THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION∗

Gongbao LI(李工宝) Yahui NIU(牛亚慧)

Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics,Central China Normal University,Wuhan 430079,China

E-mail:ligb@mail.ccnu.edu.cn;yahuniu@163.com

Abstract In the present paper,we consider the nonlocal Kirchho ffproblemwhere a,b>0,10 is a parameter.Under some assumptions on Q(x),we show the existence and local uniqueness of positive multi-peak solutions by Lyapunov-Schmidt reduction method and the local Pohozaev identity method,respectly.

Key words Kirchho ffequations;multi-peak positive solutions;local uniqueness;local Pohozaev identity;Lyapunov-Schmidt reduction

1 Introduction and Main Results

In this paper,we consider the existence and local uniqueness of muti-peak positive solutions to the following singularly perturbed Kirchho ffproblem

where a,b>0,10 is a parameter,Q:R3→ R satis fies the following assumptions

(Q2)there exist θ∈ (0,1)and k(k ≥ 2)distinct points{b1,···,bk} ⊂ R3such that for everyand

(Q3)There exist m>1,η>0,k∈N,bj=(bj,1,bj,2,bj,3)∈R3,cj,i∈R with cj,i6=0 for each i=1,2,3 and j=1,···,k such that

where x=(x1,x2,x3)∈R3.

To be precise,we first give the de finition of k-peak solutions of eq.(1.1)as usual.

De finition 1.1Let k∈N,bj∈R3,1≤j≤k.We say that uǫ∈H1(R3)is a k-peak solution of(1.1)concentrated at{b1,···,bk},if

(i)uǫhas k local maximum points xjǫ∈ R3,j=1,2,···,k,satisfying

as ǫ→ 0 for each j;

(ii)for any given τ>0,there exists R ≫ 1,such that

(iii)there exists C>0 such that

Problem(1.1)and its variants were studied extensively in the literature.To extend the classical D’Alembert’s wave equations for free vibration of elastic strings,Kirchho ff[20]proposed for the first time the following time dependent wave equation

Bernstein[4]and Pohozaev[25]studied the above type of Kirchho ffequations quite early.Since Lions[22]introduced an abstract functional framework to this problem,much attention was received,such as[1,11]and the references therein.

Problem(1.1)is a typical case of the equation

where V:R3→R is a bounded continuous function.First,we review some known results on(1.2)in the case where f(x,u)=f(u).He and Zou[18]considered problem(1.2)where V is assumed to satisfy the global condition of Rabinowitz[26]

and f:R→R is a nonlinear function with subcritical growth of type uqfor some 3

They proved the existence of multiple positive solutions to(1.2)for ǫ sufficiently small.Later,Wang et al.[27]extended the result of[18]to the case of critical growth,they established some existence and nonexistence results for(1.2)where f(u)∼ λg(u)+u5,V and g satisfy similar conditions as that of[18].By a penalization method,He,Li and Peng[17]improved an existence result of Wang et al.[27]by allowing that V only satis fies a local condition:there exists a bounded open set Ω⊂R3such that

Afterwards,by introducing a new manifold and applying a new approximation method of[13],He and Li[16]proved the existence of solutions for ǫ sufficiently small to problem(1.2)where f(u)∼uq+u5,1

On the other hand,if a=1,b=0,R3is replaced by RN,(1.1)reduces to the problem

where 1

admits a unique positive solution(up to translations)which is also nondegenerate(see e.g.[3,10]).Basing on this uniqueness and nondegeneracy property,by using the Lyapunov-Schmidt reduction method,Cao et al.[7]and Cao et al.[6,8],Noussair and Yan[24]proved the existence of solutions to eq.(1.5)for ǫ>0 sufficiently small on bounded domains and in the whole space,respectively.As far as we know,the results on the uniqueness of solutions which have the concentration phenomena are few,we only mention the recently work by Cao[5],which proved the local uniqueness of multi-peak positive solutions to equation −ǫ2∆u+V(x)u=up.For more work concerning the uniqueness of solutions with the concentration phenomena,one can also refer to[7,14].

We remark that all the results of Kirchho ffequation mentioned above were derived by variational methods.To deal with nonlinearity of type uqfor q in different subintervals of(1,5],different variational methods have to be applied.Recently,Li et al.[21]and Luo et al.[23]dealt with(1.2)(where f(x,u)=up)in the case of 1

Motivated by the work of[21]and[24],we study the existence and local uniqueness of multipeak positive solutions to the problem(1.1)by Lyapunov-Schmidt reduction method.Here,by local uniqueness,it means that ifare two solutions of equation(1.1)concentrating at the same family of concentration points,thenfor ǫ sufficiently small.

Before stating our main results,we introduce the following notation.Denote

The energy functional corresponding to eq.(1.1)is

for u∈Hǫ,where u+=max(u,0).

We call u ∈ Hǫa(weak)solution to eq.(1.1)if for any ψ ∈ Hǫ,it holds that

Similar to the proof in[23,Proposition 2.3],we can prove that the limiting problem of eq.(1.1)is given by the following system

the positive solution of(1.8)(we denote it by(U1,···,Uk))is unique and each Uiis also nondegenerate in H1(R3)in the sense that

for ϕ∈H1(R3).

Our main result is as follows.

Theorem 1.2Assume that Q(x)satis fies(Q1)and(Q2).Then,for ǫ>0 sufficiently small,equation(1.1)has a k-peak solution de fined as in De finition 1.1 concentrating around bi,1≤i≤k.

Theorem 1.3Assume that Q(x)satis fies(Q1)and(Q3).If,j=1,2,are two k-peak solutions,then for ǫ sufficiently small,we have

for i=1,···,k.

Remark 1.4If we replace condition(Q2)in Theorem 1.2 by

Remark 1.5Our main result Theorem 1.2 extend the main results in[24]which considered the Schrödinger equation(1.5)to the Kirchho ffequation(1.1).

We will follow the scheme of Cao and Peng[9]to prove Theorem 1.2 by using the Lyapunov-Schmidt reduction.The main difficulties are the appearence of the nonlocal term?R?∆u,which brings more technical difficulties than Schrödinger equation(1.5)did.Moreover,when we prove the existence and local uniqueness of the multi-peak solution of(1.1),on the one hand,there are many cross iterms,and on the other hand,the limiting equation of(1.1)is in fact a system of partial differential equations(1.8),which is different from the case of singlepeak solution and Schrödinger equation(1.5),the difference of these two aspects also make the estimate more complex.

Now,we give the main idea of the proof of Theorem 1.2.Our arguments are based on the variational method.The basic idea is to use the unique positive solution of the system(1.8)as the building block to construct solutions for(1.1).Following the scheme of Luo et al.[23],we will construct solutions of the formThen we reduce the problem we are dealing with to a finite-dimensional one by a type of Lyapunov-Schmidt reduction.The nonlocal term?RR3|∇u|2?∆u brings more delicate and complicate estimates on the orders of ǫ.

Next,we give the main idea of the proof of Theorem 1.3.We will follow the idea of Cao,Li and Luo[5].More precisely,if,i=1,2,are two distinct solutions,derived as in Theorem 1.2,then it is clear that the function

Our notations are standard.We writeRu to denote Lebesgue integrals over R3,unless otherwise stated,and.We use BR(x)(andto denote open(and close)balls in R3centered at x with radius R.

By the usual abuse of notations,we write u(x)=u(r)with r=|x|whenever u is a radial function in R3.We will use C and Cj(j∈N)to denote various positive constants,and O(t),o(t)to mean|O(t)|≤C|t|and o(t)/t→0 as t→0,respectively.

The paper is organized as follows.In Section 2,we give some notations and some preliminary estimates which play a key role in the rest of the arguments.In Section 3,we prove Theorem 1.2 and in Section 4 we prove the local uniqueness results.

2 Preliminaries

Let Iǫbe given by(1.7),we can easily check that Iǫ∈ C1(Hǫ),so its positive critical points are solutions of eq.(1.1).We will restrict ourselves to the existence of critical points of Iǫof the the following form

where

To construct solutions to eq.(1.1)in form(2.1),we will follow the scheme of Cao and Peng[9],combining reduction method and variational method.We denote

and let

Let 0<δ

Note that since the unique positive solution of equation−∆u+u=Q(bi)updecays exponentially at in fi nity,we infer that

for some σ>0.

Note that if(x1,···,xk)∈ Dδ,then|xi−xj|≥ |bi− bj|/2 ≥ 2δ with i 6=j,which implies by(2.3)that

for any given r,s>0,where γ>0 is constant.

De fine

for X=(x1,···,xk)∈ R3kand ϕ ∈ Hǫ.

Expand Jǫ(X,ϕ)near ϕ =0 for each fixed X:

where Jǫ(X,0)=Iǫ(Hǫ,X),and operators hǫ,Lǫand Rǫas follows:for ϕ,ψ ∈ Hǫ,

and

For every ǫ,δ>0 sufficiently small and for every fixed X ∈ Dδ,we will prove that Jǫ(X,·):Eǫ,X→ Eǫ,Xhas a unique critical point ϕǫ,X∈ Eǫ,X,Then,for each ǫ,δ sufficiently small,we will find a critical point Xǫfor the function Fǫ:Dδ→R induced by

It is standard to verify that(Xǫ,ϕǫ,Xǫ)is a critical point of Jǫfor ǫ sufficiently small by the chain rule.Furthermore,by Bartsch and Peng[2],we have the following lemma.

Lemma 2.1There exist ǫ0>0, δ0>0 satisfying the following property:for any ǫ∈(0,ǫ0)and δ∈ (0,δ0),Xǫ∈ Dδis a critical point of the function Fǫde fined as in(2.8)if and only if

is a critical point of Iǫ.

As a result,we obtain a solution uǫ≡ Hǫ,Xǫ+ϕǫ,Xǫto eq.(1.1).

In the rest of this section,we will estimate lǫ:Hǫ→ R and Rǫ:Hǫ→ R,which will combine with the invertibility of operator Lǫbe used to prove the existence of unique critical point ϕǫ,Xof Jǫ(X,ϕ)for every fixed X ∈ Dδ.

Lemma 2.2Assume that Q satis fies(Q1)and(Q2).Then,there exists a constant C>0,independent of ǫ,δ,such that for any X ∈ Dδthere holds

for ϕ ∈ Hǫ,here θ denotes the order of Hölder continuity of Q in the neighborhood of bj,1≤j≤k.

ProofFirst,we have

To estimate h1,using(2.4)yields

To estimate h2,note that

So

Finally,combining the above estimates gives the required estimate.

Lemma 2.3There exists a constant C>0,independent of ǫ and b such that for i ∈{0,1,2},there hold

for all ϕ∈Hǫ,where R(i)ǫis the ith derivative of Rǫ.

ProofThis lemma can be proved by the similar method as that of Lemma 3.3 in[21].

Next we consider the operator Lǫde fined as in(2.6).

Proposition 2.4For ǫ small enough and X ∈ Dδ,the operator Lǫis invertible on Eǫ,Xwith uniformly bounded inverse.In other words,there exists ǫ1,δ1and ρ >0 such that for all ǫ∈ (0,ǫ1),δ∈ (0,δ1)and all X ∈ Dδ,there holds

As the proof of Proposition 2.4 is standard,we leave it in Appendix.

In the last of this section,we give the estimate of Iǫ(Hǫ,X).

Lemma 2.5Assume that Q(x)satis fies(Q1)and(Q2).Then for ǫ>0 sufficiently small,we have

where

and

and θ is the Hölder continuity of Q(x)in the neighborhood of bj,1 ≤ j ≤ k.

ProofRecall that

Since estimates(2.4)hold for i 6=j,hence,for ǫ>0 sufficiently small,we have

for some γ>0.

As Uisatis fies eq.(1.8),thussatis fies

Combining with the following elementary inequalities

we have

The required estimate follows from(2.11)and(2.12).

3 Proof of Theorem 1.2

In this section,we prove Theorem 1.2.First,we give the result of the existence of the reduction map ϕǫ,X.

Proposition 3.1For ǫ, δ sufficiently small,there exists a C1map ϕǫ,X:Dδ→ Hǫwith∈ Eǫ,Xsatisfying

Moreover,we can choose τ∈ (0,θ/2)as small as we wish such that

This proposition can be proved by the similar arguments as that of Li et al.[21]by using Lemma 2.2,Lemma 2.3 and Proposition 2.4.We omit the details.

Now we prove Theorem 1.2.

ProofFor given X ∈Dδ,by Proposition 3.1,we will find a critical point for the function Fǫde fined as in(2.8)by Lemma 2.1.We consider the minimizing problem

Suppose Xǫ∈ Dδsatis fieswe claim that Xǫis an interior point of Dδand thus Xǫis a critical point of Fǫ,then by Lemma 2.1,uǫ≡ Hǫ,Xǫ+ϕǫ,Xǫis a solution of eq.(1.1),hence Theorem 1.2 follows.

We prove the claim by a comparison argument.First,we analyze the asymptotic behavior of Fǫwith respect to ǫ.By the Taylor expansion,we have

By using the method of the proof of Lemma 4.3 in[21],we getby direct computation.Combining(2.9),(2.10),(3.1)and Lemma 2.3,we have

Let ej∈R3(j=1,···,k)with|ej|=1,ei6=ejfor i 6=j and=bj+ǫηejwith η>1 sufficiently large such that Zǫ=(,···,)∈Dδ.By the above asymptotical formula,we have

Applying the Hölder continuity of Q,noting the fact that τ≪ θ/2,we derive that

Thus,by using F(Xǫ)≤ F(Zǫ),we deduce

If Xǫ∈∂Dδ,then by assumption(Q2),we have

for some constants 00 from Proposition 2.5 and sending→0,we infer from(3.3)thatWe reach a contradiction.

This proves the claim.Hence,we finish tne proof.

Remark 3.2If Q satis fies(Q1)and(),Theorem 1.2 can be proved by the similar arguments by replacing(3.2)with

4 Local Uniqueness Results

In this section,we prove the local uniqueness results Theorem 1.3.First,we give some important estimates:we will show that the estimates ofandin(2.2)can be improved step by step by using a Pohozaev type identity and assuming that Q(x)satis fies(Q3).

The crucial Pohozaev type identity we will use is as follows.

Proposition 4.1Let u be a positive solution of eq.(1.1).Let Ω be a bounded smooth domain in R3.Then,for each α=1,2,3,there hold

here ν =(ν1,ν2,ν3)is the unit outward normal of∂Ω.

Proposition 4.1 can be proved by multiplying both sides of equation(1.1)by ∂xαu for each 1≤α≤3 and then integrating by parts.We omit the proof,see Cao,Li and Luo[5,Proposition 2.3]for the detials.

We recall an inequality:for any 2≤q≤6,there exists a constant C>0 depending only on n,a and q but independent of ǫ such that

holds for all ϕ ∈ Hǫ.For a proof,see e.g.(3.6)of Li et al.[21].

Proposition 4.2Assume that

be a solution derived as in Theorem 1.2.Then

and

ProofFor simplicity,we denote ϕ = ϕǫ,Xin the rest of this section.First,we prove(4.3)for fi x i ∈ {1,···,k}.Applying the Pohozaev-type identity(4.1)to u=uǫwith Ω =Bd(xiǫ),where 0

for any ϕ ∈ Hǫ.We have

We estimate each side of(4.6)as follows.Using(2.2),a straightforward computation givesSo

From(4.5)and(2.3),we know

for any γ>0.Using similar arguments and choosing a suitable d if necessary,we also derive

Furthermore,since Q(x)is bounded,from(4.2)and(4.5),we have

Combing the above estimate and(4.6),we obtain

for any γ>0.

Next we estimate the left side of(4.8).Substituting the form of uǫinto the integral,and recall(2.4)and(4.2),we obtain

for any given constant γ>0.By assumption(Q3),we have

Thus,by(4.8),(4.9)and(4.10),we get

The following inequality is elementary

where a,b∈R,m>1,m∗=min{m,2},the constant C is independent of a,b.

Applying(4.12)to a= ǫxα,b=− bi,α,we have

Take α = α0such thatNote also that

Thus,by(4.11),(4.13)and Hölder’s inequality,we have

which implies

Then,from(4.11),we obtain

where tαis the αth component of t for α =1,2,3.As Ui(x)is radially symmetric decreasing,we get t=0.This yields(4.3).

Next,we will use property

Recall that

summing over i in both sides of system(1.8)for Ui,we have

as uǫis a solution of equation(1.1),we get

As a result,ϕ must satisfy

with

Hence

Furthermore,by the fact that

we have

and

where γ>0 is a constant.By the similar estimate as the proof of Proposition 5.2 in[5],we have Z

Combining the above estimate with(4.16)and(4.3),we obtain

As a result,

Thus we complete the proof of Proposition 4.2.

In the rest of this section,we devoted to prove Theorem 1.3.We argue by way of contradiction.Assumeare two distinct solutions concentrating around bj(1≤j≤k)derived as in Theorem 1.2.Set

then

and

where

It is clear that

We will prove that

to obtain a contradiction.

For fixed j ∈ 1,···,k,set

To prove(4.21),we will prove thatandholds seperately.To this end,we will establish a series of results.First,we have

Proposition 4.3There holds

ProofAdding(4.19)and(4.20)together gives

Multiply ξǫon both sides of(4.22)and integrate over R3.As the terms containing b is positive,we can throw away them,so

Since Q is bounded,we have

and

where we have used(4.2).Hence

which implies the desired estimate.The proof is complete.

Next we study the asymptotic behavior of ξǫ,j.

Proposition 4.4There exist dβ∈ R,β =1,2,3 such that(up to a subsequence)

as ǫ→ 0.

ProofWe will prove that the limiting function of ξǫ,jbelongs to the kernel of a linear operator associated to Uj.It is straightforward to deduce from(4.19)that ξǫ,jsolves

We claim that ξj∈ KerLj,that is,

Then by the fact that Ujis nondegenerate,we havefor some dβ∈ R(β =1,2,3),and thus Proposition 4.4 is proved.

To deduce(4.24),we only need to show that(4.24)is the limiting equation of eq.(4.23).Observing that

where we have used(2.3),(2.4)and(4.4),so

as ǫ→ 0.For convenience,we denote

for i=1,2.Similarly,we have

Here,we have used(4.3),which implies

Combining with

implies

for j 6=l as ǫ→ 0.From the above two formulas and ξǫ,j→ ξjin,we conclude that

in H−1(R3).Also,as

we have

Finally,from(4.25)(4.26)(4.27),we obtain(4.24).The proof is completed.

Proposition 4.5Let dβbe de fined as in Proposition 4.4.Then

ProofApplying(4.1)toandwith Ω=,where d is chosen in the same way as that of(4.5),which combine with(4.4)and Proposition 4.3 implies Z

and

We obtain

where 1≤α≤3 and

By(4.7)and(4.4),we have

by choosing γ>3+2m.

By(4.28),(4.29)and Hölder’s inequality,we get

As to Aǫ,by(2.3),we have

for any given γ>0.Hence we can deduce that

Hence by the above estimate,there holds

On the other hand,as

combining with assumption(Q3),we get

as ci,α6=0.Combining with(4.3)and Proposition 4.4,we obtain

Then dα=0 for α =1,2,3,since Uiis a radially symmetric decreasing function.

Proof of Theorem 1.3By Propositions 4.4 and 4.5,we havefor any j=1,···,k.On the other hand,by using maximum principle,we can prove

we can refer to[5,Proposition 3.5]for the similar detail proof.Consequently,we got(4.21),which contradict to=1.The proof of uniqueness is completed.

Appendix

In this section,we give the proof of Proposition 2.4.

ProofWe use a contradiction argument.Assume,on the contrary,that there exist ǫn→0,δn→ 0 and Xn=(,···,)∈ Dδnand ϕn∈ En≡ Eǫn,Xnsuch that

Since the equality is homogeneous,we may assume,with no loss of generality,that=.

To deduce contradiction,for each i0=1,···,k,we introduce

and

Note also that

and let

hn∈.Substituting hninto(A2)and send n→∞,we get,by the same argument as that of[9,Appendix],that

Claim that ϕ≡0.Indeed,since

for each j=1,2,3,sending n→∞yields

Recalling that Ui0decays exponentially,we have

where ot(1)→∞ as t→∞.So

We reach a contradiction.The proof is completed.