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Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2

  • Yuzhen Kong , Qingxuan Wang and Dun Zhao EMAIL logo
Published/Copyright: May 18, 2021
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Advanced Nonlinear Studies
From the journal Advanced Nonlinear Studies Volume 21 Issue 3

Abstract

We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ2, where the intra-component interactions μi and the inter-component interactions βi⁢j=βj⁢i (i,j=1,2,3, i≠j) are all attractive. We display the regions of μi and βi⁢j for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as βi⁢j↗βi⁢j*:=a∗+12⁢(a∗-μi)⁢(a∗-μj), where 0<μi<a∗:=∥w∥22 are fixed and w is the unique positive solution of Δ⁢w-w+w3=0 in H1⁢(ℝ2). The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.

Keywords: Ground State; 3-Component Bose–Einstein Condensate; Mass Concentration; Nondegeneracy; Uniqueness
MSC 2010: 35J50; 35Q40; 35Q55

1 Introduction

The Bose–Einstein condensate (BEC) of dilute and ultracold atomic gases trapped in a potential [3] provides unique platform for the simulation of quantum many-body systems, and it has drawn enormous interest of both physicists and mathematicians. In this paper, we are concerned with the existence and nonexistence of the ground states of the 3-component BEC in ℝ2 with harmonic-like potentials and attractive mean-field interactions between particles. We will also investigate the asymptotic behavior of the ground states. This subjects include the mass concentration phenomenon and uniqueness. Comparing with the 2-component BEC system, the 3-component BEC system becomes far more complicated, because more interactions are involved in among the components. Although the 2-component system has received massively investigation in mathematics, the study on the 3-component system seems still insufficient, especially for the system with external potentials. To the best of our knowledge, up to now, results on mass concentration and uniqueness of ground states have not been obtained for the 3-component system.

In the mean-field approximation, the dynamics of the 3-component BEC can be described by the order parameter Ψ=(ψ1⁢(x,t),ψ2⁢(x,t),ψ3⁢(x,t)) (x∈ℝd,d=1,2,3) [11]

(1.1) i ⁢ ℏ ⁢ ∂ t ⁡ ψ j = δ ⁢ E ⁢ ( Ψ ) δ ⁢ ψ j ∗ , j = 1 , 2 , 3 ,

with

(1.2) ∫ ℝ d ( | ψ 1 | 2 + | ψ 2 | 2 + | ψ 3 | 2 ) ⁢ 𝑑 x = N ,

where N is the number of atoms. In (1.1), ℏ is the Planck constant, ψj∗ denotes the complex conjugate of ψj, and E is the energy of the system, which is given by

(1.3) E ⁢ ( Ψ ) = ∑ i , j = 1 3 ∫ ℝ d ( - ℏ 2 2 ⁢ m ⁢ ψ i ∗ ⁢ Δ ⁢ ψ i ⁢ δ i ⁢ j + V ⁢ ( x ) ⁢ | ψ i | 2 ⁢ δ i ⁢ j - β i ⁢ j 2 ⁢ | ψ i | 2 ⁢ | ψ j | 2 ) ⁢ 𝑑 x ,

where m denotes the mass, V⁢(x) the external potential, and δi⁢j is the Kronecker delta symbol, i.e., δi⁢i=1, δi⁢j=0 (i≠j).

One can derive the time-evolution equations called Gross–Pitaevskii equations of the condensate state from (1.1), which read [27]

(1.4) { i ⁢ ∂ t ⁡ ψ 1 ⁢ ( x , t ) = - Δ ⁢ ψ 1 + V ⁢ ( x ) ⁢ ψ 1 - μ 1 ⁢ | ψ 1 | 2 ⁢ ψ 1 - β 12 ⁢ | ψ 2 | 2 ⁢ ψ 1 - β 13 ⁢ | ψ 3 | 2 ⁢ ψ 1 , i ⁢ ∂ t ⁡ ψ 2 ⁢ ( x , t ) = - Δ ⁢ ψ 2 + V ⁢ ( x ) ⁢ ψ 2 - μ 2 ⁢ | ψ 2 | 2 ⁢ ψ 2 - β 21 ⁢ | ψ 1 | 2 ⁢ ψ 2 - β 23 ⁢ | ψ 3 | 2 ⁢ ψ 2 , i ⁢ ∂ t ⁡ ψ 3 ⁢ ( x , t ) = - Δ ⁢ ψ 3 + V ⁢ ( x ) ⁢ ψ 3 - μ 3 ⁢ | ψ 3 | 2 ⁢ ψ 3 - β 31 ⁢ | ψ 1 | 2 ⁢ ψ 3 - β 32 ⁢ | ψ 2 | 2 ⁢ ψ 3 ,

where μi:=βi⁢i>0 (<0) denotes the attractive (repulsive) intraspecies interaction of the atoms inside each component, and βi⁢j=βj⁢i>0 (<0) for i≠j denotes the attractive (repulsive) interspecies interaction between component i and component j. This model also arises from nonlinear optics, where ψi⁢(x,t) denotes the jth component of the beam in Kerr-like photorefractive media (cf. [1]).

Note that when ψ3=0, (1.4) degenerates into a 2-component system

{ i ⁢ ∂ t ⁡ ψ 1 ⁢ ( x , t ) = - Δ ⁢ ψ 1 + V ⁢ ( x ) ⁢ ψ 1 - μ 1 ⁢ | ψ 1 | 2 ⁢ ψ 1 - β 12 ⁢ | ψ 2 | 2 ⁢ ψ 1 , i ⁢ ∂ t ⁡ ψ 2 ⁢ ( x , t ) = - Δ ⁢ ψ 2 + V ⁢ ( x ) ⁢ ψ 2 - μ 2 ⁢ | ψ 2 | 2 ⁢ ψ 2 - β 21 ⁢ | ψ 1 | 2 ⁢ ψ 2 ,

which describes the 2-component BEC (i.e. binary BEC) [33]. Recently, there has been massive literatures devoted to this system, we refer the readers to the works [2, 15, 16, 21, 25, 26, 5, 4, 6, 10, 18, 28, 31] for various topics. In particular, [15, 16] give the uniqueness and the symmetry-breaking of the ground states as β12↗a∗+(a∗-μ1)⁢(a∗-μ2) (a∗ will be defied later), and show that if μ1↗a∗, then ψ2 can either blow up or vanish, depending on how β12 approaches a∗.

In recent years, the 3-component system has been studied by many researchers. For example, in [24], the authors have obtained some theorems for the existence and nonexistence of ground state solutions of (1.4) without external potential; in [30], the authors have constructed some solutions with partial synchronization and segregation in ℝ3 with mixed couplings, in which the nondegeneracy of a positive solution of 2-component system has been used; in [12], the authors have studied existence and uniqueness of positive solutions without external potential under the assumption β12=β13=β23≥0. However, it seems that results about mass concentration and nondegeneracy of ground state for 3-component BEC with external potential are still absent.

It is known that for BEC with attractive interaction, the dynamics of the formation, decay and collapse are different in one-dimensional (1D), 2D and 3D. For the 1D system, the attractive interaction can compensate exactly for the dispersion of a wave packet, leading to an integrable, and highly stable soliton solution. In the 3D system, for the homogeneous case, all solutions are predicted to be unstable. However, for an inhomogeneous condensate, if the nonlinearity is relatively weak, the spatial localization provided by an external trap potential can stabilize the condensate against collapse [32]. In contrast, the situation is different from a 2D system. Since the mean-field interaction energy decreases with increasing condensate number, this energy cannot be balanced by the kinetic energy of atoms eventually. Therefore, the condensate tends to collapse upon itself. In [7, 19], the authors have measured and predicted the critical number of condensate atoms.

In this paper, we are interested in the ground states of (1.4) in 2D under the constraint (1.2) with the assumption μi,βi⁢j>0. We shall focus on the asymptotic behavior when every βi⁢j (i≠j) approaches their thresholds. Without loss of generality, we assume N=1.

The ground state can be obtained by minimizing E⁢(Ψ) defined by equation (1.3). Generally this solution is a stationary wave, i.e., ψj⁢(x,t)=uj⁢(x)⁢ei⁢γ⁢t (j=1,2,3) for some γ∈ℝ. Putting this form back into (1.3), we get the time-independent real energy function in terms of uj (j=1,2,3). Therefore, the GP energy functional (1.3) associated with (1.4) can be explicitly written as

E ⁢ ( u 1 , u 2 , u 3 ) = ∑ i = 1 3 ∫ ℝ 2 ( | ∇ ⁡ u i | 2 + V ⁢ ( x ) ⁢ u i 2 ) ⁢ 𝑑 x
- 1 2 ⁢ ∫ ℝ 2 ( μ 1 ⁢ u 1 4 + μ 2 ⁢ u 2 4 + μ 3 ⁢ u 3 4 + 2 ⁢ β 12 ⁢ u 1 2 ⁢ u 2 2 + 2 ⁢ β 13 ⁢ u 1 2 ⁢ u 3 2 + 2 ⁢ β 23 ⁢ u 2 2 ⁢ u 3 2 ) ⁢ 𝑑 x .

We will work in the space ℋ=ℋ1⁢(ℝ2)×ℋ1⁢(ℝ2)×ℋ1⁢(ℝ2), where ℋ1⁢(ℝ2) is defined as

ℋ 1 ⁢ ( ℝ 2 ) = { u ∈ H 1 ⁢ ( ℝ 2 ) : ∫ ℝ 2 V ⁢ ( x ) ⁢ u 2 ⁢ 𝑑 x < ∞ }

and ℋ1⁢(ℝ2) is equipped with the norm

∥ u ∥ ℋ 1 = ( ∫ ℝ 2 | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ u 2 ⁢ d ⁢ x ) 1 2 .

Using the approach similar to that in [15, 20], we know that ground states of (1.4) can be obtained equivalently by minimizers of the following L2-critical constraint variational problem

(1.5) e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) := inf ( u 1 , u 2 , u 3 ) ∈ ℳ ⁡ E μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ⁢ ( u 1 , u 2 , u 3 ) ,

where

ℳ := { ( u 1 , u 2 , u 3 ) ∈ ℋ : ∫ ℝ 2 ( u 1 2 + u 2 2 + u 3 2 ) ⁢ 𝑑 x = 1 } .

The GP energy functional in (1.5) is given by

(1.6)

E μ 1 , μ 2 , μ 3 , β 1 , β 2 , β 3 ⁢ ( u 1 , u 2 , u 3 ) = ∑ i = 1 3 ∫ ℝ 2 ( | ∇ ⁡ u i | 2 + V ⁢ ( x ) ⁢ u i 2 ) ⁢ 𝑑 x
- 1 2 ⁢ ∫ ℝ 2 ( μ 1 ⁢ u 1 4 + μ 2 ⁢ u 2 4 + μ 3 ⁢ u 3 4 + 2 ⁢ β 12 ⁢ u 1 2 ⁢ u 2 2 + 2 ⁢ β 13 ⁢ u 1 2 ⁢ u 3 2 + 2 ⁢ β 23 ⁢ u 2 2 ⁢ u 3 2 ) ⁢ 𝑑 x .

Since the GP energy functional is even in (u1,u2,u3), any minimizer (u1,u2,u3) of problem (1.5) must be nonnegative or nonpositive. Without loss of generality, we restrict our study to nonnegative minimizer of e⁢(μ1,μ2,μ3,β12,β13,β23), which satisfies the following Euler–Lagrange equation:

(1.7) { - Δ ⁢ u 1 + V ⁢ ( x ) ⁢ u 1 = γ ⁢ u 1 + μ 1 ⁢ u 1 3 + β 12 ⁢ u 2 2 ⁢ u 1 + β 13 ⁢ u 3 2 ⁢ u 1 , - Δ ⁢ u 2 + V ⁢ ( x ) ⁢ u 2 = γ ⁢ u 2 + μ 2 ⁢ u 2 3 + β 21 ⁢ u 1 2 ⁢ u 2 + β 23 ⁢ u 3 2 ⁢ u 2 , - Δ ⁢ u 3 + V ⁢ ( x ) ⁢ u 3 = γ ⁢ u 3 + μ 3 ⁢ u 3 3 + β 31 ⁢ u 1 2 ⁢ u 3 + β 32 ⁢ u 2 2 ⁢ u 3 , u j ≥ 0   in  ⁢ ℝ 2 , j = 0 , 1 , 2 ,

where γ∈ℝ is a suitable Lagrange multiplier.

We assume that the trapping potential V⁢(x) satisfies

  1. 0 ≤ V ⁢ ( x ) ∈ C loc α ⁢ ( ℝ 2 ) (0<α<1),

  2. lim | x | → ∞ ⁡ V ⁢ ( x ) = + ∞ and infx∈ℝ2⁡V⁢(x)=0 is attained,

  3. x 0 is the unique minimum point of V⁢(x), limx→0⁡V⁢(x+x0)|x|p=1 for some p>0.

Our analysis relies on the unique positive solution [22] of the following nonlinear scalar field equation:

(1.8) Δ ⁢ w - w + w 3 = 0 , w ∈ H 1 ⁢ ( ℝ 2 ) ,

where w⁢(x) decays exponentially, as shown in [13], in the sense that

(1.9) w ⁢ ( x ) , | ∇ ⁡ w ⁢ ( x ) | = O ⁢ ( | x | - 1 2 ⁢ e - | x | )   as  ⁢ | x | → ∞ .

Note that by the Pohozaev identity, one can get the following identities:

(1.10) a ∗ := ∥ w ∥ 2 2 = ∥ ∇ ⁡ w ∥ 2 2 = 1 2 ⁢ ∥ w ∥ 4 4 .

It is also known that the classical Gagliardo–Nirenberg inequality

(1.11) ∫ ℝ 2 u 4 ⁢ 𝑑 x ≤ 2 a ∗ ⁢ ∫ ℝ 2 | ∇ ⁡ u | 2 ⁢ 𝑑 x ⁢ ∫ ℝ 2 u 2 ⁢ 𝑑 x , u ∈ H 1 ⁢ ( ℝ 2 ) ∖ { 0 } ,

where the equality is attained at w [17, 34].

In this paper, our aim is to analyze the regions of μi and βi⁢j for the existence and nonexistence of ground states, and investigate the asymptotic behavior of ground states when βi⁢j↗βi⁢j∗:=a∗+12⁢(a∗-μi)⁢(a∗-μj), i,j=1,2,3, i≠j (βi⁢j↗βi⁢j∗ means βi⁢j→βi⁢j∗ with βi⁢j<βi⁢j∗).

The following Gagliardo–Nirenberg-type inequality, which has been proved in [20], will be used:

(1.12) ∫ ℝ 2 ( u 1 2 + u 2 2 + u 3 2 ) 2 ⁢ 𝑑 x ≤ 2 a ∗ ⁢ ∫ ℝ 2 ( | ∇ ⁡ u 1 | 2 + | ∇ ⁡ u 2 | 2 + | ∇ ⁡ u 3 | 2 ) ⁢ 𝑑 x ⁢ ∫ ℝ 2 ( u 1 2 + u 2 2 + u 3 2 ) ⁢ 𝑑 x ,

where (u1,u2,u3)∈H1⁢(ℝ2)×H1⁢(ℝ2)×H1⁢(ℝ2)∖{0}, 2a∗ is the best constant of inequality (1.12). The equality is attained at (w⁢sin⁡ϕ⁢cos⁡θ,w⁢sin⁡ϕ⁢sin⁡θ,w⁢cos⁡ϕ) for any ϕ,θ∈[0,2⁢π).

For simplicity, when 0<μi<a∗, we write

(1.13) α i ⁢ j ∗ = α j ⁢ i ∗ := ( a ∗ - μ i ) ⁢ ( a ∗ - μ j ) , i , j = 1 , 2 , 3 , i ≠ j ,
(1.14) β i ⁢ j ∗ = β j ⁢ i ∗ := a ∗ + 1 2 ⁢ α i ⁢ j ∗ , i , j = 1 , 2 , 3 , i ≠ j .

Our first main result can be stated as follows.

Theorem 1.1.

Suppose V⁢(x) satisfies (V1)–(V3). We have

  1. If 0 < μ i < a ∗ and 0 < β i ⁢ j < β i ⁢ j ∗ , i,j=1,2,3, i≠j, then problem (1.5) has at least one minimizer.

  2. Let 0 < μ i < a ∗ and let one of the following conditions be satisfied:

    1. β 12 ↗ β 12 ∗ (or β 12 = β 12 ∗ ), β 13 < β 13 ∗ , β23<β23∗,

    2. β 13 ↗ β 13 ∗ (or β 13 = β 13 ∗ ), β 12 < β 12 ∗ , β23<β23∗,

    3. β 23 ↗ β 23 ∗ (or β 23 = β 23 ∗ ), β 12 < β 12 ∗ , β13<β13∗,

    4. β 12 ↗ β 12 ∗ , β13↗β13∗ (or β12=β12∗, β13=β13∗), β23<β23∗,

    5. β 12 ↗ β 12 ∗ , β23↗β23∗ (or β12=β12∗, β23=β23∗), β13<β13∗,

    6. β 13 ↗ β 13 ∗ , β23↗β23∗ (or β13=β13∗, β23=β23∗), β12<β12∗.

    Then problem ( 1.5 ) has at least one minimizer.

  3. Let one of the following conditions be satisfied:

    1. one of μ k ( k ∈ { 1 , 2 , 3 } ) satisfies μ k > a ∗ ,

    2. 0 < μ i < a ∗ ( i = 1 , 2 , 3 ), β 12 > β 12 ∗ , β13>β13∗, β23>β23∗,

    3. 0 < μ i < a ∗ ( i = 1 , 2 , 3 ), β i 0 ⁢ j 0 > a ∗ + α i 0 ⁢ j 0 ∗ for some i 0 , j 0 ∈ { 1 , 2 , 3 } , i0≠j0.

    Then problem ( 1.5 ) has no minimizer.

  4. If one of μ k ( k ∈ { 1 , 2 , 3 } ) satisfies μ k = a ∗ , and β i ⁢ j ≤ β i ⁢ j ∗ holds for i , j = 1 , 2 , 3 , i≠j, then problem (1.5) has no minimizer. Furthermore, we have

    lim μ 1 ↗ a ∗ ⁡ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) = e ⁢ ( a ∗ , μ 2 , μ 3 , β 12 , β 13 , β 23 ) = 0 ,
    lim μ 2 ↗ a ∗ ⁡ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) = e ⁢ ( μ 1 , a ∗ , μ 3 , β 12 , β 13 , β 23 ) = 0 ,
    lim μ 3 ↗ a ∗ ⁡ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) = e ⁢ ( μ 1 , μ 2 , a ∗ , β 12 , β 13 , β 23 ) = 0 .

  5. If 0 < μ i < a ∗ and β i ⁢ j = β i ⁢ j ∗ holds for i , j = 1 , 2 , 3 , i≠j, then problem (1.5) has no minimizer. Furthermore,

    (1.15) lim β i ⁢ j ↗ β i ⁢ j ∗ ⁡ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) = e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 ∗ , β 13 ∗ , β 23 ∗ ) = 0 .

For convenience, we rewrite the functional Eμ1,μ2,μ3,β12,β13,β23⁢(u1,u2,u3) as

E μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ⁢ ( u 1 , u 2 , u 3 ) = ∑ i = 1 3 ∫ ℝ 2 ( | ∇ ⁡ u i | 2 + V ⁢ ( x ) ⁢ u i 2 ) ⁢ 𝑑 x - a ∗ 2 ⁢ ∫ ℝ 2 ( u 1 2 + u 2 2 + u 3 2 ) 2 ⁢ 𝑑 x
+ 1 4 ⁢ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 2 - a ∗ - μ 2 ⁢ u 2 2 ) 2 ⁢ 𝑑 x
+ 1 4 ⁢ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 2 - a ∗ - μ 3 ⁢ u 3 2 ) 2 ⁢ 𝑑 x
+ 1 4 ⁢ ∫ ℝ 2 ( a ∗ - μ 2 ⁢ u 2 2 - a ∗ - μ 3 ⁢ u 3 2 ) 2 ⁢ 𝑑 x + ( β 12 ∗ - β 12 ) ⁢ ∫ ℝ 2 u 1 2 ⁢ u 2 2 ⁢ 𝑑 x
(1.16) + ( β 13 ∗ - β 13 ) ⁢ ∫ ℝ 2 u 1 2 ⁢ u 3 2 ⁢ 𝑑 x + ( β 23 ∗ - β 23 ) ⁢ ∫ ℝ 2 u 2 2 ⁢ u 3 2 ⁢ 𝑑 x .

To get the precise blow-up behavior of minimizers as βi⁢j↗βi⁢j∗, i,j=1,2,3, i≠j, we define

(1.17) H ⁢ ( y ) := ∫ ℝ 2 | x + y | p ⁢ w 2 ⁢ ( x ) ⁢ 𝑑 x

and

(1.18) λ 0 := min y ∈ ℝ 2 ⁡ H ⁢ ( y ) = ∫ ℝ 2 | x | p ⁢ w 2 ⁢ ( x ) ⁢ 𝑑 x .

It can be verified that 𝟎 is the unique and nondegenerate critical point of H⁢(y).

Suppose that {(β12(k),β13(k),β23(k))} is a sequence such that βi⁢j(k)↗βi⁢j∗ (i,j=1,2,3, i≠j) as k→∞. Set

(1.19) e k := e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 ( k ) , β 13 ( k ) , β 23 ( k ) ) .

Let (u1⁢k,u2⁢k,u3⁢k) be a nonnegative minimizer of ek.

Through analyzing the functional (1.16), we can learn the limit behavior of nonnegative minimizers of (1.5) as β12↗β12∗, β13↗β13∗, β23↗β23∗.

Theorem 1.2.

Suppose that V⁢(x) satisfies (V1)–(V3). Let (u1⁢k,u2⁢k,u3⁢k) be a nonnegative minimizer of problem (1.19), where 0<μi<a∗, βi⁢j(k)↗βi⁢j∗ (i,j=1,2,3, i≠j) as k→∞. Then there exists a subsequence of {(β12(k),β13(k),β23(k))}, still denoted by {(β12(k),β13(k),β23(k))}, such that (u1⁢k,u2⁢k,u3⁢k) satisfies

{ lim k → ∞ ⁡ ε ¯ k ⁢ u 1 ⁢ k ⁢ ( ε ¯ k ⁢ x + z ¯ 1 ⁢ k ) = 1 a ∗ ⁢ A 1 ⁢ w , lim k → ∞ ⁡ ε ¯ k ⁢ u 2 ⁢ k ⁢ ( ε ¯ k ⁢ x + z ¯ 2 ⁢ k ) = 1 a ∗ ⁢ A 2 ⁢ w , lim k → ∞ ⁡ ε ¯ k ⁢ u 3 ⁢ k ⁢ ( ε ¯ k ⁢ x + z ¯ 3 ⁢ k ) = 1 a ∗ ⁢ A 3 ⁢ w ,   strongly in  H 1 ( ℝ 2 ) ,

where

(1.20) A 1 := α 23 ∗ A 0 , A 2 := α 13 ∗ A 0 , A 3 := α 12 ∗ A 0 ,
(1.21) A 0 := α 23 ∗ + α 13 ∗ + α 12 ∗ ,

and ε¯k>0 is defined by

(1.22) ε ¯ k := [ 4 ⁢ B k A 0 2 ⁢ p ⁢ λ 0 ] 1 p + 2

with

(1.23) B k := ( β 12 ∗ - β 12 ( k ) ) ⁢ ( a ∗ - μ 3 ) ⁢ α 12 ∗ + ( β 13 ∗ - β 13 ( k ) ) ⁢ ( a ∗ - μ 2 ) ⁢ α 13 ∗ + ( β 23 ∗ - β 23 ( k ) ) ⁢ ( a ∗ - μ 1 ) ⁢ α 23 ∗ .

Moreover, z¯i⁢k is the unique maximum point of ui⁢k satisfying

lim k → ∞ ⁡ z ¯ i ⁢ k = x 0 , where  x 0 ∈ ℝ 2 satisfies  V ⁢ ( x 0 ) = 0 ,
(1.24) lim k → ∞ ⁡ | z ¯ i ⁢ k - x 0 | ε ¯ k = 0 , lim k → ∞ ⁡ | z ¯ i ⁢ k - z ¯ j ⁢ k | ε ¯ k = 0 , i , j = 1 , 2 , 3 .

Theorem 1.2 shows that minimizers of ek must concentrate at the minimum point x0. Moreover, the structure of each component of minimizers for ek looks like a spike as k→∞, and no component of minimizers can vanish due to μi<a∗ (i=1,2,3). In fact, using the approach as that in [15], we can also prove that minimizers of ek concentrate at a flattest point of V⁢(x) if V⁢(x)=gj⁢(x)⁢∏j=1n|x-xj|pj, pj>0, n∈ℕ+, where gj⁢(x)∈Clocα⁢(ℝ2) satisfies C-1≤gj⁢(x)≤C for some positive constant C. Therefore, we assume that x0 is the unique minimum point of V⁢(x) for simplicity in the whole paper.

It is worth noting that if 0<μi<a∗ and there exists some βi⁢j such that βi⁢j<βi⁢j∗ (i,j∈{1,2,3},i≠j), but βm⁢j↗βm⁢j∗ for m≠i (for example, β23<β23∗, β12↗β12∗, β13↗β13∗), then it follows from (2) of Theorem 1.1 that minimizers of ek are bounded in ℋ, so they do not blow up as k→∞. Thus, when 0<μi<a∗ (i=1,2,3), a minimizer of (1.5) blows up only if β12↗β12∗, β13↗β13∗, β23↗β23∗.

The uniqueness of ground states for 2-component BECs has been addressed in [15, 16] by applying Pohozaev identities, and the proof relies on the fact that the positive solution (u0,v0) (see [15, (4.11)] or [16, (4.3)]) is nondegenerate. However, nondegenerate result has not been established for 3-component system.

To get the uniqueness of ground states for 3-component system, we have to prove that (U1,U2,U3) (defined in (1.26)) is nondegenerate, which is a key point for the uniqueness of nonnegative minimizers for ek, as k→∞.

For convenience, we define

(1.25)

a 1 := μ 1 a ∗ , a 2 := μ 2 a ∗ , a 3   := μ 3 a ∗ ,
b 1 := β 12 ∗ a ∗ , b 2 := β 13 ∗ a ∗ , b 3   := β 23 ∗ a ∗ .

Using the notations defined in (1.20) and (1.21), define

(1.26) ( U 1 , U 2 , U 3 ) := ( A 1 ⁢ w , A 2 ⁢ w , A 3 ⁢ w ) .

It is easy to verify that (U1,U2,U3) is a solution of the following equation:

(1.27) { - Δ ⁢ u + u = a 1 ⁢ u 3 + b 1 ⁢ v 2 ⁢ u + b 2 ⁢ z 2 ⁢ u , - Δ ⁢ v + v = a 2 ⁢ v 3 + b 1 ⁢ u 2 ⁢ v + b 3 ⁢ z 2 ⁢ v , - Δ ⁢ z + z = a 3 ⁢ z 3 + b 2 ⁢ u 2 ⁢ z + b 3 ⁢ v 2 ⁢ z .

In the case μ2=μ3, by (1.25) and (1.20), we have b1=b2,a2=a3,A2=A3, and thus U2=U3. Under this assumption, we have:

Theorem 1.3.

Suppose that (ϕ1,ϕ2,ϕ3)∈H2⁢(R2)×H2⁢(R2)×H2⁢(R2) satisfies the following eigenvalue problem:

{ Δ ⁢ ϕ 1 - ϕ 1 + 3 ⁢ a 1 ⁢ U 1 2 ⁢ ϕ 1 + b 1 ⁢ U 2 2 ⁢ ϕ 1 + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 2 + b 2 ⁢ U 3 2 ⁢ ϕ 1 + 2 ⁢ b 2 ⁢ U 3 ⁢ U 1 ⁢ ϕ 3 = 0 , Δ ⁢ ϕ 2 - ϕ 2 + 3 ⁢ a 2 ⁢ U 2 2 ⁢ ϕ 2 + b 1 ⁢ U 1 2 ⁢ ϕ 2 + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 + b 3 ⁢ U 3 2 ⁢ ϕ 2 + 2 ⁢ b 3 ⁢ U 3 ⁢ U 2 ⁢ ϕ 3 = 0 , Δ ⁢ ϕ 3 - ϕ 3 + 3 ⁢ a 3 ⁢ U 3 2 ⁢ ϕ 3 + b 2 ⁢ U 1 2 ⁢ ϕ 3 + 2 ⁢ b 2 ⁢ U 1 ⁢ U 3 ⁢ ϕ 1 + b 3 ⁢ U 2 2 ⁢ ϕ 3 + 2 ⁢ b 3 ⁢ U 3 ⁢ U 2 ⁢ ϕ 2 = 0 .

If a2=a3, then

( ϕ 1 ϕ 2 ϕ 3 ) ∈ span ⁡ { ( ∂ ⁡ U 1 ∂ ⁡ x j ∂ ⁡ U 2 ∂ ⁡ x j ∂ ⁡ U 3 ∂ ⁡ x j ) : j = 1 , 2 } .

Our proof is inspired by [10, 29]. Through more delicate analysis, we get the nondegeneracy for 3-component system if μ2=μ3. In fact, such a result on nondegeneracy holds whenever μi0=μj0 for some i0,j0∈{1,2,3} with i0≠j0; μ2=μ3 is only for convenience. Based on Theorem 1.3 and Pohozaev identities, we can establish the following uniqueness result.

Theorem 1.4.

Suppose that V⁢(x)∈C2⁢(R2) satisfies (V2)–(V3) with p≥2, and there exist C,κ,r0>0 such that

(1.28) V ⁢ ( x ) ≤ C ⁢ e κ ⁢ | x |   if  | x | is large enough .

Moreover, assume

(1.29) ∂ ⁡ V ⁢ ( x + x 0 ) ∂ ⁡ x n = ∂ ⁡ | x | p ∂ ⁡ x n + R n ⁢ ( x ) = p ⁢ | x | p - 2 ⁢ x n + R n ⁢ ( x )   𝑎𝑛𝑑   | R n ⁢ ( x ) | ≤ C ⁢ | x | q   in  ⁢ B r 0 ⁢ ( 0 ) ,

where q>p-1, n=1,2. Then under the assumptions in Theorem 1.3, for any given μi∈(0,a∗) (i=1,2,3) there exists a unique nonnegative minimizer of e⁢(μ1,μ2,μ3,β12,β13,β23) as βi⁢j↗βi⁢j∗ (j=1,2,3, j≠i).

Although we have addressed the condition μi0=μj0, Theorem 1.3 and Theorem 1.4 seem to be the first results on the nondegeneracy and uniqueness for 3-component BEC with external potential as βi⁢j↗βi⁢j∗. Comparing with 2-component system, since more interspecies interactions are involved in, we must overcome more extra difficulties. We need to consider function (ξ^1⁢k,ξ^2⁢k,ξ^3⁢k) defined in (5.23) and investigate the limit structure of (ξ^1⁢k,ξ^2⁢k,ξ^3⁢k) as k→∞.

This paper is organized as follows. The existence and nonexistence of the minimizers for problem (1.5) are classified in Section 2. Making use of (1.12), we can find the regions of μi and βi⁢j for the existence and nonexistence of ground state. Since the embedding ℋ1⁢(ℝ2)↪Lq⁢(ℝ2) is compact for 2≤q<∞, proof of the existence of minimizer for problem (1.5) is relatively simple. Section 3 is focused on the mass concentration of nonnegative minimizers for ek as k→∞. In Section 4, we prove that (U1,U2,U3) is nondegenerate. Following Theorem 1.2 and Theorem 1.3, we shall finally prove Theorem 1.4 in Section 5.

2 Existence and Nonexistence of Minimizers

Proof of (1) and (2) of Theorem 1.1.

For simplicity, similar to (1.13), when 0<μi<a, we write

α i ⁢ j = α j ⁢ i := ( a - μ i ) ⁢ ( a - μ j )   ( i , j = 1 , 2 , 3 , i ≠ j ) .

Note that for any given 0≤μj<a∗, βi⁢j<βi⁢j∗, there exists an a>0 such that

max { μ 1 , μ 2 , μ 3 } ≤ a < a ∗ , β i ⁢ j < a + 1 2 α i ⁢ j   ( i , j = 1 , 2 , 3 , i ≠ j ) .

We rewrite (1.6) as

E μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ⁢ ( u 1 , u 2 , u 3 ) = ∑ i = 1 3 ∫ ℝ 2 ( | ∇ ⁡ u i | 2 + V ⁢ ( x ) ⁢ u i 2 ) ⁢ 𝑑 x - a 2 ⁢ ∫ ℝ 2 ( u 1 2 + u 2 2 + u 3 2 ) 2 ⁢ 𝑑 x
+ 1 4 ⁢ ∫ ℝ 2 ( a - μ 1 ⁢ u 1 2 - a - μ 2 ⁢ u 2 2 ) 2 ⁢ 𝑑 x
+ 1 4 ⁢ ∫ ℝ 2 ( a - μ 1 ⁢ u 1 2 - a - μ 3 ⁢ u 3 2 ) 2 ⁢ 𝑑 x
+ 1 4 ⁢ ∫ ℝ 2 ( a - μ 2 ⁢ u 2 2 - a - μ 3 ⁢ u 3 2 ) 2 ⁢ 𝑑 x + ( a + 1 2 ⁢ α 12 - β 12 ) ⁢ ∫ ℝ 2 u 1 2 ⁢ u 2 2 ⁢ 𝑑 x
(2.1) + ( a + 1 2 ⁢ α 13 - β 13 ) ⁢ ∫ ℝ 2 u 1 2 ⁢ u 3 2 ⁢ 𝑑 x + ( a + 1 2 ⁢ α 23 - β 23 ) ⁢ ∫ ℝ 2 u 2 2 ⁢ u 3 2 ⁢ 𝑑 x .

Let {(u1⁢n,u2⁢n,u3⁢n)}∈ℋ be a minimizing sequence of e⁢(μ1,μ2,μ3,β12,β13,β23). From (2.1) and (1.12), we know that {(u1⁢n,u2⁢n,u3⁢n)} is bounded uniformly in ℋ. By [15, Lemma 2.1], we know that up to a subsequence if necessary, there exists (u10,u20,u30)∈ℋ such that

( u 1 ⁢ n , u 2 ⁢ n , u 3 ⁢ n ) → ( u 10 , u 20 , u 30 )   in ⁢ L q ⁢ ( ℝ 2 ) × L q ⁢ ( ℝ 2 ) × L q ⁢ ( ℝ 2 ) ,  2 ≤ q < ∞ , as  ⁢ n → ∞ .

Then it is easy to deduce that (u10,u20,u30)∈ℳ is a minimizer of problem (1.5).

To prove (2) of Theorem 1.1, without loss of generality, we firstly show that if β12↗β12∗, β13<β13∗, β23<β23∗ or β12=β12∗, β13<β13∗, β23<β23∗, the results hold. To this end, choosing test functions as

ζ 1 ⁢ ( x ) = θ 1 ⁢ w ⁢ ( x - x 0 ) , ζ 2 ⁢ ( x ) = 1 - θ 1 ⁢ w ⁢ ( x - x 0 ) , ζ 3 ⁢ ( x ) ≡ 0 ,

where x0∈ℝ2 satisfies V⁢(x0)=0, and

θ 1 = a ∗ - μ 1 a ∗ - μ 1 + a ∗ - μ 2 .

It is easy to confirm that Eμ1,μ2,μ3,β12,β13,β23⁢(ζ1,ζ2,ζ3)≤C for some positive C. From (1.16) and (1.12), we know that

e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) ≥ 0 .

Let {(u1⁢n,u2⁢n,u3⁢n)}∈ℋ be a minimizing sequence of e⁢(μ1,μ2,μ3,β12,β13,β23). Since β13<β13∗, β23<β23∗, there exists a positive C such that, as n→∞,

∫ ℝ 2 ( a ∗ - μ 2 ⁢ u 2 ⁢ n 2 - a ∗ - μ 3 ⁢ u 3 ⁢ n 2 ) 2 ⁢ 𝑑 x ≤ C , ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ n 2 - a ∗ - μ 3 ⁢ u 3 ⁢ n 2 ) 2 ⁢ 𝑑 x ≤ C ,
∫ ℝ 2 u 1 ⁢ n 2 ⁢ u 3 ⁢ n 2 ⁢ 𝑑 x ≤ C , ∫ ℝ 2 u 2 ⁢ n 2 ⁢ u 3 ⁢ n 2 ⁢ 𝑑 x ≤ C ,

which concludes

∫ ℝ 2 [ ( a ∗ - μ 1 ) ⁢ u 1 ⁢ n 4 + ( a ∗ - μ 3 ) ⁢ u 3 ⁢ n 4 ] ⁢ 𝑑 x ≤ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ n 2 + a ∗ - μ 3 ⁢ u 3 ⁢ n 2 ) 2 ⁢ 𝑑 x
= ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ n 2 - a ∗ - μ 3 ⁢ u 3 ⁢ n 2 ) 2 ⁢ 𝑑 x + 4 ⁢ ∫ ℝ 2 α 13 ∗ ⁢ u 1 ⁢ n 2 ⁢ u 3 ⁢ n 2 ⁢ 𝑑 x ≤ C ,

and so, {u1⁢n},{u3⁢n} are bounded uniformly in L4⁢(ℝ2). Similarly, we can prove that {u2⁢n} is bounded uniformly in L4⁢(ℝ2). Then from (1.16) it yields that {(u1⁢n,u2⁢n,u3⁢n)} is bounded uniformly in ℋ, and the results in (2) follow directly.

At last, without loss of generality, we deal with the cases β12↗β12∗, β13↗β13∗, β23<β23∗ and β12=β12∗, β13=β13∗, β23<β23∗. We choose the test functions as

ζ 1 ⁢ ( x ) = θ 2 ⁢ w ⁢ ( x - x 0 ) , ζ 2 ⁢ ( x ) ≡ 0 , ζ 3 ⁢ ( x ) = 1 - θ 2 ⁢ w ⁢ ( x - x 0 ) ,

where x0∈ℝ2 satisfies V⁢(x0)=0, and

θ 2 = a ∗ - μ 3 a ∗ - μ 1 + a ∗ - μ 3 ,

and then one can verify that Eμ1,μ2,μ3,β12,β13,β23⁢(ζ1,ζ2,ζ3)≤C for some positive C. From (1.16) and (1.12), we get

e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) ≥ 0 .

Let {(u1⁢n,u2⁢n,u3⁢n)}∈ℋ be a minimizing sequence of e⁢(μ1,μ2,μ3,β12,β13,β23). Since β23<β23∗, there exists a positive constant C such that

∫ ℝ 2 u 2 ⁢ n 2 ⁢ u 3 ⁢ n 2 ⁢ 𝑑 x ≤ C   as  ⁢ n → ∞ .

By (1.16), we have

∫ ℝ 2 ( a ∗ - μ 2 ⁢ u 2 ⁢ n 2 - a ∗ - μ 3 ⁢ u 3 ⁢ n 2 ) 2 ⁢ 𝑑 x ≤ C .

Then

∫ ℝ 2 ( a ∗ - μ 2 ) ⁢ u 2 ⁢ n 4 + ( a ∗ - μ 3 ) ⁢ u 3 ⁢ n 4 ≤ C ,

which implies that {u2⁢n},{u3⁢n} are bounded in L4⁢(ℝ2). Since

∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ n 2 - a ∗ - μ 2 ⁢ u 2 ⁢ n 2 ) 2 ⁢ 𝑑 x ≤ C ,

if ∥u1⁢n∥4→∞, then

∫ ℝ 2 ( a ∗ - μ 1 ) ⁢ u 1 ⁢ n 4 ⁢ 𝑑 x ≤ ∫ ℝ 2 ( a ∗ - μ 1 ) ⁢ u 1 ⁢ n 4 + ( a ∗ - μ 2 ) ⁢ u 2 ⁢ n 4 ⁢ d ⁢ x
≤ 2 ⁢ ∫ ℝ 2 α 12 ∗ ⁢ u 1 ⁢ n 2 ⁢ u 2 ⁢ n 2 ⁢ 𝑑 x + C
≤ 2 ⁢ { ∫ ℝ 2 ( a ∗ - μ 1 ) ⁢ u 1 ⁢ n 4 ⁢ 𝑑 x } 1 2 ⁢ { ∫ ℝ 2 ( a ∗ - μ 2 ) ⁢ u 2 ⁢ n 4 ⁢ 𝑑 x } 1 2 + C ,

which is impossible. Hence {u1⁢n} is bounded uniformly in L4⁢(ℝ2). The rest of the proof is similar to the discussion above, so we omit it here. ∎

Proof of (3)–(5) of Theorem 1.1.

Choosing a cutoff function 0≤φ∈C0∞⁢(ℝ2) such that φ⁢(x)=1 for |x|≤1, and φ⁢(x)=0 for |x|≥2. For any τ>0, set

(2.2) u 1 , θ 1 , θ 2 , τ = θ 1 ⁢ θ 2 ⁢ A τ ⁢ τ a ∗ ⁢ φ ⁢ ( x - x 0 ) ⁢ w ⁢ ( τ ⁢ ( x - x 0 ) ) , u 2 , θ 1 , θ 2 , τ = ( 1 - θ 1 ) ⁢ θ 2 ⁢ A τ ⁢ τ a ∗ ⁢ φ ⁢ ( x - x 0 ) ⁢ w ⁢ ( τ ⁢ ( x - x 0 ) ) , u 3 , θ 1 , θ 2 , τ = 1 - θ 2 ⁢ A τ ⁢ τ a ∗ ⁢ φ ⁢ ( x - x 0 ) ⁢ w ⁢ ( τ ⁢ ( x - x 0 ) ) ,

where Aτ>0 is chosen so that (u1,θ1,θ2,τ,u2,θ1,θ2,τ,u3,θ1,θ2,τ)∈ℳ, θ∈[0,1] and V⁢(x0)=0. One can verify that

1 ≤ A τ 2 ≤ 1 + C ⁢ e - 2 ⁢ τ   as ⁢ τ → ∞ .

Using argument similar to that in [20], by (1.10) and (1.9), we know that as τ→∞,

(2.3)

e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) ≤ E μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ⁢ ( u 1 , θ 1 , θ 2 , τ , u 2 , θ 1 , θ 2 , τ , u 3 , θ 1 , θ 2 , τ )
≤ τ 2 a ∗ { a ∗ - [ μ 1 θ 1 2 θ 2 2 + μ 2 ( 1 - θ 1 ) 2 θ 2 2 + μ 3 ( 1 - θ 2 ) 2 + 2 β 12 θ 1 θ 2 2 ( 1 - θ 1 )
+ 2 β 13 θ 1 θ 2 ( 1 - θ 2 ) + 2 β 23 ( 1 - θ 1 ) θ 2 ( 1 - θ 2 ) ] } + C e - τ .

If μ1>a∗, take θ1=θ2=1. From (1.16) and (2.3), we have

e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) ≤ - ∞   as ⁢ τ → ∞ .

If μ2>a∗ or μ3>a∗, taking θ1=0, θ2=1, or θ2=0, respectively, we can conclude that there is no minimizer of (1.5).

If β12>a∗+α12∗, setting θ1=a∗-μ2⁢(a∗-μ1+a∗-μ2)-1 and θ2=1, then

μ 1 ⁢ θ 1 2 + μ 2 ⁢ ( 1 - θ 1 ) 2 + 2 ⁢ β 12 ⁢ θ 1 ⁢ ( 1 - θ 1 ) > a ∗ .

It yields from (2.3) that e⁢(μ1,μ2,μ3,β12,β13,β23)≤-∞ as τ→∞. Similarly, if β13>a∗+α13∗, setting θ1=1, θ2=a∗-μ3⁢(a∗-μ1+a∗-μ3)-1, then

μ 1 ⁢ θ 2 2 + μ 3 ⁢ ( 1 - θ 2 ) 2 + 2 ⁢ β 13 ⁢ θ 2 ⁢ ( 1 - θ 2 ) > a ∗ .

If β23>a∗+α23∗, setting θ1=0, θ2=a∗-μ3⁢(a∗-μ2+a∗-μ3)-1, then

μ 2 ⁢ θ 2 2 + μ 3 ⁢ ( 1 - θ 2 ) 2 + 2 ⁢ β 23 ⁢ θ 2 ⁢ ( 1 - θ 2 ) > a ∗ .

Hence there is no minimizer for (1.5).

If μ1=a∗, μ2, μ3≤a∗ and βi⁢j≤βi⁢j∗ (i,j=1,2,3), we can take θ1=θ2=1, and then from (1.16) and (2.3), we have

0 ≤ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) ≤ 0 .

Using argument similar to that in [15], we deduce that there is no minimizer of (1.5). If μ1↗a∗, then taking θ1=θ2=1 and τ=(a∗-μ1)-14 in (2.3), it yields that

lim μ 1 ↗ a ∗ ⁡ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 ) = 0 .

If μ2=a∗ (μ3=a∗), taking θ1=0, θ2=1 (θ2=0), the results follow easily.

If β12=β12∗,β13=β13∗,β23=β23∗,0<μi<a∗, setting

(2.4) θ 1 = ( a ∗ - μ 2 ) 1 2 ( a ∗ - μ 1 ) 1 2 + ( a ∗ - μ 2 ) 1 2 , θ 2 = 1 A 0 ⁢ ( α 23 ∗ + α 13 ∗ ) ,

then a directly calculation yields that

μ 1 ⁢ θ 1 2 ⁢ θ 2 2 + μ 2 ⁢ ( 1 - θ 1 ) 2 ⁢ θ 2 2 + μ 3 ⁢ ( 1 - θ 2 ) 2 + 2 ⁢ β 12 ⁢ θ 1 ⁢ θ 2 2 ⁢ ( 1 - θ 1 )
+ 2 ⁢ β 13 ⁢ θ 1 ⁢ θ 2 ⁢ ( 1 - θ 2 ) + 2 ⁢ β 23 ⁢ ( 1 - θ 1 ) ⁢ θ 2 ⁢ ( 1 - θ 2 ) = a ∗ .

Applying (1.16) and (2.3), we conclude that (1.5) has no minimizer.

If β12>β12∗, β13>β13∗, β23>β23∗, 0<μi<a∗, taking θ1,θ2 as in (2.4), we get

(2.5)

a * - [ μ 1 θ 1 2 θ 2 2 + μ 2 ( 1 - θ 1 ) 2 θ 2 2 + μ 3 ( 1 - θ 2 ) 2 + 2 β 12 θ 1 θ 2 2 ( 1 - θ 1 ) + 2 β 13 θ 1 θ 2 ( 1 - θ 2 )
+ 2 β 23 ( 1 - θ 1 ) θ 2 ( 1 - θ 2 ) ]
= - [ 2 ⁢ ( β 12 - β 12 ∗ ) ⁢ θ 1 ⁢ θ 2 2 ⁢ ( 1 - θ 1 ) + 2 ⁢ ( β 13 - β 13 ∗ ) ⁢ θ 1 ⁢ θ 2 ⁢ ( 1 - θ 2 ) + 2 ⁢ ( β 23 - β 23 ∗ ) ⁢ ( 1 - θ 1 ) ⁢ θ 2 ⁢ ( 1 - θ 2 ) ] < 0 ,

and then, from (2.5) and (2.3), we have e⁢(μ1,μ2,μ3,β12,β13,β23)≤-∞, so problem (1.5) has no minimizer.

If β12↗β12∗,β13↗β13∗,β23↗β23∗, taking θ1,θ2 as in (2.4), let

τ 2 = { 2 ⁢ ( β 12 ∗ - β 12 ) ⁢ θ 1 ⁢ θ 2 2 ⁢ ( 1 - θ 1 ) + 2 ⁢ ( β 13 ∗ - β 13 ) ⁢ θ 1 ⁢ θ 2 ⁢ ( 1 - θ 2 ) + 2 ⁢ ( β 23 ∗ - β 23 ) ⁢ ( 1 - θ 1 ) ⁢ θ 2 ⁢ ( 1 - θ 2 ) } - 1 2 ,

then

0 ≤ lim β i ⁢ j ↗ β i ⁢ j ∗ ⁡ e ⁢ ( μ 1 , μ 2 , μ 3 , β 12 , β 13 , β 23 )
= lim β i ⁢ j ∗ ↗ β i ⁢ j ⁡ 1 a ∗ ⁢ { 2 ⁢ ( β 12 ∗ - β 12 ) ⁢ θ 1 ⁢ θ 2 2 ⁢ ( 1 - θ 1 ) + 2 ⁢ ( β 13 ∗ - β 13 ) ⁢ θ 1 ⁢ θ 2 ⁢ ( 1 - θ 2 ) + 2 ⁢ ( β 23 ∗ - β 23 ) ⁢ ( 1 - θ 1 ) ⁢ θ 2 ⁢ ( 1 - θ 2 ) } 1 2 = 0 ,

which yields the result. ∎

3 Mass Concentration

In this section, we analyze the asymptotic behavior of nonnegative minimizers of problem (1.5) as βi⁢j↗βi⁢j∗ (i,j=1,2,3, i≠j), where every μi∈(0,a∗) is fixed.

Lemma 3.1.

Let βi⁢j(k)↗βi⁢j∗ (i,j=1,2,3, i≠j) as k→∞ and (u1⁢k,u2⁢k,u3⁢k) be a nonnegative minimizer of problem (1.19). Then we have:

  1. ( u 1 ⁢ k , u 2 ⁢ k , u 3 ⁢ k ) blows up in the sense that for i = 1 , 2 , 3 ,

    (3.1) lim k → ∞ ⁡ ∫ ℝ 2 | ∇ ⁡ u i ⁢ k | 2 ⁢ 𝑑 x = + ∞   𝑎𝑛𝑑   lim k → ∞ ⁡ ∫ ℝ 2 u i ⁢ k 4 ⁢ 𝑑 x = + ∞ .

  2. ( u 1 ⁢ k , u 2 ⁢ k , u 3 ⁢ k ) also satisfies

    (3.2)

    lim k → ∞ ⁡ ∫ ℝ 2 V ⁢ ( x ) ⁢ u 1 ⁢ k 2 ⁢ 𝑑 x = lim k → ∞ ⁡ ∫ ℝ 2 V ⁢ ( x ) ⁢ u 2 ⁢ k 2 ⁢ 𝑑 x = lim k → ∞ ⁡ ∫ ℝ 2 V ⁢ ( x ) ⁢ u 3 ⁢ k 2 ⁢ 𝑑 x = 0 ,
    lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ k 2 - a ∗ - μ 2 ⁢ u 2 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0 ,
    lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ k 2 - a ∗ - μ 3 ⁢ u 3 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0 ,
    lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 2 ⁢ u 2 ⁢ k 2 - a ∗ - μ 3 ⁢ u 3 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0

    and

    (3.3) lim k → ∞ ⁡ ∫ ℝ 2 ( | ∇ ⁡ u 1 ⁢ k | 2 + | ∇ ⁡ u 2 ⁢ k | 2 + | ∇ ⁡ u 3 ⁢ k | 2 ) ⁢ 𝑑 x ∫ ℝ 2 ( u 1 ⁢ k 2 + u 2 ⁢ k 2 + u 3 ⁢ k 2 ) 2 ⁢ 𝑑 x = a ∗ 2 ,
    (3.4) lim k → ∞ ⁡ ∫ ℝ 2 u 1 ⁢ k 4 ⁢ 𝑑 x ∫ ℝ 2 u 2 ⁢ k 4 ⁢ 𝑑 x = a ∗ - μ 2 a ∗ - μ 1 , lim k → ∞ ⁡ ∫ ℝ 2 u 1 ⁢ k 4 ⁢ 𝑑 x ∫ ℝ 2 u 3 ⁢ k 4 ⁢ 𝑑 x = a ∗ - μ 3 a ∗ - μ 1 , lim k → ∞ ⁡ ∫ ℝ 2 u 2 ⁢ k 4 ⁢ 𝑑 x ∫ ℝ 2 u 3 ⁢ k 4 ⁢ 𝑑 x = a ∗ - μ 3 a ∗ - μ 2 .

Proof.

Firstly, (3.2) follows directly from (1.15) and (1.16). Now we claim that

(3.5) lim k → ∞ ⁡ ∫ ℝ 2 ( | ∇ ⁡ u 1 ⁢ k | 2 + | ∇ ⁡ u 2 ⁢ k | 2 + | ∇ ⁡ u 3 ⁢ k | 2 ) ⁢ 𝑑 x = + ∞ .

Indeed, suppose on the contrary that

lim k → ∞ ⁡ ∑ i = 1 3 ∫ ℝ 2 | ∇ ⁡ u i ⁢ k | 2 ⁢ 𝑑 x ≤ C

holds for some positive constant C, then there exists (U1,U2,U3)∈ℋ such that, up to a subsequence if necessary, (u1⁢k,u2⁢k,u3⁢k)→(U1,U2,U3) in L2⁢(ℝ2)×L2⁢(ℝ2)×L2⁢(ℝ2). And so,

0 ≤ E μ 1 , μ 2 , μ 3 , β 12 ∗ , β 13 ∗ , β 23 ∗ ⁢ ( U 1 , U 2 , U 3 ) ≤ lim inf k → ∞ ⁡ E μ 1 , μ 2 , μ 3 , β 12 k , β 13 k , β 23 k ⁢ ( u 1 ⁢ k , u 2 ⁢ k , u 3 ⁢ k ) = 0 ,

which implies that (U1,U2,U3) is a minimizer of e⁢(μ1,μ2,μ3,β12∗,β13∗,β23∗), a contradiction to (4) of Theorem 1.1. It yields from (1.16), (3.2) and (3.5) that (3.3) holds and ∥∇⁡ui⁢k∥L2⁢(ℝ2)→∞ for at least one i∈{1,2,3}. By Hölder’s inequality and (3.2), we have

lim k → ∞ ⁡ [ a * - μ 1 ⁢ ∥ u 1 ⁢ k ∥ L 4 ⁢ ( ℝ 2 ) 2 - a * - μ 2 ⁢ ∥ u 2 ⁢ k ∥ L 4 ⁢ ( ℝ 2 ) 2 ] 2 ≤ lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ u 1 ⁢ k 2 - a ∗ - μ 2 ⁢ u 2 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0 ,

which implies that (3.4) holds, and ∥ui⁢k∥4→∞ (i=1,2,3) as k→∞. Moreover, from (1.11) we see that (3.1) holds. ∎

Write

ε k := ( ∫ ℝ 2 ( | ∇ ⁡ u 1 ⁢ k | 2 + | ∇ ⁡ u 2 ⁢ k | 2 + | ∇ ⁡ u 3 ⁢ k | 2 ) ⁢ 𝑑 x ) - 1 2 > 0 ,

and by Lemma 3.1 we know that εk→0 as k→∞.

Lemma 3.2.

Under the assumptions of Lemma 3.1, we have:

  1. There exist a sequence { y ε k } ∈ ℝ 2 and positive constants R 0 and η such that

    lim inf k → ∞ ⁡ ∫ B R 0 ⁢ ( 0 ) w i ⁢ k 2 ⁢ 𝑑 x ≥ η > 0 , i = 1 , 2 , 3 ,

    where w i ⁢ k ⁢ ( x ) is defined as

    (3.6) w i ⁢ k ⁢ ( x ) := ε k ⁢ u i ⁢ k ⁢ ( ε k ⁢ x + ε k ⁢ y ε k ) .

    Furthermore,

    (3.7) ∫ ℝ 2 ( | ∇ ⁡ w 1 ⁢ k | 2 + | ∇ ⁡ w 2 ⁢ k | 2 + | ∇ ⁡ w 3 ⁢ k | 2 ) ⁢ 𝑑 x = 1 , lim k → ∞ ⁡ ∫ ℝ 2 ( w 1 ⁢ k 2 + w 2 ⁢ k 2 + w 3 ⁢ k 2 ) 2 ⁢ 𝑑 x = 2 a ∗ ,

    and

    (3.8)

    lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ w 1 ⁢ k 2 - a ∗ - μ 2 ⁢ w 2 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0 ,
    lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 1 ⁢ w 1 ⁢ k 2 - a ∗ - μ 3 ⁢ w 3 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0 ,
    lim k → ∞ ⁡ ∫ ℝ 2 ( a ∗ - μ 2 ⁢ w 2 ⁢ k 2 - a ∗ - μ 3 ⁢ w 3 ⁢ k 2 ) 2 ⁢ 𝑑 x = 0 .

  2. For any sequence { ( β 12 ( k ) , β 13 ( k ) , β 23 ( k ) ) } with β i ⁢ j ( k ) ↗ β i ⁢ j ∗ ( i , j = 1 , 2 , 3 , i≠j) as k→∞, there exists a subsequence, which is still denoted by {(β12(k),β13(k),β23(k))}, such that εk⁢yεk→x0, where x0∈ℝ2 satisfies V⁢(x0)=0.

  3. For any sequence { ( β 12 ( k ) , β 13 ( k ) , β 23 ( k ) ) } with β i ⁢ j ( k ) ↗ β i ⁢ j ∗ ( i , j = 1 , 2 , 3 , i≠j) as k→∞, there exist a subsequence, which is still denoted by {(β12(k),β13(k),β23(k))}, and an x1∈ℝ2 such that

    (3.9) { lim k → ∞ ⁡ w 1 ⁢ k ⁢ ( x ) = 1 a ∗ ⁢ A 1 ⁢ w ⁢ ( x - x 1 ) , lim k → ∞ ⁡ w 2 ⁢ k ⁢ ( x ) = 1 a ∗ ⁢ A 2 ⁢ w ⁢ ( x - x 1 ) , lim k → ∞ ⁡ w 3 ⁢ k ⁢ ( x ) = 1 a ∗ ⁢ A 3 ⁢ w ⁢ ( x - x 1 ) ,   strongly in  H 1 ( ℝ 2 ) ,

    where A i ( i = 1 , 2 , 3 ) is defined in ( 1.20 ).

Proof.

The proof of (1) and (2) in Lemma 3.2 is similar to that in [18], so here we only focus on (3). Note that the nonnegative minimizer (u1⁢k,u2⁢k,u3⁢k) solves the system

(3.10) { - Δ ⁢ u 1 ⁢ k + V ⁢ ( x ) ⁢ u 1 ⁢ k = γ k ⁢ u 1 ⁢ k + μ 1 ⁢ u 1 ⁢ k 3 + β 12 ( k ) ⁢ u 2 ⁢ k 2 ⁢ u 1 ⁢ k + β 13 ( k ) ⁢ u 3 ⁢ k 2 ⁢ u 1 ⁢ k , - Δ ⁢ u 2 ⁢ k + V ⁢ ( x ) ⁢ u 2 ⁢ k = γ k ⁢ u 2 ⁢ k + μ 2 ⁢ u 2 ⁢ k 3 + β 21 ( k ) ⁢ u 1 ⁢ k 2 ⁢ u 2 ⁢ k + β 23 ( k ) ⁢ u 3 ⁢ k 2 ⁢ u 2 ⁢ k , - Δ ⁢ u 3 ⁢ k + V ⁢ ( x ) ⁢ u 3 ⁢ k = γ k ⁢ u 3 ⁢ k + μ 3 ⁢ u 3 ⁢ k 3 + β 31 ( k ) ⁢ u 1 ⁢ k 2 ⁢ u 3 ⁢ k + β 32 ( k ) ⁢ u 2 ⁢ k 2 ⁢ u 3 ⁢ k ,

where γk is a suitable Lagrange multiplier. By (3.6) and (3.10), we know that (w1⁢k,w2⁢k,w3⁢k) satisfies

(3.11) { - Δ ⁢ w 1 ⁢ k + ε k 2 ⁢ V ⁢ ( ε k ⁢ x + ε k ⁢ y ε k ) ⁢ w 1 ⁢ k = ε k 2 ⁢ γ k ⁢ w 1 ⁢ k + μ 1 ⁢ w 1 ⁢ k 3 + β 12 ( k ) ⁢ w 2 ⁢ k 2 ⁢ w 1 ⁢ k + β 13 ( k ) ⁢ w 3 ⁢ k 2 ⁢ w 1 ⁢ k , - Δ ⁢ w 2 ⁢ k + ε k 2 ⁢ V ⁢ ( ε k ⁢ x + ε k ⁢ y ε k ) ⁢ w 2 ⁢ k = ε k 2 ⁢ γ k ⁢ w 2 ⁢ k + μ 2 ⁢ w 2 ⁢ k 3 + β 12 ( k ) ⁢ w 1 ⁢ k 2 ⁢ w 2 ⁢ k + β 23 ( k ) ⁢ w 3 ⁢ k 2 ⁢ w 2 ⁢ k , - Δ ⁢ w 3 ⁢ k + ε k 2 ⁢ V ⁢ ( ε k ⁢ x + ε k ⁢ y ε k ) ⁢ w 3 ⁢ k = ε k 2 ⁢ γ k ⁢ w 3 ⁢ k + μ 3 ⁢ w 3 ⁢ k 3 + β 13 ( k ) ⁢ w 1 ⁢ k 2 ⁢ w 3 ⁢ k + β 23 ( k ) ⁢ w 2 ⁢ k 2 ⁢ w 3 ⁢ k .

There exist a subsequence {(β12(k),β13(k),β23(k))} with βi⁢j(k)↗βi⁢j∗ as k→∞, and (w10,w20,w30)∈ℋ, such that (w1⁢k,w2⁢k,w3⁢k)⇀(w10,w20,w30) in ℋ, and wi⁢0≥0(i,j=1,2,3). By (3.2), (3.7) and (3.10), the identities

γ k = ∫ ℝ 2 [ | ∇ ⁡ u 1 ⁢ k | 2 + | ∇ ⁡ u 2 ⁢ k | 2 + | ∇ ⁡ u 3 ⁢ k | 2 + V ⁢ ( x ) ⁢ ( u 1 ⁢ k 2 + u 2 ⁢ k 2 + u 3 ⁢ k 2 ) ] ⁢ 𝑑 x
- ∫ ℝ 2 [ μ 1 ⁢ u 1 ⁢ k 4 + μ 2 ⁢ u 2 ⁢ k 4 + μ 3 ⁢ u 3 ⁢ k 4 + 2 ⁢ β 12 ( k ) ⁢ u 1 ⁢ k 2 ⁢ u 2 ⁢ k 2 + 2 ⁢ β 13 ( k ) ⁢ u 1 ⁢ k 2 ⁢ u 3 ⁢ k 2 + 2 ⁢ β 23 ( k ) ⁢ u 2 ⁢ k 2 ⁢ u 3 ⁢ k 2 ] ⁢ 𝑑 x
= 2 ⁢ e k - ∫ ℝ 2 ( | ∇ ⁡ u 1 ⁢ k | 2 + | ∇ ⁡ u 2 ⁢ k | 2 + | ∇ ⁡ u 3 ⁢ k | 2 ) ⁢ 𝑑 x - ∫ ℝ 2 V ⁢ ( x ) ⁢ ( u 1 ⁢ k 2 + u 2 ⁢ k 2 + u 3 ⁢ k 2 ) ⁢ 𝑑 x
= 2 ⁢ e k - 1 ε k 2 ⁢ ∫ ℝ 2 ( | ∇ ⁡ w 1 ⁢ k | 2 + | ∇ ⁡ w 2 ⁢ k | 2 + | ∇ ⁡ w 3 ⁢ k | 2 ) ⁢ 𝑑 x - ∫ ℝ 2 V ⁢ ( ε k ⁢ x + ε k ⁢ y ε k ) ⁢ ( w 1 ⁢ k 2 + w 2 ⁢ k 2 + w 3 ⁢ k 2 ) ⁢ 𝑑 x

hold, and then

ε k 2 ⁢ γ k = 2 ⁢ ε k 2 ⁢ e k - ∫ ℝ 2 ( | ∇ ⁡ w 1 ⁢ k | 2 + | ∇ ⁡ w 2 ⁢ k | 2 + | ∇ ⁡ w 3 ⁢ k | 2 ) ⁢ 𝑑 x
- ε k 2 ⁢ ∫ ℝ 2 V ⁢ ( ε k ⁢ x + ε k ⁢ y ε k ) ⁢ ( w 1 ⁢ k 2 + w 2 ⁢ k 2 + w 3 ⁢ k 2 ) ⁢ 𝑑 x → - 1   as  ⁢ k → ∞ .

It follows from (3.8) that

a ∗ - μ 1 ⁢ w 1 ⁢ k 2 = a ∗ - μ 2 ⁢ w 2 ⁢ k 2 = a ∗ - μ 3 ⁢ w 3 ⁢ k 2   a.e. in  ⁢ ℝ 2 ,

and then w10 satisfies

(3.12) - Δ ⁢ w 10 + w 10 = { μ 1 + β 12 ∗ ⁢ a ∗ - μ 1 a ∗ - μ 2 + β 13 ∗ ⁢ a ∗ - μ 1 a ∗ - μ 3 } ⁢ w 10 3 = a ∗ ⁢ 1 A 1 2 ⁢ w 10 3   a.e. in  ⁢ ℝ 2 .

From the uniqueness (up to a translation) of positive solution of (1.8), we get

(3.13) w 10 ⁢ ( x ) = A 1 a ∗ ⁢ w ⁢ ( x - x 1 )   for some ⁢ x 1 ∈ ℝ 2 .

Similarly, we have

w 20 ⁢ ( x ) = A 2 a ∗ ⁢ w ⁢ ( x - x 1 ) , w 30 ⁢ ( x ) = A 3 a ∗ ⁢ w ⁢ ( x - x 1 ) ,

where A1,A2,A3 are defined by (1.20). This indicates that wi⁢k→wi⁢0 in L2⁢(ℝ2) (i=1,2,3). By the interpolation inequalities, from (3.11) and (3.12), we conclude that (3.9) holds. ∎

Proof of Theorem 1.2.

Firstly, we claim that

(3.14) w ¯ i ⁢ k ⁢ ( x ) := ε k ⁢ u i ⁢ k ⁢ ( ε k ⁢ x + z ¯ i ⁢ k ) → A i a * ⁢ w ⁢ ( x ) ,

where z¯i⁢k is a global maximum point of ui⁢k, i=1,2,3.

Indeed, by De Giorgi–Nash–Moser theory [14, 23], we know that wi⁢k⁢(x) admits at least one global maximum point. Moreover,

w i ⁢ k ⁢ ( x ) → 0   as ⁢ | x | → ∞ ⁢ uniformly on ⁢ k .

Set zk:=εk⁢yεk, and then wi⁢k⁢(x) attains its global maximum at the point x=z¯i⁢k-zkεk. One can deduce that

| z ¯ i ⁢ k - z k ε k | ≤ C   as ⁢ k → ∞ .

According to (3.14), we can also have

(3.15) w ¯ i ⁢ k ⁢ ( x ) = w i ⁢ k ⁢ ( x + z ¯ i ⁢ k - z k ε k ) .

By (3) of Lemma 3.2, it yields that

w ¯ i ⁢ k ⁢ ( x ) → w ¯ i ⁢ 0 ⁢ ( x ) = w i ⁢ 0 ⁢ ( x + y i )   strongly in ⁢ H 1 ⁢ ( ℝ 2 ) ⁢ as ⁢ k → ∞ ,

where wi⁢0 is defined by (3.13), and

y i = lim k → ∞ ⁡ z ¯ i ⁢ k - z k ε k .

Thus we get

w ¯ i ⁢ 0 ⁢ ( x ) = A i a ∗ ⁢ w ⁢ ( x + y i - x 1 ) .

The rest of the proof is similar to that in [15] and we omit it here. It then follows that

lim k → ∞ ⁡ | z ¯ i ⁢ k - z ¯ j ⁢ k | ε k = 0 , i , j = 1 , 2 , 3 .

Secondly, we claim that

e k ≤ p + 2 p ⁢ a ∗ ⁢ ( p ⁢ λ 0 2 ) 2 p + 2 ⁢ ( 2 A 0 2 + o ⁢ ( 1 ) ) p p + 2 ⁢ B k p p + 2   as ⁢ k → ∞ ,

where A0,Bk,λ0 are defined by (1.21), (1.23) and (1.18), respectively.

Let (u1,θ1,θ2,τ,u2,θ1,θ2,τ,u3,θ1,θ2,τ) be the test function defined by (2.2). Set θ1,θ2 as in (2.4). Similar to (2.3), we have

∫ ℝ 2 ( | ∇ u 1 , θ 1 , θ 2 , τ | 2 + | ∇ u 2 , θ 1 , θ 2 , τ | 2 + | ∇ u 3 , θ 1 , θ 2 , τ | 2 ) d x - 1 2 ∫ ℝ 2 ( μ 1 u 1 , θ 1 , θ 2 , τ 4 + μ 2 u 2 , θ 1 , θ 2 , τ 4
+ μ 3 u 3 , θ 1 , θ 2 , τ 4 + 2 β 12 ( k ) u 1 , θ 1 , θ 2 , τ 2 u 2 , θ 1 , θ 2 , τ 2 + 2 β 13 ( k ) u 1 , θ 1 , θ 2 , τ 2 u 3 , θ 1 , θ 2 , τ 2 + 2 β 23 ( k ) u 1 , θ 2 , θ 2 , τ 2 u 3 , θ 1 , θ 2 , τ 2 ) d x
≤ τ 2 a ∗ { a ∗ - [ μ 1 θ 1 2 θ 2 2 + μ 2 ( 1 - θ 1 ) 2 θ 2 2 + μ 3 ( 1 - θ 2 ) 2
+ 2 β 12 ( k ) θ 1 θ 2 2 ( 1 - θ 1 ) + 2 β 13 ( k ) θ 1 θ 2 ( 1 - θ 2 ) + 2 β 23 ( k ) ( 1 - θ 1 ) θ 2 ( 1 - θ 2 ) ] } + C e - τ
= τ 2 a ∗ ⁢ A 0 2 { a ∗ [ α 12 ∗ 2 + α 23 ∗ 2 + α 13 ∗ 2 + 2 α 13 ∗ ( a ∗ - μ 2 ) + 2 α 23 ∗ ( a ∗ - μ 1 ) + 2 α 12 ∗ ( a ∗ - μ 3 ) ]
- [ μ 1 α 23 ∗ 2 + μ 2 α 13 ∗ 2 + μ 3 α 12 ∗ 2 + 2 β 12 ( k ) α 12 ∗ ( a ∗ - μ 3 ) + 2 β 13 ( k ) α 13 ∗ ( a ∗ - μ 2 )
+ 2 β 23 ( k ) α 23 ∗ ( a ∗ - μ 1 ) ] } + C e - τ
= τ 2 a ∗ ⁢ A 0 2 { [ α 12 ∗ + 2 ( a ∗ - β 12 ( k ) ) ] α 12 ∗ ( a ∗ - μ 3 ) + [ α 13 ∗ + 2 ( a ∗ - β 13 ( k ) ) ] α 13 ∗ ( a ∗ - μ 2 )
+ [ α 23 ∗ + 2 ( a ∗ - β 23 ( k ) ) ] α 23 ∗ ( a ∗ - μ 1 ) } + C e - τ
= 2 ⁢ τ 2 a ∗ ⁢ A 0 2 ⁢ B k + C ⁢ e - τ ,

where (1.14) has been used. The fact that

∑ i = 1 3 ∫ ℝ 2 V ⁢ ( x ) ⁢ u i , θ 1 , θ 2 , τ 2 ⁢ 𝑑 x ≤ ( 1 + o ⁢ ( 1 ) ) ⁢ λ 0 a ∗ ⁢ τ p

implies

(3.16) e k ≤ 2 ⁢ τ 2 a ∗ ⁢ A 0 2 ⁢ B k + ( 1 + o ⁢ ( 1 ) ) ⁢ λ 0 a ∗ ⁢ τ p .

Putting τ=(A02⁢p⁢(1+o⁢(1))⁢λ04⁢Bk)1p+2 into (3.16), we get

(3.17) e k ≤ p + 2 p ⁢ a ∗ ⁢ ( p ⁢ λ 0 2 ) 2 p + 2 ⁢ ( 2 A 0 2 + o ⁢ ( 1 ) ) p p + 2 ⁢ B k p p + 2 .

Finally, we claim that

lim k → ∞ ⁡ e k B k p p + 2 ≥ p + 2 p ⁢ a ∗ ⁢ ( p ⁢ λ 0 2 ) 2 p + 2 ⁢ ( 2 A 0 2 ) p p + 2 .

Let (u1⁢k,u2⁢k,u3⁢k) be the convergence subsequence obtained before. We deduce from (1.16) and (3.14) that

e k ≥ ( β 12 ∗ - β 12 ( k ) ) ⁢ ∫ ℝ 2 u 1 ⁢ k 2 ⁢ u 2 ⁢ k 2 ⁢ 𝑑 x + ( β 13 ∗ - β 13 ( k ) ) ⁢ ∫ ℝ 2 u 1 ⁢ k 2 ⁢ u 3 ⁢ k 2 ⁢ 𝑑 x + ( β 23 ∗ - β 23 ( k ) ) ⁢ ∫ ℝ 2 u 2 ⁢ k 2 ⁢ u 3 ⁢ k 2 ⁢ 𝑑 x + ∑ i = 1 3 ∫ ℝ 2 V ⁢ ( x ) ⁢ u i ⁢ k 2 ⁢ 𝑑 x
= ∑ i = 1 3 ∫ ℝ 2 V ⁢ ( ε k ⁢ x + z ¯ i ⁢ k ) ⁢ w ¯ i ⁢ k 2 ⁢ 𝑑 x + ( β 12 ∗ - β 12 ( k ) ) ⁢ 1 ε k 2 ⁢ ∫ ℝ 2 w ¯ 1 ⁢ k 2 ⁢ w ¯ 2 ⁢ k 2 ⁢ 𝑑 x + ( β 13 ∗ - β 13 ( k ) ) ⁢ 1 ε k 2 ⁢ ∫ ℝ 2 w ¯ 1 ⁢ k 2 ⁢ w ¯ 3 ⁢ k 2 ⁢ 𝑑 x
+ ( β 23 ∗ - β 23 ( k ) ) ⁢ 1 ε k 2 ⁢ ∫ ℝ 2 w ¯ 2 ⁢ k 2 ⁢ w ¯ 3 ⁢ k 2 ⁢ 𝑑 x .

From (3.15), (3.9) and (1.20) we have

(3.18) lim k → ∞ ⁡ 1 ε k 2 ⁢ ∫ ℝ 2 [ ( β 12 ∗ - β 12 ( k ) ) ⁢ w ¯ 1 ⁢ k 2 ⁢ w ¯ 2 ⁢ k 2 + ( β 13 ∗ - β 13 ( k ) ) ⁢ w ¯ 1 ⁢ k 2 ⁢ w ¯ 3 ⁢ k 2 + ( β 23 ∗ - β 23 ( k ) ) ⁢ w ¯ 2 ⁢ k 2 ⁢ w ¯ 3 ⁢ k 2 ] ⁢ 𝑑 x = lim k → ∞ ⁡ 2 ⁢ B k ε k 2 ⁢ a ∗ ⁢ A 0 2 .

Using an argument similar to that in [15], we have

lim k → ∞ ⁡ z ¯ i ⁢ k - x 0 ε k = z ¯ 0 , i = 1 , 2 , 3 , and    | z ¯ 0 | ≤ C .

Thus we obtain

(3.19)

lim inf k → ∞ ε k - p ∑ i = 1 3 ∫ ℝ 2 V ( ε k x + z ¯ i ⁢ k ) w ¯ i ⁢ k 2 d x = lim inf k → ∞ { ∫ ℝ 2 V ⁢ ( ε k ⁢ x + z ¯ 1 ⁢ k ) | ε k ⁢ x + z ¯ 1 ⁢ k - x 0 | p | x + z ¯ 1 ⁢ k - x 0 ε k | p α 23 ∗ a ∗ ⁢ A 0 w 2 d x
+ ∫ ℝ 2 V ⁢ ( ε k ⁢ x + z ¯ 2 ⁢ k ) | ε k ⁢ x + z ¯ 2 ⁢ k - x 0 | p ⁢ | x + z ¯ 2 ⁢ k - x 0 ε k | p ⁢ α 13 ∗ a ∗ ⁢ A 0 ⁢ w 2 ⁢ 𝑑 x
+ ∫ ℝ 2 V ⁢ ( ε k ⁢ x + z ¯ 3 ⁢ k ) | ε k ⁢ x + z ¯ 3 ⁢ k - x 0 | p | x + z ¯ 3 ⁢ k - x 0 ε k | p α 12 ∗ a ∗ ⁢ A 0 w 2 d x } ≥ λ 0 a ∗ .

Combining (3.18), (3.19) and (3.17), we get

λ 0 a * ⁢ ε k p ≤ C ⁢ B k p p + 2 , 2 ⁢ B k ε k 2 ⁢ a * ⁢ A 0 2 ≤ C ⁢ B k p p + 2   as ⁢ k → ∞ ,

which guarantee that there exist positive constants C1, C2 such that C1⁢Bk1p+2≤εk≤C2⁢Bk1p+2 as k→∞. Up to a subsequence, we may assume that

lim k → ∞ ⁡ ε k B k 1 p + 2 = M > 0 .

Also, from (3.18) and (3.19) we have

(3.20) lim inf k → ∞ ⁡ e k B k p p + 2 ≥ λ 0 ⁢ M p a ∗ + 2 M 2 ⁢ a ∗ ⁢ A 0 2 ≥ p + 2 p ⁢ a ∗ ⁢ ( p ⁢ λ 0 2 ) 2 p + 2 ⁢ ( 2 A 0 2 ) p p + 2 ,

where we take the infimum over M>0 on the right side of (3.20). The equality holds if and only if z¯0=0 and limk→∞⁡ε¯kεk=1, where ε¯k is defined by (1.22). The proof is completed. ∎

4 Nondegeneracy

Let ai,bi(i=1,2,3) be defined by (1.25) and μ2=μ3 ( which implies b1=b2,A2=A3,U2=U3). It is easy to verify that (U1,U2,U3) is a solution of (1.27). We say that (U1,U2,U3) is nondegenerate if the solution of the linearized equation

(4.1) { Δ ⁢ ϕ 1 - ϕ 1 + 3 ⁢ a 1 ⁢ U 1 2 ⁢ ϕ 1 + b 1 ⁢ U 2 2 ⁢ ϕ 1 + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 2 + b 2 ⁢ U 3 2 ⁢ ϕ 1 + 2 ⁢ b 2 ⁢ U 3 ⁢ U 1 ⁢ ϕ 3 = 0 , Δ ⁢ ϕ 2 - ϕ 2 + 3 ⁢ a 2 ⁢ U 2 2 ⁢ ϕ 2 + b 1 ⁢ U 1 2 ⁢ ϕ 2 + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 + b 3 ⁢ U 3 2 ⁢ ϕ 2 + 2 ⁢ b 3 ⁢ U 2 ⁢ U 3 ⁢ ϕ 3 = 0 , Δ ⁢ ϕ 3 - ϕ 3 + 3 ⁢ a 3 ⁢ U 3 2 ⁢ ϕ 3 + b 2 ⁢ U 1 2 ⁢ ϕ 3 + 2 ⁢ b 2 ⁢ U 1 ⁢ U 3 ⁢ ϕ 1 + b 3 ⁢ U 2 2 ⁢ ϕ 3 + 2 ⁢ b 3 ⁢ U 2 ⁢ U 3 ⁢ ϕ 2 = 0 .

is exactly two-dimensional, namely,

( ϕ 1 ϕ 2 ϕ 3 ) = ∑ j = 1 2 c j ⁢ ( ∂ ⁡ U 1 ∂ ⁡ x j ∂ ⁡ U 2 ∂ ⁡ x j ∂ ⁡ U 3 ∂ ⁡ x j )

for some constants cj. We claim firstly that this solution is nondegenerate in the space of radial functions.

Notice that if b1=b2, then by (1.25) and (1.20), we can deduce that a2=a3, A2=A3.

Lemma 4.1.

If b1=b2, μi<a∗ (i=1,2,3), then the solution (U1,U2,U3) is nondegenerate in the space of radial functions.

Proof.

Since (U1,U2,U3)=(A1⁢w,A2⁢w,A2⁢w) is a solution of (1.27), system (4.1) becomes

(4.2) { Δ ⁢ ϕ 1 - ϕ 1 + ( 3 ⁢ a 1 ⁢ A 1 2 + 2 ⁢ b 1 ⁢ A 2 2 ) ⁢ w 2 ⁢ ϕ 1 + 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 ⁢ w 2 ⁢ ϕ 2 + 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 ⁢ w 2 ⁢ ϕ 3 = 0 , Δ ⁢ ϕ 2 - ϕ 2 + ( 3 ⁢ a 2 ⁢ A 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ A 2 2 ) ⁢ w 2 ⁢ ϕ 2 + 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 ⁢ w 2 ⁢ ϕ 1 + 2 ⁢ b 3 ⁢ A 2 2 ⁢ w 2 ⁢ ϕ 3 = 0 , Δ ⁢ ϕ 3 - ϕ 3 + ( 3 ⁢ a 2 ⁢ A 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ A 2 2 ) ⁢ w 2 ⁢ ϕ 3 + 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 ⁢ w 2 ⁢ ϕ 1 + 2 ⁢ b 3 ⁢ A 2 2 ⁢ w 2 ⁢ ϕ 2 = 0 , ϕ i = ϕ i ⁢ ( r ) , i = 1 , 2 , 3 .

By an orthonormal transformation, (4.2) can be reduced to three decoupling equations

Δ ⁢ Φ 1 - Φ 1 + 3 ⁢ w 2 ⁢ Φ 1 = 0 , Φ 1 = Φ 1 ⁢ ( r ) , Δ ⁢ Φ 2 - Φ 2 + λ 1 ⁢ w 2 ⁢ Φ 2 = 0 , Φ 2 = Φ 2 ⁢ ( r ) , Δ ⁢ Φ 3 - Φ 3 + λ 2 ⁢ w 2 ⁢ Φ 3 = 0 , Φ 3 = Φ 3 ⁢ ( r ) ,

where

(4.3) λ 1 = 3 - 2 ⁢ b 1 , λ 2 = 3 - 2 ⁢ b 1 ⁢ A 1 2 + 2 ⁢ b 3 ⁢ A 1 2 - 2 ⁢ b 3 .

This will be proved in Remark 4.2. It follows from [35, Lemma 4.1] that the eigenvalues of

Δ ⁢ Φ - Φ + ν ⁢ w 2 ⁢ Φ = 0 , Φ ∈ H 1 ⁢ ( ℝ 2 )

are

ν 1 = 1 , ν 2 = ν 3 = 3 , ν 4 > 3 ,

where the eigenfunction corresponding to ν1 is cw, and the eigenfunction corresponding to ν2 are spanned by ∂⁡w∂⁡xj (j=1,2). Hence Φ1=0. It follows from (1.25) and (1.14) that

λ 1 = 3 - 2 ⁢ b 1 = 3 - 2 a ∗ ⁢ β 12 ∗ = 1 - 1 a ∗ ⁢ α 12 ∗ < 1 ,

which implies Φ2=0. Now, by a direct computation we have

λ 2 = 3 - 2 ⁢ b 1 ⁢ A 1 2 + 2 ⁢ b 3 ⁢ A 1 2 - 2 ⁢ b 3 < 1 ,

which results Φ3=0. ∎

Remark 4.2.

In this remark we present the proof of (4.3). We claim that λ1≠λ2 unless b1=b3. Indeed, if λ1=λ2, then

(4.4) b 1 ⁢ ( A 1 2 - 1 ) = b 3 ⁢ ( A 1 2 - 1 ) .

Since A12<1, equation (4.4) holds if and only if b1=b3.

We claim that 3,λ1,λ2 are eigenvalues of the matrix

(4.5) A := ( 3 ⁢ a 1 ⁢ A 1 2 + 2 ⁢ b 1 ⁢ A 2 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 3 ⁢ a 2 ⁢ A 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ A 2 2 2 ⁢ b 3 ⁢ A 2 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ b 3 ⁢ A 2 2 3 ⁢ a 2 ⁢ A 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ A 2 2 ) .

Indeed, by use of (1.27), (1.25) and (1.20), we have

(4.6) a 1 ⁢ A 1 2 + 2 ⁢ b 1 ⁢ A 2 2 = 1 , a 2 ⁢ A 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ A 2 2 = 1 .

Then

A = ( 2 ⁢ a 1 ⁢ A 1 2 + 1 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ a 2 ⁢ A 2 2 + 1 2 ⁢ b 3 ⁢ A 2 2 2 ⁢ b 1 ⁢ A 1 ⁢ A 2 2 ⁢ b 3 ⁢ A 2 2 2 ⁢ a 2 ⁢ A 2 2 + 1 ) .

Set D:=det⁡(A-λ⁢I), and we have

(4.7)

D = ( 2 ⁢ a 1 ⁢ A 1 2 + 1 - λ ) ⁢ ( 2 ⁢ a 2 ⁢ A 2 2 + 1 - λ ) 2 + 16 ⁢ b 1 2 ⁢ b 3 ⁢ A 1 2 ⁢ A 2 4
- 8 ⁢ b 1 2 ⁢ A 1 2 ⁢ A 2 2 ⁢ ( 2 ⁢ a 2 ⁢ A 2 2 + 1 - λ ) - ( 2 ⁢ a 1 ⁢ A 1 2 + 1 - λ ) ⁢ 4 ⁢ b 3 2 ⁢ A 2 4 .

Taking b3⁢A22=1-a2⁢A22-b1⁢A12 and λ=3, put them into (4.7), then

D = 8 ⁢ b 1 ⁢ A 1 2 ⁢ ( 2 - 2 ⁢ a 2 ⁢ A 2 2 - b 1 ⁢ A 1 2 ) ⁢ ( a 1 ⁢ A 1 2 - 1 + 2 ⁢ b 1 ⁢ A 2 2 ) .

According to (4.6), we have D=0. Thus λ=3 is an eigenvalue of (4.5).

Taking b3⁢A22=1-a2⁢A22-b1⁢A12 and λ=3-2⁢b1 and put them into (4.7), then

D = 8 ⁢ { ( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ ( a 2 ⁢ A 2 2 + b 1 - 1 ) 2 + 2 ⁢ b 1 2 ⁢ b 3 ⁢ A 1 2 ⁢ A 2 4 - 2 ⁢ b 1 2 ⁢ A 1 2 ⁢ A 2 2 ⁢ ( a 2 ⁢ A 2 2 + b 1 - 1 ) - ( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ b 3 2 ⁢ A 2 4 }
= 8 { ( a 1 A 1 2 + b 1 - 1 ) ( a 2 A 2 2 + b 1 - 1 ) 2 + 2 b 1 2 A 1 2 A 2 2 ( 1 - a 2 A 2 2 - b 1 A 1 2 )
- 2 b 1 2 A 1 2 A 2 2 ( a 2 A 2 2 + b 1 - 1 ) - ( a 1 A 1 2 + b 1 - 1 ) ( 1 - a 2 A 2 2 - b 1 A 1 2 ) 2 }
= 8 ⁢ { ( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ [ ( a 2 ⁢ A 2 2 + b 1 - 1 ) 2 - ( 1 - a 2 ⁢ A 2 2 - b 1 ⁢ A 1 2 ) 2 ] + 2 ⁢ b 1 2 ⁢ A 1 2 ⁢ A 2 2 ⁢ ( 2 - 2 ⁢ a 2 ⁢ A 2 2 - b 1 ⁢ A 1 2 - b 1 ) }
= 8 ⁢ { ( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ b 1 ⁢ ( A 1 2 - 1 ) ⁢ ( 2 - 2 ⁢ a 2 ⁢ A 2 2 - b 1 ⁢ A 1 2 - b 1 ) + 2 ⁢ b 1 2 ⁢ A 1 2 ⁢ A 2 2 ⁢ ( 2 - 2 ⁢ a 2 ⁢ A 2 2 - b 1 ⁢ A 1 2 - b 1 ) }
= [ ( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ ( A 1 2 - 1 ) + 2 ⁢ b 1 ⁢ A 1 2 ⁢ A 2 2 ] ⁢ 8 ⁢ b 1 ⁢ ( 2 - 2 ⁢ a 2 ⁢ A 2 2 - b 1 ⁢ A 1 2 - b 1 ) .

Since A2=1-A122, by (4.6), we have a1⁢A12+b1⁢(1-A12)=1. So

( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ ( A 1 2 - 1 ) + 2 ⁢ b 1 ⁢ A 1 2 ⁢ A 2 2 = ( a 1 ⁢ A 1 2 + b 1 - 1 ) ⁢ ( A 1 2 - 1 ) + b 1 ⁢ A 1 2 ⁢ ( 1 - A 1 2 )
= ( a 1 ⁢ A 1 2 + b 1 - 1 - b 1 ⁢ A 1 2 ) ⁢ ( A 1 2 - 1 ) = 0 ,

which implies D=0, and then λ1 is an eigenvalue of A.

If 3,λ1,λ2 are eigenvalues of A, then

D = ( 3 - λ ) ⁢ ( λ 1 - λ ) ⁢ ( λ 2 - λ ) = 0 .

By virtue of (4.7), the coefficient of λ2 is

3 + λ 1 + λ 2 = 2 ⁢ a 1 ⁢ A 1 2 + 1 + 2 ⁢ ( 2 ⁢ a 2 ⁢ A 2 2 + 1 ) ,

and then

λ 2 = 2 ⁢ a 1 ⁢ A 1 2 + 4 ⁢ a 2 ⁢ A 2 2 - 3 + 2 ⁢ b 1
= - 4 ⁢ b 1 ⁢ A 2 2 + 4 ⁢ a 2 ⁢ A 2 2 + 2 ⁢ b 2 - 1
= - 2 ⁢ b 1 ⁢ ( 1 - A 1 2 ) + 2 ⁢ a 2 ⁢ ( 1 - A 1 2 ) + 2 ⁢ b 1 - 1
= 2 ⁢ b 1 ⁢ A 1 2 + 2 ⁢ a 2 - 2 ⁢ a 2 ⁢ A 1 2 - 1
= 2 ⁢ ( 1 - a 2 ⁢ A 1 2 - b 3 ⁢ A 2 2 ) + 2 ⁢ a 2 - 2 ⁢ a 2 ⁢ A 1 2 - 1
= 1 - a 2 ⁢ ( 1 - A 1 2 ) - b 3 ⁢ ( 1 - A 1 2 ) + 2 ⁢ a 2 - 2 ⁢ a 2 ⁢ A 1 2 .

If λ2=3-2⁢b1⁢A12+2⁢b3⁢A12-2⁢b3, then

1 - a 2 ⁢ ( 1 - A 1 2 ) - b 3 ⁢ ( 1 - A 1 2 ) + 2 ⁢ a 2 - 2 ⁢ a 2 ⁢ A 1 2 = 3 - 2 ⁢ b 1 ⁢ A 1 2 + 2 ⁢ b 3 ⁢ A 1 2 - 2 ⁢ b 3
⇔ a 2 - a 2 ⁢ A 1 2 = 2 - 2 ⁢ b 1 ⁢ A 1 2 + b 3 ⁢ A 1 2 - b 3 .

By use of (4.6), we have

0 = a 2 ⁢ A 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ A 2 2 - 1 = a 2 ⁢ 1 - A 1 2 2 + b 1 ⁢ A 1 2 + b 3 ⁢ 1 - A 1 2 2 - 1
= 2 ⁢ { a 2 - a 2 ⁢ A 1 2 + 2 ⁢ b 1 ⁢ A 1 2 + b 3 - b 3 ⁢ A 1 2 - 2 } .

This ends the proof of (4.3).

Proof of Theorem 1.3.

Inspired by [10], let ej⁢(θ) be the eigenfunctions of the Laplace-Beltrami operator ΔS1 on S1, and then the corresponding eigenvalues satisfy (see [9, 29, 35])

μ 0 = 0 , μ 1 = μ 2 = 1 , μ 2 < μ 3 , … .

For any solution (ϕ1,ϕ2,ϕ3) of (4.1), set

ϕ i ⁢ j ⁢ ( r ) = ∫ S 1 ϕ i ⁢ ( r , θ ) ⁢ e j ⁢ ( θ ) ⁢ 𝑑 θ , i = 1 , 2 , 3 , j = 0 , 1 , … .

According to [9], for a function ϕ,

Δ ⁢ ϕ = 1 r ⁢ ∂ r ⁡ ( r ⁢ ∂ r ⁡ ϕ ) + Δ S 1 ⁢ ϕ r 2 .

Then (ϕ1⁢j,ϕ2⁢j,ϕ3⁢j) satisfies

(4.8) { Δ ⁢ ϕ 1 ⁢ j - ϕ 1 ⁢ j - μ j r 2 ⁢ ϕ 1 ⁢ j + 3 ⁢ a 1 ⁢ U 1 2 ⁢ ϕ 1 ⁢ j + b 1 ⁢ U 2 2 ⁢ ϕ 1 ⁢ j + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 2 ⁢ j + b 1 ⁢ U 2 2 ⁢ ϕ 1 ⁢ j + 2 ⁢ b 1 ⁢ U 2 ⁢ U 1 ⁢ ϕ 3 ⁢ j = 0 , Δ ⁢ ϕ 2 ⁢ j - ϕ 2 ⁢ j - μ j r 2 ⁢ ϕ 2 ⁢ j + 3 ⁢ a 2 ⁢ U 2 2 ⁢ ϕ 2 ⁢ j + b 1 ⁢ U 1 2 ⁢ ϕ 2 ⁢ j + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 ⁢ j + b 3 ⁢ U 2 2 ⁢ ϕ 2 ⁢ j + 2 ⁢ b 3 ⁢ U 2 2 ⁢ ϕ 3 ⁢ j = 0 , Δ ⁢ ϕ 3 ⁢ j - ϕ 3 ⁢ j - μ j r 2 ⁢ ϕ 3 ⁢ j + 3 ⁢ a 2 ⁢ U 2 2 ⁢ ϕ 3 ⁢ j + b 1 ⁢ U 1 2 ⁢ ϕ 3 ⁢ j + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 ⁢ j + b 3 ⁢ U 2 2 ⁢ ϕ 3 ⁢ j + 2 ⁢ b 3 ⁢ U 2 2 ⁢ ϕ 2 ⁢ j = 0 .

Notice that

(4.9) { Δ ⁢ U 1 ′ - U 1 ′ + 3 ⁢ a 1 ⁢ U 1 2 ⁢ U 1 ′ + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ U 2 ′ + b 1 ⁢ U 2 2 ⁢ U 1 ′ + 2 ⁢ b 2 ⁢ U 1 ⁢ U 3 ⁢ U 3 ′ + b 2 ⁢ U 3 2 ⁢ U 1 ′ = 1 r 2 ⁢ U 1 ′ , Δ ⁢ U 2 ′ - U 2 ′ + 3 ⁢ a 2 ⁢ U 2 2 ⁢ U 2 ′ + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ U 1 ′ + b 1 ⁢ U 1 2 ⁢ U 2 ′ + 2 ⁢ b 3 ⁢ U 2 ⁢ U 3 ⁢ U 3 ′ + b 3 ⁢ U 3 2 ⁢ U 2 ′ = 1 r 2 ⁢ U 2 ′ , Δ ⁢ U 3 ′ - U 3 ′ + 3 ⁢ a 3 ⁢ U 3 2 ⁢ U 3 ′ + 2 ⁢ b 2 ⁢ U 1 ⁢ U 3 ⁢ U 1 ′ + b 2 ⁢ U 1 2 ⁢ U 3 ′ + 2 ⁢ b 3 ⁢ U 2 ⁢ U 3 ⁢ U 2 ′ + b 3 ⁢ U 2 2 ⁢ U 3 ′ = 1 r 2 ⁢ U 3 ′ ,

where u′=∂⁡u∂⁡r, a2=a3, U2=U3, b1=b2.

Step 1. We claim that if j≥3, then ϕi⁢j≡0. Firstly, we claim that ψj:=ϕ2⁢j-ϕ3⁢j≡0 for j≥1. Indeed, it yields from the second and third equations of (4.8) that

(4.10) Δ ⁢ ψ j - ψ j - μ j r 2 ⁢ ψ j + ( 3 ⁢ a 2 ⁢ U 2 2 + b 1 ⁢ U 1 2 - b 3 ⁢ U 2 2 ) ⁢ ψ j = 0 .

Multiplying (4.10) by U2′, and integrating over the ball BR, which is centered at the origin with radius R, we get

(4.11) ∫ ∂ ⁡ B R ∂ ⁡ ψ j ∂ ⁡ ν ⁢ U 2 ′ ⁢ 𝑑 S - ∫ B R ψ j ′ ⁢ U 2 ′′ ⁢ 𝑑 x - ∫ B R ψ j ⁢ U 2 ′ ⁢ 𝑑 x - ∫ B R μ j r 2 ⁢ ψ j ⁢ U 2 ′ ⁢ 𝑑 x + ∫ B R ( 3 ⁢ a 2 ⁢ U 2 2 + b 1 ⁢ U 1 2 - b 3 ⁢ U 2 2 ) ⁢ ψ j ⁢ U 2 ′ ⁢ 𝑑 x = 0 ,

where ν=(ν1,ν2) denotes the outward unit normal of ∂⁡BR. Multiplying the second equation in (4.9) by ψj, and integrating over the ball BR, we obtain

(4.12)

∫ ∂ ⁡ B R ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ψ j ⁢ 𝑑 S - ∫ B R U 2 ′′ ⁢ ψ j ′ ⁢ 𝑑 x - ∫ B R U 2 ′ ⁢ ψ j ⁢ 𝑑 x
+ ∫ B R ( 3 ⁢ a 2 ⁢ U 2 2 ⁢ U 2 ′ ⁢ ψ j + 2 ⁢ b 1 ⁢ U 2 ⁢ U 1 ⁢ U 1 ′ ⁢ ψ j + b 1 ⁢ U 1 2 ⁢ U 2 ′ ⁢ ψ j + 3 ⁢ b 3 ⁢ U 2 2 ⁢ U 2 ′ ⁢ ψ j ) ⁢ 𝑑 x = ∫ B R 1 r 2 ⁢ U 2 ′ ⁢ ψ j ⁢ 𝑑 x .

It yields from (4.11) subtracting (4.12) that

(4.13) ∫ ∂ ⁡ B R ( ∂ ⁡ ψ j ∂ ⁡ ν ⁢ U 2 ′ - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ψ j ) ⁢ 𝑑 S + ∫ B R 1 - μ j r 2 ⁢ ψ j ⁢ U 2 ′ ⁢ 𝑑 x - 2 ⁢ b 1 ⁢ ∫ B R U 2 ⁢ U 1 ⁢ U 1 ′ ⁢ ψ j ⁢ 𝑑 x - 4 ⁢ b 3 ⁢ ∫ B R U 2 2 ⁢ U 2 ′ ⁢ ψ j ⁢ 𝑑 x = 0 .

By definition we have

ϕ i ⁢ j ( 0 ) = ϕ i ⁢ j ′ ( 0 ) = 0   ( i = 1 , 2 , 3 ) .

If ψj≡0, then the claim is proved. Thus, without loss of generality, we first assume that there is some r0⁢j>0 such that ψj>0 for 0<r<r0⁢j and ψj⁢(r0⁢j)=0 (If ψj>0 for r∈(0,+∞), choose r0⁢j=+∞). By the standard ODE theory we know that ψj′⁢(r0⁢j)<0. Let R=r0⁢j, by (4.13), and we have

∫ ∂ ⁡ B r 0 ⁢ j ( ∂ ⁡ ψ j ∂ ⁡ ν ⁢ U 2 ′ - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ψ j ) ⁢ 𝑑 S > 0 , ∫ B r 0 ⁢ j 1 - μ j r 2 ⁢ ψ j ⁢ U 2 ′ ⁢ 𝑑 x ≥ 0 ,
- 2 ⁢ b 1 ⁢ ∫ B r 0 ⁢ j U 2 ⁢ U 1 ⁢ U 1 ′ ⁢ ψ j ⁢ 𝑑 x > 0 , - 4 ⁢ b 3 ⁢ ∫ B r 0 ⁢ j U 2 2 ⁢ U 2 ′ ⁢ ψ j ⁢ 𝑑 x > 0 ,

which contradicts to (4.13). Similarly, we can assume that there is some r¯0⁢j>0 such that ψj<0 for 0<r<r¯0⁢j and ψj⁢(r¯0⁢j)=0 ( If ψj<0 for (0,+∞), choose r¯0⁢j=+∞). Again by the standard ODE theory, ψj′⁢(r¯0⁢j)>0. Let R=r¯0⁢j, by (4.13), and we have

∫ ∂ ⁡ B r ¯ 0 ⁢ j ( ∂ ⁡ ψ j ∂ ⁡ ν ⁢ U 2 ′ - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ψ j ) ⁢ 𝑑 S < 0 , ∫ B r ¯ 0 ⁢ j 1 - μ j r 2 ⁢ ψ j ⁢ U 2 ′ ⁢ 𝑑 x ≤ 0 ,
- 2 ⁢ b 1 ⁢ ∫ B r ¯ 0 ⁢ j U 2 ⁢ U 1 ⁢ U 1 ′ ⁢ ψ j ⁢ 𝑑 x < 0 , - 4 ⁢ b 3 ⁢ ∫ B r ¯ 0 ⁢ j U 2 2 ⁢ U 2 ′ ⁢ ψ j ⁢ 𝑑 x < 0 ,

which contradicts to (4.13), and so ϕ2⁢j=ϕ3⁢j holds for j=1,2,….

If ϕ1⁢j≡0, by the first equation of (4.8), we get

ϕ 2 ⁢ j + ϕ 3 ⁢ j = 0 ,

then ϕ2⁢j=ϕ3⁢j≡0. Using the same arguments as above, without loss of generality, we assume that there is some r1⁢j>0 such that ϕ1⁢j<0 for 0<r<r1⁢j and ϕ1⁢j⁢(r1⁢j)=0. By the standard ODE theory, ϕ1⁢j′⁢(r1⁢j)>0.

We claim that ϕ2⁢j=ϕ3⁢j<0 for r small enough. Suppose that there is some r2⁢j>0 such that ϕ2⁢j>0 for 0<r<r2⁢j, and ϕ2⁢j⁢(r2⁢j)=0, r1⁢j≤r2⁢j. By the standard ODE theory, ϕ2⁢j′⁢(r2⁢j)<0. Multiplying the first equation of (4.8) by U1′, and integrating it over the ball BR, we have

(4.14)

J 1 := ∫ ∂ ⁡ B R ( U 1 ′ ⁢ ∂ ⁡ ϕ 1 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 1 ′ ∂ ⁡ ν ⁢ ϕ 1 ⁢ j ) ⁢ 𝑑 S + ∫ B R 1 - μ j r 2 ⁢ U 1 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x
+ 4 ⁢ b 1 ⁢ ∫ B R U 1 ⁢ U 2 ⁢ U 1 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x - 4 ⁢ b 1 ⁢ ∫ B R U 1 ⁢ U 2 ⁢ U 2 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x = 0 .

On the other hand,

∫ ∂ ⁡ B r 1 ⁢ j ( U 1 ′ ⁢ ∂ ⁡ ϕ 1 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 1 ′ ∂ ⁡ ν ⁢ ϕ 1 ⁢ j ) ⁢ 𝑑 S < 0 , ∫ B r 1 ⁢ j 1 - μ j r 2 ⁢ U 1 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x < 0 ,
4 ⁢ b 1 ⁢ ∫ B r 1 ⁢ j U 1 ⁢ U 2 ⁢ U 1 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x < 0 , - 4 ⁢ b 1 ⁢ ∫ B r 1 ⁢ j U 1 ⁢ U 2 ⁢ U 2 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x < 0 ,

which contradicts to (4.14). If r1⁢j>r2⁢j, multiplying the second equation of (4.8) by U2′, and integrating over the ball BR, we get

(4.15)

J 2 := ∫ ∂ ⁡ B R ( U 2 ′ ⁢ ∂ ⁡ ϕ 2 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ϕ 2 ⁢ j ) ⁢ 𝑑 S + ∫ B R 1 - μ j r 2 ⁢ U 2 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x
+ 2 ⁢ b 1 ⁢ ∫ B R U 1 ⁢ U 2 ⁢ U 2 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x - 2 ⁢ b 1 ⁢ ∫ B R U 1 ⁢ U 2 ⁢ U 1 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x = 0 .

However, we have

∫ ∂ ⁡ B r 2 ⁢ j ( U 2 ′ ⁢ ∂ ⁡ ϕ 2 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ϕ 2 ⁢ j ) ⁢ 𝑑 S > 0 , ∫ B r 2 ⁢ j 1 - μ j r 2 ⁢ U 2 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x > 0 ,
2 ⁢ b 1 ⁢ ∫ B r 2 ⁢ j U 1 ⁢ U 2 ⁢ U 2 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x > 0 , - 2 ⁢ b 1 ⁢ ∫ B r 2 ⁢ j U 1 ⁢ U 2 ⁢ U 1 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x > 0 ,

which contradicts to (4.15). Therefore there is some r2⁢j>0 such that ϕ2⁢j=ϕ3⁢j<0 for 0<r<r2⁢j and ϕ2⁢j⁢(r2⁢j)=0. Recall that ϕ2⁢j′⁢(r2⁢j)>0, from (4.14) and (4.15), we get

J 1 + 2 ⁢ J 2 = 0 = ∫ ∂ ⁡ B R ( U 1 ′ ⁢ ∂ ⁡ ϕ 1 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 1 ′ ∂ ⁡ ν ⁢ ϕ 1 ⁢ j ) ⁢ 𝑑 S + 2 ⁢ ∫ ∂ ⁡ B R ( U 2 ′ ⁢ ∂ ⁡ ϕ 2 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ϕ 2 ⁢ j ) ⁢ 𝑑 S + ∫ B R 1 - μ j r 2 ⁢ ( U 1 ′ ⁢ ϕ 1 ⁢ j + 2 ⁢ U 2 ′ ⁢ ϕ 2 ⁢ j ) ⁢ 𝑑 x
(4.16) = : I 1 ( r ) + 2 I 2 ( r ) + I 3 ( r ) .

Let us deal with the above terms case by case.

Case 1: r1⁢j=r2⁢j. In this case, by a direct calculation, we have

I 1 ⁢ ( r 1 ⁢ j ) < 0 , I 2 ⁢ ( r 1 ⁢ j ) < 0 , I 3 ⁢ ( r 1 ⁢ j ) < 0 ,

which contradicts to (4.16).

Case 2: r2⁢j<r1⁢j. In this case, we can deduce easily that I3⁢(r2⁢j)<0, I2⁢(r2⁢j)<0. Define

Ψ ⁢ ( r ) := r ⁢ ϕ 1 ⁢ j ′ ⁢ ( r ) ⁢ U 1 ′ ⁢ ( r ) - r ⁢ U 1 ′′ ⁢ ( r ) ⁢ ϕ 1 ⁢ j ⁢ ( r ) ,

then for r2⁢j<r<r1⁢j,

Ψ ′ ⁢ ( r ) = ( r ⁢ ϕ 1 ⁢ j ′ ⁢ ( r ) ) ′ ⁢ U 1 ′ ⁢ ( r ) - ( r ⁢ U 1 ′′ ⁢ ( r ) ) ⁢ ϕ 1 ⁢ j ⁢ ( r ) .

Using the equalities

1 r ⁢ ( r ⁢ ϕ 1 ⁢ j ′ ) ′ - ϕ 1 ⁢ j - μ j r 2 ⁢ ϕ 1 ⁢ j + ( 3 ⁢ a 1 ⁢ U 1 2 + 2 ⁢ b 1 ⁢ U 2 2 ) ⁢ ϕ 1 ⁢ j + 4 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 2 ⁢ j = 0 ,
1 r ⁢ ( r ⁢ U 1 ′′ ) ′ - U 1 ′ - 1 r 2 ⁢ U 1 ′ + ( 3 ⁢ a 1 ⁢ U 1 2 + 2 ⁢ b 1 ⁢ U 2 2 ) ⁢ U 1 ′ + 4 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ U 2 ′ = 0 ,

we get

(4.17) Ψ ′ ⁢ ( r ) = μ j - 1 r ⁢ U 1 ′ ⁢ ϕ 1 ⁢ j + 4 ⁢ r ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ U 2 ′ ⁢ ϕ 1 ⁢ j - 4 ⁢ r ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ U 1 ′ ⁢ ϕ 2 ⁢ j .

If there is an r3⁢j∈(r2⁢j,r1⁢j) such that ϕ2⁢j>0 in r2⁢j<r<r3⁢j and ϕ2⁢j⁢(r3⁢j)=0, by (4.15), we have

0 = ∫ ∂ ⁡ B r 3 ⁢ j ( U 2 ′ ⁢ ∂ ⁡ ϕ 2 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ϕ 2 ⁢ j ) ⁢ 𝑑 S - ∫ ∂ ⁡ B r 2 ⁢ j ( U 2 ′ ⁢ ∂ ⁡ ϕ 2 ⁢ j ∂ ⁡ ν - ∂ ⁡ U 2 ′ ∂ ⁡ ν ⁢ ϕ 2 ⁢ j ) ⁢ 𝑑 S + ∫ B r 3 ⁢ j ∖ B r 2 ⁢ j 1 - μ j r 2 ⁢ U 2 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x
+ 2 ⁢ b 1 ⁢ ∫ B r 3 ⁢ j ∖ B r 2 ⁢ j U 1 ⁢ U 2 ⁢ U 2 ′ ⁢ ϕ 1 ⁢ j ⁢ 𝑑 x - 2 ⁢ b 1 ⁢ ∫ B r 3 ⁢ j ∖ B r 2 ⁢ j U 1 ⁢ U 2 ⁢ U 1 ′ ⁢ ϕ 2 ⁢ j ⁢ 𝑑 x > 0 ,

which is a contradiction. Therefore, we can assume that ϕ2⁢j>0 in r2⁢j<r<r1⁢j. Hence according to (4.17), Ψ′⁢(r)>0 for r2⁢j<r<r1⁢j, which yields

Ψ ⁢ ( r 1 ⁢ j ) = r 1 ⁢ j ⁢ ( ϕ 1 ⁢ j ′ ⁢ ( r 1 ⁢ j ) ⁢ U 1 ′ ⁢ ( r 1 ⁢ j ) - U 1 ′′ ⁢ ( r 1 ⁢ j ) ⁢ ϕ 1 ⁢ j ⁢ ( r 1 ⁢ j ) ) < 0 .

So we get

0 > Ψ ⁢ ( r 1 ⁢ j ) > Ψ ⁢ ( r 2 ⁢ j ) = 1 | S 1 | ⁢ I 1 ⁢ ( r 2 ⁢ j ) ,

which contradicts to (4.16).

Case 3: r1⁢j<r2⁢j. In this case, we can deduce easily that I1⁢(r1⁢j), I3⁢(r1⁢j)<0. Similar to the above arguments, we can assume that ϕ1⁢j>0 for r1⁢j<r<r2⁢j. Then Ψ′⁢(r)<0 for r1⁢j<r<r2⁢j, which implies that Ψ⁢(r2⁢j)<Ψ⁢(r1⁢j)<0. Then we can get I2⁢(r1⁢j)<0, which leads a contradiction to (4.16).

In conclusion, for j≥3, ϕ1⁢j=ϕ2⁢j=ϕ3⁢j≡0.

Step 2. We claim that (ϕ1⁢j,ϕ2⁢j,ϕ3⁢j)=cj⁢(U1′,U2′,U3′) holds for j=1,2, where cj is a positive number. Note that (ϕ1⁢j,ϕ2⁢j,ϕ3⁢j) satisfies the following equations for j=1,2 and μj=1:

(4.18) { Δ ⁢ ϕ 1 ⁢ j - ϕ 1 ⁢ j - 1 r 2 ⁢ ϕ 1 ⁢ j + 3 ⁢ a 1 ⁢ U 1 2 ⁢ ϕ 1 ⁢ j + b 1 ⁢ U 2 2 ⁢ ϕ 1 ⁢ j + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 2 ⁢ j + b 1 ⁢ U 2 2 ⁢ ϕ 1 ⁢ j + 2 ⁢ b 1 ⁢ U 2 ⁢ U 1 ⁢ ϕ 3 ⁢ j = 0 , Δ ⁢ ϕ 2 ⁢ j - ϕ 2 ⁢ j - 1 r 2 ⁢ ϕ 2 ⁢ j + 3 ⁢ a 2 ⁢ U 2 2 ⁢ ϕ 2 ⁢ j + b 1 ⁢ U 1 2 ⁢ ϕ 2 ⁢ j + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 ⁢ j + b 3 ⁢ U 2 2 ⁢ ϕ 2 ⁢ j + 2 ⁢ b 3 ⁢ U 2 2 ⁢ ϕ 3 ⁢ j = 0 , Δ ⁢ ϕ 3 ⁢ j - ϕ 3 ⁢ j - 1 r 2 ⁢ ϕ 3 ⁢ j + 3 ⁢ a 2 ⁢ U 2 2 ⁢ ϕ 3 ⁢ j + b 1 ⁢ U 1 2 ⁢ ϕ 3 ⁢ j + 2 ⁢ b 1 ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 ⁢ j + b 3 ⁢ U 2 2 ⁢ ϕ 3 ⁢ j + 2 ⁢ b 3 ⁢ U 2 2 ⁢ ϕ 2 ⁢ j = 0 .

Similar to the proof as above, we can deduce that at least one of the relations of ϕ1⁢j=ϕ2⁢j=ϕ3⁢j=0 or ϕ1⁢j,ϕ2⁢j,ϕ3⁢j<0 or ϕ1⁢j,ϕ2⁢j,ϕ3⁢j>0 must hold for all r>0. Since (U1′,U2′,U3′) is a solution of (4.18), we know that (ϕ1⁢j,ϕ2⁢j,ϕ3⁢j)-c⁢(U1′,U2′,U3′) also satisfies (4.18). Let c0=ϕ1⁢j⁢(1)U1′⁢(1). Then ϕ1⁢j-c0⁢U1′=0, And so (ϕ1⁢j,ϕ2⁢j,ϕ3⁢j)=c0⁢(U1′,U2′,U3′).

By the discussion as in [13], since ϕi⁢0⁢(r)∈Lr2⁢(ℝ2), it yields from Lemma 4.1 that ϕi⁢0≡0 (i=1,2,3). Therefore, ϕi⁢(r,θ)=∑j=12ϕi⁢j⁢(r)⁢ej⁢(θ). The proof is completed. ∎

Remark 4.3.

With slight changes, this result on nondegeneracy can be established in ℝ3.

5 Asymptotic Uniqueness of Minimizers

In this section, we prove the uniqueness of nonnegative minimizers of ek as k→∞, where μ1,μ2,μ3∈(0,a∗) are fixed.

Lemma 5.1.

Suppose that V⁢(x)∈C2⁢(R2) satisfies (V2)–(V3) and (1.28). For any given 0<μ1, μ2, μ3<a∗, assume (β12(k),β13(k),β23(k))↗(β12∗,β13∗,β23∗) as k→∞, and (u1⁢k,u2⁢k,u3⁢k) is a nonnegative minimizer of the functional ek. Then there exists a subsequence of {(β12(k),β13(k), β23(k))}, still denoted by {(β12(k),β13(k),β23(k))} such that:

  1. The minimizer ( u 1 ⁢ k , u 2 ⁢ k , u 3 ⁢ k ) satisfies

    u ~ i ⁢ k ( x ) := ε ¯ k u i ⁢ k ( ε ¯ k x + z ¯ i ⁢ k ) → 1 a ∗ A i w ( x )   ( i = 1 , 2 , 3 )

    uniformly in ℝ 2 as k → ∞ , where A i is given by ( 1.20 ) and ε ¯ k = [ 4 ⁢ B k A 0 2 ⁢ p ⁢ λ 0 ] 1 p + 2 . Furthermore, the unique maximum point ( z ¯ 1 ⁢ k , z ¯ 2 ⁢ k , z ¯ 3 ⁢ k ) of ( u 1 ⁢ k , u 2 ⁢ k , u 3 ⁢ k ) satisfies ( 1.24 ).

  2. The minimizer ( u 1 ⁢ k , u 2 ⁢ k , u 3 ⁢ k ) decays exponentially in the sense that

    (5.1) u ~ i ⁢ k ⁢ ( x ) ≤ C ⁢ e - | x | 2 , | ∇ ⁡ u ~ i ⁢ k ⁢ ( x ) | ≤ C ⁢ e - | x | 4   𝑎𝑠 ⁢ | x | → ∞ , i = 1 , 2 , 3 .

Proof.

The proof can follow the idea in [15, Lemma 4.1], we omit it here. ∎

From Lemma 5.1 we know that (U1,U2,U3)=(A1⁢w,A2⁢w,A3⁢w) is a positive solution of the following system:

(5.2) { Δ ⁢ u 1 - u 1 + μ 1 a ∗ ⁢ u 1 3 + β 12 ∗ a ∗ ⁢ u 2 2 ⁢ u 1 + β 13 ∗ a ∗ ⁢ u 3 2 ⁢ u 1 = 0 , Δ ⁢ u 2 - u 2 + μ 2 a ∗ ⁢ u 2 3 + β 12 ∗ a ∗ ⁢ u 1 2 ⁢ u 2 + β 23 ∗ a ∗ ⁢ u 3 2 ⁢ u 2 = 0 , Δ ⁢ u 3 - u 3 + μ 3 a ∗ ⁢ u 3 3 + β 13 ∗ a ∗ ⁢ u 1 2 ⁢ u 3 + β 23 ∗ a ∗ ⁢ u 2 2 ⁢ u 3 = 0 .

It follows from Theorem 1.3 that the positive solution is nondegenerate if μ2=μ3, in the sense that the set of the solutions of the following linearized system of (5.2) about (U1,U2,U3):

ℒ 1 ⁢ ( ϕ 1 , ϕ 2 , ϕ 3 ) := Δ ⁢ ϕ 1 - ϕ 1 + 3 ⁢ μ 1 a ∗ ⁢ U 1 2 ⁢ ϕ 1 + β 12 ∗ a ∗ ⁢ U 2 2 ⁢ ϕ 1 + 2 ⁢ β 12 ∗ a ∗ ⁢ U 2 ⁢ U 1 ⁢ ϕ 2 + β 13 ∗ a ∗ ⁢ U 3 2 ⁢ ϕ 1 + 2 ⁢ β 13 ∗ a ∗ ⁢ U 3 ⁢ U 1 ⁢ ϕ 3 = 0 ,
ℒ 2 ⁢ ( ϕ 1 , ϕ 2 , ϕ 3 ) := Δ ⁢ ϕ 2 - ϕ 2 + 3 ⁢ μ 2 a ∗ ⁢ U 2 2 ⁢ ϕ 2 + β 12 ∗ a ∗ ⁢ U 1 2 ⁢ ϕ 2 + 2 ⁢ β 12 ∗ a ∗ ⁢ U 1 ⁢ U 2 ⁢ ϕ 1 + β 23 ∗ a ∗ ⁢ U 3 2 ⁢ ϕ 2 + 2 ⁢ β 23 ∗ a ∗ ⁢ U 3 ⁢ U 2 ⁢ ϕ 3 = 0 ,
ℒ 3 ⁢ ( ϕ 1 , ϕ 2 , ϕ 3 ) := Δ ⁢ ϕ 3 - ϕ 3 + 3 ⁢ μ 3 a ∗ ⁢ U 3 2 ⁢ ϕ 3 + β 13 ∗ a ∗ ⁢ U 1 2 ⁢ ϕ 3 + 2 ⁢ β 13 ∗ a ∗ ⁢ U 1 ⁢ U 3 ⁢ ϕ 1 + β 23 ∗ a ∗ ⁢ U 2 2 ⁢ ϕ 3 + 2 ⁢ β 23 ∗ a ∗ ⁢ U 3 ⁢ U 2 ⁢ ϕ 2 = 0 ,

is exactly two-dimensional, namely,

(5.3) ( ϕ 1 ϕ 2 ϕ 3 ) = ∑ j = 1 2 c j ⁢ ( ∂ ⁡ U 1 ∂ ⁡ x j ∂ ⁡ U 2 ∂ ⁡ x j ∂ ⁡ U 3 ∂ ⁡ x j )

for some constants cj. Recall that from (1.14), if μ2=μ3, then β12∗=β13∗, U2=U3.

Under the same assumptions of Lemma 5.1 with μ2=μ3, let (u1⁢k,u2⁢k,u3⁢k) be the convergent subsequence obtained in Theorem 1.2, and then we claim that u2⁢k=u3⁢k in ℝ2. Indeed, define

Ω + = { x ∈ ℝ 2 : u 2 ⁢ k - u 3 ⁢ k > 0 } ;

if Ω+≠∅, then Ω+ is a piecewise C1 smooth domain. Multiplying the second equation of (3.10) by u3⁢k and the third equation of (3.10) by u2⁢k, integrating by parts on Ω+ and subtracting one from the other, we have

∫ ∂ ⁡ Ω + ( ∂ ⁡ u 3 ⁢ k ∂ ⁡ ν ⁢ u 2 ⁢ k - ∂ ⁡ u 2 ⁢ k ∂ ⁡ ν ⁢ u 3 ⁢ k ) ⁢ 𝑑 S = μ 2 ⁢ ∫ Ω + u 2 ⁢ k ⁢ u 3 ⁢ k ⁢ ( u 2 ⁢ k 2 - u 3 ⁢ k 2 ) ⁢ 𝑑 x + β 23 ( k ) ⁢ ∫ Ω + u 2 ⁢ k ⁢ u 3 ⁢ k ⁢ ( u 3 ⁢ k 2 - u 2 ⁢ k 2 ) ⁢ 𝑑 x
= ( μ 2 - β 23 ( k ) ) ⁢ ∫ Ω + u 2 ⁢ k ⁢ u 3 ⁢ k ⁢ ( u 2 ⁢ k 2 - u 3 ⁢ k 2 ) ⁢ 𝑑 x < 0 ,

where β23(k)↗β23*>a*>μ2 has been used, and ν denotes the unit outward normal vector of ∂⁡Ω+. Because u2⁢k=u3⁢k on ∂⁡Ω+ and ∂⁡(u3⁢k-u2⁢k)∂⁡ν≥0, we have

∫ ∂ ⁡ Ω + ( ∂ ⁡ u 3 ⁢ k ∂ ⁡ ν ⁢ u 2 ⁢ k - ∂ ⁡ u 2 ⁢ k ∂ ⁡ ν ⁢ u 3 ⁢ k ) ⁢ 𝑑 S = ∫ ∂ ⁡ Ω + [ ∂ ⁡ ( u 3 ⁢ k - u 2 ⁢ k ) ∂ ⁡ ν + ∂ ⁡ u 2 ⁢ k ∂ ⁡ ν ⁢ ( u 2 ⁢ k - u 3 ⁢ k ) ] ⁢ 𝑑 S ≥ 0 ,

which is a contradiction, and thus Ω+=∅, i.e., u2⁢k≤u3⁢k in ℝ2. Similarly, one can deduce that u3⁢k≤u2⁢k in ℝ2, so u2⁢k=u3⁢k holds in ℝ2.

Let μ1, μ2, μ3∈(0,a∗) be given and μ2=μ3. Suppose β12(k)↗β12∗, β13(k)↗β13∗, β23(k)↗β23∗ as k→∞. Assume that there are two different nonnegative minimizers (u1⁢k,u2⁢k,u3⁢k) and (v1⁢k,v2⁢k,v3⁢k) of ek. Let (z¯1⁢k,z¯2⁢k,z¯3⁢k) and (x2⁢k,y2⁢k,z2⁢k) be the unique maximum point of (u1⁢k,u2⁢k,u3⁢k) and (v1⁢k,v2⁢k,v3⁢k), respectively. We can show that the nonnegative minimizer (u1⁢k,u2⁢k,u3⁢k) solves system (3.10) with γk=γ1⁢k, and (v1⁢k,v2⁢k,v3⁢k) solves system (3.10) with γk=γ2⁢k, where γ1⁢k,γ2⁢k are suitable Lagrange multipliers given by

γ 1 ⁢ k = e k - μ 1 2 ⁢ ∫ ℝ 2 u 1 ⁢ k 4 ⁢ 𝑑 x - μ 2 2 ⁢ ∫ ℝ 2 u 2 ⁢ k 4 ⁢ 𝑑 x - μ 3 2 ⁢ ∫ ℝ 2 u 3 ⁢ k 4 ⁢ 𝑑 x - β 12 ( k ) ⁢ ∫ ℝ 2 u 1 ⁢ k 2 ⁢ u 2 ⁢ k 2 ⁢ 𝑑 x
(5.4) - β 13 ( k ) ⁢ ∫ ℝ 2 u 1 ⁢ k 2 ⁢ u 3 ⁢ k 2 ⁢ 𝑑 x - β 23 ( k ) ⁢ ∫ ℝ 2 u 2 ⁢ k 2 ⁢ u 3 ⁢ k 2 ⁢ 𝑑 x ,
γ 2 ⁢ k = e k - μ 1 2 ⁢ ∫ ℝ 2 v 1 ⁢ k 4 ⁢ 𝑑 x - μ 2 2 ⁢ ∫ ℝ 2 v 2 ⁢ k 4 ⁢ 𝑑 x - μ 3 2 ⁢ ∫ ℝ 2 v 3 ⁢ k 4 ⁢ 𝑑 x - β 12 ( k ) ⁢ ∫ ℝ 2 v 1 ⁢ k 2 ⁢ v 2 ⁢ k 2 ⁢ 𝑑 x
(5.5) - β 13 ( k ) ⁢ ∫ ℝ 2 v 1 ⁢ k 2 ⁢ v 3 ⁢ k 2 ⁢ 𝑑 x - β 23 ( k ) ⁢ ∫ ℝ 2 v 2 ⁢ k 2 ⁢ v 3 ⁢ k 2 ⁢ 𝑑 x ,

where limk→∞⁡ε¯k2⁢γi⁢k=-1 (i=1,2). Define

u ¯ i ⁢ k ( x ) := a ∗ ε ¯ k u i ⁢ k ( ε ¯ k x + x 2 ⁢ k ) , v ¯ i ⁢ k ( x ) := a ∗ ε ¯ k v i ⁢ k ( ε ¯ k x + x 2 ⁢ k )   ( i = 1 , 2 , 3 ) .

Following the arguments similar to that in [15], it yields from Lemma 5.1 that

(5.6) u ¯ i ⁢ k → U i = A i w , v ¯ i ⁢ k → U i = A i w   ( i = 1 , 2 , 3 ) ,

uniformly in ℝ2 as k→∞. Notice that u2⁢k=u3⁢k, v2⁢k=v3⁢k in ℝ2 as k→∞, and since μ2=μ3, we have U2=U3. Moreover, (u¯1⁢k,u¯2⁢k,u¯3⁢k) satisfies

(5.7) { - Δ ⁢ u ¯ 1 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ u ¯ 1 ⁢ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ u ¯ 1 ⁢ k + μ 1 a ∗ ⁢ u ¯ 1 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ u ¯ 2 ⁢ k 2 ⁢ u ¯ 1 ⁢ k + β 13 ( k ) a ∗ ⁢ u ¯ 3 ⁢ k 2 ⁢ u ¯ 1 ⁢ k , - Δ ⁢ u ¯ 2 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ u ¯ 2 ⁢ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ u ¯ 2 ⁢ k + μ 2 a ∗ ⁢ u ¯ 2 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ u ¯ 1 ⁢ k 2 ⁢ u ¯ 2 ⁢ k + β 23 ( k ) a ∗ ⁢ u ¯ 3 ⁢ k 2 ⁢ u ¯ 2 ⁢ k , - Δ ⁢ u ¯ 3 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ u ¯ 3 ⁢ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ u ¯ 3 ⁢ k + μ 3 a ∗ ⁢ u ¯ 3 ⁢ k 3 + β 13 ( k ) a ∗ ⁢ u ¯ 1 ⁢ k 2 ⁢ u ¯ 3 ⁢ k + β 23 ( k ) a ∗ ⁢ u ¯ 2 ⁢ k 2 ⁢ u ¯ 3 ⁢ k ,

and (v¯1⁢k,v¯2⁢k,v¯3⁢k) satisfies

(5.8) { - Δ ⁢ v ¯ 1 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ v ¯ 1 ⁢ k = ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ v ¯ 1 ⁢ k + μ 1 a ∗ ⁢ v ¯ 1 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ v ¯ 2 ⁢ k 2 ⁢ v ¯ 1 ⁢ k + β 13 ( k ) a ∗ ⁢ v ¯ 3 ⁢ k 2 ⁢ v ¯ 1 ⁢ k , - Δ ⁢ v ¯ 2 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ v ¯ 2 ⁢ k = ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ v ¯ 2 ⁢ k + μ 2 a ∗ ⁢ v ¯ 2 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ v ¯ 1 ⁢ k 2 ⁢ v ¯ 2 ⁢ k + β 23 ( k ) a ∗ ⁢ v ¯ 3 ⁢ k 2 ⁢ v ¯ 2 ⁢ k , - Δ ⁢ v ¯ 3 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ v ¯ 3 ⁢ k = ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ v ¯ 3 ⁢ k + μ 3 a ∗ ⁢ v ¯ 3 ⁢ k 3 + β 13 ( k ) a ∗ ⁢ v ¯ 1 ⁢ k 2 ⁢ v ¯ 3 ⁢ k + β 23 ( k ) a ∗ ⁢ v ¯ 2 ⁢ k 2 ⁢ v ¯ 3 ⁢ k .

Lemma 5.2.

Assume μ1, μ2, μ3∈(0,a∗) and μ2=μ3. We have

C 1 ⁢ ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ≤ ∥ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ≤ C 2 ⁢ ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ,
C 3 ⁢ ∥ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ≤ ∥ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ≤ C 4 ⁢ ∥ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 )

as k→∞, where the positive constants Ci (i=1,2,3,4) are independent of k.

Proof.

Since u¯2⁢k=u¯3⁢k, v¯2⁢k=v¯3⁢k in ℝ2 as k→∞, for a positive constant C, when k is large enough, we have

C - 1 ⁢ ∥ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ≤ ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ≤ C ⁢ ∥ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) .

Firstly, we prove that neither of the following equalities can hold:

(5.9) lim k → ∞ ⁡ ∥ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) = + ∞ , lim k → ∞ ⁡ ∥ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ∥ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) = + ∞ .

Indeed, by use of (5.7) and (5.8), we have

(5.10)

- Δ ⁢ ( u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ) + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ( u ¯ 1 ⁢ k - v ¯ 1 ⁢ k )
= γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ u ¯ 1 ⁢ k - γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ v ¯ 1 ⁢ k + γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ v ¯ 1 ⁢ k - γ 2 ⁢ k ⁢ ε ¯ k 2 ⁢ v ¯ 1 ⁢ k
+ μ 1 a ∗ ⁢ ( u ¯ 1 ⁢ k 3 - v ¯ 1 ⁢ k 3 ) + β 12 ( k ) a ∗ ⁢ ( u ¯ 2 ⁢ k 2 ⁢ u ¯ 1 ⁢ k - v ¯ 2 ⁢ k 2 ⁢ v ¯ 1 ⁢ k ) + β 13 ( k ) a ∗ ⁢ ( u ¯ 3 ⁢ k 2 ⁢ u ¯ 1 ⁢ k - v ¯ 3 ⁢ k 2 ⁢ v ¯ 1 ⁢ k )
= γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ( u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ) + ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) ⁢ v ¯ 1 ⁢ k + μ 1 a ∗ ⁢ ( u ¯ 1 ⁢ k 3 - v ¯ 1 ⁢ k 3 )
+ β 12 ( k ) a ∗ ⁢ [ u ¯ 2 ⁢ k 2 ⁢ ( u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ) + v ¯ 1 ⁢ k ⁢ ( u ¯ 2 ⁢ k 2 - v ¯ 2 ⁢ k 2 ) ] + β 13 ( k ) a ∗ ⁢ [ u ¯ 3 ⁢ k 2 ⁢ ( u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ) + v ¯ 1 ⁢ k ⁢ ( u ¯ 3 ⁢ k 2 - v ¯ 3 ⁢ k 2 ) ] ,

where ε¯k2⁢(γ1⁢k-γ2⁢k) satisfies

(5.11)

ε ¯ k 2 ( γ 1 ⁢ k - γ 2 ⁢ k ) = ε ¯ k 2 { - μ 1 2 ∫ ℝ 2 ( u 1 ⁢ k 4 - v 1 ⁢ k 4 ) d x - μ 2 2 ∫ ℝ 2 ( u 2 ⁢ k 4 - v 2 ⁢ k 4 ) d x - μ 3 2 ∫ ℝ 2 ( u 3 ⁢ k 4 - v 3 ⁢ k 4 ) d x
- β 12 ( k ) ⁢ ∫ ℝ 2 [ u 1 ⁢ k 2 ⁢ ( u 2 ⁢ k 2 - v 2 ⁢ k 2 ) + v 2 ⁢ k 2 ⁢ ( u 1 ⁢ k 2 - v 1 ⁢ k 2 ) ] ⁢ 𝑑 x
- β 13 ( k ) ⁢ ∫ ℝ 2 [ u 1 ⁢ k 2 ⁢ ( u 3 ⁢ k 2 - v 3 ⁢ k 2 ) + v 3 ⁢ k 2 ⁢ ( u 1 ⁢ k 2 - v 1 ⁢ k 2 ) ] ⁢ 𝑑 x
- β 23 ( k ) ∫ ℝ 2 [ u 2 ⁢ k 2 ( u 3 ⁢ k 2 - v 3 ⁢ k 2 ) + v 3 ⁢ k 2 ( u 2 ⁢ k 2 - v 2 ⁢ k 2 ) ] d x }
= - 1 2 ⁢ ( a * ) 2 ⁢ { μ 1 ⁢ ∫ ℝ 2 ( u ¯ 1 ⁢ k 4 - v ¯ 1 ⁢ k 4 ) ⁢ 𝑑 x + μ 2 ⁢ ∫ ℝ 2 ( u ¯ 2 ⁢ k 4 - v ¯ 2 ⁢ k 4 ) ⁢ 𝑑 x + μ 3 ⁢ ∫ ℝ 2 ( u ¯ 3 ⁢ k 4 - v ¯ 3 ⁢ k 4 ) ⁢ 𝑑 x }
- β 12 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ¯ 1 ⁢ k 2 ⁢ ( u ¯ 2 ⁢ k 2 - v ¯ 2 ⁢ k 2 ) + v ¯ 2 ⁢ k 2 ⁢ ( u ¯ 1 ⁢ k 2 - v ¯ 1 ⁢ k 2 ) ] ⁢ 𝑑 x
- β 13 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ¯ 1 ⁢ k 2 ⁢ ( u ¯ 3 ⁢ k 2 - v ¯ 3 ⁢ k 2 ) + v ¯ 3 ⁢ k 2 ⁢ ( u ¯ 1 ⁢ k 2 - v ¯ 1 ⁢ k 2 ) ] ⁢ 𝑑 x
- β 23 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ¯ 2 ⁢ k 2 ⁢ ( u ¯ 3 ⁢ k 2 - v ¯ 3 ⁢ k 2 ) + v ¯ 3 ⁢ k 2 ⁢ ( u ¯ 2 ⁢ k 2 - v ¯ 2 ⁢ k 2 ) ] ⁢ 𝑑 x ,

where (5.4) and (5.5) have been used. Set

ξ k ⁢ ( x ) := u ¯ 1 ⁢ k ⁢ ( x ) - v ¯ 1 ⁢ k ∥ u ¯ 1 ⁢ k ⁢ ( x ) - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) .

From (5.10) and (5.11), we know that ξk satisfies

(5.12)

- Δ ⁢ ξ k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ξ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ k + v ¯ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ γ 1 ⁢ k - γ 2 ⁢ k ∥ u ¯ 1 ⁢ k ⁢ ( x ) - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) + μ 1 a ∗ ⁢ ( u ¯ 1 ⁢ k 2 + v ¯ 1 ⁢ k 2 + u ¯ 1 ⁢ k ⁢ v ¯ 1 ⁢ k ) ⁢ ξ k
+ β 12 ( k ) a ∗ ⁢ [ u ¯ 2 ⁢ k 2 ⁢ ξ k + v ¯ 1 ⁢ k ⁢ ( u ¯ 2 ⁢ k 2 - v ¯ 2 ⁢ k 2 ) ∥ u ¯ 1 ⁢ k ⁢ ( x ) - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ]
+ β 13 ( k ) a ∗ ⁢ [ u ¯ 3 ⁢ k 2 ⁢ ξ k + v ¯ 1 ⁢ k ⁢ ( u ¯ 3 ⁢ k 2 - v ¯ 3 ⁢ k 2 ) ∥ u ¯ 1 ⁢ k ⁢ ( x ) - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ] .

Following the ideas in [15], we deduce that there exist a subsequence of {(β12(k),β13(k),β23(k))}, still denoted by {(β12(k),β13(k),β23(k))}, and a function ξ⁢(x) such that ξk→ξ in Cloc1⁢(ℝ2) as k→∞.

Applying (5.6), (5.9) and (5.11) to (5.12), we deduce that ξ satisfies

(5.13) 0 = Δ ⁢ ξ - ξ + ( 3 ⁢ μ 1 a ∗ ⁢ U 1 2 + β 12 ∗ a ∗ ⁢ U 2 2 + β 13 ∗ a ∗ ⁢ U 3 2 ) ⁢ ξ - [ 2 ⁢ μ 1 ( a ∗ ) 2 ⁢ ∫ ℝ 2 U 1 3 ⁢ ξ ⁢ 𝑑 x + 2 ⁢ β 12 ∗ ( a ∗ ) 2 ⁢ ∫ ℝ 2 U 1 ⁢ U 2 2 ⁢ ξ ⁢ 𝑑 x + 2 ⁢ β 13 ∗ ( a ∗ ) 2 ⁢ ∫ ℝ 2 U 1 ⁢ U 3 2 ⁢ ξ ⁢ 𝑑 x ] ⁢ U 1 .

Employing (1.25) and (4.6), we see that (5.13) can be simplified as

(5.14) Δ ⁢ ξ - ξ + ( 1 + 2 ⁢ a 1 ⁢ A 1 2 ) ⁢ w 2 ⁢ ξ = 2 ⁢ A 1 2 a ∗ ⁢ w ⁢ ∫ ℝ 2 w 3 ⁢ ξ ⁢ 𝑑 x .

Multiplying (5.14) by w and multiplying (1.8) by ξ, and integrating by parts, respectively, it yields

∫ ℝ 2 ( ∇ ⁡ w ⁢ ∇ ⁡ ξ + w ⁢ ξ ) ⁢ 𝑑 x = ∫ ℝ 2 ( 1 + 2 ⁢ A 1 2 ⁢ ( a 1 - 1 ) ) ⁢ w 3 ⁢ ξ ⁢ 𝑑 x = ∫ ℝ 2 w 3 ⁢ ξ ⁢ 𝑑 x .

Since a1<1, it follows that ∫ℝ2w3⁢ξ⁢𝑑x=0. Then equation (5.14) becomes

Δ ⁢ ξ - ξ + ( 1 + 2 ⁢ a 1 ⁢ A 1 2 ) ⁢ w 2 ⁢ ξ = 0 .

However, since 1<1+2⁢a1⁢A12<3, by the arguments as in [10], we deduce that ξ≡0. Similar to the proof of [15, Lemma 4.2], we can prove that neither of the equalities in (5.9) is true.

On the other hand, suppose

(5.15) lim k → ∞ ⁡ ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ∥ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) = + ∞ , lim k → ∞ ⁡ ∥ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ∥ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) = + ∞ .

From (5.7) and (5.8), and we have

(5.16)

- Δ ⁢ ( u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ) + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ( u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ) = γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ( u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ) + ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) ⁢ v ¯ 2 ⁢ k + μ 2 a ∗ ⁢ ( u ¯ 2 ⁢ k 3 - v ¯ 2 ⁢ k 3 )
+ β 12 ( k ) a ∗ ⁢ [ u ¯ 1 ⁢ k 2 ⁢ ( u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ) + v ¯ 2 ⁢ k ⁢ ( u ¯ 1 ⁢ k 2 - v ¯ 1 ⁢ k 2 ) ]
+ β 23 ( k ) a ∗ ⁢ [ u ¯ 3 ⁢ k 2 ⁢ ( u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ) + v ¯ 2 ⁢ k ⁢ ( u ¯ 3 ⁢ k 2 - v ¯ 3 ⁢ k 2 ) ] .

Setting

l k ⁢ ( x ) := u ¯ 2 ⁢ k ⁢ ( x ) - v ¯ 2 ⁢ k ⁢ ( x ) ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) ,

it yields from (5.16) and (5.11) that

(5.17)

- Δ ⁢ l k + ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ l k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ l k + v ¯ 2 ⁢ k ⁢ ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) + μ 2 a ∗ ⁢ ( u ¯ 2 ⁢ k 2 + u ¯ 2 ⁢ k ⁢ v ¯ 2 ⁢ k + v ¯ 2 ⁢ k 2 ) ⁢ l k
+ β 12 ( k ) a ∗ ⁢ u ¯ 1 ⁢ k 2 ⁢ l k + β 12 ( k ) a ∗ ⁢ v ¯ 2 ⁢ k ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ u ¯ 1 ⁢ k - v ¯ 1 ⁢ k ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 )
+ β 23 ( k ) a ∗ ⁢ u ¯ 3 ⁢ k 2 ⁢ l k + β 23 ( k ) a ∗ ⁢ v ¯ 2 ⁢ k ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) .

By the standard elliptic regularity theory, there exist a subsequence of {(β12(k),β13(k),β23(k))}, still denoted by {(β12(k),β13(k),β23(k))}, and a function ξ⁢(x) such that lk→l in Cloc1⁢(ℝ2) as k→∞. Since u¯2⁢k-v¯2⁢k=u¯3⁢k-v¯3⁢k as k→∞, we obtain

u ¯ 3 ⁢ k - v ¯ 3 ⁢ k ∥ u ¯ 2 ⁢ k - v ¯ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) → l   in ⁢ C loc 1 ⁢ ( ℝ 2 ) ⁢ as ⁢ k → ∞ .

Applying (5.6), (5.11) and (5.15) to (5.17), we conclude that l satisfies

(5.18)

0 = Δ ⁢ l - l + ( 3 ⁢ μ 2 a ∗ ⁢ U 2 2 + β 12 ∗ a ∗ ⁢ U 1 2 + β 23 ∗ a ∗ ⁢ U 3 2 + 2 ⁢ β 23 ∗ a ∗ ⁢ U 2 ⁢ U 3 ) ⁢ l
- 4 ⁢ [ μ 2 ( a ∗ ) 2 ⁢ ∫ ℝ 2 U 2 3 ⁢ l ⁢ 𝑑 x + β 12 ∗ ( a ∗ ) 2 ⁢ ∫ ℝ 2 U 1 2 ⁢ U 2 ⁢ l ⁢ 𝑑 x + β 23 ∗ ( a ∗ ) 2 ⁢ ∫ ℝ 2 U 3 3 ⁢ l ⁢ 𝑑 x ] ⁢ U 2 .

Employing (1.25) and (4.6), we see that(5.18) can be simplified to

(5.19) Δ ⁢ l - l + ( 2 ⁢ a 2 ⁢ A 2 2 + 1 + 2 ⁢ b 3 ⁢ A 2 2 ) ⁢ w 2 ⁢ l = 4 ⁢ A 2 2 a ∗ ⁢ w ⁢ ∫ ℝ 2 w 3 ⁢ l ⁢ 𝑑 x .

Multiplying (5.19) by w and multiplying (1.8) by l, and integrating by parts, respectively, it yields that

∫ ℝ 2 ( ∇ ⁡ w ⁢ ∇ ⁡ l + w ⁢ l ) ⁢ 𝑑 x = ( 2 ⁢ a 2 ⁢ A 2 2 + 1 + 2 ⁢ b 3 ⁢ A 2 2 - 4 ⁢ A 2 2 ) ⁢ ∫ ℝ 2 w 3 ⁢ l ⁢ 𝑑 x = ∫ ℝ 2 w 3 ⁢ l ⁢ 𝑑 x .

However,

a 2 + b 3 - 2 = μ 2 a ∗ + 1 a ∗ ⁢ [ a ∗ + 1 2 ⁢ ( a ∗ - μ 2 ) ] - 2 = 1 2 ⁢ a ∗ ⁢ ( μ 2 - a ∗ ) < 0 ,

which implies that ∫ℝ2w3⁢l⁢𝑑x=0. Then (5.19) becomes

Δ ⁢ l - l + ( 2 ⁢ a 2 ⁢ A 2 2 + 1 + 2 ⁢ b 3 ⁢ A 2 2 ) ⁢ w 2 ⁢ l = 0 .

Now we prove that 1<2⁢a2⁢A22+1+2⁢b3⁢A22<3. It is easy to see that 2⁢a2⁢A22+1+2⁢b3⁢A22>1, and thus we need only to prove that the inequality on the right-hand side holds. Indeed,

(5.20)

2 ⁢ a 2 ⁢ A 2 2 + 1 + 2 ⁢ b 3 ⁢ A 2 2 < 3 ⇔ a 2 ⁢ A 2 2 + b 3 ⁢ A 2 2 < 1
⇔ μ 2 + a ∗ + 1 2 ⁢ a ∗ - 1 2 ⁢ μ 2 a ∗ ⋅ α 12 ∗ A 0 < 1
⇔ ( 3 2 + μ 2 2 ⁢ a ∗ ) ⁢ α 12 ∗ A 0 < 1 .

Since μ2=μ3, we have

A 0 = a ∗ - μ 2 + 2 ⁢ α 12 ∗ .

Then by use of (5.20), we have

a 2 ⁢ A 2 2 + b 3 ⁢ A 2 2 < 1 ⇔ ( 3 2 + μ 2 2 ⁢ a ∗ ) ⁢ α 12 ∗ < a ∗ - μ 2 + 2 ⁢ α 12 ∗
⇔ ( 3 2 + μ 2 2 ⁢ a ∗ ) ⁢ ( a ∗ - μ 1 ) 1 2 < ( a ∗ - μ 2 ) 1 2 + 2 ⁢ ( a ∗ - μ 1 ) 1 2
⇔ ( μ 2 2 ⁢ a ∗ - 1 2 ) ⁢ ( a ∗ - μ 1 ) 1 2 < ( a ∗ - μ 2 ) 1 2 .

Since μ2<a∗, we have μ22⁢a∗-12<0, which implies that (5.20) holds. Again by the arguments as in [10], we can obtain l≡0, so (5.15) cannot hold. This completes the proof. ∎

If ui⁢k≡vi⁢k (i∈{1,2,3}) in ℝ2, it then follows from Lemma 5.2 that uj⁢k≡vj⁢k (j=1,2,3, j≠i), and the proof of Theorem 1.4 is done. Therefore, we need only consider the case ui⁢k≢vi⁢k (i=1,2,3). Set

(5.21) u ^ i ⁢ k ⁢ ( x ) := a ∗ ⁢ ε ¯ k ⁢ u i ⁢ k ⁢ ( x ) , v ^ i ⁢ k ⁢ ( x ) := a ∗ ⁢ ε ¯ k ⁢ v i ⁢ k ⁢ ( x ) .

It follows from Lemma 5.1 that

u ¯ i ⁢ k ( x ) = u ^ i ⁢ k ( ε ¯ k x + x 2 ⁢ k ) → U i ( x ) , v ¯ i ⁢ k ( x ) = v ^ i ⁢ k ( ε ¯ k x + x 2 ⁢ k ) → U i ( x )   ( i = 1 , 2 , 3 ) ,

uniformly in ℝ2 as k→∞. Moreover, (u^1⁢k,u^2⁢k,u^3⁢k) satisfies

(5.22) { - ε ¯ k 2 ⁢ Δ ⁢ u ^ 1 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ u ^ 1 ⁢ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ u ^ 1 ⁢ k + μ 1 a ∗ ⁢ u ^ 1 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ u ^ 2 ⁢ k 2 ⁢ u ^ 1 ⁢ k + β 13 ( k ) a ∗ ⁢ u ^ 3 ⁢ k 2 ⁢ u ^ 1 ⁢ k , - ε ¯ k 2 ⁢ Δ ⁢ u ^ 2 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ u ^ 2 ⁢ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ u ^ 2 ⁢ k + μ 2 a ∗ ⁢ u ^ 2 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ u ^ 1 ⁢ k 2 ⁢ u ^ 2 ⁢ k + β 23 ( k ) a ∗ ⁢ u ^ 3 ⁢ k 2 ⁢ u ^ 2 ⁢ k , - ε ¯ k 2 ⁢ Δ ⁢ u ^ 3 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ u ^ 3 ⁢ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ u ^ 3 ⁢ k + μ 3 a ∗ ⁢ u ^ 3 ⁢ k 3 + β 13 ( k ) a ∗ ⁢ u ^ 1 ⁢ k 2 ⁢ u ^ 3 ⁢ k + β 23 ( k ) a ∗ ⁢ u ^ 2 ⁢ k 2 ⁢ u ^ 3 ⁢ k ,

and (v^1⁢k,v^2⁢k,v^3⁢k) satisfies

{ - ε ¯ k 2 ⁢ Δ ⁢ v ^ 1 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ v ^ 1 ⁢ k = ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ v ^ 1 ⁢ k + μ 1 a ∗ ⁢ v ^ 1 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ v ^ 2 ⁢ k 2 ⁢ v ^ 1 ⁢ k + β 13 ( k ) a ∗ ⁢ v ^ 3 ⁢ k 2 ⁢ v ^ 1 ⁢ k , - ε ¯ k 2 ⁢ Δ ⁢ v ^ 2 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ v ^ 2 ⁢ k = ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ v ^ 2 ⁢ k + μ 2 a ∗ ⁢ v ^ 2 ⁢ k 3 + β 12 ( k ) a ∗ ⁢ v ^ 1 ⁢ k 2 ⁢ v ^ 2 ⁢ k + β 23 ( k ) a ∗ ⁢ v ^ 3 ⁢ k 2 ⁢ v ^ 2 ⁢ k , - ε ¯ k 2 ⁢ Δ ⁢ v ^ 3 ⁢ k + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ v ^ 3 ⁢ k = ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ v ^ 3 ⁢ k + μ 3 a ∗ ⁢ v ^ 3 ⁢ k 3 + β 13 ( k ) a ∗ ⁢ v ^ 1 ⁢ k 2 ⁢ v ^ 3 ⁢ k + β 23 ( k ) a ∗ ⁢ v ^ 2 ⁢ k 2 ⁢ v ^ 3 ⁢ k .

As ui⁢k≢vi⁢k (i=1,2,3) in ℝ2, we define

(5.23) ξ ^ i ⁢ k ⁢ ( x ) := u i ⁢ k ⁢ ( x ) - v i ⁢ k ⁢ ( x ) ∥ u 1 ⁢ k - v 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u 2 ⁢ k - v 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u 2 ⁢ k - v 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ,

by the definition, and we also have equivalently

(5.24) ξ ^ i ⁢ k ⁢ ( x ) = u ^ i ⁢ k ⁢ ( x ) - v ^ i ⁢ k ⁢ ( x ) ∥ u ^ 1 ⁢ k - v ^ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u ^ 2 ⁢ k - v ^ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u ^ 2 ⁢ k - v ^ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 .

For simplicity, we set

(5.25) D k := ∥ u 1 ⁢ k - v 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u 2 ⁢ k - v 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u 2 ⁢ k - v 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ,
(5.26) D ^ k := ∥ u ^ 1 ⁢ k - v ^ 1 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u ^ 2 ⁢ k - v ^ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 ⁢ ∥ u ^ 2 ⁢ k - v ^ 2 ⁢ k ∥ L ∞ ⁢ ( ℝ 2 ) 1 3 .

From Lemma 5.2, we deduce that there exists a positive constant C independent of k such that

(5.27) 0 ≤ | ξ ^ i ⁢ k ⁢ ( x ) | ≤ C < ∞   and   | ξ ^ 1 ⁢ k ⁢ ( x ) ⁢ ξ ^ 2 ⁢ k ⁢ ( x ) ⁢ ξ ^ 3 ⁢ k ⁢ ( x ) | ≤ 1   in ⁢ ℝ 2 .

Furthermore, we have the following local estimates of (ξ^1⁢k,ξ^2⁢k,ξ^3⁢k).

Lemma 5.3.

Suppose that μ1,μ2,μ3∈(0,a∗) are given. Then for any x0∈R2, there exists a small constant δ>0 such that

(5.28) ∫ ∂ ⁡ B δ ⁢ ( x 0 ) ( ε ¯ k 2 ⁢ | ∇ ⁡ ξ ^ i ⁢ k | 2 + 1 2 ⁢ ξ ^ i ⁢ k 2 + ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ ξ ^ i ⁢ k 2 ) ⁢ 𝑑 S = O ⁢ ( ε ¯ k 2 )   𝑎𝑠 ⁢ k → ∞ , i = 1 , 2 , 3 .

Proof.

It follows from (1.7), (5.21) and (5.24) that (ξ^1⁢k,ξ^2⁢k,ξ^3⁢k) satisfies

(5.29) { ε ¯ k 2 ⁢ Δ ⁢ ξ ^ 1 ⁢ k - ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ ξ ^ 1 ⁢ k + ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) D ^ k ⁢ v ^ 1 ⁢ k + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ ^ 1 ⁢ k + μ 1 a ∗ ⁢ ( u ^ 1 ⁢ k 2 + v ^ 1 ⁢ k 2 + u ^ 1 ⁢ k ⁢ v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k + β 12 ( k ) a ∗ ⁢ [ u ^ 2 ⁢ k 2 ⁢ ξ ^ 1 ⁢ k + v ^ 1 ⁢ k ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ] + β 13 ( k ) a ∗ ⁢ [ u ^ 3 ⁢ k 2 ⁢ ξ ^ 1 ⁢ k + v ^ 1 ⁢ k ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ] = 0 , ε ¯ k 2 ⁢ Δ ⁢ ξ ^ 2 ⁢ k - ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ ξ ^ 2 ⁢ k + ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) D ^ k ⁢ v ^ 2 ⁢ k + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ ^ 2 ⁢ k + μ 2 a ∗ ⁢ ( u ^ 2 ⁢ k 2 + v ^ 2 ⁢ k 2 + u ^ 2 ⁢ k ⁢ v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + β 12 ( k ) a ∗ ⁢ [ u ^ 1 ⁢ k 2 ⁢ ξ ^ 2 ⁢ k + v ^ 2 ⁢ k ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] + β 23 ( k ) a ∗ ⁢ [ u ^ 3 ⁢ k 2 ⁢ ξ ^ 2 ⁢ k + v ^ 2 ⁢ k ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ] = 0 , ε ¯ k 2 ⁢ Δ ⁢ ξ ^ 3 ⁢ k - ε ¯ k 2 ⁢ V ⁢ ( x ) ⁢ ξ ^ 3 ⁢ k + ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) D ^ k ⁢ v ^ 3 ⁢ k + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ ^ 3 ⁢ k + μ 3 a ∗ ⁢ ( u ^ 3 ⁢ k 2 + v ^ 3 ⁢ k 2 + u ^ 3 ⁢ k ⁢ v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k + β 13 ( k ) a ∗ ⁢ [ u ^ 1 ⁢ k 2 ⁢ ξ ^ 3 ⁢ k + v ^ 3 ⁢ k ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] + β 23 ( k ) a ∗ ⁢ [ u ^ 2 ⁢ k 2 ⁢ ξ ^ 3 ⁢ k + v ^ 3 ⁢ k ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ] = 0 .

By use of (5.11) and (5.26), we have

(5.30)

ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) D ^ k = 1 ε ¯ k 2 { - μ 1 2 ⁢ ( a ∗ ) 2 ∫ ℝ 2 ( u ^ 1 ⁢ k 2 + v ^ 1 ⁢ k 2 ) ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ξ ^ 1 ⁢ k d x
- μ 2 2 ⁢ ( a ∗ ) 2 ⁢ ∫ ℝ 2 ( u ^ 2 ⁢ k 2 + v ^ 2 ⁢ k 2 ) ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ⁢ 𝑑 x
- μ 3 2 ⁢ ( a ∗ ) 2 ⁢ ∫ ℝ 2 ( u ^ 3 ⁢ k 2 + v ^ 3 ⁢ k 2 ) ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ⁢ 𝑑 x
- β 12 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + v ^ 2 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 x
- β 13 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k + v ^ 3 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 x
- β 23 ( k ) ( a ∗ ) 2 ∫ ℝ 2 [ u ^ 2 ⁢ k 2 ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ξ ^ 3 ⁢ k + v ^ 3 ⁢ k 2 ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ξ ^ 2 ⁢ k ] d x } .

Since |ξ^i⁢k| is bounded uniformly in k, we have

(5.31) | ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) D ^ k | ≤ C 5 ⁢ 1 ε ¯ k 2 ⁢ ∫ ℝ 2 w 3 ⁢ ( x ε ¯ k ) ⁢ 𝑑 x ≤ C ,

where C and C5 are positive constants independent of k. By setting

c k := ε ¯ k 2 ⁢ ( γ 1 ⁢ k - γ 2 ⁢ k ) D ^ k ,

c k is bounded uniformly in k. Multiplying the first equation of (5.29) by ξ^1⁢k and integrating over ℝ2, we have

ε ¯ k 2 ⁢ ∫ ℝ 2 | ∇ ⁡ ξ ^ 1 ⁢ k | 2 ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ ℝ 2 V ⁢ ( x ) ⁢ ξ ^ 1 ⁢ k 2 ⁢ 𝑑 x - ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ξ ^ 1 ⁢ k 2 ⁢ 𝑑 x
  = μ 1 a ∗ ⁢ ∫ ℝ 2 ( u ^ 1 ⁢ k 2 + v ^ 1 ⁢ k 2 + u ^ 1 ⁢ k ⁢ v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k 2 ⁢ 𝑑 x + β 12 ( k ) a ∗ ⁢ ∫ ℝ 2 [ u ^ 2 ⁢ k 2 ⁢ ξ ^ 1 ⁢ k 2 + v ^ 1 ⁢ k ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 x
      + β 13 ( k ) a ∗ ⁢ ∫ ℝ 2 [ u ^ 3 ⁢ k 2 ⁢ ξ ^ 1 ⁢ k 2 + v ^ 1 ⁢ k ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 x + c k ⁢ ∫ ℝ 2 v ^ 1 ⁢ k ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x
  ≤ μ 1 ⁢ ε ¯ k 2 a ∗ ⁢ ∫ ℝ 2 [ u ¯ 1 ⁢ k 2 + v ¯ 1 ⁢ k 2 + u ¯ 1 ⁢ k ⁢ v ¯ 1 ⁢ k ] ⁢ 𝑑 x + β 12 ( k ) ⁢ ε ¯ k 2 a ∗ ⁢ ∫ ℝ 2 [ u ¯ 2 ⁢ k 2 + v ¯ 1 ⁢ k ⁢ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ] ⁢ 𝑑 x
      + β 13 ( k ) ⁢ ε ¯ k 2 a ∗ ⁢ ∫ ℝ 2 [ u ¯ 3 ⁢ k 2 + v ¯ 1 ⁢ k ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ] ⁢ 𝑑 x + c k ⁢ ε ¯ k 2 ⁢ ∫ ℝ 2 v ¯ 1 ⁢ k ⁢ 𝑑 x
  ≤ C ⁢ ε ¯ k 2   as ⁢ k → ∞ ,

where the exponential decay of u¯i⁢k,v¯i⁢k(i=1,2,3) has been used. This implies that there exists a positive constant C6 such that

ε ¯ k 2 ⁢ ∫ ℝ 2 | ∇ ⁡ ξ ^ 1 ⁢ k | 2 ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ ℝ 2 V ⁢ ( x ) ⁢ ξ ^ 1 ⁢ k 2 ⁢ 𝑑 x - ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ξ ^ 1 ⁢ k 2 ⁢ 𝑑 x < C 6 ⁢ ε ¯ k 2   as ⁢ k → ∞ .

Following the discussion in [8], we can obtain (5.28). ∎

Based on the above estimates and Theorem 1.3, by virtue of Pohozaev identities, we can now prove Theorem 1.4.

Proof of Theorem 1.4.

We proceed the proof in two steps.

Step 1. Set

(5.32) ξ i ⁢ k ( x ) := ξ ^ i ⁢ k ( ε ¯ k x + x 2 ⁢ k ) ( i = 1 , 2 , 3 ) .

From (5.29), one can get

{ Δ ⁢ ξ 1 ⁢ k - ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ξ 1 ⁢ k + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ 1 ⁢ k + μ 1 a ∗ ⁢ ( u ¯ 1 ⁢ k 2 + v ¯ 1 ⁢ k 2 + u ¯ 1 ⁢ k ⁢ v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k + β 12 ( k ) a ∗ ⁢ [ u ¯ 2 ⁢ k 2 ⁢ ξ 1 ⁢ k + v ¯ 1 ⁢ k ⁢ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k ] + β 13 ( k ) a ∗ ⁢ [ u ¯ 3 ⁢ k 2 ⁢ ξ 1 ⁢ k + v ¯ 1 ⁢ k ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k ] + c k ⁢ v ¯ 1 ⁢ k = 0 , Δ ⁢ ξ 2 ⁢ k - ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ξ 2 ⁢ k + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ 2 ⁢ k + μ 2 a ∗ ⁢ ( u ¯ 2 ⁢ k 2 + v ¯ 2 ⁢ k 2 + u ¯ 2 ⁢ k ⁢ v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k + β 12 ( k ) a ∗ ⁢ [ u ¯ 1 ⁢ k 2 ⁢ ξ 2 ⁢ k + v ¯ 2 ⁢ k ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ] + β 23 ( k ) a ∗ ⁢ [ u ¯ 3 ⁢ k 2 ⁢ ξ 2 ⁢ k + v ¯ 2 ⁢ k ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k ] + c k ⁢ v ¯ 2 ⁢ k = 0 , Δ ⁢ ξ 3 ⁢ k - ε ¯ k 2 ⁢ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ξ 3 ⁢ k + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ξ 3 ⁢ k + μ 3 a ∗ ⁢ ( u ¯ 3 ⁢ k 2 + v ¯ 3 ⁢ k 2 + u ¯ 3 ⁢ k ⁢ v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k + β 13 ( k ) a ∗ ⁢ [ u ¯ 1 ⁢ k 2 ⁢ ξ 3 ⁢ k + v ¯ 3 ⁢ k ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ] + β 23 ( k ) a ∗ ⁢ [ u ¯ 2 ⁢ k 2 ⁢ ξ 3 ⁢ k + v ¯ 3 ⁢ k ⁢ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k ] + c k ⁢ v ¯ 3 ⁢ k = 0 .

By use of (5.30), we have

(5.33)

c k = - μ 1 2 ⁢ ( a ∗ ) 2 ⁢ ∫ ℝ 2 ( u ¯ 1 ⁢ k 2 + v ¯ 1 ⁢ k 2 ) ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ⁢ 𝑑 x - μ 2 2 ⁢ ( a ∗ ) 2 ⁢ ∫ ℝ 2 ( u ¯ 2 ⁢ k 2 + v ¯ 2 ⁢ k 2 ) ⁢ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k ⁢ 𝑑 x
- μ 3 2 ⁢ ( a ∗ ) 2 ⁢ ∫ ℝ 2 ( u ¯ 3 ⁢ k 2 + v ¯ 3 ⁢ k 2 ) ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k ⁢ 𝑑 x
- β 12 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ¯ 1 ⁢ k 2 ⁢ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k + v ¯ 2 ⁢ k 2 ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ] ⁢ 𝑑 x
- β 13 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ¯ 1 ⁢ k 2 ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k + v ¯ 3 ⁢ k 2 ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ] ⁢ 𝑑 x
- β 23 ( k ) ( a ∗ ) 2 ⁢ ∫ ℝ 2 [ u ¯ 2 ⁢ k 2 ⁢ ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k + v ¯ 3 ⁢ k 2 ⁢ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k ] ⁢ 𝑑 x .

It yields from (5.27) and (5.32) that ξi⁢k⁢(x) (i=1,2,3) are all bounded uniformly in ℝ2. According to (5.31), since ck is bounded, there is a constant b0 such that

lim k → ∞ ⁡ c k = - 2 ⁢ b 0 .

By the standard elliptic regularity theory, it follows from equation (5.33) that ∥ξi⁢k∥Cloc1,α⁢(ℝ2)≤C (i=1,2,3) for some α∈(0,1), where C>0 is independent of k. Therefore, up to a subsequence if necessary, we have (ξ1⁢k,ξ2⁢k,ξ3⁢k)→(ξ10,ξ20,ξ30) in Cloc1⁢(ℝ2), where (ξ10,ξ20,ξ30) satisfies

{ Δ ⁢ ξ 10 - ξ 10 + 3 ⁢ μ 1 a ∗ ⁢ U 1 2 ⁢ ξ 10 + β 12 ∗ a ∗ ⁢ ( U 2 2 ⁢ ξ 10 + 2 ⁢ U 1 ⁢ U 2 ⁢ ξ 20 ) + β 13 ∗ a ∗ ⁢ ( U 3 2 ⁢ ξ 10 + 2 ⁢ U 1 ⁢ U 3 ⁢ ξ 30 ) = 2 ⁢ b 0 ⁢ U 1 , Δ ⁢ ξ 20 - ξ 20 + 3 ⁢ μ 2 a ∗ ⁢ U 2 2 ⁢ ξ 20 + β 12 ∗ a ∗ ⁢ ( U 1 2 ⁢ ξ 20 + 2 ⁢ U 1 ⁢ U 2 ⁢ ξ 10 ) + β 23 ∗ a ∗ ⁢ ( U 3 2 ⁢ ξ 20 + 2 ⁢ U 2 ⁢ U 3 ⁢ ξ 30 ) = 2 ⁢ b 0 ⁢ U 2 , Δ ⁢ ξ 30 - ξ 30 + 3 ⁢ μ 3 a ∗ ⁢ U 3 2 ⁢ ξ 30 + β 13 ∗ a ∗ ⁢ ( U 1 2 ⁢ ξ 30 + 2 ⁢ U 1 ⁢ U 3 ⁢ ξ 10 ) + β 23 ∗ a ∗ ⁢ ( U 2 2 ⁢ ξ 30 + 2 ⁢ U 2 ⁢ U 3 ⁢ ξ 20 ) = 2 ⁢ b 0 ⁢ U 3 .

From (5.3), one can deduce that (ξ10,ξ20,ξ30) satisfies

(5.34) ( ξ 10 ξ 20 ξ 30 ) = b 0 ⁢ ( U 1 + x ⋅ ∇ ⁡ U 1 U 2 + x ⋅ ∇ ⁡ U 2 U 3 + x ⋅ ∇ ⁡ U 3 ) + ∑ j = 1 2 b j ⁢ ( ∂ ⁡ U 1 ∂ ⁡ x j ∂ ⁡ U 2 ∂ ⁡ x j ∂ ⁡ U 3 ∂ ⁡ x j )

for some constants bj with j=0,1,2.

Step 2. Firstly, we show that the following Pohozaev-type identity holds:

0 = b 0 ⁢ ∫ ℝ 2 p ⁢ | x | p - 2 ⁢ x j ⁢ ( x ⋅ ∇ ⁡ U 1 2 + x ⋅ ∇ ⁡ U 2 2 + x ⋅ ∇ ⁡ U 3 2 ) ⁢ 𝑑 x + 2 ⁢ b 0 ⁢ ∫ ℝ 2 p ⁢ | x | p - 2 ⁢ x j ⁢ ( U 1 2 + U 2 2 + U 3 2 ) ⁢ 𝑑 x
(5.35) - ∑ i = 1 2 b i ∫ ℝ 2 p ∂ ⁡ | x | p - 2 ⁢ x j ∂ ⁡ x i ( U 1 2 + U 2 2 + U 3 2 ) d x   ( j = 1 , 2 ) .

Multiplying the first equation of (5.22) by ∂⁡u^1⁢k∂⁡xj and integrating over Bδ⁢(x2⁢k), where δ is given by Lemma 5.3, we get the following identity:

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ Δ ⁢ u ^ 1 ⁢ k ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) V ⁢ ( x ) ⁢ u ^ 1 ⁢ k ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ 𝑑 x
= γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ 𝑑 x + μ 1 a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 3 ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ 𝑑 x
+ β 12 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ u ^ 1 ⁢ k ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ 𝑑 x + β 13 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ u ^ 1 ⁢ k ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ 𝑑 x .

Following the arguments in [15], we can get

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ Δ ⁢ u ^ 1 ⁢ k ⁢ 𝑑 x = - ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ ν ⁢ 𝑑 S + 1 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) | ∇ ⁡ u ^ 1 ⁢ k | 2 ⁢ ν j ⁢ 𝑑 S

and

ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) V ⁢ ( x ) ⁢ u ^ 1 ⁢ k ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ 𝑑 x = ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ V ⁢ ( x ) ⁢ ν j ⁢ 𝑑 S - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ 𝑑 x .

And so

(5.36) ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ 𝑑 x + β 12 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x + β 13 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x = - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ ν ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) | ∇ ⁡ u ^ 1 ⁢ k | 2 ⁢ ν j ⁢ 𝑑 S - γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S - μ 1 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 4 ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ V ⁢ ( x ) ⁢ ν j ⁢ 𝑑 S .

Similarly, we have

(5.37) ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ 𝑑 x + β 12 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x + β 23 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x = - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 2 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ u ^ 2 ⁢ k ∂ ⁡ ν ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) | ∇ ⁡ u ^ 2 ⁢ k | 2 ⁢ ν j ⁢ 𝑑 S - γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S - μ 2 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 4 ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ V ⁢ ( x ) ⁢ ν j ⁢ 𝑑 S ,
(5.38) ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ 𝑑 x + β 13 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ u ^ 3 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x + β 23 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ∂ ⁡ u ^ 3 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x = - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 3 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ u ^ 3 ⁢ k ∂ ⁡ ν ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) | ∇ ⁡ u ^ 3 ⁢ k | 2 ⁢ ν j ⁢ 𝑑 S - γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S - μ 3 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 4 ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ V ⁢ ( x ) ⁢ ν j ⁢ 𝑑 S .

Note that (5.36), (5.37) and (5.38) still hold when γ1⁢k is substituted by γ2⁢k and u^1⁢k,u^2⁢k,u^3⁢k are substituted by v^1⁢k,v^2⁢k,v^3⁢k, respectively. Thus we have

ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ D ^ k ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x + β 12 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k 2 ⁢ ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j - v ^ 2 ⁢ k 2 ⁢ ∂ ⁡ v ^ 1 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x
      + β 13 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k 2 ⁢ ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j - v ^ 3 ⁢ k 2 ⁢ ∂ ⁡ v ^ 1 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x
  = - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ ν - ∂ ⁡ v ^ 1 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ v ^ 1 ⁢ k ∂ ⁡ ν ) ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( | ∇ ⁡ u ^ 1 ⁢ k | 2 - | ∇ ⁡ v ^ 1 ⁢ k | 2 ) ⁢ ν j ⁢ 𝑑 S
      - γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S + γ 2 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S
      - μ 1 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 4 - v ^ 1 ⁢ k 4 ) ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 - v ^ 1 ⁢ k 2 ) ⁢ V ⁢ ( x ) ⁢ ν j ⁢ 𝑑 S ,

and then

ε ¯ k 2 ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ξ ^ 1 ⁢ k d x + 1 D ^ k { β 12 ( k ) a ∗ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k 2 ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j - v ^ 2 ⁢ k 2 ∂ ⁡ v ^ 1 ⁢ k 2 ∂ ⁡ x j ) d x
+ β 13 ( k ) a ∗ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k 2 ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j - v ^ 3 ⁢ k 2 ∂ ⁡ v ^ 1 ⁢ k 2 ∂ ⁡ x j ) d x }
= - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ ν ⁢ ∂ ⁡ ξ ^ 1 ⁢ k ∂ ⁡ x j + ∂ ⁡ v ^ 1 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ ξ ^ 1 ⁢ k ∂ ⁡ ν ) ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ ∇ ⁡ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⋅ ∇ ⁡ ξ ^ 1 ⁢ k ⁢ d ⁢ S
- γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ ν j ⁢ 𝑑 S - c k ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S
- μ 1 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 + v ^ 1 ⁢ k 2 ) ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ V ⁢ ( x ) ⁢ ν j ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 S
= : II 1 .

Similarly,

ε ¯ k 2 ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ξ ^ 2 ⁢ k d x + 1 D ^ k { β 12 ( k ) a ∗ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j - v ^ 1 ⁢ k 2 ∂ ⁡ v ^ 2 ⁢ k 2 ∂ ⁡ x j ) d x
+ β 23 ( k ) a ∗ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k 2 ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j - v ^ 3 ⁢ k 2 ∂ ⁡ v ^ 2 ⁢ k 2 ∂ ⁡ x j ) d x }
= - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ∂ ⁡ u ^ 2 ⁢ k ∂ ⁡ ν ⁢ ∂ ⁡ ξ ^ 2 ⁢ k ∂ ⁡ x j + ∂ ⁡ v ^ 2 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ ξ ^ 2 ⁢ k ∂ ⁡ ν ) ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ ∇ ⁡ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⋅ ∇ ⁡ ξ ^ 2 ⁢ k ⁢ d ⁢ S
- γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ⁢ ν j ⁢ 𝑑 S - c k ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S
- μ 2 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k 2 + v ^ 2 ⁢ k 2 ) ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ V ⁢ ( x ) ⁢ ν j ⁢ ξ ^ 2 ⁢ k ⁢ 𝑑 S
= : II 2

and

ε ¯ k 2 ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ξ ^ 3 ⁢ k d x + 1 D ^ k { β 13 ( k ) a ∗ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 ∂ ⁡ u ^ 3 ⁢ k 2 ∂ ⁡ x j - v ^ 1 ⁢ k 2 ∂ ⁡ v ^ 3 ⁢ k 2 ∂ ⁡ x j ) d x
+ β 23 ( k ) a ∗ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k 2 ∂ ⁡ u ^ 3 ⁢ k 2 ∂ ⁡ x j - v ^ 2 ⁢ k 2 ∂ ⁡ v ^ 3 ⁢ k 2 ∂ ⁡ x j ) d x }
= - 2 ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ∂ ⁡ u ^ 3 ⁢ k ∂ ⁡ ν ⁢ ∂ ⁡ ξ ^ 3 ⁢ k ∂ ⁡ x j + ∂ ⁡ v ^ 3 ⁢ k ∂ ⁡ x j ⁢ ∂ ⁡ ξ ^ 3 ⁢ k ∂ ⁡ ν ) ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ ∇ ⁡ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⋅ ∇ ⁡ ξ ^ 3 ⁢ k ⁢ d ⁢ S
- γ 1 ⁢ k ⁢ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ⁢ ν j ⁢ 𝑑 S - c k ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 3 ⁢ k 2 ⁢ ν j ⁢ 𝑑 S
- μ 3 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k 2 + v ^ 3 ⁢ k 2 ) ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ⁢ ν j ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ V ⁢ ( x ) ⁢ ν j ⁢ ξ ^ 3 ⁢ k ⁢ 𝑑 S
= : II 3 .

Since we have

∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j ⁢ 𝑑 x = - ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j ⁢ u ^ 2 ⁢ k 2 ⁢ 𝑑 x + ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ u ^ 1 ⁢ k 2 ⁢ u ^ 2 ⁢ k 2 ⁢ 𝑑 S ,

and then

1 D ^ k ⁢ { β 12 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k 2 ⁢ ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j - v ^ 2 ⁢ k 2 ⁢ ∂ ⁡ v ^ 1 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x + β 13 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k 2 ⁢ ∂ ⁡ u ^ 1 ⁢ k 2 ∂ ⁡ x j - v ^ 3 ⁢ k 2 ⁢ ∂ ⁡ v ^ 1 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x } + 1 D ^ k ⁢ { β 12 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j - v ^ 1 ⁢ k 2 ⁢ ∂ ⁡ v ^ 2 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x + β 23 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 3 ⁢ k 2 ⁢ ∂ ⁡ u ^ 2 ⁢ k 2 ∂ ⁡ x j - v ^ 3 ⁢ k 2 ⁢ ∂ ⁡ v ^ 2 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x } + 1 D ^ k ⁢ { β 13 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 ⁢ ∂ ⁡ u ^ 3 ⁢ k 2 ∂ ⁡ x j - v ^ 1 ⁢ k 2 ⁢ ∂ ⁡ v ^ 3 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x + β 23 ( k ) a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 2 ⁢ k 2 ⁢ ∂ ⁡ u ^ 3 ⁢ k 2 ∂ ⁡ x j - v ^ 2 ⁢ k 2 ⁢ ∂ ⁡ v ^ 3 ⁢ k 2 ∂ ⁡ x j ) ⁢ 𝑑 x } = β 12 ( k ) a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + v ^ 2 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 S + β 13 ( k ) a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ [ u ^ 3 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k + v ^ 1 ⁢ k 2 ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ] ⁢ 𝑑 S + β 23 ( k ) a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ν j ⁢ [ u ^ 2 ⁢ k 2 ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k + v ^ 3 ⁢ k 2 ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ] ⁢ 𝑑 S = : II 4 .

Hence

(5.39) ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ⁢ 𝑑 x = II 1 + II 2 + II 3 - II 4 .

By Lemma 5.28 and (5.31), by the exponential decay of ∇⁡u^i⁢k⁢(ε¯k⁢x+x2⁢k),∇⁡v^i⁢k⁢(ε¯k⁢x+x2⁢k) (i=1,2,3), we have

(5.40) II 1 + II 2 + II 3 - II 4 = o ⁢ ( e - C ⁢ δ ε ¯ k )

as k→∞. Moreover,

(5.41)

∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x
= ∫ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x j ⁢ [ u ¯ 1 ⁢ k ⁢ ( x - x 2 ⁢ k ε ¯ k ) + v ¯ 1 ⁢ k ⁢ ( x - x 2 ⁢ k ε ¯ k ) ] ⁢ ξ 1 ⁢ k ⁢ ( x - x 2 ⁢ k ε ¯ k ) ⁢ 𝑑 x
= ε ¯ k ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) ∂ ⁡ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ∂ ⁡ x j ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ⁢ 𝑑 x
= ε ¯ k ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) ∂ ⁡ V ⁢ ( ε ¯ k ⁢ ( x + x 2 ⁢ k - x 0 ε ¯ k ) + x 0 ) ∂ ⁡ x j ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ⁢ 𝑑 x
= ε ¯ k ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) ε ¯ k ⁢ ∂ ⁡ V ⁢ ( ε ¯ k ⁢ ( x + x 2 ⁢ k - x 0 ε ¯ k ) + x 0 ) ∂ ⁡ ε ¯ k ⁢ x j ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ⁢ 𝑑 x
= ε ¯ k ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) { ε ¯ k ⁢ ∂ ⁡ | ε ¯ k ⁢ ( x + x 2 ⁢ k - x 0 ε ¯ k ) | p ∂ ⁡ ε ¯ k ⁢ x j + ε ¯ k ⁢ R j ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k - x 0 ) } ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ⁢ 𝑑 x
= ε ¯ k 2 ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) { ε ¯ k p - 1 ⁢ ∂ ⁡ | x + x 2 ⁢ k - x 0 ε ¯ k | p ∂ ⁡ x j + R j ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k - x 0 ) } ⁢ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k ⁢ 𝑑 x .

It yields from (1.29), Lemma 5.1 and the boundedness of ξi⁢k (i=1,2,3) that

| ∫ B δ ε ¯ k ⁢ ( 0 ) R j ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k - x 0 ) ⁢ ( u ¯ i ⁢ k + v ¯ i ⁢ k ) ⁢ ξ i ⁢ k ⁢ 𝑑 x | ≤ C ⁢ ε ¯ k q .

On the other hand, we have

∫ B δ ε ¯ k c ⁢ ( 0 ) ∂ ⁡ | x + x 2 ⁢ k - x 0 ε ¯ k | p ∂ ⁡ x j ⁢ { ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k + ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k + ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k } ⁢ 𝑑 x = o ⁢ ( e - C ⁢ δ ε ¯ k ) .

It follows from (5.39), (5.40) and (5.41) that

(5.42) o ⁢ ( e - C ⁢ δ ε ¯ k ) = ∫ ℝ 2 ∂ ⁡ | x + x 2 ⁢ k - x 0 ε ¯ k | p ∂ ⁡ x j ⁢ { ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ⁢ ξ 1 ⁢ k + ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ⁢ ξ 2 ⁢ k + ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ⁢ ξ 3 ⁢ k } ⁢ 𝑑 x

Letting k→∞, we deduce from Lemma 5.1, equations (5.6) and (5.42) that

0 = 2 ⁢ ∫ ℝ 2 ∂ ⁡ | x | p ∂ ⁡ x j ⁢ ( U 1 ⁢ ξ 10 + U 2 ⁢ ξ 20 + U 3 ⁢ ξ 30 ) ⁢ 𝑑 x
= 2 ∫ ℝ 2 ( ∂ ⁡ | x | p ∂ ⁡ x j U 1 [ b 0 ( U 1 + x ⋅ ∇ U 1 ) + ∑ i = 1 2 b i ∂ ⁡ U 1 ∂ ⁡ x i ] + ∂ ⁡ | x | p ∂ ⁡ x j U 2 [ b 0 ( U 2 + x ⋅ ∇ U 2 ) + ∑ i = 1 2 b i ∂ ⁡ U 2 ∂ ⁡ x i ]
+ ∂ ⁡ | x | p ∂ ⁡ x j U 3 [ b 0 ( U 3 + x ⋅ ∇ U 3 ) + ∑ i = 1 2 b i ∂ ⁡ U 3 ∂ ⁡ x i ] ) d x
= 2 ⁢ b 0 ⁢ ∫ ℝ 2 ∂ ⁡ | x | p ∂ ⁡ x j ⁢ ( U 1 2 + U 2 2 + U 3 2 ) ⁢ 𝑑 x + b 0 ⁢ ∫ ℝ 2 ∂ ⁡ | x | p ∂ ⁡ x j ⁢ ( x ⋅ ∇ ⁡ U 1 2 + x ⋅ ∇ ⁡ U 2 2 + x ⋅ ∇ ⁡ U 3 2 ) ⁢ 𝑑 x
- ∑ i = 1 2 b i ⁢ ∫ ℝ 2 ∂ 2 ⁡ | x | p ∂ ⁡ x j ⁢ ∂ ⁡ x i ⁢ ( U 1 2 + U 2 2 + U 3 2 ) ⁢ 𝑑 x ,

which guarantees the Pohozaev-type identity (5.35).

Now we prove that b0=0 in (5.35). Multiplying the first equation of (5.22) by (x-x2⁢k)⋅∇⁡u^1⁢k, and integrating over Bδ⁢(x2⁢k), where δ>0 is small enough, then

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ] ⁢ Δ ⁢ u ^ 1 ⁢ k
  = ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( γ 1 ⁢ k - V ⁢ ( x ) ) ⁢ u ^ 1 ⁢ k ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ] ⁢ 𝑑 x + μ 1 a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 3 ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ] ⁢ 𝑑 x
(5.43)       + β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k 2 ] ⁢ 𝑑 x + β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k 2 ] ⁢ 𝑑 x .

Using argument similar to that in [15], we have

(5.44) - ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ] ⁢ Δ ⁢ u ^ 1 ⁢ k ⁢ 𝑑 x = - ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ⁢ d ⁢ S + ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ u ^ 1 ⁢ k | 2 ⁢ 𝑑 S ,
(5.45) ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) γ 1 ⁢ k ⁢ u ^ 1 ⁢ k ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ⁢ d ⁢ x = 1 2 ⁢ ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( x - x 2 ⁢ k ) ⋅ ν ⁢ u ^ 1 ⁢ k 2 ⁢ 𝑑 S - ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 u ^ 1 ⁢ k 2 ⁢ 𝑑 x + ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ 𝑑 x ,
(5.46) - ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) V ⁢ ( x ) ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ ( 1 2 ⁢ u ^ 1 ⁢ k 2 ) ⁡ d ⁢ x = - ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) V ⁢ ( x ) ⁢ 1 2 ⁢ u ^ 1 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) 1 2 ⁢ u ^ 1 ⁢ k 2 ⁢ [ ∇ ⁡ V ⋅ ( x - x 2 ⁢ k ) + 2 ⁢ V ⁢ ( x ) ] ⁢ 𝑑 x ,
(5.47) μ 1 a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) 1 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k 4 ⁢ d ⁢ x = μ 1 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( x - x 2 ⁢ k ) ⋅ ν ⁢ u ^ 1 ⁢ k 4 ⁢ 𝑑 S - μ 1 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 4 ⁢ 𝑑 x ,
(5.48) β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k 2 ⁢ d ⁢ x = β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ u ^ 2 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν - β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 2 ⁢ k 2 + 2 ⁢ u ^ 2 ⁢ k 2 ] ⁢ 𝑑 S ,
(5.49) β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k 2 ⁢ d ⁢ x = β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S - β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k 2 + 2 ⁢ u ^ 3 ⁢ k 2 ] ⁢ 𝑑 S .

So we can get

(5.50)

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ] ⁢ Δ ⁢ u ^ 1 ⁢ k = - ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 u ^ 1 ⁢ k 2 ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ 𝑑 x - μ 1 2 ⁢ a ∗ ⁢ ∫ ℝ 2 u ^ 1 ⁢ k 4 ⁢ 𝑑 x
- β 12 ( k ) a ∗ ⁢ ∫ ℝ 2 u ^ 1 ⁢ k 2 ⁢ u ^ 2 ⁢ k 2 ⁢ 𝑑 x - β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 2 ⁢ k 2 ⁢ d ⁢ x
- β 13 ( k ) a ∗ ⁢ ∫ ℝ 2 u ^ 1 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ 𝑑 x - β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k 2 ⁢ d ⁢ x + T 1 ,

where

(5.51)

T 1 := ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ 𝑑 x + μ 1 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 4 ⁢ 𝑑 x - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ d ⁢ x
+ ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ [ γ 1 ⁢ k - V ⁢ ( x ) ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + μ 1 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
+ β 12 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ u ^ 2 ⁢ k 2 ⁢ 𝑑 x + β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ u ^ 2 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
+ β 13 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ 𝑑 x + β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S .

Similarly, we have

(5.52)

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 2 ⁢ k ] ⁢ Δ ⁢ u ^ 2 ⁢ k = - ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 u ^ 2 ⁢ k 2 ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ 𝑑 x
- μ 2 2 ⁢ a ∗ ⁢ ∫ ℝ 2 u ^ 2 ⁢ k 4 ⁢ 𝑑 x + β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 2 ⁢ k 2 ⁢ d ⁢ x
- β 23 ( k ) a ∗ ⁢ ∫ ℝ 2 u ^ 2 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ 𝑑 x - β 23 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k 2 ⁢ d ⁢ x + T 2 ,

where

(5.53)

T 2 := ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ 𝑑 x + μ 2 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 4 ⁢ 𝑑 x - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ d ⁢ x
+ ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ [ γ 1 ⁢ k - V ⁢ ( x ) ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + μ 2 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
+ β 23 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ 𝑑 x + β 23 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ u ^ 3 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S .

We also have

(5.54)

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k ] ⁢ Δ ⁢ u ^ 3 ⁢ k = - ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 u ^ 3 ⁢ k 2 ⁢ 𝑑 x + ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ 𝑑 x
- μ 3 2 ⁢ a ∗ ⁢ ∫ ℝ 2 u ^ 3 ⁢ k 4 ⁢ 𝑑 x + β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k 2 ⁢ d ⁢ x
+ β 23 ( k ) 2 ⁢ a ∗ ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 2 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k 2 ⁢ d ⁢ x + T 3 ,

where

(5.55)

T 3 := ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ 𝑑 x + μ 3 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 4 ⁢ 𝑑 x - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ d ⁢ x
+ ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 2 ⁢ [ γ 1 ⁢ k - V ⁢ ( x ) ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + μ 3 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 3 ⁢ k 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S .

Moreover, it follows from (5.4) and (5.21) that

(5.56)

e k = γ 1 ⁢ k a ∗ ⁢ ε ¯ k 2 ∫ ℝ 2 ( u ^ 1 ⁢ k 2 + u ^ 2 ⁢ k 2 + u ^ 3 ⁢ k 2 ) d x + 1 ( a ∗ ) 2 ⁢ ε ¯ k 4 ∫ ℝ 2 ( μ 1 2 u ^ 1 ⁢ k 4 + μ 2 2 u ^ 2 ⁢ k 4 + μ 3 2 u ^ 3 ⁢ k 4
+ β 12 ( k ) u ^ 1 ⁢ k 2 u ^ 2 ⁢ k 2 + β 13 ( k ) u ^ 1 ⁢ k 2 u ^ 3 ⁢ k 2 + β 23 ( k ) u ^ 2 ⁢ k 2 u ^ 3 ⁢ k 2 ) d x .

From (5.43)–(5.56), we obtain

(5.57)

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 + u ^ 2 ⁢ k 2 + u ^ 3 ⁢ k 2 ) ⁢ ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ 𝑑 x + a ∗ ⁢ ε ¯ k 4 ⁢ e k
= T 1 + T 2 + T 3 + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 1 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 1 ⁢ k ⁢ d ⁢ S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ u ^ 1 ⁢ k | 2 ⁢ 𝑑 S
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 2 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 2 ⁢ k ⁢ d ⁢ S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ u ^ 2 ⁢ k | 2 ⁢ 𝑑 S
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ u ^ 3 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ u ^ 3 ⁢ k ⁢ d ⁢ S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ u ^ 3 ⁢ k | 2 ⁢ 𝑑 S .

Similarly, for v^1⁢k,v^2⁢k,v^3⁢k, we have

(5.58)

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( v ^ 1 ⁢ k 2 + v ^ 2 ⁢ k 2 + v ^ 3 ⁢ k 2 ) ⁢ ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ 𝑑 x + a ∗ ⁢ ε ¯ k 4 ⁢ e k
= T ¯ 1 + T ¯ 2 + T ¯ 3 + ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ v ^ 1 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ v ^ 1 ⁢ k ⁢ d ⁢ S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ v ^ 1 ⁢ k | 2 ⁢ 𝑑 S
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ v ^ 2 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ v ^ 2 ⁢ k ⁢ d ⁢ S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ v ^ 2 ⁢ k | 2 ⁢ 𝑑 S
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ∂ ⁡ v ^ 3 ⁢ k ∂ ⁡ ν ⁢ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ v ^ 3 ⁢ k ⁢ d ⁢ S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ | ∇ ⁡ v ^ 3 ⁢ k | 2 ⁢ 𝑑 S ,

where

T ¯ 1 := ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ 𝑑 x + μ 1 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 4 ⁢ 𝑑 x - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ d ⁢ x + ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ [ γ 2 ⁢ k - V ⁢ ( x ) ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + μ 1 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + β 12 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ v ^ 2 ⁢ k 2 ⁢ 𝑑 x + β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ v ^ 2 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + β 13 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ v ^ 3 ⁢ k 2 ⁢ 𝑑 x + β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ v ^ 3 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S ,
T ¯ 2 := ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 2 ⁢ 𝑑 x + μ 2 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 4 ⁢ 𝑑 x - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 2 ⁢ x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ d ⁢ x + ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 2 ⁢ [ γ 2 ⁢ k - V ⁢ ( x ) ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + μ 2 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + β 23 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 2 ⁢ v ^ 3 ⁢ k 2 ⁢ 𝑑 x + β 23 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 2 ⁢ k 2 ⁢ v ^ 3 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S ,

and

T ¯ 3 := ε ¯ k 2 ⁢ γ 2 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 3 ⁢ k 2 ⁢ 𝑑 x + μ 3 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 3 ⁢ k 4 ⁢ 𝑑 x - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) v ^ 3 ⁢ k 2 ⁢ x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ d ⁢ x
+ ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 3 ⁢ k 2 ⁢ [ γ 2 ⁢ k - V ⁢ ( x ) ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S + μ 3 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) v ^ 3 ⁢ k 4 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S .

By combining (5.57) and (5.58), it yields that

- ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ [ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k + ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ] ⁢ 𝑑 x
= T 1 - T ¯ 1 + T 2 - T ¯ 2 + T 3 - T ¯ 3 D ^ k
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ( ∇ ⁡ u ^ 1 ⁢ k ⋅ ν ) ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ ξ ^ 1 ⁢ k ] + ( ∇ ⁡ ξ ^ 1 ⁢ k ⋅ ν ) ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ v ^ 1 ⁢ k ] ) ⁢ 𝑑 S
- ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ ( ∇ ⁡ u ^ 1 ⁢ k + ∇ ⁡ v ^ 1 ⁢ k ) ⋅ ξ ^ 1 ⁢ k ⁢ 𝑑 S
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ( ∇ ⁡ u ^ 2 ⁢ k ⋅ ν ) ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ ξ ^ 2 ⁢ k ] + ( ∇ ⁡ ξ ^ 2 ⁢ k ⋅ ν ) ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ v ^ 2 ⁢ k ] ) ⁢ 𝑑 S
- ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ ( ∇ ⁡ u ^ 2 ⁢ k + ∇ ⁡ v ^ 2 ⁢ k ) ⋅ ξ ^ 2 ⁢ k ⁢ 𝑑 S
+ ε ¯ k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( ( ∇ ⁡ u ^ 3 ⁢ k ⋅ ν ) ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ ξ ^ 3 ⁢ k ] + ( ∇ ⁡ ξ ^ 3 ⁢ k ⋅ ν ) ⁢ [ ( x - x 2 ⁢ k ) ⋅ ∇ ⁡ v ^ 3 ⁢ k ] ) ⁢ 𝑑 S
- ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ ( x - x 2 ⁢ k ) ⋅ ν ] ⁢ ( ∇ ⁡ u ^ 3 ⁢ k + ∇ ⁡ v ^ 3 ⁢ k ) ⋅ ξ ^ 3 ⁢ k ⁢ 𝑑 S .

It can be obtained by Lemma 5.1 and equation (5.28) that

(5.59) - ε ¯ k 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ [ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k + ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ] ⁢ 𝑑 x = T 1 - T ¯ 1 + T 2 - T ¯ 2 + T 3 - T ¯ 3 D ^ k + o ⁢ ( e - C ⁢ δ ε ¯ k )   as  ⁢ k → ∞ .

By the same arguments, we get

T 1 - T ¯ 1 D ^ k = ε ¯ k 2 ⁢ γ 1 ⁢ k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x + c k ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) v ^ 1 ⁢ k 2 ⁢ 𝑑 x + μ 1 2 ⁢ a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) ( u ^ 1 ⁢ k 2 + v ^ 1 ⁢ k 2 ) ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x
- ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ ( x ) ) ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x + c k 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) u ^ 1 ⁢ k 2 ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
+ ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) γ 2 ⁢ k ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S - ε ¯ k 2 2 ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) V ⁢ ( x ) ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
+ μ 1 4 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) ( x - x 2 ⁢ k ) ⋅ ν ⁢ ( u ^ 1 ⁢ k 2 + v ^ 1 ⁢ k 2 ) ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 S
+ β 12 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + v ^ 2 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 x
+ β 12 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + v ^ 2 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
+ β 13 ( k ) a ∗ ⁢ ∫ ℝ 2 ∖ B δ ⁢ ( x 2 ⁢ k ) [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k + v ^ 3 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ 𝑑 x
+ β 13 ( k ) 2 ⁢ a ∗ ⁢ ∫ ∂ ⁡ B δ ⁢ ( x 2 ⁢ k ) [ u ^ 1 ⁢ k 2 ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k + v ^ 3 ⁢ k 2 ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ] ⁢ ( x - x 2 ⁢ k ) ⋅ ν ⁢ 𝑑 S
(5.60) = - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ ( x ) ) ⁢ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k ⁢ 𝑑 x + o ⁢ ( e - C ⁢ δ ε ¯ k ) ,

and

(5.61) T 2 - T ¯ 2 D ^ k = - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ ( x ) ) ⁢ ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k ⁢ 𝑑 x + o ⁢ ( e - C ⁢ δ ε ¯ k ) ,
(5.62) T 3 - T ¯ 3 D ^ k = - ε ¯ k 2 2 ⁢ ∫ B δ ⁢ ( x 2 ⁢ k ) ( x 2 ⁢ k ⋅ ∇ ⁡ V ⁢ ( x ) ) ⁢ ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ⁢ 𝑑 x + o ⁢ ( e - C ⁢ δ ε ¯ k ) .

Now, from (5.39), (5.40), (5.60), (5.61) and (5.62), we conclude that

T 1 - T ¯ 1 + T 2 - T ¯ 2 + T 3 - T ¯ 3 D ^ k = o ⁢ ( e - C ⁢ δ ε ¯ k )   as ⁢ k → ∞ .

Hence, with the help of (5.59),

(5.63)

o ⁢ ( e - C ⁢ δ ε ¯ k ) = ∫ B δ ⁢ ( x 2 ⁢ k ) ( V ⁢ ( x ) + x ⋅ ∇ ⁡ V 2 ) ⁢ [ ( u ^ 1 ⁢ k + v ^ 1 ⁢ k ) ⁢ ξ ^ 1 ⁢ k + ( u ^ 2 ⁢ k + v ^ 2 ⁢ k ) ⁢ ξ ^ 2 ⁢ k + ( u ^ 3 ⁢ k + v ^ 3 ⁢ k ) ⁢ ξ ^ 3 ⁢ k ] ⁢ 𝑑 x
= ε ¯ k 2 ∫ B δ ε ¯ k ⁢ ( 0 ) ( V ( ε ¯ k x + x 2 ⁢ k ) + ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⋅ ∇ ⁡ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) 2 ) ⋅ [ ( u ¯ 1 ⁢ k + v ¯ 1 ⁢ k ) ξ 1 ⁢ k
+ ( u ¯ 2 ⁢ k + v ¯ 2 ⁢ k ) ξ 2 ⁢ k + ( u ¯ 3 ⁢ k + v ¯ 3 ⁢ k ) ξ 3 ⁢ k ] d x .

By (V3), Lemma 5.1 and (5.6), we can deduce that

(5.64)

lim k → ∞ ⁡ ε ¯ k - p ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ( u ¯ i ⁢ k + v ¯ i ⁢ k ) ⁢ ξ i ⁢ k ⁢ 𝑑 x
= lim k → ∞ ⁡ ∫ B δ ε ¯ k ⁢ ( 0 ) V ⁢ ( ε ¯ k ⁢ ( x + x 2 ⁢ k - x 0 ε ¯ k ) + x 0 ) | ε ¯ k ⁢ ( x + x 2 ⁢ k - x 0 ε ¯ k ) | p ⁢ | x + x 2 ⁢ k - x 0 ε ¯ k | p ⁢ ( u ¯ i ⁢ k + v ¯ i ⁢ k ) ⁢ ξ i ⁢ k ⁢ 𝑑 x
= 2 ∫ ℝ 2 | x | p U i ξ i ⁢ 0 d x   ( i = 1 , 2 , 3 ) .

As x⋅∇⁡|x|p=p⁢|x|p, so

(5.65) lim k → ∞ ⁡ ε ¯ k - p 2 ⁢ ∫ B δ ε ¯ k ⁢ ( 0 ) ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⋅ ∇ ⁡ V ⁢ ( ε ¯ k ⁢ x + x 2 ⁢ k ) ⁢ ( u ¯ i ⁢ k + v ¯ i ⁢ k ) ⁢ ξ i ⁢ k ⁢ 𝑑 x = p ⁢ ∫ ℝ 2 | x | p ⁢ U i ⁢ ξ i ⁢ 0 ⁢ 𝑑 x , i = 1 , 2 , 3 .

From (5.63), (5.64) and (5.65), it yields that

∫ ℝ 2 | x | p U i ξ i ⁢ 0 d x = 0   ( i = 1 , 2 , 3 ) .

Together with (5.34), we can obtain

0 = 2 ⁢ ∫ ℝ 2 | x | p ⁢ U 1 ⁢ [ b 0 ⁢ ( U 1 + x ⋅ ∇ ⁡ U 1 ) + ∑ i = 1 2 b i ⁢ ∂ ⁡ U 1 ∂ ⁡ x i ] ⁢ 𝑑 x + 2 ⁢ ∫ ℝ 2 | x | p ⁢ U 2 ⁢ [ b 0 ⁢ ( U 2 + x ⋅ ∇ ⁡ U 2 ) + ∑ i = 1 2 b i ⁢ ∂ ⁡ U 2 ∂ ⁡ x i ] ⁢ 𝑑 x
+ 2 ⁢ ∫ ℝ 2 | x | p ⁢ U 3 ⁢ [ b 0 ⁢ ( U 3 + x ⋅ ∇ ⁡ U 3 ) + ∑ i = 1 2 b i ⁢ ∂ ⁡ U 3 ∂ ⁡ x i ] ⁢ 𝑑 x .

Then, from (1.17) and (5.6),

2 ⁢ ∫ ℝ 2 ( | x | p ⁢ U 1 ⁢ ∑ i = 1 2 b i ⁢ ∂ ⁡ U 1 ∂ ⁡ x i + | x | p ⁢ U 2 ⁢ ∑ i = 1 2 b i ⁢ ∂ ⁡ U 2 ∂ ⁡ x i + | x | p ⁢ U 3 ⁢ ∑ i = 1 2 b i ⁢ ∂ ⁡ U 3 ∂ ⁡ x i ) ⁢ 𝑑 x
= ∫ ℝ 2 | x | p ⁢ ( b 1 ⁢ ∂ ⁡ w 2 ∂ ⁡ x 1 + b 2 ⁢ ∂ ⁡ w 2 ∂ ⁡ x 2 ) ⁢ 𝑑 x = - b 1 ⁢ ∫ ℝ 2 w 2 ⁢ p ⁢ | x | p - 2 ⁢ x 1 ⁢ 𝑑 x - b 2 ⁢ ∫ ℝ 2 w 2 ⁢ p ⁢ | x | p - 2 ⁢ x 2 ⁢ 𝑑 x
= - ( b 1 , b 2 ) ⋅ ∇ ⁡ H ⁢ ( 𝟎 ) = 0 .

Therefore

0 = - ( b 1 , b 2 ) ⋅ ∇ ⁡ H ⁢ ( 𝟎 ) + 2 ⁢ b 0 ⁢ { H ⁢ ( 𝟎 ) + 1 2 ⁢ ∫ ℝ 2 | x | p ⁢ ( x ⋅ ∇ ⁡ U 1 2 + x ⋅ ∇ ⁡ U 2 2 + x ⋅ ∇ ⁡ U 3 2 ) ⁢ 𝑑 x } .

Since ∇⁡H⁢(𝟎)=0 and

∫ ℝ 2 | x | p ⁢ x ⋅ ∇ ⁡ U i 2 ⁢ d ⁢ x = - ∫ ℝ 2 U i 2 ⁢ [ p ⁢ | x | p + 2 ⁢ | x | p ] ⁢ 𝑑 x , i = 1 , 2 , 3 ,

we get

0 = - b 0 ⁢ p ⁢ ∫ ℝ 2 | x | p ⁢ ( U 1 2 + U 2 2 + U 3 2 ) ⁢ 𝑑 x = - b 0 ⁢ p ⁢ ∫ ℝ 2 | x | p ⁢ | w | 2 ⁢ 𝑑 x ,

which implies that b0=0. Putting it into (5.35), we have

0 = b 1 ⁢ ∫ ℝ 2 [ ( p - 2 ) ⁢ | x | p - 4 ⁢ x 1 2 + | x | p - 2 ] ⁢ w 2 ⁢ 𝑑 x + b 2 ⁢ ∫ ℝ 2 ( p - 2 ) ⁢ x 1 ⁢ x 2 ⁢ | x | p - 4 ⁢ w 2 ⁢ 𝑑 x ,
0 = b 1 ⁢ ∫ ℝ 2 ( p - 2 ) ⁢ x 1 ⁢ x 2 ⁢ | x | p - 4 ⁢ w 2 ⁢ 𝑑 x + b 2 ⁢ ∫ ℝ 2 [ ( p - 2 ) ⁢ | x | p - 4 ⁢ x 2 2 + | x | p - 2 ] ⁢ w 2 ⁢ 𝑑 x .

Due to

(5.66)

( ∫ ℝ 2 ( p - 2 ) ⁢ x 1 ⁢ x 2 ⁢ | x | p - 4 ⁢ w 2 ⁢ 𝑑 x ) 2 ≤ ∫ ℝ 2 ( p - 2 ) ⁢ x 1 2 ⁢ | x | p - 4 ⁢ w 2 ⁢ 𝑑 x ⁢ ∫ ℝ 2 ( p - 2 ) ⁢ x 2 2 ⁢ | x | p - 4 ⁢ w 2 ⁢ 𝑑 x
< ∫ ℝ 2 [ ( p - 2 ) ⁢ | x | p - 4 ⁢ x 1 2 + | x | p - 2 ] ⁢ w 2 ⁢ 𝑑 x ⁢ ∫ ℝ 2 [ ( p - 2 ) ⁢ | x | p - 4 ⁢ x 2 2 + | x | p - 2 ] ⁢ w 2 ⁢ 𝑑 x ,

we get b1=b2=0. Notice that (5.66) is equivalent to

| ∂ 2 ⁡ H ⁢ ( y ) ∂ ⁡ y 1 2 ∂ 2 ⁡ H ⁢ ( y ) ∂ ⁡ y 1 ⁢ ∂ ⁡ y 2 ∂ 2 ⁡ H ⁢ ( y ) ∂ ⁡ y 1 ⁢ ∂ ⁡ y 2 ∂ 2 ⁡ H ⁢ ( y ) ∂ ⁡ y 2 2 | ( 0 , 0 ) > 0 ,

i.e., (0,0) is a nondegenerate critical point of H⁢(y).

Since b0=b1=b2=0, we confirm that ξi⁢0=0, i=1,2,3. Let (xk,yk,zk) be the point which satisfies |ξ1⁢k⁢(xk)⁢ξ2⁢k⁢(yk)⁢ξ3⁢k⁢(zk)|=∥ξ1⁢k⁢ξ2⁢k⁢ξ3⁢k∥L∞⁢(ℝ2)=1. By use of the exponential decay presented by (5.1), applying the maximum principle, we know that xk,yk,zk are bounded uniformly in ℝ2. Thus, ξi⁢k→ξi⁢0≢0 uniformly in ℝ2 as k→∞, which contradicts to ξi⁢0=0 (i=1,2,3). This ends the proof. ∎

Communicated by Zhi-Qiang Wang

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12075102

Award Identifier / Grant number: 11971212

Award Identifier / Grant number: 11801519

Funding statement: This work is supported by NSFC under the Grant Nos. 12075102, 11971212 and 11801519.

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Received: 2021-02-18
Revised: 2021-04-21
Accepted: 2021-04-21
Published Online: 2021-05-18
Published in Print: 2021-08-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Frontmatter
  2. The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems
  3. Multiplicity and Concentration of Solutions for Kirchhoff Equations with Magnetic Field
  4. Anisotropic 𝑝-Laplacian Evolution of Fast Diffusion Type
  5. Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers
  6. Infinitely Many Solutions for the Nonlinear Schrödinger–Poisson System with Broken Symmetry
  7. Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2
  8. Periodic Solutions to Klein–Gordon Systems with Linear Couplings
  9. Existence and Uniqueness of Multi-Bump Solutions for Nonlinear Schrödinger–Poisson Systems
  10. Connecting and Closed Geodesics of a Kropina Metric
  11. Maximum Principles and ABP Estimates to Nonlocal Lane–Emden Systems and Some Consequences
  12. Existence of Ground States of Fractional Schrödinger Equations
https://doi.org/10.1515/ans-2021-2131
Keywords for this article
Ground State; 3-Component Bose–Einstein Condensate; Mass Concentration; Nondegeneracy; Uniqueness
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Articles in the same Issue

  1. Frontmatter
  2. The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems
  3. Multiplicity and Concentration of Solutions for Kirchhoff Equations with Magnetic Field
  4. Anisotropic 𝑝-Laplacian Evolution of Fast Diffusion Type
  5. Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers
  6. Infinitely Many Solutions for the Nonlinear Schrödinger–Poisson System with Broken Symmetry
  7. Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2
  8. Periodic Solutions to Klein–Gordon Systems with Linear Couplings
  9. Existence and Uniqueness of Multi-Bump Solutions for Nonlinear Schrödinger–Poisson Systems
  10. Connecting and Closed Geodesics of a Kropina Metric
  11. Maximum Principles and ABP Estimates to Nonlocal Lane–Emden Systems and Some Consequences
  12. Existence of Ground States of Fractional Schrödinger Equations
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