Stable solitary waves for a class of nonlinear Schrödinger system with quadratic interaction

Stable solitary waves for a class of nonlinear Schrödinger system with quadratic interaction

Guoqing Zhang

and Tongmo Gu

College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China.

Received 29 June 2018, appeared 4 December 2018 Communicated by Dimitri Mugnai

Abstract. We consider the existence and orbital stability of bound state solitary waves and ground state solitary waves for a class of nonlinear Schrödinger system with quadratic interaction in Rⁿ (n = 2, 3). The existence of bound state and ground state solitary waves are studied by variational arguments and Concentration-compactness Lemma. In additional, we also prove the orbital stability of bound state and ground state solitary waves.

Keywords: bound (ground) state solitary waves, quadratic interaction, variational ar- guments.

2010 Mathematics Subject Classification: 35J20, 35J60.

1 Introduction

In this paper, we consider the following system of nonlinear Schrödinger equations







i∂tu+ ¹

2m∆u= _λvu, (_x,t)∈_Rⁿ⁺¹, i∂_tv+ ¹

2M∆v=µu², (x,t)∈_Rⁿ⁺¹,

(1.1)

where u andv are complex-valued wave fields, mand M are positive constants, λandµare complex constants, anduis the complex conjugate ofu.

Such systems have interesting applications in several branches of physics, such as in the study of interactions of waves with different polarizations [1,11]. The Cauchy problem for System 1.1 has been studied from the point of view of small data scattering [6,7]. In 2013, Hayashi, Ozawa and Tanaka [8] studied the well-posedness of Cauchy problem for System1.1 with large data. In particular, System 1.1 is regarded as a non-relativistic limit of the system of nonlinear Klein–Gordon equations









 1

2c²m∂²_tu− ¹

2m∆u+ ^mc

2

2 u= −λvu, (x,t)∈_Rⁿ⁺¹, 1

2c²M∂²_tv− ¹

2M∆v+ ^Mc

2

2 v= −µu², (x,t)∈_Rⁿ⁺¹,

(1.2)

BCorresponding author. Email: zgqw2001@usst.edu.cn

under the mass resonance condition M=2m, wherecis the speed of light.

Assumeλ=cµ,c>0, λ6=0 andµ6=0, we introduce new functions (u,_e ve)defined by ue(x,t) =

rc 2|µ|u

r 1 2mx,t

!

, ve(x,t) =−^λ 2v

r 1 2mx,t

! , and System (1.1) satisfies







i∂_tue+_∆u_e=−2veu,e (x,t)∈_Rⁿ⁺¹, i∂_tve+ ^m

M∆ev= −u_e², (x,t)∈_Rⁿ⁺¹, (1.3) Using the ansatz (u_e(x,t),ve(x,t)) = (e^iωtφ(x),e^i2ωtψ(x)), φ(x),ψ(x) 6≡0 with ω > 0, System (1.3) becomes

(−_∆φ+ωφ=_2φψ, x∈_Rⁿ,

−κ∆ψ+2ωψ=φ², x∈_Rⁿ, (1.4)

whereκ= _M^m_.

Let L^p(_Rⁿ) denote the usual Lebesgue space with the norm |u|_p = (R

Rⁿ|u|^pdx)^1p. The space H¹(_Rⁿ) _:= {u ∈ L²(_Rⁿ)_,∇u ∈ L²(_Rⁿ)} with the corresponding norm kuk = (R

Rⁿ|∇u|²+|u|²dx)¹²_{, and}H_r¹(_Rⁿ)_:={u∈ H¹(_Rⁿ)_;uis radially symmetric}.

Recently, as 2≤ n ≤5, Hayashi, Ozawa and Tanaka [8] obtained the existence of radially symmetric ground states for System (1.4) by using rearrangement method, Pohozaev identity and the Sobolev compact embeddingH¹_r(_Rⁿ)⊂L³(_Rⁿ)_.

In this paper, firstly, we prove the existence of bound states for System (1.4) by using the Concentration-compactness Lemma and direct methods in the critical points theory. Secondly, we discuss the general case for System (1.4), i.e.,

(−_∆φ+λ₁φ=2φψ, x∈_Rⁿ,

−κ∆ψ+λ₂ψ=φ², x∈_Rⁿ, (1.5) where(λ₁,λ₂)∈_R². By using the Concentration-compactness Lemma, variational arguments and rearrangement result of Shibata [13], we obtain the existence of ground states for System (1.5). In particular, ifλ₁= ¹₂λ₂ >0, then System (1.5) can be reduced to System (1.4) and the existence of ground states for System (1.4) is obtained in [8]. Furthermore, we also prove the orbital stability of bound states and ground states.

Remark 1.1. In contrast to results in [8], we obtain the existence of bound states in the whole space H¹(_Rⁿ). Since the embedding H¹(_Rⁿ) ⊂ L³(_Rⁿ) is only continuous, we ap- ply the Concentration-compactness Lemma and variational arguments to obtain the existence of bound states.

2 Preliminaries and main results

In this section, we state our main results in this paper.

Now, we define the functionalsI,J andQ: H¹(_Rⁿ)×H¹(_Rⁿ)→_Rby I(φ,ψ) = ¹

2 Z

Rⁿ(|∇φ|²+κ|∇ψ|²)dx−

Z

Rⁿφ²ψdx, ∀(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ), J(φ,ψ) = ¹

2 Z

Rⁿ(|∇φ|²+κ|∇ψ|²)dx, ∀(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ),

and

Q(φ,ψ) = ^ω 2

Z

Rⁿ|φ|²dx+2 Z

Rⁿ|ψ|²dx

, ∀(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ).

It is obvious that I,J andQ ∈ C¹(H¹(_Rⁿ)×H¹(_Rⁿ),R). Hence, (φ,ψ)is a weak solution of System (1.4) if and only if(φ,ψ)is a critical point of the functionalS:= I+Q.

LetM_N ={(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ):Q(φ,ψ) =N, |φ|²₂,|ψ|²₂>0}for someN>0, and the minimizing problem

IN =inf{I(φ,ψ);(φ,ψ)∈ MN}. (2.1) Besides, for everyN >_{0, let}P_N denote the set of bound states of System (1.4), that is,

P_N ={(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ);I(φ,ψ) =I_N and(φ,ψ)∈ M_N}, which generates the solitary waves of System (1.1).

Theorem 2.1. Let n=2, 3. Then we have:

(1) For all N >0,there exists(φ_N,ψ_N)∈H¹(_Rⁿ)×H¹(_Rⁿ)a solution of (φ_N,ψ_N)∈ M_N,

I(φN,ψN) =min{I(φ,ψ);(φ,ψ)∈ MN}. (2.2) (2) If (φ_N,ψ_N)is a solution of the minimizing problem(2.2), then there exists a Lagrange multiplier

σ_N >0such that

(−_∆φ+σ_Nωφ=2φψ, x∈_Rⁿ,

−_κ∆ψ+2σ_Nωψ=φ², x∈_Rⁿ, (2.3) whereσ_N is given by

σ_N =

2

nJ(φ_N,ψ_N)−I_N

N . (2.4)

(3) The set

Σ:= {(N,σN);N>0,σN is a Lagrange multiplier of the minimizing problem(2.2)}

is a closed graph in(0,+_∞)×(0,+_∞). In particular, ifΣis a function, then it is continuous and there exists N₀ >₀such thatσ_N0 =1. So,(φ_N0,ψ_N0)is a bound state of System(1.4).

Next, we define the set

M_α,β ={(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ): |φ|²₂ =α, |ψ|²₂= β} for any α,β>0, and the minimizing problem

I_α,β =inf{I(φ,ψ);(φ,ψ)∈ M_α,β}. Besides, for anyα,β>_{0, let}

G_α,β ={(φ,ψ)∈ M_α,β;I(φ,ψ) =I_α,β}, which denotes the set of ground states of System (1.5).

Theorem 2.2.

(1) For anyα,β> 0, any minimizing sequence{(φn,ψn)}_n≥1 ⊂ H¹(_Rⁿ)×H¹(_Rⁿ)with respect to I_α,β is pre-compact. That is, taking a subsequence, there exist(φ,ψ) ∈ M_α,β and{y_n}_n≥1 ⊂ _Rⁿ such thatφ_n(· −y_n)→φ,ψ_n(· −y_n)→ψin H¹(_Rⁿ)as n →_∞.

(2) Let(λ₁,λ₂)be the Lagrange multiplier associated with(_φ,ψ)on M_α,β, we haveλ₁ >0.

(3) If(_φ,ψ)∈G_α,β, we have(|φ|,|ψ|)∈G_α,β. One also has(φ^∗,ψ^∗)∈G_α,β whenever(_φ,ψ)∈ G_α,β andφ^∗,ψ^∗ >0, where f^∗represents the symmetric decreasing rearrangement of the function f . Definition 2.3. For anyN>0, the setP_N is stable if for any ε>0 there exists aδ(ε)>0 such that if(φ0,ψ0)∈ H¹(_Rⁿ)×H¹(_Rⁿ)verifies

inf

(φ_N,ψN)∈PN

k(φ₀,ψ₀)−(φ_N,ψ_N)k_H1(_Rⁿ)×H¹(_Rⁿ)< δ(ε),

then the solution(φ(t),ψ(t))of the System (1.1) withφ(0) =φ₀, ψ(0) =ψ₀satisfies sup

t∈_R

(φN,ψinfN)∈PN

k(φ(t),ψ(t))−(φ_N,ψ_N)k_H1(_Rⁿ)×H¹(_Rⁿ) <ε.

Besides, we can also define the setG_α,β is stable in the same way.

Theorem 2.4. Let n=2, 3, the sets P_N and G_α,β are stable.

Now, we recall the rearrangement results of Shibata [13] as presented in [9]. Letube a Borel measureable function onRⁿ. Thenuis said to vanish at infinity if|{x∈ _Rⁿ;|u(x)|>s}|<_∞ for every s > 0. Here | · | stands for then-dimensional Lebesgue measure. Considering two Borel functions u,v which vanish at infinity in Rⁿ, we define for s > 0, set A^?(u,v;s) := {x∈_Rⁿ;|x|<r}wherer≥0 is chosen so that

|B_r(0)|=|{x∈_Rⁿ;|u(x)|> s}|+|x∈_Rⁿ;|v(x)|>s}|, and{u,v}^? by

{u,v}^?(x):=

Z _∞

0 χ_A^?_(u,v;s)(x)ds, whereχ_A(x)is a characteristic function of the set A⊂_Rⁿ. Lemma 2.5([9, Lemma A.1]).

(1) The function{u,v}^?(x)is radially symmetric, non-increasing and lower semi-continuous. More- over, for each s>₀there holds{x∈_Rⁿ;{u,v}^? >_s}= _A^?(_u,v;s)_.

(2) LetΦ:[0,∞)→[0,∞)be non-decreasing, lower semi-continuous, continuous at 0 andΦ(0) =0.

Then{_Φ(u),Φ(v)}^?= _Φ({u,v}^?). (3) |{u,v}^?|^p_p =|u|^p_p+|v|^p_pfor1≤ p<_∞.

(4) If u,v ∈ H¹(_Rⁿ), then {u,v}^? ∈ H¹(_Rⁿ) and|∇{u,v}^?|²₂ ≤ |∇u|²₂+|∇v|²₂. In addition, if u,v ∈ (H¹(_Rⁿ)∩C¹(_Rⁿ))\{0} are radially symmetric, positive and non-increasing, then we

have _Z

Rⁿ|∇{u,v}^?|²dx <

Z

Rⁿ|∇u|²dx+

Z

Rⁿ|∇v|²dx.

(5) Let u₁,u₂,v₁,v₂ ≥0be Borel measurable functions which vanish at infinity, then we have Z

Rⁿ(u₁u₂+v₁v₂)dx≤

Z

Rⁿ{u₁,v₁}^?{u₂,v₂}^?dx.

3 Bound states

Let{(φ_n,ψ_n)}_n≥1 be a minimizing sequence for the minimizing problem (2.1), that is, the sequence {(φ_n,ψ_n)}_n≥1 ∈ H¹(_Rⁿ)×H¹(_Rⁿ)satisfies Q(φ_n,ψ_n)→ N and I(φ_n,ψ_n) → I_N, as n→∞. Then, we have

Lemma 3.1. As n = 2, 3, there exists B > 0 such thatk(φn,ψn)k_H1(_Rⁿ)×H¹(_Rⁿ) ≤ B for all n, and the functional I is bounded below on M_N.

Proof. By the Gagliardo–Nirenberg inequality, we have _Z

Rⁿ|φ|³dx ¹₃

≤C _Z

Rⁿ|∇φ|²dx ₁₂ⁿ _Z

Rⁿ|φ|²dx ¹₂−₁₂ⁿ

. Hence, we have

Z

Rⁿφ²ψdx≤ _Z

Rⁿ(φ²)³²dx ²_{3 Z}

Rⁿ|ψ|³dx ¹₃

= _Z

Rⁿ|φ|³dx ²_{3 Z}

Rⁿ|ψ|³dx ¹₃

≤C Z

Rⁿ|∇φ|²dx ⁿ₆ Z

Rⁿ|∇ψ|²dx ₁₂ⁿ

.

Since n = 2, 3, we have ⁿ₆ + ₁₂ⁿ < _{1. Thus,} I is coercive and in particular I_N > −_{∞. By the} coerciveness of I on M_N, the sequence{(φn,ψn)}_n≥1 is bounded in H¹(_Rⁿ)×H¹(_Rⁿ). Thus, there exists B>0 such thatk(φ_n,ψ_n)k_H1(_Rⁿ)×H¹(_Rⁿ) ≤Bfor all n.

Lemma 3.2. For any N>0, IN <0and IN is continuous with respect to N.

Proof. Let A(φ) = ¹₂R

Rⁿ|∇φ|²dx,B(ψ) = ^κ₂R

Rⁿ|∇ψ|²dx, andC(φ,ψ) =R

Rⁿφ²ψdx, hence, I(φ,ψ) =A(φ) +B(ψ)−C(φ,ψ).

Now let (φ(x),ψ(x)) ∈ M_N be fixed. For any b > 0, we define φ_θ(x) = θ^bn2 φ(θ^bx), ψ_θ(x) = θ

bn

2 ψ(θ^bx), then(φ_θ(x),ψ_θ(x))∈ MN as well. We have the following scaling laws:

A(φ_θ(x)) = ¹ 2

Z

Rⁿ|θ

bn2 ∇φ(θ^bx)|²dx =θ^2bA(φ(x)), B(ψ_θ(x)) = ^κ

2 Z

Rⁿ|θ

bn

2∇ψ(θ^bx)|²dx=θ^2bB(ψ(x)), and

C(φ_θ(x),ψ_θ(x)) =

Z

Rⁿθ^bnφ²(θ^bx)θ

bn2 ψ(θ^bx)dx=θ

bn2 C(φ(x),ψ(x)). So, we get

I(φ_θ(x),ψ_θ(x)) =θ^2bA+θ^2bB−θ

bn 2 C.

Since n =2, 3, we have ^bn₂ <2b. Lettingθ →0, then I(φ_θ(x),ψ_θ(x))→ 0⁻. Hence, we prove IN <0.

In order to prove that I_N is a continuous function, we assume N_n = N+o(1). From the definition of INn, for anyε >0, there exists(φn,ψn)∈ MNn such that

I(φ_n,ψ_n)≤ I_Nn+_{ε. (3.1)}

Setting

(u_n,v_n):=

rN N_nφ_n,

r N N_nψ_n

! , we have that(un,vn)∈ MN and

I_N ≤ I(un,vn) = I(φn,ψn) +o(1). (3.2) Combining (3.1) and (3.2), we obtain

I_N ≤ I_Nn+ε+o(1). Reversing the argument, we obtain similarly that

I_Nn ≤ I_N+ε+o(1).

Therefore, sinceε>0 is arbitrary, we deduce that I_Nn = I_N+o(1). Lemma 3.3. ^I_N^N is decreasing in(0,+_∞).

Proof. For (φ,ψ) ∈ H¹(_Rⁿ)× H¹(_Rⁿ), we define (φ_θ(x),ψ_θ(x)) := θ^bφ(θ^ax),θ^bψ(θ^ax),

∀θ > 0. Choosinga,b> 0, such that 2b−na = 1, it follows that Q(φ_θ(x)_,ψ_θ(x)) = θQ(_φ,ψ) and we can write

I(φ_θ(x),ψ_θ(x)) =θ^2a+1I(φ,ψ) +θ^2a+1 Z

Rⁿφ²ψdx−θ^b+1 Z

Rⁿφ²ψdx. (3.3) We can choosea,b>0 such that 2b−na=1, b>2aand it follows from (3.3) that

I(φ_θ(x)_,ψ_θ(x))<θ^2a+1I(_φ,ψ)_, ∀θ >_1.

Since(φ(x),ψ(x))∈ MN ⇔(φ_θ(x),ψ_θ(x))∈ M_θN, ∀θ,N>0, it follows that I_θN < θ^2a+1I_N <θI_N, ∀θ >1.

Thus,

I_θN θN < ^IN

N, ∀θ >1.

Lemma 3.4. For any N>0andλ∈(0,N), we have I_N < I_λ+I_N−λ. Proof. Thanks to the following well-known inequality: ∀a,b,A, B>0,

min a

A, b B

≤ ^a+b

A+B ≤max a

A, b B

, where the equalities hold if and only if _A^a = _B^b, we get

(−I_λ) + (−I_N−λ)

λ+N−λ ≤max −I_λ

λ ,−I_N−λ

N−λ

. Without loss of generality, we assume ⁻_λ^Iλ is larger than ⁻_N^IN₋⁻_λ^λ, then

(−I_λ) + (−I_N−λ)

N ≤ −I_λ

λ . By Lemma3.3, we have

I_λ+I_N−λ ≥ ^N

λI_λ > I_N.

Proof of Theorem2.1. Our proof is divided into five steps:

Step 1. The minimizing problem (2.2) has a solution. By Lemma3.1, the sequence{(φn,ψn)}

is bounded in H¹(_Rⁿ)×H¹(_Rⁿ). If sup

y∈_Rⁿ Z

BR(y)

(|φ_n|²+|ψ_n|²)dx= o(1),

for some R > 0, the φn → 0, ψn → 0 in L^p(_Rⁿ) for 2 < p < 2^∗, see [11,12]. This is in- compatible with the fact that I_N < 0, see Lemma 3.2. Thus, the vanishing of minimizing sequence {(φ_n,ψ_n)} does not exist. Besides, Lemma 3.4 prevents their dichotomy. Accord- ing to Concentration-compactness Lemma, only concentration exists, and we get a solution (φ_N,ψ_N)of the minimizing problem (2.2).

Step 2. There exists a positive Lagrange multiplier σN. Let (φN,ψN)a solution of the mini- mizing problem (2.2). From the Lagrange Multiplier Theorem, there exists θ ∈ _R such that I⁰(φ_N,ψ_N) =θQ⁰(φ_N,ψ_N), that means

−_∆φN−2φ_Nψ_N =θωφ_N,

−_κ∆ψN−φ²_N =2θωψ_N. (3.4)

By multiply the above equations respectively byφ_N, ψ_N and integrating onRⁿ, we get I_N−¹

2 Z

Rⁿφ²_Nψ_Ndx= θN. (3.5)

Since I_N <0,∀N>0, we obtain easily from (3.5) thatθ <0.

For anyλ,c>0, we consider

(φ_λ(x),ψ_λ(x)):=λ

cn

2 φ_N(λ^cx),λ

cn

2ψ_N(λ^cx),

then(φ_λ(x),ψ_λ(x))∈ M_N and I(φ_N,ψ_N) =min_λ>0I(φ_λ(x),ψ_λ(x)). In particular, 0= ^d

dλI(φ_λ(x),ψ_λ(x)) λ=1

=2cJ(φN,ψN)− ^cn 2

Z

Rⁿφ²_NψNdx. (3.6) Merging (3.5) and (3.6), we get

I_N− ²

nJ(φ_N,ψ_N) =θN, which implies thatθ <0 and the Lagrange multiplier

σN =−θ =

2

nJ(φ_N,ψ_N)−I_N

N >0. (3.7)

Step 3. There existγ(n)>0 such that

− ^IN

N <σ_N <γ(n)− ^IN

N. (3.8)

SinceI(φ_N,ψ_N)<0, we get from Hölder’s inequality and the Gagliardo–Nirenberg inequality that

J(φ_N,ψ_N)<

Z

Rⁿφ²_Nψ_Ndx≤ ¹ 2

|φ_N|⁴₃+|ψ_N|³₂

≤ C

|∇φ_N|₂²ⁿ³ |φ_N|⁴₂⁻²ⁿ³ +|∇ψ_N|₂ⁿ³|ψ_N|²₂⁻ⁿ³

≤ C

J(φ_N,ψ_N)ⁿ³ +J(φ_N,ψ_N)ⁿ⁶ρ(N),

(3.9)

whereC>0 andρ(N):=maxn

N⁽²⁻ⁿ³⁾, N⁽¹⁻ⁿ⁶⁾o . Let f :(_0,∞)→_Rthe function defined by

f(s):= ^s sⁿ³ +sⁿ⁶ ,

and we know f⁰(s)>0,∀s>0 and lim_s→0⁺ f(s) =0. So, we can rewrite (3.9) as

J(φ_N,ψ_N)< f⁻¹(Cρ(N)). (3.10) Note that

ρ(s) =s⁽¹⁻ⁿ⁶⁾ ifs ≤1, and f(s)≥ ¹

2s⁽¹⁻ⁿ⁶⁾ ifs≤1, ρ(_s) =_s⁽²⁻ⁿ³⁾ _ifs ≥1, and f(_s)≥ ¹

2s⁽¹⁻ⁿ³⁾ ifs≥1.

By a straightforward calculation we see that there existsC₁>0 such that f⁻¹(Cρ(N))≤C₁N ifN ≤1,

f⁻¹(Cρ(N))≤C₁N^(63−−nn) ifN≥1.

Hence, we obtain from (3.10) that

J(φN,ψN)<C₁N, ∀N> 0.

Letγ(n) = ^2C_n¹, (3.8) holds.

Step 4. Σis closed in(0,+_∞)×(0,+_∞). For all(φ_N,ψ_N)solution of the minimizing problem (2.2), we define

σ(φ_N,ψ_N)_:= ¹ N

2

nJ(φ_N,ψ_N)−I_N

,

ΣN :={σ(φ_N,ψ_N); (φ_N,ψ_N)solution of the minimizing problem(2.2)}. Then it is easy to see thatΣ={(N,σ_N);N>0,σ_N ∈_ΣN}.

Let(N_n,σ_n)∈_Σ such that(N_n,σ_n)→ (N,σ), N >0. By definition, there exists(φ_n,ψ_n)∈ H¹(_Rⁿ)×H¹(_Rⁿ)_{such that}Q(φ_n,ψ_n) =N_n, I(φ_n,ψ_n) =I_Nn and

σ_n= ¹ Nn

2

nJ(φ_n,ψ_n)−I_Nn

.

By Lemmas3.1and3.2,{(φn,ψn)}is bounded in H¹(_Rⁿ)×H¹(_Rⁿ). If we define (un,vn):=

rN Nnφn,

r N Nnψn

! ,

then {(u_n,v_n)} is also bounded in H¹(_Rⁿ)×H¹(_Rⁿ) and Q(u_n,v_n) = N. By using the Concentration-compactness Lemma, there exists a subsequence satisfying only one of the following three cases: 1) concentration; 2) vanishing; 3) dichotomy.

By using the argument as in step 1, only concentration exists. Therefore, there exists {y_n}_n≥1⊂_Rⁿand(φ,ψ)∈ H¹(_Rⁿ)×H¹(_Rⁿ)such that

φn(· −yn)*φ, ψn(· −yn)*ψ weakly in H¹(_Rⁿ), φ_n(· −y_n)→φ, ψ_n(· −y_n)→ψ in L²(_Rⁿ),

Z

Rⁿφ²_n(· −yn)ψn(· −yn)dx=

Z

Rⁿφ²_nψndx→

Z

Rⁿφ²ψdx.

In particular,Q(φ,ψ) = NandI(φ,ψ)≥ I_N. On the other hand, I(φ_N,ψ_N)≤lim inf

n→_∞ I(φ_n(· −yⁿ),ψ_n(· −yⁿ)) = lim

n→_∞I(φ_n,ψ_n) = I_N.

So, I(φ_N,ψ_N) = I_N and (φ_N,ψ_N) is a solution of the minimizing problem (2.2). Moreover, since

J(φ_n,ψ_n) = I(φ_n,ψ_n) +

Z

Rⁿφ²_nψ_ndx→ I(φ_N,ψ_N) +

Z

Rⁿφ²_Nψ_N = J(φ_N,ψ_N), we conclude that

σ = ¹ N

2

nJ(φ_N,ψ_N)−I_N

∈_ΣN.

Step 5. IfΣis a function, then it is continuous and there exists N₀ > 0 such thatσ_N0 = 1. In particular, (φN₀,ψN₀) is a bound state of System (1.4). This follows easily from Step 4, (3.8) and Lemma3.3.

4 Ground states

Lemma 4.1. The energy I_α,β satisfies that (i) For anyα,β>0,−_∞< I_α,β <0.

(ii) I_α,β is continuous with respect toα,β≥0.

(iii) I_α+α⁰,β+β⁰ ≤ I_α,β+I_α⁰_,β⁰ forα,α⁰,β,β⁰ ≥0.

Proof. The proofs of (i) and (ii) use the same arguments as in Lemmas 3.1and 3.2. Next, we prove (iii). Indeed, forε>0, there exists(u,v)∈ M_α,β∩C₀^∞(_Rⁿ)and(φ,ψ)∈ M_α⁰_,β⁰∩C^∞₀ (_Rⁿ). By using parallel transformation, we can assume that(suppu∪suppv)∩(suppφ∪suppψ) =

∅. Therefore(u+φ,v+ψ)∈ M_α+α⁰,β+β⁰ and

I_α+α⁰,β+β⁰ ≤ I(u+φ,v+ψ) =I(u,v) +I(φ,ψ)≤ I_α,β+I_α⁰_,β⁰+2ε.

Sinceε>0 is arbitrarily, it asserts (iii).

Lemma 4.2. For any minimizing sequence {(φ_n,ψ_n)}_n≥1 of I_α,β, if (φ_n,ψ_n) * (φ,ψ) weakly in H¹(_Rⁿ)×H¹(_Rⁿ), then

Z

Rⁿφ²_nψ_n−(φ_n−φ)²(ψ_n−ψ)dx=

Z

Rⁿφ²ψdx+o(1).

Proof. The idea of its proof comes from [5] (see also Lemma 2.3 of [4]). For any a₁, a₂, b₁, b₂ ∈_Randε>0, we deduce from the mean value theorem and Young’s inequality that

|(a₁+a₂)²(b₁+b₂)−a²₁b₁| ≤Cε(|a₁|³+|a₂|³+|b₁|³+|b₂|³) +Cε(|a₂|³+|b₂|³). Denote a₁ :=φn−φ, b₁:=ψn−ψ, a₂:=φ, b₂ :=ψ. Then

f_n^ε := |φ_n²ψ_n−(φ_n−φ)²(ψ_n−ψ)−φ²ψ| −Cε(|φ_n−φ|³+|φ|³+|ψ_n−ψ|³+|ψ|³|)₊

≤ |φ²ψ|+C_ε(|φ|³+|ψ|³)_,

and the dominated convergence theorem yields Z

Rⁿ f_n^εdx→0, asn→_∞. (4.1)

Since

|φ_n²ψ_n−(φ_n−φ)²(ψ_n−ψ)−φ²ψ| ≤ f_n^ε+Cε(|φ_n−φ|³+|ψ_n−ψ|³+|φ|³+|ψ|³|), by the boundedness of{(φ_n,ψ_n)}_n≥1 inH¹(_Rⁿ)×H¹(_Rⁿ)and (4.1), it follows that

Z

Rⁿφ²_nψ_n−(φ_n−φ)²(ψ_n−ψ)dx=

Z

Rⁿφ²ψdx+o(1).

Lemma 4.3. Any minimizing sequence{(φ_n,ψ_n)}_n≥1 ⊂ H¹(_Rⁿ)×H¹(_Rⁿ)with respect to I_α,β is, up to translation, strongly convergent in L^p(_Rⁿ)×L^p(_Rⁿ)for2< p<2^∗.

Proof. Similar to the Step 1 of the proof of Theorem2.1, we can know that there exists aβ₀ >0 and a sequence{yn} ⊂_Rⁿ such that

sup

y∈_Rⁿ Z

BR(yn)

(|φ_n|²+|ψ_n|²)dx≥β₀>0,

and we deduce from the weak convergence inH¹(_Rⁿ)×H¹(_Rⁿ)and the local compactness in L^p(_Rⁿ)×L^p(_Rⁿ)_that(φ_n(_x−yn)_,ψ_n(_x−yn))*(_φ,ψ)6= (0, 0)_{weakly inH}¹(_Rⁿ)×H¹(_Rⁿ)_. In order to prove thatun(x) := φn(x)−φ(x+yn) → 0, vn(x) := ψn(x)−ψ(x+yn) → 0 in L^p(_Rⁿ)for 2< p<2^∗, we suppose that there exists a 2<q<2^∗ such that(u_n,v_n)9(0, 0)in L^p(_Rⁿ)×L^p(_Rⁿ). Note that under this assumption by contradiction there exists a sequence {z_n} ⊂_Rⁿsuch that

(u_n(x−z_n),v_n(x−z_n))*(u,v)6= (0, 0) weakly inH¹(_Rⁿ)×H¹(_Rⁿ).

Now, combining the Brézis–Lieb Lemma ([10]), Lemma 4.2 and the translational invari- ance, we conclude

I(φ_n,ψ_n) = I(un(x−yn),vn(x−yn)) +I(φ,ψ) +o(1)

= I(u_n(x−z_n)−u,v_n(x−z_n)−v) +I(u,v) +I(φ,ψ) +o(1), (4.2)

|φ_n(x−y_n)|²₂=|u_n(x−z_n)−u|²₂+|u|²₂+|φ|²₂+o(1), and

|ψ_n(x−y_n)|²₂= |v_n(x−z_n)−v|²₂+|v|²₂+|ψ|²₂+o(1). Letα⁰ := α− |u|²₂− |φ|²₂,β⁰ :=α− |v|²₂− |ψ|²₂, then

|u_n(x−z_n)−u|²₂ =α⁰+o(1), |v_n(x−z_n)−v|²₂= β⁰+o(1). (4.3) Noting that

|u|²₂≤lim inf

n→_∞ |u_n(x−z_n)|²₂ =lim inf

n→_∞ |φ_n(x−y_n)−φ|²₂= α− |φ|²₂,

then α⁰ ≥ 0. Similarly, β⁰ ≥ 0. Recording that I(φ_n,ψ_n) → I_α,β, in consideration of (4.3), Lemma4.1 (ii) and (4.2), we get

I_α,β ≥ I_α⁰_,β⁰+I(u,v) +I(_φ,ψ)_{. (4.4)} We know from the front that(φ,ψ)6= (0, 0)and (u,v) 6= (0, 0). As forφ,ψ,u,v, if one of them is identically zero, we have

I_α,β ≥ I_α⁰_,β⁰+I(u,v) +I(φ,ψ)> I_α⁰_,β⁰+I_|u|2

2,|v|²₂+I_|φ|2

2,|ψ|²₂ ≥ I_α,β,

which is impossible. So,φ,ψ,u,v6≡0. If I(u,v)> I_|u|2

2,|v|²₂ or I(φ,ψ)> I_|φ|2

2,|ψ|²₂, we also have a contradiction. Hence I(u,v) = I_|u|2

2,|v|²₂ and I(φ,ψ) = I_|φ|2

2,|ψ|²₂. We denote by φ^∗,ψ^∗, u^∗, v^∗ the classical Schwarz symmetric-decreasing rearrangement ofφ,ψ,u,v. Since

|φ^∗|²₂ =|φ|²₂, |ψ^∗|²₂ =|ψ|²₂, |u^∗|²₂ =|u|²₂, |v^∗|²₂= |v|²₂, I(φ^∗,ψ^∗)≤ I(φ,ψ), I(u^∗,v^∗)≤ I(u,v)

see [10], we conclude that

I(φ^∗,ψ^∗) =I_|φ|2

2,|ψ|²₂, I(u^∗,v^∗) =I_|u|2 2,|v|²₂.

Therefore, (φ^∗,ψ^∗), (u^∗,v^∗) are solutions of the System (1.1) and from standard regularity results we have thatφ^∗,ψ^∗,u^∗,v^∗ ∈C²(_Rⁿ)_.

By Lemma2.5, we have Z

Rⁿ

∇ {φ^∗,u^∗}^?

2dx<

Z

Rⁿ |∇φ^∗|²+|∇u^∗|²dx≤

Z

Rⁿ |∇φ|²+|∇u|²dx, Z

Rⁿ

∇ {ψ^∗,v^∗}^?

2dx<

Z

Rⁿ |∇ψ^∗|²+|∇v^∗|²dx≤

Z

Rⁿ |∇ψ|²+|∇v|²dx, and

Z

Rⁿ {φ^∗,u^∗}^?2{ψ^∗,v^∗}^?dx≥

Z

Rⁿ

(φ^∗)²ψ^∗+ (u^∗)²v^∗ dx ≥

Z

Rⁿ φ²ψ+u²v dx.

Thus,

I(φ,ψ) +I(u,v)> I {φ^∗,u^∗}^?,{ψ^∗,v^∗}^?, (4.5)

and _Z

Rⁿ

{φ^∗,u^∗}^?

2dx=

Z

Rⁿ

|φ^∗|²+|u^∗|²dx=

Z

Rⁿ |φ|²+|u|²dx =α−α⁰, Z

Rⁿ

{ψ^∗,v^∗}^?

2dx=

Z

Rⁿ

|ψ^∗|²+|v^∗|²dx=

Z

Rⁿ |ψ|²+|v|²dx= β−β⁰.

(4.6) Taking (4.4)–(4.6) and Lemma4.1(iii) into consideration, one obtains the contradiction

I_α,β > I_α⁰_,β⁰+I_α−α⁰,β−β⁰ ≥ I_α,β.

The contradiction indicates that u_n(x) _:= φ_n(x)−φ(x+y_n) → _{0 and} v_n(x) _:= ψ_n(x)−

ψ(x+yn)→0 in L^p(_Rⁿ)for 2< p<2^∗.

Proof of Theorem2.2. (1) Let{(φn,ψn)}be a minimizing sequence for the functional I on M_α,β. In light of Lemma 4.3, we know that there exists {y_n} ⊂ _Rⁿ such that φ_n(x−y_n) → φ, ψ_n(x−y_n)→ψin L^p(_Rⁿ)for 2< p <2^∗. Hence, by weak convergence, we get

I(φ,ψ)≤ I_α,β. (4.7)

Now, we let |φ|²₂ = α⁰, |ψ|²₂ = β⁰. To show that |φ|²₂ = α and |ψ|²₂ = β, we assume by contradiction that α⁰ < αor β⁰ < β. We consider the following three cases: (1) 0 ≤ α⁰ < α, 0≤ β⁰ < βandα⁰+β⁰ 6=0; (2) 0≤α⁰ <_α, β⁰ =β; and (3) 0≤β⁰ <_β,α⁰ = _α.

Case 1. 0≤α⁰ <α, 0≤β⁰ <βandα⁰+β⁰ 6=0. By definitionI(φ,ψ)≥ I_α⁰_,β⁰and thus it results from (4.7) thatI_α⁰_,β⁰ ≤ I_α,β. From Lemma4.1(iii), I_α,β ≤ I_α⁰_,β⁰+I_α−α⁰_,β−β⁰and by Lemma4.1(i), I_α−α⁰_,β−β⁰<0, we obtain I_α,β < I_α⁰_,β⁰ and it is a contradiction.