAN[3]and ZHAO[22]proved infinitely many solutions with the more general sublinear nonlinearities(where,f(x,t)≤)by using the genus properties in critical point theory(see[13]).Very recently,WANG and HAN[17]studied the following Kirchhoffequation

wherea,b>0 andN ≥3.Without any growth conditions for the nonlinear termf(x,u)onuat infinity,they obtained the existence of multiple solutions of equation using the symmetric mountain pass theorem,as follows:

Theorem 1.1[17]Suppose that the following conditions are satisfied,

(V) infRN V(x)>0 and there are constantsr >0,α>Nsuch that,for anyb>0,

wherem(·) denotes the Lebesgue measure;

(f1) There exists a constantδ >0 such thatf(x,t)∈C(RN ×[?δ,δ],R) andf(x,?t)=f(x,t) for all|t|≤δandx ∈RN;

(f2) There is a ballBr0(x0) such that

(f3) There is a constantτ >0 and a functiong(t)∈C([?τ,τ],R+) such that|f(x,t)| ≤g(t) for all|t|≤τandx ∈RN.

Then the equation (KE) possesses a sequence of weak solutions{un}inX ∩L∞(RN) asn →∞.

Under the coercive condition (V),the authors obtained a new compact embedding theorem as follows:

Lemma 1.1[17]Suppose that (V) holds.ThenXis compact embedded intoLp(RN)forp ∈[1,6).

In the same year,LI and ZHONG[9]proposed the following conditions:

(V0) is weaker than (V).Whenf(x,t) satisfied (f1)-(f3),they obtained infinity many weak solutions for the equation(K)and showed that these solutions tend to zero by a variant of the symmetric mountain pass lemma.It is worth noting thatf(x,t) is only locally defined for|t|small.

Motivated by the above papers,we consider some new sublinear nonlinearities onf(x,t)of the equation (K),which differ from previous studies (where,f(x,t) is not symmetrical and not integrable inx),and establish the existence of nontrivial solution.

Before stating our result,we use a continuous functional space F such that,for anyθ(t)∈F,there exists constants0>0 such that

Denote the usualLq-norm with the norm‖·‖qfor 1≤q ≤+∞.Cstands for various positive constants.

To get the critical point of the corresponding functional,we consider the following function space

equipped with the inner product and the norm

From Lemma 1.1,for anyp ∈[1,6),there exists a constantsp >0 such that

By (1.6),we have

In order to prove that the infimum is achieved,we consider a minimizing sequence{un}?Xsuch that

Going if necessary to a subsequence,we assumeun ?inX.By Lemma 1.1 and the Br′ezis-Lieb lemma[16],we have

Moreover,for the nonlinearityf,we make the following conditions:

Now,we start our main result of this paper.

Theorem 1.2Suppose that the conditions (V) and (F1)-(F4) hold.Then the equation(K) has at least one nontrivial solution.

Remark 1.1(F1)was introduced by WANG and XIAO in[18]to show the existence of periodic solutions for subquadratic second order non-autonomous Hamiltonian systems.As far as we know,this is the first time to use in the Kirchhoffequation (K).

Remark 1.2Let

whereQ(t)∈C(R),H(x)∈C(R3)such thatF(x,t)is aC1class function.Simple calculation shows thatF(x,t) satisfies the conditions (F1)-(F4),but not satisfies (f2).

2.Preliminaries

Definition 2.1[16]LetXbe a Banach space,I ∈C1(X,R) andc ∈R.The functionIsatisfies the(PS)condition if any sequence(un)?Xsuch that{I(un)}is bounded,I′(un)→0 has a convergent subsequence.

Combining (2.1) with (F1),we have

which is a first order linear ordinary differential equation.By solving (2.3),we can get

Sinceg(s)≤0,G(r|t|)>0,we have

Hence,we obtain

This,together with (F4),completes the proof.

Lemma 2.2Suppose that (V),(F1),(F3) and (F4) hold.Then,for anyε >0 andx ∈R3,t ∈R,there existsdε >0,such that

ProofFrom Lemma 2.1 and (F4),there existsd3>0 such that

It follows from (F1) that

We can easily obtain (2.7) from (F3) and (2.10).Hence,we also obtain (2.8) from (2.7).

It follows from (V) and (F1)-(F4) that the corresponging energy functional onXgiven by

By standard arguments,we can easily obtain thatIis of classC1(X,R),and the critical points ofIare solutions of equation (K).In order to obtain the critical points ofI,we also need to use the following lemma.

Lemma 2.3[16]LetXbe a real Banach space andI ∈C1(X,R) satisfy the (PS)-condition.IfIis bounded from below,thenc=infX Iis a critical value ofI.

3.Proof of Theorem 1.2

Now we can prove our main result.

Proof of Theorem 1.2By the definition ofG,we have

By the condition (V) and (3.1),there existsl∞≥t∞such that

First,we show thatIis bounded from below onX.By (1.6),(2.11),(3.2),Lemma 2.1 and Lemma 2.2,we have

which implies thatI(un)→+∞as‖un‖→+∞.Consequently,Iis bounded from below.

Next,we prove thatIsatisfies the (PS)-condition.Assume that{un}is a sequence such that{I(un)}is bounded andI′(un)→0 asn →+∞.Then by (3.3),there exists a constantA >0 such that‖un‖2≤‖un‖≤A,n ∈N.So passing to a subsequence if necessary,it can be assumed thatun ?u0inX.By Lemma 1.1,un →u0inLp(1≤p<6).

Thus,we have proved that

In addition,sinceun ?u0,then we have

asn →+∞.In view of the definition of weak convergence,we have

It follows from (3.4),(3.5),(3.6) and (3.7) that‖un0‖→0 asn →∞.Hence,Isatisfies the (PS)-condition.By Lemma 2.3,we know thatc=infX I(u) is a critical value ofI,that is,there existssuch thatI()=c.

Finally,we show that=0,Letu?∈X{0}(where,‖u?‖2=1,‖u?‖=),andt >0 be small enough,then,by (F2),and (1.10),we have

Since 1≤μ <6,it follows from (F2) thatI(tu?)<0 ford1>0 small enough.Hence,I()=c<0,therefore,is a nontrivial solution of equation (K).The proof is finished.