Ground states of a system of three Schr?dinger equations

Chen Xinyan,Liu Haidong,Liu Zhaoli

(1.School of Mathematical Sciences,Capital Normal University,Beijing 100048,China;2.Institute of Mathematics,Jiaxing University,Zhejiang 314001,China)

(Communicated by Guo Zhenhua)

Abstract：In this paper we consider a system of three coupled nonlinear Schr?dinger equations,which comes from nonlinear optics and Bose-Einstein condensates.We deal with two types:systems with periodic potentials,and systems with trapping potentials.Using the generalized Nehari manifold and delicate energy estimates,we establish the existence of a positive ground state for either type provided that the interacting potentials are suitably small.

Keywords：ground state,nonlinear Schr?dinger system,variational methods

1 Introduction

In this paper we consider the nonlinear Schr?dinger system of three equations

A solution(u1,u2,u3)of(1.1)is said to be nontrivial if all the three components are nonzero.Only nontrivial solutions are of physical interest since(1.1)describes wave functions of three species and each component stands for a wave function of a species.Existence of nontrivial solutions is mathematically challenging since there is no standard technique to distinguish them from semitrivial solutions which we mean solutions with exactly one or two components being nonzero.In this paper,we are interested in the existence of a ground state of(1.1)which,by definition,we mean a nontrivial solution which has the least energy among all nontrivial solutions.

Nonlinear Schr?dinger system has attracted much attention in the last twenty years.While most of the existing papers have been devoted to the existence,multiplicity and quantitative properties of nontrivial solutions for systems of two coupled Schr?dinger equations in various different parameter regimes of nonlinear couplings(see References[7-19]),very few papers focus on systems of k(k≥3)coupled equations(see References[7],[13],[20-26]).The reason is that many techniques which can be applied to systems of two coupled equations are not easy to be adapted to systems of at least three coupled equations.It has been turned out that systems of at least three coupled equations are much more complicated than systems of two coupled equations.

The first motivation of the present paper is to generalize a result in Reference[27]for a system of two coupled equations to system(1.1).The authors of Reference[27]proved that,among other things,the system of two coupled equations

has a positive ground state if β12is suitably small,assuming that Vjand βijare either periodic potentials or trapping potentials.We shall extend this result to(1.1).

Until now,almost all the studies in the literature have been conducted on systems with constant coefficients,that is,systems with Vjand βijbeing all constants.All the above mentioned papers except Reference[27]concern systems with constant coefficients.To exhibit a new existence result for a system of three equations with nonconstant potentials is the second motivation of our paper.

We assume that Vjand βijare either periodic potentials or trapping potentials.Note that constants Vjand βijare both periodic potentials and trapping potentials.The precise assumptions on Vjand βijare as follows.

(A1)For i,j=1,2,3,Vjand βijare positive functions and are τk-periodic in xk,τk>0,k=1,···,N.

and

It is known that,under the assumption(A1)or(A2),the scalar equation

possesses a positive ground state.Let wjbe a positive ground state of(1.3).It is clear that the infimum

is attained by wjand

(A3)For 1≤i

(A4)For 1≤i

The first main result of this paper is for the periodic case and is as follows.

Theorem 1.1 If(A1)and(A3)hold,then system(1.1)has a positive ground state.

This theorem has an immediate corollary.

Corollary 1.1 If(A4)holds,then the system

has a positive ground state.

Our second theorem is for the trapping potential case.

Theorem 1.2 If(A2),(A3)and(A4)hold,then system(1.1)has a positive ground state.

Note that Corollary 1.1 is also an immediate consequence of Theorem 1.2.Theorems 1.1 and 1.2 coincide if in particular Vjand βijare constants.However,our proof of Theorem 1.2 is based on Corollary 1.1.

The paper is organized as follows.We prove some useful lemmas in Section 2.Section 3 is devoted to the proof of Theorem 1.1 while Section 4 is devoted to the proof of Theorem 1.2.

2 Preliminaries

We introduce some notations first.Throughout this paper,we shall use the equivalent norms

The symbols?and→denote the weak and strong convergence,respectively,and o(1)stands for a quantity tending to 0.

Solutions of system(1.1)correspond to critical points of the functional

Clearly,nontrivial solutions of system(1.1)are contained in the generalized Nehari manifold

where

Since we are interested in selecting nontrivial solutions from all solutions of(1.1),we need to avoid solutions with only one or only two components being nonzero which are in fact solutions of scalar equations and systems of two equations.The functional and the Nehari manifold associated with the scalar equation(1.3)is denoted by Ijand Njrespectively,j=1,2,3,i.e.,

and

For 1≤i

That is

and

Define

which are the least energy of(1.3),(2.1)and(1.1)respectively.To prove Theorems 1.1 and 1.2,we need to build suitable relationships among these quantities.

In what follows,we always assume that either the assumptions of Theorem 1.1 or the assumptions of Theorem 1.2 hold.We recall that,under the assumption(A1)or(A2),wjis a positive ground state of(1.3).It is easy to see that

where Sjis the infimum defined in the introduction.

Lemma 2.1 For 1≤i

Proof It follows from(A3)that

Since the function

is strictly increasing in η∈(0,min{Si,Sj}),we have

The proof is complete.

Remark 2.1 Let 1≤i

Lemma 2.2 If(Ui,Uj)is a ground state of system(2.1),then

Proof By Lemmas 3.1 and 3.2 in Reference[27],we have

Then Lemma 2.1 yields

Similarly,

The proof is complete.

Lemma 2.3 For 1≤i

Proof Letting(Ui,Uj)be a positive ground state of(2.1),we have

Denote

Using(A3)we see that

This inequality combined with(2.2)and(2.3)implies ci

Since

we deduce from Lemma 3.1 in Reference[27]that

The proof is complete.

Lemma 2.4 c

We shall only prove c

and

To prove Lemma 2.4,we need to show that the linear system

has a solution(r,s,t)with three positive components,where

Set

and

Lemma 2.5We have

Proof This is just direct and elementary computation.

Lemma 2.6 The following inequalities hold:

Proof In view of Remark 1.1 one easily obtains the first three inequalities.We estimate b2+b3as

Then,by(A3)and Remark 1.1,

and

The proof is complete.

Lemma 2.7 ?>0,?j>0 for j=1,2,3,and

Proof As a consequence of Lemma 2.5 and the first three inequalities in Lemma 2.6,we have

By Lemma 2.5 and the first and the fourth inequalities in Lemma 2.6,we see that

The following estimate for?2uses Lemma 2.5 and the first,the second,the fourth,and the sixth inequalities in Lemma 2.6

In the same way,we also have?3>0.To prove the last inequality of the lemma,observe from Lemma 2.5 that

Then using the first,the fourth,the fifth and the sixth inequalities in Lemma 2.6 yields

The proof is complete.

We now prove Lemma 2.4.

Proof of Lemma 2.4 By Lemma 2.7,(2.4)has a unique solution(r,s,t)with

Then

and

the infimum c=infNI can be estimated as

We then use Lemma 2.7 again to obtain c

Proof This is a direct consequence of Lemmas 2.3 and 2.4.

then

and

ProofDenote

that is

By(A3),we see that

and

Then

The estimate of the determinant is as follows

The proof is complete.

3 Proof of Theorem 1.1

In this section we prove the first main result.

where

where

We claim that

It remains to prove(3.2).Since,by Lemma 2.9,

the P.L.Lions lemma implies that,for any r>0,

from which it follows

Passing to a subsequence,we may assume that the limits

exist.We shall prove that

It remains to prove(3.4).We use an argument of contradiction and assume that(3.4)is false.Then we have the following three cases.

Case 1:γj>0 for j=1,2,3.In this case,we consider the linear system

where

By(3.3),the linear system(3.5)can be rewritten as

We see from the arguments in the proof of Lemma 2.9 that,for large m,

Then the linear system(3.5)has a unique solution(rm,sm,tm)and

Consider the linear system

where

By(3.6),the linear system(3.7)can be rewritten as

Since γ2>0 and γ3>0,we have

for large m.Then the linear system(3.7)has a unique solution(sm,tm)and

sm=1+o(1),tm=1+o(1).

which is impossible.In the case where u2=u3=0,we have

which also contradicts Lemma 2.4.

which is impossible.In the case where u3=0,we have

which contradicts Lemma 2.4.

In each case we have come to a contradiction.This proves(3.4)and(3.2)in turn.The proof is complete.

4 Proof of Theorem 1.2

In the trapping potential case,we need to consider the limit systems of(1.1),(1.3)and(2.1).The functional associated with(1.4)is

and the generalized Nehari manifold is given by

where

The functional and the Nehari manifold associated with the scalar equation

respectively.

Define

which are the least energy of(4.1),(4.2)and(1.4)respectively.

Proof The method of the proof of the first three inequalities is standard(see Reference[27]).Now we use the idea in Reference[27]to prove that if at least one of the nine functions Vjand βijis not constant then c

Note that(u1,u2,u3)satisfies

By the assumption(A2)we see that,for i,j=1,2,3,

For simplicity of symbols,we denote

and

By(A2),we see that

where

Then

Note that

After a lengthy but elementary calculation,we expand Ayas

where high order terms mean the summation of the square,the cubic and the fourth order terms of χijand ψj.By(A2),if at least one of the nine functions Vj(j=1,2,3)and βij(1 ≤i≤j≤3)is not constant then

Similar to the proof of?y>0,we have

According to(4.4),if|y|is sufficiently large,then each of χij(y)and|ψj(y)|is sufficiently small.Hence we arrive at c∞>c.The proof is complete.

We are in a position to prove the second main result.

from which it follows

Assume up to a subsequence that the limits

Therefore,it remains to prove

We assume,by contradiction,that(4.7)is false and we divide the discussion into three cases.

where

By(4.6),the linear system(4.8)can be rewritten as

We see from arguments in the proof of Lemma 2.9 that,for large m,

Then the linear system(4.8)has a unique solution(rm,sm,tm)and

Consider the linear system

where

By(4.9),the linear system(4.10)can be rewritten as

for large m.Then the linear system(4.10)has a unique solution(sm,tm)and

which also contradicts Lemma 2.4.

which is impossible.In the case where u3=0,we use Lemma 4.1 to deduce

which contradicts Lemma 2.4.

In each case we have come to a contradiction.The proof is complete.