Segregated and synchronized vector solutions to linearly coupled systems of Schrödinger equations

Abstract

In this paper, we study the following linearly coupled system $-{\epsilon }^2\mathrm{\Delta }{u}_i+P_i\left(x\right)u_i=u_i^3+\sum _{j\ne i}^N{\lambda }_{ij}{u}_j,u_i\in H^1\left({\mathbb{R}}^3\right),i=1,\dots ,N$, where ε > 0 is a small parameter, P_i(x) are positive potentials, and λ_ij = λ_ji > 0 (i ≠ j) are coupling constants for i, j = 1, …, N. We investigate the effect of potentials to the structure of the solutions. More precisely, we construct multi-spikes solutions concentrating near the local maximum point $x_0^i$ of P_i(x). When $x_0^i=x_0^j$, $P_i\left(x_0^i\right)=P_j\left(x_0^j\right)=a,i\ne j,\hspace{0.17em}i,j=1,\dots ,N$, the components have spikes clustering at the same point as ε → 0⁺. When $x_0^i\ne x_0^j,\hspace{0.17em}i\ne j$, the components have spikes clustering at the different points as ε → 0⁺.

I. INTRODUCTION

This paper is concerned with the following nonlinear Schrödinger (NLS) system in ℝ³:

$${\displaystyle \{\begin{array}{l}-{\epsilon }^2\mathrm{\Delta }{u}_i+P_i\left(x\right)u_i=u_i^3+\sum _{i\ne j}^N{\lambda }_{ij}{u}_j,x\in {\mathbb{R}}^3,\hfill \\ u_i\in H^1\left({\mathbb{R}}^3\right),i=1,\dots ,N,\hfill \end{array}}$$ (1.1)

where ε > 0 is a small parameter, P_i(x) are positive potentials, and λ_ij are coupling constants satisfying λ_ij = λ_ji > 0 for i, j = 1, …, N. This type of system arises when one considers stationary pulselike (standing wave) solutions of the time-dependent N-coupled Schrödinger system of the form

$${\displaystyle \{\begin{array}{l}-i\mathit{ħ}\frac{\partial {\psi }_j}{\partial t}-\frac{{\mathit{ħ}}^2}{2}\mathrm{\Delta }{\psi }_j+P_j\left(x\right){\psi }_j={\left|{\psi }_j\right|}^2{\psi }_j+\sum _{k\ne j}^N{\lambda }_{jk}{\psi }_k,x\in {\mathbb{R}}^3,t>0,\hfill \\ {\psi }_j={\psi }_j(x,t)\in \mathbb{C},j=1,\dots ,N,\hfill \end{array}}$$ (1.2)

where i denotes the imaginary unit and ħ is the Plank constant. In fact, system (1.2) appears in nonlinear optics (cf. Ref. 1). For example, the propagation of optical pulses in nonlinear N-core directional coupler can be described by N linearly coupled nonlinear Schrödinger system (1.2). Here, ψ_j (j = 1, 2…, N) are envelope functions and λ_jk, which is the normalized coupling coefficient between the cores, is equal to the linear coupling coefficient times the dispersion length. Generally speaking, for system (1.2) with constant potentials, the sign of λ_jk determines whether the interactions of fiber couplers are repulsive or attractive. In the attractive case, the components of a vector solution tend to go along with each other leading to synchronization, and in the repulsive case, the components tend to segregate with each other leading to phase separations. These phenomena have been documented in numeric simulations (e.g., Ref. 1 and references therein) and in mathematical proofs (e.g., Ref. 18 and references therein). However, if the potentials are not constants, what will happen? That is the main motivation of this paper. In the rest of this paper, we will investigate whether the interactions of fiber couplers are repulsive or attractive in the case of the potentials being not constants.

In particular for N = 1, problem (1.1) is the following classic Schrödinger equation

$${\displaystyle -{\epsilon }^2\mathrm{\Delta }u+P\left(x\right)u=u^p,\hspace{0.17em}x\in {\mathbb{R}}^n,u>0,}$$ (1.3)

where $1<p<\frac{n+2}{n-2}$. The Equation (1.3) arises in the study of solitary waves in nonlinear equations of the Klein-Gordon or Schrödinger type and has been under extensive studies in recent years (see, e.g., Refs. 8–10, 16, 17, 25, 27, 33, and 34 and references therein).

Recently, the system of Schrödinger equations has been broadly investigated in many aspects, such as the existence and qualitative properties of solutions, asymptotic behavior, concentration, and multi-bump phenomena for semiclassical states. For results in this direction, the readers can refer Refs. 3, 4, 14, 15, 17, 20–22, 26, and 30 and the references therein. Mathematical work on systems with the nonlinearly coupling terms (e.g., the term $\sum _{i\ne j}{u}_i$ in (1.1) being replaced by $\sum _{i\ne j}^N\underset{i}{\overset{2}{u}}{u}_j$) has been studied extensively in recent years, for example, Refs. 6, 7, 12, 15, 19, 23, 24, 28, 29, 31, 32, and 35 and references therein, where phase separation or synchronization vector solutions have been proved in several cases. From these papers, we know that the existence of multi-bump solutions requires the presence of a suitable potential depending upon x which breaks the symmetry invariance of the autonomous NLS.

However, as far as we know, it seems that there are few results for linearly coupled system (1.1). In Ref. 18, for ε fixed and sufficiently small λ_ij = λ(i ≠ j) > 0, Lin and Peng construct non-radial vector solutions of segregated type with two components having exactly l positive bumps, where l is a positive integer. In Ref. 13, Chen and Zou get positive solutions of the coupled system for sufficiently small ε > 0, which have concentration phenomenon as ε → 0. In Ref. 2, by using concentration compactness principle, Ambrosetti, Cerami, and Ruiz studied the existence of both positive ground and bound states under some decaying assumptions of the potentials at infinity for N = 2. In Ref. 5, Ambrosetti et al. showed that there exists a solution with one multi-bump component having bumps located near some points, while the other component has one negative peak when N = 2.

The main purpose of this paper is to investigate how potentials P_i(x) and coupling constants λ_ij influence the structure of solutions to Equation (1.1). Under some assumptions on λ_ij and P_i(x), we construct multi-peak positive vector solutions for both the segregate case and the synchronize case for problem (1.1).

More precisely, we construct multi-spike solutions concentrating near the local maximum point $x_0^i$ of P_i(x). When $x_0^i=x_0^j$, $P_i\left(x_0^i\right)=P_j\left(x_0^j\right)=a,i\ne j,\hspace{0.17em}i,j=1,\dots ,N$, the components have spikes clustering at the same point as ε → 0⁺. When $x_0^i\ne x_0^j,\hspace{0.17em}i\ne j$, the components have spikes clustering at the different points as ε → 0⁺.

As for i = 1, …, N, we assume that P_i(x) satisfy the following hypotheses.

(H₀): P_i(x) ∈ C¹(ℝ³, ℝ) and there exist positive constants b_i such that $\underset{{\mathbb{R}}^N}{inf}{P}_i\left(x\right)\ge b_i$ for all x ∈ ℝ³.

(H₁): Suppose that A₁, …, A_h ⊂ {1, …, N} disjoint mutually such that $\bigcup _{s=1}^h{A}_s=\{1,\dots ,N\}$. For each A_s(s = 1, …, h), there exist δ_s > 0 and $x_0^s\in {\mathbb{R}}^3$ such that $P_j\left(x\right)<P_j\left(x_0^s\right)$ for $x\in B_{{\delta }_s}\left(x_0^s\right)\setminus \left\{x_0^s\right\}$ and j ∈ A_s. For every s, we denote $P_{k_1}\left(x_0^s\right)=P_{k_2}\left(x_0^s\right)=a_s$ for k₁, k₂ ∈ A_s. Moreover, for every k₁ ∈ A_s1,  k₂ ∈ A_s2 (s₁ ≠ s₂), we assume $x_0^{s_1}\ne x_0^{s_2}$.

(H₂): There exist positive constants L_i and θ_i such that $\left|P_i\left(x\right)-P_i\left(y\right)\right|\le L_i{\left|x-y\right|}^{{\theta }_i}$ for all x, y ∈ ℝ³, θ_i > 0.

Our main result is as follows.

Theorem 1.1.

Suppose that (H₀), (H₁), and (H₂) hold. There exists ε₀ > 0 such that for any ε ∈ (0, ε₀), problem (1.1) has a positive solution (u_1ε, …, u_Nε), with u_jε (for every j ∈ {1, …, N}) having multiple spikes clustering near $x_0^s(s=1,\dots ,h)$ provided that

• (1)${\lambda }_{\left|A_s\right|-1}^{\ast }<{\lambda }_{ij}≔\lambda <\frac{\underset{i\in A_s}{min}{b}_i}{\left|A_s\right|-1}$, if   i, j belong to the same A_s,
• (2)λ_ij ∼ ε^α, if  i, j do not belong to the same A_s,

where ${\lambda }_{\left|A_s\right|-1}^{\ast }$, α are constants, $\alpha \ge \frac{\underset{i}{min}\{{\theta }_i,1\}}{2}$, and $\left|A_s\right|$ denotes the number of the set A_s’s elements.

In particular, when h = 1, we rewrite Theorem 1.1 in the following form.

Theorem 1.2.

Suppose that (H₀), (H₁), and (H₂) hold and h = 1. Then there is λ^∗ > 0 such that ${\lambda }_{N-1}^{\ast }<{\lambda }_{ij}≔\lambda <\frac{\underset{i\in \{1,\dots ,N\}}{min}{b}_i}{N-1}(i\ne j,\hspace{0.17em}i,j=1,\dots ,N)$. For any positive integer k ∈ ℤ⁺, problem (1.1) has a positive solution (u_1ε, …, u_Nε), with k spikes concentrating near x₀ and (u_1ε, …, u_Nε) is a synchronized vector solution for ε sufficient small.

Let us outline the main idea in the proof of Theorem 1.1. Without loss of generality, we only consider the case N = 3, i.e., problem (1.1) can be written as

$${\displaystyle \{\begin{array}{l}-{\epsilon }^2\mathrm{\Delta }u+P_1\left(x\right)u=u^3+{\lambda }_{1}v+{\lambda }_{2}w,x\in {\mathbb{R}}^3,\hfill \\ -{\epsilon }^2\mathrm{\Delta }v+P_2\left(x\right)v=v^3+{\lambda }_{1}u+{\lambda }_{3}w,x\in {\mathbb{R}}^3,\hfill \\ -{\epsilon }^2\mathrm{\Delta }w+P_3\left(x\right)w=w^3+{\lambda }_{2}u+{\lambda }_{3}v,x\in {\mathbb{R}}^3,\hfill \end{array}}$$ (1.4)

where λ₁ > 0,  λ₂ > 0,  λ₃ > 0 are coupling constants. Suppose that

$${\displaystyle \underset{B_{\delta }\left(x_0^1\right)\setminus \left\{x_0^1\right\}}{max}{P}_i\left(x\right)=P_i\left(x_0^1\right)=a_1,i=1,2}$$

and

$${\displaystyle \underset{B_{\delta }\left(x_0^2\right)\setminus \left\{x_0^2\right\}}{max}{P}_3\left(x\right)=P_3\left(x_0^2\right)=a_2,x_0^1\ne x_0^2.}$$

We denote θ₁ = min{θ₁, θ₂, θ₃} .

Hereafter,

$${\displaystyle H=\left\{(u,v,w)\in H^1\left({\mathbb{R}}^3\right)\times H^1\left({\mathbb{R}}^3\right)\times H^1\left({\mathbb{R}}^3\right):\right.{\int }_{{\mathbb{R}}^3}{P}_1\left(x\right)u^2<\infty ,\hspace{0.17em}{\int }_{{\mathbb{R}}^3}{P}_2\left(x\right)v^2<\infty ,}$$

$${\displaystyle \left.{\int }_{{\mathbb{R}}^3}{P}_3\left(x\right)w^2<\infty \right\}}$$

endowed with following norm:

$${\displaystyle {‖(u,v,w)‖}^2=〈(u,v,w),(u,v,w)〉={‖u‖}_{\epsilon ,P_1}^2+{‖v‖}_{\epsilon ,P_2}^2+{‖w‖}_{\epsilon ,P_3}^2,\forall (u,v,w)\in H,}$$

where

$${\displaystyle {‖u‖}_{\epsilon ,P_i}^2={〈u,u〉}_{\epsilon ,P_i}={\int }_{{\mathbb{R}}^3}\left({\epsilon }^2{\left|\nabla u\right|}^2+P_i\left(x\right)u^2\right),i=1,2,3.}$$

Let W_ai denote the solution of the equation

$${\displaystyle \{\begin{array}{l}-\mathrm{\Delta }w+a_{i}w=w^3,\hspace{0.17em}w>0,\hspace{0.17em}x\in {\mathbb{R}}^3,\hfill \\ w\left(0\right)=\underset{x\in {\mathbb{R}}^3}{max}w\left(x\right),\hspace{0.17em}\hspace{0.17em}w\left(x\right)\in H^1\left({\mathbb{R}}^3\right).\hfill \end{array}}$$ (1.5)

Particularly, we set W = W₁. It is well-known that $W\left(x\right)=W\left(\left|x\right|\right)$ is nondegeneracy and satisfies

$${\displaystyle W^{\prime }\left(r\right)<0,\hspace{0.17em}\hspace{0.17em}\underset{r\to \infty }{lim}{r}^{\frac{N-1}{2}}{e}^{r}W\left(r\right)=C>0,\hspace{0.17em}\hspace{0.17em}\underset{r\to \infty }{lim}\frac{W^{\prime }\left(r\right)}{W\left(r\right)}=-1.}$$

We will use (U, V, W_a2) = (W_a1−λ1, W_a1−λ1, W_a2) to build up the approximate solutions for (1.4), where (U, V) = (W_a1−λ1, W_a1−λ1) is the solution of following equation:

$${\displaystyle \{\begin{array}{l}-\mathrm{\Delta }u+a_{1}u=u^3+{\lambda }_{1}v,x\in {\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }v+a_{1}v=v^3+{\lambda }_{1}u,x\in {\mathbb{R}}^3.\hfill \end{array}}$$ (1.6)

We define

$${\displaystyle D_{\epsilon ,\delta }^{k,y}=\left\{x=(y^1,\dots ,y^k)\in {\left({\mathbb{R}}^3\right)}^k:y^j\in B_{\delta }\left(x_0^1\right),\hspace{0.17em}\frac{\left|y^i-y^j\right|}{\epsilon }\ge {\left|ln\epsilon \right|}^{\frac{1}{2}},\hspace{0.17em}i\ne j,\hspace{0.17em}\hspace{0.17em}i,j=1,\dots ,k\right\}}$$

and

$${\displaystyle D_{\epsilon ,\delta }^{m,z}=\left\{z=(z^1,\dots ,z^m)\in {\left({\mathbb{R}}^3\right)}^m:z^j\in B_{\delta }\left(x_0^2\right),\hspace{0.17em}\frac{\left|z^i-z^j\right|}{\epsilon }\ge {\left|ln\epsilon \right|}^{\frac{1}{2}},\hspace{0.17em}i\ne j,\hspace{0.17em}\hspace{0.17em}i,j=1,\dots ,m\right\}.}$$

Fix $y=(y^1,\dots ,y^k)\in D_{\epsilon ,\delta }^{k,y}$,  $z=(z^1,\dots ,z^m)\in D_{\epsilon ,\delta }^{m,z}$,  we set

$${\displaystyle U_{\epsilon ,y}=\sum _{i=1}^k{U}_{\epsilon ,y^i}\left(x\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{V}_{\epsilon ,y}=\sum _{i=1}^k{V}_{\epsilon ,y^i}\left(x\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{W}_{\epsilon ,z}=\sum _{i=1}^m{W}_{\epsilon ,z^i}\left(x\right),}$$

where

$${\displaystyle U_{\epsilon ,y^i}=W_{a_1-{\lambda }_1}\left(\frac{x-y^i}{\epsilon }\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{V}_{\epsilon ,y^i}=W_{a_1-{\lambda }_1}\left(\frac{x-y^i}{\epsilon }\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{W}_{\epsilon ,z^i}=W_{a_2}\left(\frac{x-z^i}{\epsilon }\right).}$$

We write

$${\displaystyle E_{k,m}^{y,z}=\left\{(\phi ,\psi ,\varphi )\in H:{〈\left(\frac{\partial U_{\epsilon ,y^i}}{\partial y_l^i},\frac{\partial V_{\epsilon ,y_l^i}}{\partial y_l^i}\right),\left(\phi ,\psi \right)〉}_{P_1{P}_2}=0,{〈\frac{\partial W_{\epsilon ,z^j}}{\partial z_l^j},\varphi 〉}_{\epsilon ,P_3}=0,\right.}$$

$${\displaystyle \left.i=1,\dots ,k,\hspace{0.17em}j=1,\dots ,m,\hspace{0.17em}l=1,2,3\right\}}$$

and

$${\displaystyle M_{k,m}^{y,z}=\left\{(y^1,\dots ,y^k,z^1,\dots ,z^m,\phi ,\psi ,\varphi ):(y^1,\dots ,y^k)\in D_{\epsilon ,\delta }^{k,y},(z^1,\dots ,z^m)\in D_{\epsilon ,\delta }^{m,z},\right.}$$

$${\displaystyle \left.(\phi ,\psi ,\varphi )\in E_{k,m}^{y,z}\right\}.}$$

Here,

$${\displaystyle {〈\left(\frac{\partial U_{\epsilon ,y^i}}{\partial y_l^i},\frac{\partial V_{\epsilon ,y^i}}{\partial y_l^i}\right),\left(\phi ,\psi \right)〉}_{P_1{P}_2}={〈\frac{\partial U_{\epsilon ,y^i}}{\partial y_l^i},\phi 〉}_{\epsilon ,P_1}+{〈\frac{\partial V_{\epsilon ,y_l^i}}{\partial y_l^i},\psi 〉}_{\epsilon ,P_2}.}$$

It is easy to see that Theorem 1.1 is a direct consequence of the following result in the case of N = 3.

Theorem 1.3.

Under the assumptions of Theorem 1.1, λ^∗ < λ₁ < a₁, λ₂, λ₃ ∼ ε^α, $\alpha \ge \frac{min\{{\theta }_1,1\}}{2}$, for ε sufficiently small, problem (1.4) has a solution of the form

$${\displaystyle (u_{\epsilon },v_{\epsilon },w_{\epsilon })=(U_{\epsilon ,y}\left(x\right)+{\phi }_{\epsilon },\hspace{0.17em}{V}_{\epsilon ,y}\left(x\right)+{\psi }_{\epsilon },W_{\epsilon ,z}\left(x\right)+{\varphi }_{\epsilon }),}$$

where $({\phi }_{\epsilon },{\psi }_{\epsilon },{\varphi }_{\epsilon })\in E_{k,m}^{y,z}$.

Remark 1.4.

In Theorem (1.3), taking w_ε = 0. In a similar way, we can obtain that (u_ε, v_ε) = (U_ε,y(x) + φ_ε,  V_ε,y(x) + ψ_ε) is a synchronized vector solution of Equation (1.1) for N = 2. Similarly, Theorem (1.2) is proved.

To prove Theorem 1.3, we will use the well-known Lyapunov-Schmidt reduction in dealing with the singularly perturbed elliptic problems. For Lyapunov-Schmidt reduction, one needs to use the nondegeneracy of solution of (2.1). So, we will adopt the idea of Ambrosetti, Cerami, and Ruiz in Ref. 2 to research the nondegeneracy of the solutions for the l-linearly coupled Schrödinger equations.

The main difficulty in constructing solutions to the linearly coupled system is to get a better control of the error terms. Compared with non-linearly coupled system, the linear term ∑u_j ’s attenuation speed is not enough quick. To overcome this difficulty, we need to adjust the parameter λ_ij to increase the speed of attenuation.

Our paper is organized as follows. In Section II, we will give nondegeneracy of the solution. In Sec. III, we will establish some preliminary estimate. We will carry out a reduction procedure and study the reduced one dimensional problem to prove Theorem 1.3 in Sec. IV. In the Appendix, an energy expansion for the functional corresponding to problem (1.1) will be established.

II. NONDEGENERACY OF THE SOLUTION

First, we will introduce the nondegeneracy of the solutions (u₁, …, u_l) for the l ≥ 2 coupled Schrödinger equations,

$${\displaystyle \{\begin{array}{l}-\mathrm{\Delta }{u}_i+a{u}_i=u_i^3+\lambda \sum _{i\ne j}^l{u}_j,x\in {\mathbb{R}}^3,\hfill \\ u_i\in H^1\left({\mathbb{R}}^3\right),i=1,\dots ,l,\hfill \end{array}}$$ (2.1)

where u₁(x) = ⋯ = u_l(x) = W_{a−(l−1) λ}(x).

Proposition 2.1.

If $\lambda \in ({\lambda }_{l-1}^{\ast },\frac{a}{l-1})$, where ${\lambda }_{l-1}^{\ast }$ is a positive constant, (u₁, …, u_l) is nondegeneracy for system (2.1) in (H¹(ℝ³))^l in the sense that the kernel is given by span $\left\{(\frac{\partial W}{\partial x_i},\dots ,\frac{\partial W}{\partial x_i})|i=1,2,3\right\}$.

Proof.

Following from Ref. 2, let us denote

$${\displaystyle Z_{\lambda }≕\left\{\left(W_{a-(l-1)\lambda }(x-\xi ),\dots ,W_{a-(l-1)\lambda }(x-\xi )\right):\xi \in {\mathbb{R}}^3\right\}}$$

and

$${\displaystyle F_{\lambda }(u_1,\dots ,u_l)≕(-\mathrm{\Delta }{u}_1+a{u}_1-u_1^3-\lambda \sum _{j\ne 1}{u}_j,\dots ,-\mathrm{\Delta }{u}_i+a{u}_i-u_i^3-\lambda \sum _{j\ne i}{u}_j,\dots ,-\mathrm{\Delta }{u}_l+a{u}_l-u_l^3-\lambda \sum _{j\ne l}{u}_j).}$$ (2.2)

We may take ξ = 0. It is easy to compute

$${\displaystyle T_{(W_{a-(l-1)\lambda },\dots ,W_{a-(l-1)\lambda })}Z=span\left\{(\frac{\partial W}{\partial x_i},\dots ,\frac{\partial W}{\partial x_i}),\hspace{0.17em}\hspace{0.17em}i=1,2,3\right\}.}$$

On the other hand,

$${\displaystyle F_{\lambda }^{\prime }(W_{a-(l-1)\lambda },\dots ,W_{a-(l-1)\lambda })({\varphi }_1,\dots ,{\varphi }_l)=\left(-\mathrm{\Delta }{\varphi }_1+a{\varphi }_1-3W_{a-(l-1)\lambda }^2{\varphi }_1-\lambda \sum _{j\ne 1}{\varphi }_j\right.}$$

$${\displaystyle \cdots }$$

$${\displaystyle -\mathrm{\Delta }{\varphi }_i+a{\varphi }_i-3W_{a-(l-1)\lambda }^2{\varphi }_i-\lambda \sum _{j\ne i}{\varphi }_j}$$

$${\displaystyle \cdots }$$

$${\displaystyle \left.-\mathrm{\Delta }{\varphi }_l+a{\varphi }_l-3W_{a-(l-1)\lambda }^2{\varphi }_l-\lambda \sum _{j\ne l}{\varphi }_j\right).}$$ (2.3)

Letting (ϕ₁, …, ϕ_l) belong to its kernel, by the subtraction of the l equations with each other, we get

$${\displaystyle -\mathrm{\Delta }({\varphi }_i-{\varphi }_j)+a({\varphi }_i-{\varphi }_j)+(\lambda -3W_{a-(l-1)\lambda }^2)({\varphi }_i-{\varphi }_j)=0,\hspace{0.17em}\hspace{0.17em}(i\ne j,\hspace{0.17em}i,j=1,\dots ,l).}$$

Taking into account the choice of λ, we have $\lambda -3W_{a-(l-1)\lambda }^2\ge 0$. Hence, ϕ_i − ϕ_j = 0. Moreover, ϕ_j satisfies the equation

$${\displaystyle -\mathrm{\Delta }{\varphi }_j+(a-(l-1)\lambda ){\varphi }_j-3W_{a-(l-1)\lambda }^2{\varphi }_j=0\hspace{0.17em}\hspace{0.17em}(j=1,\dots ,l).}$$

We conclude that ϕ_j ∈ span $\left\{\frac{\partial W}{\partial x_i},i=1,2,3\right\}$. □

III. SOME PRELIMINARIES

Let

$${\displaystyle I_{\epsilon }(u,v,w)≔\frac{1}{2}{\int }_{{\mathbb{R}}^3}\left({\epsilon }^2{\left|\nabla u\right|}^2+P_1\left(x\right)u^2+{\epsilon }^2{\left|\nabla v\right|}^2+P_2\left(x\right)v^2+{\epsilon }^2{\left|\nabla w\right|}^2+P_3\left(x\right)w^2\right)}$$

$${\displaystyle -\frac{1}{4}{\int }_{{\mathbb{R}}^3}\left(u^4+v^4+w^4\right)-{\lambda }_1{\int }_{{\mathbb{R}}^3}uv-{\lambda }_2{\int }_{{\mathbb{R}}^3}uw-{\lambda }_3{\int }_{{\mathbb{R}}^3}vw}$$

and

$${\displaystyle J_{\epsilon }(\phi ,\psi ,\varphi )=I_{\epsilon }(U_{\epsilon ,y}+\phi ,V_{\epsilon ,y}+\psi ,W_{\epsilon ,z}+\varphi ),\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}(\phi ,\psi ,\varphi )\in E_{k,m}^{y,z}.}$$

Expand J_ε(φ, ψ, ϕ) as follows:

$${\displaystyle J_{\epsilon }(\phi ,\psi ,\varphi )=J_{\epsilon }(0,0,0)+{\ell }_{\epsilon }(\phi ,\psi ,\varphi )+\frac{1}{2}〈L_{\epsilon }(\phi ,\psi ,\varphi ),(\phi ,\psi ,\varphi )〉+R_{\epsilon }(\phi ,\psi ,\varphi ),}$$ (3.1)

where

$${\displaystyle {\ell }_{\epsilon }(\phi ,\psi ,\varphi )=\left.{\int }_{{\mathbb{R}}^3}\left(\right.P_1\left(x\right)-a_1\left.\right)U_{\epsilon ,y}\phi +(P_2\left(x\right)-a_1)V_{\epsilon ,y}\psi +(P_3\left(x\right)-a_2)W_{\epsilon ,z}\varphi \right)}$$

$${\displaystyle \left.+{\int }_{{\mathbb{R}}^3}\left(\right.\sum _{i=1}^k{U}_{\epsilon ,y^i}^3-U_{\epsilon ,y}^3\left.\right)\phi +(\sum _{i=1}^k{V}_{\epsilon ,y^i}^3-V_{\epsilon ,y}^3)\psi +(\sum _{i=1}^m{W}_{\epsilon ,z^i}^3-W_{\epsilon ,z}^3)\varphi \right)}$$

$${\displaystyle -{\lambda }_2{\int }_{{\mathbb{R}}^3}(U_{\epsilon ,y}\varphi +W_{\epsilon ,z}\phi )-{\lambda }_3{\int }_{{\mathbb{R}}^3}(V_{\epsilon ,y}\varphi +W_{\epsilon ,z}\psi )}$$ (3.2)

and

$${\displaystyle L_{\epsilon }(\phi ,\psi ,\varphi )={\int }_{{\mathbb{R}}^3}\left({\epsilon }^2{\left|\nabla \phi \right|}^2+P_1\left(x\right){\phi }^2+{\epsilon }^2{\left|\nabla \psi \right|}^2+P_2\left(x\right){\psi }^2+{\epsilon }^2{\left|\nabla \varphi \right|}^2+P_3\left(x\right){\varphi }^2\right)}$$

$${\displaystyle -{\int }_{{\mathbb{R}}^3}\left(3U_{\epsilon ,y}^2{\phi }^2+3V_{\epsilon ,y}^2{\psi }^2+3W_{\epsilon ,z}^2{\varphi }^2\right)-2{\lambda }_1{\int }_{{\mathbb{R}}^3}\phi \psi }$$

$${\displaystyle -\hspace{0.17em}2{\lambda }_2{\int }_{{\mathbb{R}}^3}\phi \varphi -2{\lambda }_3{\int }_{{\mathbb{R}}^3}\psi \varphi .}$$ (3.3)

In order to find a critical point for J_ε(φ, ψ, ϕ), we need to discuss each term in expansion (3.1).

Lemma 3.1.

There exist a constant C > 0 independent of ε such that

$${\displaystyle R_{\epsilon }^{\left(i\right)}(\phi ,\psi ,\varphi )\le C({\epsilon }^{-\frac{3}{2}}{‖(\phi ,\psi ,\varphi )‖}^{3-i}+{\epsilon }^{-3}{‖(\phi ,\psi ,\varphi )‖}^{4-i}),i=0,1,2.}$$

Proof.

By (3.1)–(3.3), we know that

$${\displaystyle \left|R(\phi ,\psi ,\varphi )\right|\le \left|{\int }_{{\mathbb{R}}^3}\sum _{i=1}^k(U_{\epsilon ,y^i}{\phi }^3+V_{\epsilon ,y^i}{\psi }^3)+\sum _{i=1}^m{W}_{\epsilon ,z^i}{\varphi }^3+\frac{1}{4}{\phi }^4+\frac{1}{4}{\psi }^4+\frac{1}{4}{\varphi }^4\right|.}$$

Let $\tilde{\phi }\left(x\right)=\phi \left(\epsilon x\right)$. Since $\underset{{\mathbb{R}}^3}{inf}{P}_i\left(x\right)\ge b_i>0$, we see

$${\displaystyle {\int }_{{\mathbb{R}}^3}{\left|\phi \right|}^{4}dx={\epsilon }^3{\int }_{{\mathbb{R}}^3}{\left|\tilde{\phi }\right|}^{4}dx\le C{\epsilon }^3\left(\right.{\int }_{{\mathbb{R}}^3}{\left|\nabla \tilde{\phi }\right|}^2+{\left|\tilde{\phi }\right|}^2{\left.\right)}^2}$$

$${\displaystyle =C{\epsilon }^3\left(\right.{\epsilon }^{-3}{\int }_{{\mathbb{R}}^3}({\epsilon }^2{\left|\nabla \phi \right|}^2+P_1\left(x\right){\left|\phi \right|}^2)dx{\left.\right)}^2}$$

$${\displaystyle \le C{\epsilon }^{-3}{‖\phi ‖}_{\epsilon ,P_1}^4.}$$ (3.4)

Similar to (3.4), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^3}{\left|\psi \right|}^{4}dx\le C{\epsilon }^{-3}{‖\psi ‖}_{\epsilon ,P_2}^4,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\int }_{{\mathbb{R}}^3}{\left|\varphi \right|}^{4}dx\le C{\epsilon }^{-3}{‖\varphi ‖}_{\epsilon ,P_3}^4.}$$ (3.5)

Using the Hölder inequality, we get

$${\displaystyle {\int }_{{\mathbb{R}}^3}\sum _{i=1}^k{U}_{\epsilon ,y^i}{\phi }^3\le \sum _{i=1}^k{\left({\int }_{{\mathbb{R}}^3}{U}_{\epsilon ,y^i}^4\right)}^{\frac{1}{4}}{\left({\int }_{{\mathbb{R}}^3}{\phi }^4\right)}^{\frac{3}{4}}}$$

$${\displaystyle \le C\sum _{i=1}^k{\left({\epsilon }^3{\int }_{{\mathbb{R}}^3}{W}^4\right)}^{\frac{1}{4}}{\left({\epsilon }^{-3}{‖\phi ‖}_{\epsilon ,P_1}^4\right)}^{\frac{3}{4}}}$$

$${\displaystyle \le C{\epsilon }^{-\frac{3}{2}}{‖\phi ‖}_{\epsilon ,P_1}^3,}$$ (3.6)

and analogously,

$${\displaystyle {\int }_{{\mathbb{R}}^3}\sum _{i=1}^k{V}_{\epsilon ,y^i}{\psi }^3\le C{\epsilon }^{-\frac{3}{2}}{‖\psi ‖}_{\epsilon ,P_2}^3,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\int }_{{\mathbb{R}}^3}\sum _{i=1}^m{W}_{\epsilon ,y^i}{\varphi }^3\le C{\epsilon }^{-\frac{3}{2}}{‖\varphi ‖}_{\epsilon ,P_3}^3.}$$ (3.7)

Combining (3.4)–(3.7), we find

$${\displaystyle ‖R_{\epsilon }(\phi ,\psi ,\varphi )‖\le C({\epsilon }^{-\frac{3}{2}}{‖(\phi ,\psi ,\varphi )‖}^3+{\epsilon }^{-3}{‖(\phi ,\psi ,\varphi )‖}^4).}$$

Similarly, we deduce that

$${\displaystyle \left|〈R_{\epsilon }^{\prime }(\phi ,\psi ,\varphi ),(g,h,f)〉\right|\le C({\epsilon }^{-\frac{3}{2}}{‖(\phi ,\psi ,\varphi )‖}^2+{\epsilon }^{-3}{‖(\phi ,\psi ,\varphi )‖}^3)‖(g,h,f)‖}$$

and

$${\displaystyle \left|〈R_{\epsilon }^{″}(\phi ,\psi ,\varphi )(g,h,f),(w_1,w_2,w_3)〉\right|}$$

$${\displaystyle \le C({\epsilon }^{-\frac{3}{2}}‖(\phi ,\psi ,\varphi )‖+{\epsilon }^{-3}{‖(\phi ,\psi ,\varphi )‖}^2)‖(g,h,f)‖‖(w_1,w_2,w_3)‖.}$$

□

Lemma 3.2.

There exist a constant C > 0 independent of ε such that

$${\displaystyle ‖{\ell }_{\epsilon }‖\le C\left(({\lambda }_2+{\lambda }_3){\epsilon }^{\frac{3}{2}}+{\epsilon }^{\frac{3}{2}+{\theta }_1}+{\epsilon }^{\frac{3}{2}}\sum _{i=1}^k\left|(P_1\left(y^i\right)-P_1\left(x_0^1\right))\right|+{\epsilon }^{\frac{3}{2}}\sum _{i=1}^k\left|(P_2\left(y^i\right)-P_2\left(x_0^1\right))\right|\right.}$$

$${\displaystyle \left.+{\epsilon }^{\frac{3}{2}}\sum _{i=1}^m\left|(P_3\left(z^i\right)-P_3\left(x_0^2\right))\right|+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }}\right).}$$

Proof.

Directly calculating, we have

$${\displaystyle \left|\sum _{i=1}^k{\int }_{{\mathbb{R}}^3}\left(P_1\left(x\right)-P_1\left(x_0^1\right)\right)U_{\epsilon ,y^i}\phi \right|}$$

$${\displaystyle =\left|\sum _{i=1}^k{\int }_{{\mathbb{R}}^3}\left(P_1\left(x\right)-P_1\left(y^i\right)+P_1\left(y^i\right)-P_1\left(x_0^1\right)\right)U_{\epsilon ,y^i}\phi \right|}$$

$${\displaystyle \le C\left(\sum _{i=1}^k{\epsilon }^{\frac{3}{2}}\left|P_1\left(x_0^1\right)-P_1\left(y^i\right)\right|+{\epsilon }^{\frac{3}{2}+{\theta }_1}\right){‖\phi ‖}_{\epsilon ,P_1}.}$$ (3.8)

Analogously,

$${\displaystyle \left|\sum _{i=1}^k{\int }_{{\mathbb{R}}^3}\left(P_2\left(x\right)-P_2\left(x_0^1\right)\right)V_{\epsilon ,y^i}\psi \right|\le C\left(\sum _{i=1}^k{\epsilon }^{\frac{3}{2}}\left|P_2\left(x_0^1\right)-P_2\left(y^i\right)\right|+{\epsilon }^{\frac{3}{2}+{\theta }_2}\right){‖\psi ‖}_{\epsilon ,P_2}}$$ (3.9)

and

$${\displaystyle \left|\sum _{i=1}^m{\int }_{{\mathbb{R}}^3}\left(P_3\left(x\right)-P_3\left(x_0^2\right)\right)W_{\epsilon ,z^i}\varphi \right|\le C\left(\sum _{i=1}^m{\epsilon }^{\frac{3}{2}}\left|P_3\left(x_0^2\right)-P_3\left(z^i\right)\right|+{\epsilon }^{\frac{3}{2}+{\theta }_3}\right){‖\varphi ‖}_{\epsilon ,P_3}.}$$ (3.10)

By direct computation, we obtain

$${\displaystyle \left|{\int }_{{\mathbb{R}}^3}(\sum _{j=1}^k{U}_{\epsilon ,y^j}^3-U_{\epsilon ,y}^3)\phi \right|\le C\left|{\int }_{{\mathbb{R}}^3}\sum _{i\ne j}^k{U}_{\epsilon ,y^i}^2{U}_{\epsilon ,y^j}\phi \right|\le C{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}{‖\phi ‖}_{\epsilon ,P_1}.}$$ (3.11)

Similar to (3.12), we get the following result:

$${\displaystyle \left|{\int }_{{\mathbb{R}}^3}(\sum _{j=1}^k{V}_{\epsilon ,y^j}^3-V_{\epsilon ,y}^3)\psi \right|\le C{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}{‖\psi ‖}_{\epsilon ,P_2}}$$ (3.12)

and

$${\displaystyle \left|{\int }_{{\mathbb{R}}^3}(\sum _{j=1}^m{W}_{\epsilon ,z^j}^3-W_{\epsilon ,z}^3)\varphi \right|\le C{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }}{‖\varphi ‖}_{\epsilon ,P_3}.}$$ (3.13)

We know

$${\displaystyle {\int }_{{\mathbb{R}}^3}{U}_{\epsilon ,y}\varphi \le C{\epsilon }^{\frac{3}{2}}{‖\varphi ‖}_{\epsilon ,P_3}^2,\hspace{0.17em}\hspace{0.17em}{\int }_{{\mathbb{R}}^3}{V}_{\epsilon ,y}\varphi \le C{\epsilon }^{\frac{3}{2}}{‖\varphi ‖}_{\epsilon ,P_3}^2,}$$

$${\displaystyle {\int }_{{\mathbb{R}}^3}{W}_{\epsilon ,z}\phi \le C{\epsilon }^{\frac{3}{2}}{‖\phi ‖}_{\epsilon ,P_1}^2,{\int }_{{\mathbb{R}}^3}{W}_{\epsilon ,z}\psi \le C{\epsilon }^{\frac{3}{2}}{‖\psi ‖}_{\epsilon ,P_2}^2.}$$ (3.14)

From (3.8)–(3.14) and (3.2), we obtain

$${\displaystyle ‖{\ell }_{\epsilon }‖\le C\left({\epsilon }^{\frac{3}{2}+{\theta }_1}+{\epsilon }^{\frac{3}{2}}\sum _{i=1}^k\left(\right.\left|P_1\left(y^i\right)-P_1\left(x_0^1\right)\right|+\left|P_2\left(y^i\right)-P_2\left(x_0^1\right)\right|\left.\right)\right.}$$

$${\displaystyle \left.+{\epsilon }^{\frac{3}{2}}\sum _{i=1}^m\left|P_3\left(z^i\right)-P_3\left(x_0^2\right)\right|\right)+({\lambda }_2+{\lambda }_3){\epsilon }^{\frac{3}{2}}+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}}$$

$${\displaystyle +\left.{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }}\right).}$$

□

IV. THE FINITE-DIMENSIONAL REDUCTION AND PROOF OF THE MAIN RESULTS

In this section, we intend to prove the main theorem by the Lyapunov-Schmidt reduction.

First, we discuss the invertibility of L_ε.

Lemma 4.1.

For ${\lambda }_1\in ({\lambda }_1^{\ast },a_1)$, λ₂, λ₃ ∼ ε^α and ε sufficiently small, there is a constant ϱ^∗ > 0 independent of ε such that

$${\displaystyle L_{\epsilon }(\phi ,\psi ,\varphi )\ge {\varrho }^{\ast }‖(\phi ,\psi ,\varphi )‖,\hspace{0.17em}\hspace{0.17em}(\phi ,\psi ,\varphi )\in E_{k,m}^{y,z},}$$

where ${\lambda }_1^{\ast }$ is defined in Proposition 2.1.

Proof.

Arguing by contradiction, we suppose there are $(y^n,z^n)=(y^{1,n},\dots ,y^{k,n},z^{1,n},\dots ,z^{m,n})\in D_{{\epsilon }_n,{\delta }_n}^{k,y^n}\times D_{{\epsilon }_n,{\delta }_n}^{m,z^n}$ and $({\phi }_n,{\psi }_n,{\varphi }_n)\in E_{k,m}^{y^n,z^n}$, with

$${\displaystyle 〈L_{\epsilon }({\phi }_n,{\psi }_n,{\varphi }_n),(g,h,f)〉=o_n\left(1\right)‖({\phi }_n,{\psi }_n,{\varphi }_n)‖‖(g,h,f)‖,\hspace{0.17em}\forall (g,h,f)\in E_{k,m}^{y^n,z^n}.}$$ (4.1)

We may assume that ${‖({\phi }_n,{\psi }_n,{\varphi }_n)‖}^2={\epsilon }_n^3$. Fix i(i ∈ {1, …, k}),  j(j ∈ {1, …, m}) and let

$${\displaystyle {\tilde{\phi }}_{n,i}={\phi }_n({\epsilon }_{n}x+y^{i,n}),\hspace{0.17em}{\tilde{\psi }}_{n,i}={\phi }_n({\epsilon }_{n}x+y^{i,n}),\hspace{0.17em}\hspace{0.17em}{\tilde{\varphi }}_{n,j}={\varphi }_n({\epsilon }_{n}x+z^{j,n}).}$$

We have

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left({\left|\nabla {\tilde{\phi }}_{n,i}\right|}^2+P_1({\epsilon }_{n}x+y^{i,n}){\tilde{\phi }}_{n,i}^2+{\left|\nabla {\tilde{\psi }}_{n,i}\right|}^2+P_2({\epsilon }_{n}x+y^{i,n}){\tilde{\psi }}_{n,i}^2\right)\le C}$$

and

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left({\left|\nabla {\tilde{\varphi }}_{n,j}\right|}^2+P_3({\epsilon }_{n}x+z^{j,n}){\tilde{\varphi }}_{n,j}^2\right)\le C.}$$

So, we suppose the existence of (φ, ψ, ϕ) ∈ (H¹(ℝ³))³ such that

$${\displaystyle {\tilde{\phi }}_{n,i}\to \phi ,\text{weakly in}{H}_{\mathit{loc}}^1\left({\mathbb{R}}^3\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{\phi }}_{n,i}\to \phi ,\text{strongly in}{L}_{\mathit{loc}}^2\left({\mathbb{R}}^3\right),}$$

$${\displaystyle {\tilde{\psi }}_{n,i}\to \psi ,\text{weakly in}{H}_{\mathit{loc}}^1\left({\mathbb{R}}^3\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{\psi }}_{n,i}\to \psi ,\text{strongly in}{L}_{\mathit{loc}}^2\left({\mathbb{R}}^3\right),}$$

$${\displaystyle {\tilde{\varphi }}_{n,j}\to \varphi ,\text{weakly in}{H}_{\mathit{loc}}^1\left({\mathbb{R}}^3\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{\varphi }}_{n,j}\to \varphi ,\text{strongly in}{L}_{\mathit{loc}}^2\left({\mathbb{R}}^3\right).}$$

Moreover, (φ, ψ, ϕ) satisfy

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left(\nabla \frac{\partial U}{\partial y_l}\nabla \phi +\frac{\partial U}{\partial y_l}\phi +\nabla \frac{\partial V}{\partial y_l}\nabla \psi +\frac{\partial V}{\partial y_l}\psi \right)=0,}$$

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left(\nabla \frac{\partial W}{\partial y_l}\nabla \varphi +\frac{\partial W}{\partial y_l}\varphi \right)=0,\hspace{0.17em}\hspace{0.17em}l=1,2,3.}$$

We claim that

$${\displaystyle \phi =0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\psi =0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\varphi =0.}$$

For any $(g,h)\in C_0^{\infty }\left({\mathbb{R}}^3\right)\times C_0^{\infty }\left({\mathbb{R}}^3\right)$, we decompose

$${\displaystyle (g_n,h_n)=(g,h)-\sum _{i=1}^k\sum _{l=1}^3{a}_{n,i,l}(\frac{\partial {\tilde{U}}_{{\epsilon }_n,y_l^{t,n}}}{\partial y_l^{t,n}},\frac{\partial {\tilde{V}}_{{\epsilon }_n,y_l^{t,n}}}{\partial y_l^{t,n}}),}$$

where

$${\displaystyle {\tilde{U}}_{{\epsilon }_n,y^{t,n}}=U_{{\epsilon }_n,y^{t,n}}({\epsilon }_{n}x+y^{i,n}),{\tilde{V}}_{{\epsilon }_n,y^{t,n}}=V_{{\epsilon }_n,y^{t,n}}({\epsilon }_{n}x+y^{i,n}).}$$

Then, with the same argument as in Ref. 11, it is not difficult to check that a_n,t,l → a_t,l, a_n,i,l → 0, i ≠ t,  l = 1, 2, 3 as n → ∞.

Taking $(g_n\left(\frac{x-y^{t,n}}{\epsilon }\right),h_n\left(\frac{x-y^{t,n}}{\epsilon }\right),0)$ into (4.1), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^3}(\nabla \phi \nabla g+\phi g-3U^2\phi g)+{\int }_{{\mathbb{R}}^3}(\nabla \psi \nabla h+\psi h-3V^2\psi h)}$$

$${\displaystyle +\sum _{l=1}^3{a}_{t,l}\left({\int }_{{\mathbb{R}}^3}(\nabla \frac{\partial U}{\partial x_l}\nabla \phi +\frac{\partial U}{\partial x_l}\phi -3U^2\frac{\partial U}{\partial x_l}\phi )+(\nabla \frac{\partial V}{\partial x_l}\nabla \psi +\frac{\partial V}{\partial x_l}\psi -3V^2\frac{\partial V}{\partial x_l}\psi )\right.}$$

$${\displaystyle \left.-{\int }_{{\mathbb{R}}^3}{\lambda }_1(\frac{\partial U}{\partial x_l}\phi +\frac{\partial V}{\partial x_l}\psi )\right)-{\lambda }_1{\int }_{{\mathbb{R}}^3}(\phi h+\psi g)=0,l=1,2,3.}$$ (4.2)

Note that

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left(\nabla \frac{\partial U}{\partial x_l}\nabla \phi +\frac{\partial U}{\partial x_l}\phi +\nabla \frac{\partial V}{\partial x_l}\nabla \psi +\frac{\partial V}{\partial x_l}\psi \right.}$$

$${\displaystyle \left.-3U^2\frac{\partial U}{\partial x_l}\phi -3V^2\frac{\partial V}{\partial x_l}\psi \right)\left.-{\lambda }_1(\frac{\partial U}{\partial x_l}\phi +\frac{\partial V}{\partial x_l}\psi )\right)=0.}$$

Therefore, we have

$${\displaystyle {\int }_{{\mathbb{R}}^3}(\nabla \phi \nabla g+\phi g-3U^2\phi g)+{\int }_{{\mathbb{R}}^3}(\nabla \psi \nabla h+\psi h-3V^2\psi h)}$$

$${\displaystyle -{\lambda }_1{\int }_{{\mathbb{R}}^3}(\phi h+\psi g)=0.}$$

Then, we get (φ, ψ) that satisfies

$${\displaystyle \{\begin{array}{l}-\mathrm{\Delta }\phi +a_1\phi -3U^2\phi -{\lambda }_1\psi =0,x\in {\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }\psi +a_1\psi -3V^2\psi -{\lambda }_1\phi =0,x\in {\mathbb{R}}^3.\hfill \end{array}}$$

From Proposition 2.1 for l = 2, we have (U, V) = (W_a1−λ1, W_a1−λ1) which is nondegeneracy, with λ^∗ < λ₁ < a₁, which implies that (φ, ψ) = (0, 0).

Next, we denote

$${\displaystyle f_n=f-\sum _{j=1}^m\sum _{l=1}^3{a}_{n,j,l}\frac{\partial {\tilde{W}}_{{\epsilon }_n,z^{r,n}}}{\partial z_l^{r,n}},}$$

where ${\tilde{W}}_{{\epsilon }_n,z^{r,n}}=W_{{\epsilon }_n,z^{r,n}}({\epsilon }_{n}x+z^{j,n})$ and taking $(0,0,f_n\left(\frac{x-z^{r,n}}{\epsilon }\right))$ into (4.1), we get

$${\displaystyle {\int }_{{\mathbb{R}}^3}(\nabla \varphi \nabla f+\varphi f-3W^2\varphi f)=0.}$$

We get that ϕ = 0.

Hence, by ${\lambda }_2,{\lambda }_3\sim {\epsilon }^{\alpha },\hspace{0.17em}\alpha >\frac{min\{{\theta }_1,1\}}{2}$, we obtain

$${\displaystyle o_n\left(1\right){\epsilon }_n^3={\int }_{{\mathbb{R}}^3}\left({\epsilon }_n^2{\left|\nabla {\phi }_n\right|}^2+P_1\left(x\right){\phi }_n^2+{\epsilon }_n^2{\left|\nabla {\psi }_n\right|}^2+P_2\left(x\right){\psi }_n^2+{\epsilon }_n^2{\left|\nabla {\varphi }_n\right|}^2+P_3\left(x\right){\varphi }_n^2\right)}$$

$${\displaystyle -{\int }_{{\mathbb{R}}^3}\left(3U_{{\epsilon }_n,y^n}^2{\phi }_n^2+3V_{{\epsilon }_n,y^n}^2{\psi }_n^2+3W_{{\epsilon }_n,z^n}^2{\varphi }_n^2\right)-2{\lambda }_1{\int }_{{\mathbb{R}}^3}{\phi }_n{\psi }_n}$$

$${\displaystyle -2{\lambda }_2{\int }_{{\mathbb{R}}^3}{\phi }_n{\varphi }_n-2{\lambda }_3{\int }_{{\mathbb{R}}^3}{\psi }_n{\varphi }_n}$$

$${\displaystyle =(1-\frac{{\lambda }_1}{b}){‖({\phi }_n,{\psi }_n,{\varphi }_n)‖}^2-{\int }_{{\mathbb{R}}^3}\left(3U_{{\epsilon }_n,y^n}^2{\phi }_n^2+3V_{{\epsilon }_n,y^n}^2{\psi }_n^2+3W_{{\epsilon }_n,z^n}^2{\varphi }_n^2\right)}$$

$${\displaystyle +O\left({\epsilon }_n^{\alpha }\right){‖({\phi }_n,{\psi }_n,{\varphi }_n)‖}^2}$$

$${\displaystyle \ge (1-\frac{{\lambda }_1}{b}){‖({\phi }_n,{\psi }_n,{\varphi }_n)‖}^2+o\left(e^{-R}\right){\epsilon }_n^3+O\left({\epsilon }_n^{\alpha }\right){\epsilon }_n^3.}$$

Since b > λ₁, we get contradiction. Then, we complete the result. □

Proposition 4.2.

For ε > 0 small enough, there is a C¹ map from $D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{m,z}$ to H such that $(\phi ,\psi ,\varphi )\in E_{k,m}^{y,z}$ satisfying

$${\displaystyle 〈\frac{\partial J_{\epsilon }(y,z,\phi ,\psi ,\varphi )}{\partial (\phi ,\psi ,\varphi )},(f_1,f_2,f_3)〉=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}\hspace{0.17em}(f_1,f_2,f_3)\in E_{k,m}^{y,z}.}$$

Moreover,

Proof.

We know that ℓ_ε(φ, ψ, ϕ) is a bounded linear function in $E_{k,m}^{y,z}$ by Lemma 3.2. Thus, there is ${\overline{\ell }}_{\epsilon }$ such that

$${\displaystyle {\ell }_{\epsilon }(\phi ,\psi ,\varphi )=〈{\overline{\ell }}_{\epsilon },(\phi ,\psi ,\varphi )〉.}$$

So, finding a critical point for J_ε(φ, ψ, ϕ) in $E_{k,m}^{y,z}$ is equivalent to solving

$${\displaystyle {\overline{\ell }}_{\epsilon }-L_{\epsilon }(\phi ,\psi ,\varphi )+R_{\epsilon }^{\prime }(\phi ,\psi ,\varphi )=0.}$$

Since L_ε is invertible by Lemma 4.1, we get

$${\displaystyle A(\phi ,\psi ,\varphi )=-L_{\epsilon }^{-1}{\overline{\ell }}_{\epsilon }-L_{\epsilon }^{-1}{R}_{\epsilon }^{\prime }(\phi ,\psi ,\varphi ).}$$

Set

$${\displaystyle D=\left\{(\phi ,\psi ,\varphi )\in E_{k,m}^{y,z}:‖(\phi ,\psi ,\varphi )‖\right.\le {\epsilon }^{\frac{3}{2}+{\theta }_1-\tau }+({\lambda }_2+{\lambda }_3){\epsilon }^{\frac{3}{2}-\tau }+{\epsilon }^{\frac{3}{2}}\left(\sum _{i=1}^k\left({\left|P_1\left(y^i\right)-P_1\left(x_0^1\right)\right|}^{1-\tau }\right)\right.}$$

$${\displaystyle +\left.\sum _{i=1}^k\left({\left|P_2\left(z^i\right)-P_2\left(x_0^1\right)\right|}^{1-\tau }\right)+\sum _{i=1}^m\left({\left|P_3\left(z^i\right)-P_3\left(x_0^2\right)\right|}^{1-\tau }\right)\right)}$$

$${\displaystyle \left.+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}(1-\tau )\left|y^i-y^j\right|}{\epsilon }}+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}(1-\tau )\left|z^i-z^j\right|}{\epsilon }}\right\}.}$$

Now, we verify that A is a contraction from D to D. Indeed, for any (φ, ψ, ϕ) ∈ D, by Lemmas 3.1 and 3.2, we obtain

$${\displaystyle ‖A(\phi ,\psi ,\varphi )‖\le C(‖{\overline{\ell }}_{\epsilon }‖+‖R_{\epsilon }^{\prime }‖)\le C(‖{\overline{\ell }}_{\epsilon }‖+{\epsilon }^{-\frac{3}{2}}{‖(\phi ,\psi ,\varphi )‖}^2)}$$

$${\displaystyle \le C\left({\epsilon }^{\frac{3}{2}+{\theta }_1}+({\lambda }_2+{\lambda }_3){\epsilon }^{\frac{3}{2}-\tau }+{\epsilon }^{\frac{3}{2}}\left(\sum _{i=1}^k{\left|(P_1\left(y^i\right)-P_1\left(x_0^1\right))\right|}^{2(1-\tau )}\right.\right.}$$

$${\displaystyle \left.+\sum _{i=1}^k{\left|(P_2\left(y^i\right)-P_2\left(x_0^1\right))\right|}^{2(1-\tau )}+\sum _{i=1}^m{\left|(P_3\left(z^i\right)-P_3\left(x_0^2\right))\right|}^{2(1-\tau )}\right)}$$

$${\displaystyle \left.+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{2\sqrt{a_1-{\lambda }_1}(1-\tau )\left|y^i-y^j\right|}{\epsilon }}+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^m{e}^{-\frac{2\sqrt{a_2}(1-\tau )\left|z^i-z^j\right|}{\epsilon }}\right)}$$

$${\displaystyle \le {\epsilon }^{\frac{3}{2}+{\theta }_1-\tau }+({\lambda }_2+{\lambda }_3){\epsilon }^{\frac{3}{2}-\tau }+{\epsilon }^{\frac{3}{2}}\left(\sum _{i=1}^k{\left|(P_1\left(y^i\right)-P_1\left(x_0^1\right))\right|}^{1-\tau }\right.}$$

$${\displaystyle \left.+\sum _{i=1}^k{\left|(P_2\left(y^i\right)-P_2\left(x_0^1\right))\right|}^{1-\tau }+\sum _{i=1}^m{\left|(P_3\left(y^i\right)-P_3\left(x_0^2\right))\right|}^{1-\tau }\right)}$$

$${\displaystyle +{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}(1-\tau )\left|y^i-y^j\right|}{\epsilon }}+{\epsilon }^{\frac{3}{2}}\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}(1-\tau )\left|z^i-z^j\right|}{\epsilon }},}$$

and we get A maps from D to D. On the other hand, for (φ₁, ψ₁, ϕ₁), (φ₂, ψ₂, ϕ₂) ∈ D and ε sufficiently small, we have

$${\displaystyle ‖A({\phi }_1,{\psi }_1,{\varphi }_1)-A({\phi }_2,{\psi }_2,{\varphi }_2)‖=‖L_{\epsilon }^{-1}{R}_{\epsilon }^{\prime }({\phi }_1,{\psi }_1,{\varphi }_1)-L_{\epsilon }^{-1}{R}_{\epsilon }^{\prime }({\phi }_2,{\psi }_2,{\varphi }_2)‖}$$

$${\displaystyle \le C‖R_{\epsilon }^{″}\left(\right.({\phi }_1,{\psi }_1,{\varphi }_1)+s({\phi }_2,{\psi }_2,{\varphi }_2)\left.\right)‖‖({\phi }_1,{\psi }_1,{\varphi }_1)-({\phi }_2,{\psi }_2,{\varphi }_2)‖}$$

$${\displaystyle \le C\left({\epsilon }^{-\frac{3}{2}}‖({\phi }_1,{\psi }_1,{\varphi }_1)‖+{\epsilon }^{-3}{‖({\phi }_1,{\psi }_1,{\varphi }_1)‖}^2\right)‖({\phi }_1,{\psi }_1,{\varphi }_1)-({\phi }_2,{\psi }_2,{\varphi }_2)‖}$$

$${\displaystyle \le \frac{1}{2}‖({\phi }_1,{\psi }_1,{\varphi }_1)-({\phi }_2,{\psi }_2,{\varphi }_2)‖.}$$

Therefore, A is a contraction map from D to D. By the contraction mapping theorem, there exist $(\phi ,\psi ,\varphi )\in E_{k,m}^{y,z}$ such that A(φ, ψ, ϕ) = (φ, ψ, ϕ).

Moreover,

□

Proof of Theorem 1.3.

It follows from Lemma 4.1 and Propositions 4.2 and A.2 that

$${\displaystyle F(y,z)=J_{\epsilon }(\phi (y,z),\psi (y,z),\varphi (y,z))=I_{\epsilon }(U_{\epsilon ,y}+\phi ,V_{\epsilon ,y}+\psi ,W_{\epsilon ,z}+\varphi )=J_{\epsilon }(0,0,0)+{\ell }_{\epsilon }(\phi ,\psi ,\varphi )+\frac{1}{2}〈L_{\epsilon }(\phi ,\psi ,\varphi ),(\phi ,\psi ,\varphi )〉+R_{\epsilon }(\phi ,\psi ,\varphi )=I_{\epsilon }(U_{\epsilon ,y},V_{\epsilon ,y},W_{\epsilon ,z})+O(‖{\ell }_{\epsilon }‖‖(\phi ,\psi ,\varphi )‖+{‖(\phi ,\psi ,\varphi )‖}^2)=\frac{1}{4}A{\epsilon }^3-{\epsilon }^3(C_1\sum _{i=1}^k(P_1\left(x_0^1\right)-P_1\left(y^i\right))+C_1\sum _{i=1}^k(P_2\left(x_0^1\right)-P_2\left(y^i\right))+C_2\sum _{i=1}^m(P_3\left(x_0^2\right)-P_3\left(z^i\right)))-{\epsilon }^3{C}_3\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}-{\epsilon }^3{C}_4\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }}+O\left({\epsilon }^{3+{\theta }_1}\right),}$$

since λ₂,  λ₃ ∼ ε^α, $\alpha \ge \frac{min\{{\theta }_1,1\}}{2}$.

Consider the following maximization problem:

$${\displaystyle \underset{(y,z)\in D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{m,z}}{max}F(y,z).}$$ (4.3)

Assume (4.3) is achieved by some $(y_{\epsilon },z_{\epsilon })\in D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{m,z}$, next we claim that (y_ε, z_ε) is an interior point of $D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{m,z}$.

Let us choose ${\overline{y}}^j=x_0^1+L_1\epsilon \left|ln\epsilon \right|e_j$, ${\overline{z}}^j=x_0^2+L_2\epsilon \left|ln\epsilon \right|e_j$ for L₁, L₂ > 0, and some vectors e₁, e₂, …, e_n, with $\left|e_i-e_j\right|=1$ for i ≠ j. Then, $\frac{\left|{\overline{y}}^i-{\overline{y}}^j\right|}{\epsilon }=L_1\left|ln\epsilon \right|$, $\frac{\left|{\overline{z}}^i-{\overline{z}}^j\right|}{\epsilon }=L_2\left|ln\epsilon \right|$, which means that $\overline{y}≔({\overline{y}}^1,\dots ,{\overline{y}}^k)\in D_{\epsilon ,\delta }^{k,y}$, $\overline{z}≔({\overline{z}}^1,\dots ,{\overline{z}}^m)\in D_{\epsilon ,\delta }^{m,z}$ for ε sufficiently small.

Thus, for L₁, L₂ large enough, we have

$${\displaystyle \frac{1}{4}A{\epsilon }^3-C{\epsilon }^3{\left(\epsilon \left|ln\epsilon \right|\right)}^{min\{{\theta }_1,1\}}\le F(\overline{y},\overline{z})\le F(y_{\epsilon },z_{\epsilon })}$$

$${\displaystyle \le \frac{1}{4}A{\epsilon }^3-{\epsilon }^3\sum _{i=1}^k\left(\right.(P_1\left(x_0^1\right)-P_1\left(y^i\right))+(P_2\left(x_0^1\right)-P_2\left(y^i\right))\left.\right)}$$

$${\displaystyle -{\epsilon }^3\sum _{i=1}^m\left(\right.(P_3\left(x_0^2\right)-P_3\left(z^i\right))\left.\right)-{\epsilon }^3\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^i\right|}{\epsilon }}}$$

$${\displaystyle -{\epsilon }^3\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }},}$$

that is,

Therefore, we can deduce

$${\displaystyle \sum _{i=1}^k(P_1\left(x_0^1\right)-P_1\left(y^i\right))\le C{\left(\epsilon \left|ln\epsilon \right|\right)}^{min\{{\theta }_1,1\}},\sum _{i=1}^k(P_2\left(x_0^1\right)-P_2\left(y^i\right))\le C{\left(\epsilon \left|ln\epsilon \right|\right)}^{min\{{\theta }_1,1\}},}$$

$${\displaystyle \sum _{i=1}^m(P_3\left(x_0^2\right)-P_3\left(z^i\right))\le C{\left(\epsilon \left|ln\epsilon \right|\right)}^{min\{{\theta }_1,1\}},}$$

$${\displaystyle \sum _{i\ne j}^k\frac{\left|y^i-y^j\right|}{\epsilon }\ge C\left|ln\epsilon \right|>{\left|ln\epsilon \right|}^{\frac{1}{2}},\hspace{0.17em}\hspace{0.17em}\sum _{i\ne j}^m\frac{\left|z^i-z^j\right|}{\epsilon }\ge C\left|ln\epsilon \right|>{\left|ln\epsilon \right|}^{\frac{1}{2}},}$$

which imply that (y_ε, z_ε) is an interior point of $D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{m,z}$. Then, the corresponding vector function (u_ε, v_ε, w_ε) is a critical point of I_ε, where $u_{\epsilon }=\sum _{i=1}^k{U}_{\epsilon ,y^i}+{\phi }_{\epsilon }$, $v_{\epsilon }=\sum _{i=1}^k{V}_{\epsilon ,y^i}+{\psi }_{\epsilon }$, $v_{\epsilon }=\sum _{i=1}^m{W}_{\epsilon ,z^i}+{\varphi }_{\epsilon }$.

That u_ε,  v_ε,  w_ε are all positive and can be proved following the argument in Ref. 22. We get (y_ε, z_ε) which is an interior point $D_{\epsilon ,\delta }^{k,y}\times D_{\epsilon ,\delta }^{m,z}$. Hence, (U_ε,y + φ_ε, V_ε,y + ψ_ε, W_ε,y + ϕ_ε) is the solution of problem (1.4). □

APPENDIX: ENERGY EXPANSIONS

In this section, we will expand the energies I_ε(U_ε,y, V_ε,y, W_ε,z), where

$${\displaystyle I_{\epsilon }(u,v,w)=\frac{1}{2}{\int }_{{\mathbb{R}}^3}\left({\epsilon }^2{\left|\nabla u\right|}^2+P_1\left(x\right)u^2+{\epsilon }^2{\left|\nabla v\right|}^2+P_2\left(x\right)v^2+{\epsilon }^2{\left|\nabla w\right|}^2+P_3\left(x\right)w^2\right)}$$

$${\displaystyle -\frac{1}{4}{\int }_{{\mathbb{R}}^3}\left(u^4+v^4+w^4\right)-{\lambda }_1{\int }_{{\mathbb{R}}^3}uv-{\lambda }_2{\int }_{{\mathbb{R}}^3}uw-{\lambda }_3{\int }_{{\mathbb{R}}^3}vw.}$$

Proposition A.1 see Ref. 5 —

Suppose that $u\left(x\right),\hspace{0.17em}v\left(x\right)\in H_r^1\left({\mathbb{R}}^N\right)$ satisfy

$${\displaystyle u\left(r\right)\backsim r^{\alpha }{e}^{-\beta r},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v\left(r\right)\backsim r^{\gamma }{e}^{-\eta r}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(r\to +\infty ),}$$

where α, γ ∈ ℝ, β > 0, η > 0. Let y ∈ ℝ^N, with $\left|y\right|\to +\infty$. We have the following:

• (i)
  if β < η, then

  $${\displaystyle {\int }_{{\mathbb{R}}^3}{u}_{y}v\sim {\left|y\right|}^{\alpha }{e}^{-\beta \left|y\right|}.}$$
• (ii)
  If β = η, suppose for simplicity that α ≥ γ, then,

  $${\displaystyle {\int }_{{\mathbb{R}}^3}{u}_{y}v\sim \left\{\begin{array}{ll}{e}^{-\beta \left|y\right|}{\left|y\right|}^{\alpha +\gamma +\frac{1+N}{2}},\hfill & if\gamma >-\frac{1+N}{2},\hfill \\ e^{-\beta \left|y\right|}{\left|y\right|}^{\alpha }ln\left|y\right|,\hfill & if\gamma =-\frac{N+1}{2},\hfill \\ e^{-\beta \left|y\right|}{\left|y\right|}^{\alpha },\hfill & if\gamma <-\frac{N+1}{2}.\hfill \end{array}\right.}$$ (A1)

Proposition A.2.

There is a small constant σ > 0 and positive constants C₁,  C₂,  C₃,  C₄ such that

$${\displaystyle I_{\epsilon }(U_{\epsilon ,y}\left(x\right),V_{\epsilon ,y}\left(x\right),W_{\epsilon ,z}\left(x\right))=\frac{1}{4}A{\epsilon }^3-{\epsilon }^3\left(C_1\sum _{i=1}^k(P_1\left(x_0^1\right)-P_1\left(y^i\right))\right.}$$

$${\displaystyle \left.+C_1\sum _{i=1}^k(P_2\left(x_0^1\right)-P_2\left(y^i\right))+C_2\sum _{i=1}^m(P_3\left(x_0^2\right)-P_3\left(z^i\right))\right)}$$

$${\displaystyle -{\epsilon }^3{C}_3\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}-{\epsilon }^3{C}_4\sum _{i\ne j}^m{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }}}$$

$${\displaystyle +O\left({\epsilon }^{3+{\theta }_1}\right),}$$ (A2)

where A = (2k + m)∫_ℝ³W⁴.

Proof.

By the definition of I_ε, we get

$${\displaystyle I_{\epsilon }(U_{\epsilon ,y},V_{\epsilon ,y},W_{\epsilon ,z})=\frac{1}{2}{\int }_{{\mathbb{R}}^3}\left({\epsilon }^2{\left|\nabla U_{\epsilon ,y}\right|}^2+P_1\left(x\right)U_{\epsilon ,y}^2+{\epsilon }^2{\left|\nabla V_{\epsilon ,y}\right|}^2+P_2\left(x\right)V_{\epsilon ,y}^2\right.}$$

$${\displaystyle +\left.{\epsilon }^2{\left|\nabla W_{\epsilon ,z}\right|}^2+P_3\left(x\right)W_{\epsilon ,z}^2\right)-\frac{1}{4}{\int }_{{\mathbb{R}}^3}(U_{\epsilon ,y}^4+V_{\epsilon ,y}^4+W_{\epsilon ,z}^4)}$$

$${\displaystyle -{\lambda }_1{\int }_{{\mathbb{R}}^3}{U}_{\epsilon ,y}{V}_{\epsilon ,y}-{\lambda }_2{\int }_{{\mathbb{R}}^3}{U}_{\epsilon ,y}{W}_{\epsilon ,z}-{\lambda }_3{\int }_{{\mathbb{R}}^3}{V}_{\epsilon ,y}{W}_{\epsilon ,z}}$$

$${\displaystyle =\frac{1}{2}{\int }_{{\mathbb{R}}^3}\left(\sum _{i=1}^k(P_1\left(x\right)-P_1\left(x_0^1\right))U_{\epsilon ,y^i}^2+\sum _{i\ne j}^k(P_1\left(x\right)-P_1\left(x_0^1\right))U_{\epsilon ,y^i}{U}_{\epsilon ,y^j}\right.}$$

$${\displaystyle +\sum _{i=1}^k(P_2\left(x\right)-P_2\left(x_0^1\right))V_{\epsilon ,y^i}^2+\sum _{i\ne j}^k(P_2\left(x\right)-P_2\left(x_0^1\right))V_{\epsilon ,y^i}{V}_{\epsilon ,y^j}}$$

$${\displaystyle \left.+\sum _{i=1}^m(P_3\left(x\right)-P_3\left(x_0^2\right))W_{\epsilon ,z^i}^2+\sum _{i\ne j}^m(P_3\left(x\right)-P_3\left(x_0^2\right))W_{\epsilon ,z^i}{W}_{\epsilon ,z^j}\right)}$$

$${\displaystyle -\frac{1}{4}{\int }_{{\mathbb{R}}^3}\left(\right.{\left(\sum _{i=1}^k{U}_{\epsilon ,y^i}\right)}^4-\sum _{i=1}^k{U}_{\epsilon ,y^i}^4-2\sum _{i\ne j}^k{U}_{\epsilon ,y^i}^3{U}_{\epsilon ,y^j}\left.\right)}$$

$${\displaystyle -\frac{1}{4}{\int }_{{\mathbb{R}}^3}\left(\right.{\left(\sum _{i=1}^k{V}_{\epsilon ,y^i}\right)}^4-\sum _{i=1}^k{V}_{\epsilon ,y^i}^4-2\sum _{i\ne j}^k{V}_{\epsilon ,y^i}^3{V}_{\epsilon ,y^j}\left.\right)}$$

$${\displaystyle -\frac{1}{4}{\int }_{{\mathbb{R}}^3}\left(\right.{\left(\sum _{i=1}^m{W}_{\epsilon ,z^i}\right)}^4-\sum _{i=1}^m{W}_{\epsilon ,z^i}^4-2\sum _{i\ne j}^m{W}_{\epsilon ,z^i}^3{W}_{\epsilon ,z^j}\left.\right)}$$

$${\displaystyle +\frac{1}{4}{\int }_{{\mathbb{R}}^3}\sum _{i=1}^k\left(\right.U_{\epsilon ,y^i}^4+\sum _{i=1}^k{V}_{\epsilon ,y^i}^4+\sum _{i=1}^m{W}_{\epsilon ,z^i}^4\left.\right)}$$

$${\displaystyle -{\lambda }_2{\int }_{{\mathbb{R}}^3}{U}_{\epsilon ,y}{W}_{\epsilon ,z}-{\lambda }_3{\int }_{{\mathbb{R}}^3}{V}_{\epsilon ,y}{W}_{\epsilon ,z}.}$$ (A3)

By direct computation, we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^3}\sum _{i=1}^k(P_1\left(x\right)-P_1\left(x_0^1\right))U_{\epsilon ,y^i}^2+\sum _{i\ne j}^k(P_1\left(x\right)-P_1\left(x_0^1\right))U_{\epsilon ,y^i}{U}_{\epsilon ,y^j}}$$

$${\displaystyle ={\int }_{{\mathbb{R}}^3}\left(\sum _{i=1}^k(P_1\left(x\right)-P_1\left(y^i\right)+P_1\left(y^i\right)-P_1\left(x_0^1\right))U_{\epsilon ,y^i}^2\right.}$$

$${\displaystyle \left.+\sum _{i\ne j}^k(P_1\left(x\right)-P_1\left(y^i\right)+P_1\left(y^i\right)-P_1\left(x_0^1\right))U_{\epsilon ,y^i}{U}_{\epsilon ,y^j}\right)}$$

$${\displaystyle =-{\epsilon }^3{C}_1\sum _{i=1}^k(P_1\left(x_0^1\right)-P_1\left(y^i\right))+O\left({\epsilon }^{3+{\theta }_1}\right).}$$ (A4)

Similar to (A4), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left(\right.\sum _{i=1}^k(P_2\left(x\right)-P_2\left(x_0^1\right))V_{\epsilon ,y^i}^2+\sum _{i\ne j}^k(P_2\left(x\right)-P_2\left(x_0^1\right))V_{\epsilon ,y^i}{V}_{\epsilon ,y^j}\left.\right)}$$

$${\displaystyle =-{\epsilon }^3{C}_1\sum _{i=1}^k(P_2\left(x_0^1\right)-P_2\left(y^i\right))+O\left({\epsilon }^{3+{\theta }_2}\right)}$$ (A5)

and

$${\displaystyle {\int }_{{\mathbb{R}}^3}\left(\right.\sum _{i=1}^m(P_3\left(x\right)-P_3\left(x_0^2\right))W_{\epsilon ,z^i}^2+\sum _{i\ne j}^m(P_3\left(x\right)-P_3\left(x_0^2\right))W_{\epsilon ,z^i}{W}_{\epsilon ,z^j}\left.\right)}$$

$${\displaystyle =-{\epsilon }^3{C}_1\sum _{i=1}^m(P_3\left(x_0^2\right)-P_3\left(z^i\right))+O\left({\epsilon }^{3+{\theta }_3}\right).}$$ (A6)

With elementary computations, we have

$${\displaystyle {\int }_{{\mathbb{R}}^3}{\left(\sum _{i=1}^k{U}_{\epsilon ,y^i}\right)}^4-\sum _{i=1}^k{U}_{\epsilon ,y^i}^4-2\sum _{i\ne j}^k{U}_{\epsilon ,y^i}^3{U}_{\epsilon ,y^j}}$$

$${\displaystyle =2{\int }_{{\mathbb{R}}^3}\sum _{i\ne j}^k{U}_{\epsilon ,y^i}^3{U}_{\epsilon ,y^j}+O\left({\int }_{{\mathbb{R}}^3}\sum _{i\ne j}^k{U}_{\epsilon ,y^i}^2{U}_{\epsilon ,y^j}^2\right)}$$

$${\displaystyle ={\epsilon }^3{C}_2\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}+O({\epsilon }^3\sum _{i\ne j}^k{e}^{-\frac{2\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}{\left|\frac{\epsilon }{\left|y^i-y^j\right|}\right|}^{2}ln\left|y^i-y^j\right|)}$$

$${\displaystyle ={\epsilon }^3{C}_2\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}+O\left({\epsilon }^{3+{\theta }_1}\right).}$$ (A7)

Similar to (A7), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^3}{\left(\sum _{i=1}^k{V}_{\epsilon ,y^i}\right)}^4-\sum _{i=1}^k{V}_{\epsilon ,y^i}^4-2\sum _{i\ne j}^k{V}_{\epsilon ,y^i}^3{V}_{\epsilon ,y^j}={\epsilon }^3{C}_2\sum _{i\ne j}^k{e}^{-\frac{\sqrt{a_1-{\lambda }_1}\left|y^i-y^j\right|}{\epsilon }}+O\left({\epsilon }^{3+{\theta }_2}\right),}$$ (A8)

$${\displaystyle {\int }_{{\mathbb{R}}^3}{\left(\sum _{i=1}^m{W}_{\epsilon ,z^i}\right)}^4-\sum _{i=1}^m{W}_{\epsilon ,z^i}^4-2\sum _{i\ne j}^m{W}_{\epsilon ,z^i}^3{W}_{\epsilon ,z^j}={\epsilon }^3{C}_2\sum _{i\ne j}{e}^{-\frac{\sqrt{a_2}\left|z^i-z^j\right|}{\epsilon }}+O\left({\epsilon }^{3+{\theta }_3}\right),}$$ (A9)

$${\displaystyle {\int }_{{\mathbb{R}}^3}{U}_{\epsilon ,y^i}{W}_{\epsilon ,z^j}=C{\epsilon }^3{e}^{-\frac{min\{\sqrt{a_1-{\lambda }_1},\sqrt{a_2}\}\left|y^i-z^j\right|}{\epsilon }}=O\left({\epsilon }^{3+{\theta }_1}\right).}$$ (A10)

Inserting (A4)–(A10) into (A3), we get (A2). □