Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-07T02:37:51.436Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-degeneracy of synchronized vector solutions for weakly coupled nonlinear schrödiner systems

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  22 April 2022

Qing Guo  and
Leiga Zhao
Qing Guo
Affiliation:
College of Science, Minzu University of China, Beijing 100081, China (guoqing0117@163.com)
Leiga Zhao
Affiliation:
School of Mathematics and Statistics, Beijing Technology and Business University, (zhaoleiga@163.com)
Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]
First, we revisit the proof of the existence of an unbounded sequence of non-radial positive vector solutions of synchronized type obtained in S. Peng and Z. Wang [Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal. 208 (2013), 305–339] to give a point-wise estimate of the solutions. Taking advantage of these estimates, we then show a non-degeneracy result of the synchronized solutions in some suitable symmetric space by use of the locally Pohozaev identities. The main difficulties of BEC systems come from the interspecies interaction between the components, which never appear in the study of single equations. The idea used to estimate the coupling terms is inspired by the characterization of the Fermat points in the famous Fermat problem, which is the main novelty of this paper.

Keywords

Schrödinger systemsNon-degeneracyPopozaev identitiesLyapunov–Schmidt reductionSynchronized solutions

MSC classification

Primary: 35B25: Singular perturbations
Secondary: 35J47: Second-order elliptic systems 35J60: Nonlinear elliptic equations

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 441 - 459
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

1. Introduction

In this paper, we consider the following coupled nonlinear Schrödinger system

(1.1)\begin{equation} \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & ~\text{in }\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & ~\text{in }\mathbb{R}^{3}, \end{cases} \end{equation}

where $V_1(x)$ and $V_2(x)$ are continuous positive radial functions, $\mu _1>0$, $\mu _2>0$ and $\beta \in \mathbb {R}$ are coupling constants. System (1.1) appears in describing two-component Bose–Einstein condensates (BEC) with potentials $V_i(x)$, $i=1,\,2$ [Reference Bao and Cai2–Reference Guo, Zeng and Zhou5, Reference Guo, Lin and Wei7–Reference Guo, Li, Wei and Zeng9, Reference Lin and Wei11, Reference Long and Peng12, Reference Peng and Wang15, Reference Royo-Letelier16] and the references therein). Here we assume $(V_i)$ satisfies the following condition ($V$) There are constants $m_i>2,\,b_i\in \mathbb {R}$, $i=1,\,2$ and $\theta >0$ such that as $r\rightarrow +\infty$

\[ V_1(r)=1+\displaystyle\frac{a_1}{r^{m_1}}+O\left(\frac 1{r^{m_1+\theta}}\right);\quad V_2(r)=1+\frac{a_2}{r^{m_2}}+O\left(\frac 1{r^{m_2+\theta}}\right), \]

which give the asymptotic behaviour at infinity on the potentials. It is known that $a_i$ in $(V)$ represents the intraspecies interaction of the atoms inside each component, $\beta$ denotes that the interspecies interaction between two components and $\mu _1,\,\mu _2$ are chemical potentials.

If we denote the unique solution of the following problem by $W$,

\[ \begin{cases} -\Delta w + w = w^{3}, & \text{in } \mathbb{R}^{3},\\ w(0)=\displaystyle\max_{x\in\mathbb{R}^{2}}w(x), & w\in H^{1}(\mathbb{R}^{3}), \end{cases} \]

then one find that $(Q_1,\,Q_2)=(\gamma _1 W,\,\gamma _2 W)$ solves

\[ \begin{cases} -\Delta Q_1 + Q_1= \mu_1 Q_1^{3} +\beta Q_1Q_2^{2} & \text{in }\mathbb{R}^{3},\\ -\Delta Q_2+ Q_2= \mu_2 Q_2^{3} +\beta Q_1^{2}Q_2 & \text{in }\mathbb{R}^{3} \end{cases} \]

provided $\beta \in (-\sqrt {\mu _1\mu _2},\,\min \{\mu _1,\,\mu _2\}) \cup (\max \{\mu _1,\,\mu _2\},\,+\infty )$, where

\[ \gamma_1=\sqrt{\frac{\mu_2-\beta}{\mu_1\mu_2-\beta^{2}}},\quad \gamma_2=\sqrt{\frac{\mu_1-\beta}{\mu_1\mu_2-\beta^{2}}}. \]

Set

\[ x^{j}=\left(r\cos\displaystyle\frac{2(j-1)\pi}{k},r\sin\frac{2(j-1)\pi}{k},0\right), \quad j=1,\ldots,k, \]

where $r\in [r_0klnk,\,r_1klnk]$ for some $r_1>r_0>0$. Let

\[ U_{1,r}(x)=\sum_{j=1}^{k}Q_{1,x^{j}}(x),\quad U_{2,r}(x)=\sum_{j=1}^{k}Q_{2,x^{j}}(x) \]

with $Q_{i,x^{j}}(x)=Q_{i}(x-x^{j}),\, i=1,\,2.$ We define

\begin{align*} H_s& =\left\{u:u\ is\ even\ in\ y_2,y_3,\ u(r\cos\theta,r\sin\theta,y'')\vphantom{\left(r\cos\left(\theta+\frac{2\pi j}k\right),r\sin\left(\theta+\frac{2\pi j}k\right),y''\right)}\right.\\& \left.\quad=u\left(r\cos\left(\theta+\frac{2\pi j}k\right),r\sin\left(\theta+\frac{2\pi j}k\right),y''\right)\right\}.\end{align*}

Theorem A [c.f [Reference Peng and Wang15]] Suppose $V_i(r),\,i=1,\,2$ satisfy $(V)$. Then there exists an integer $k_0>0$, such that for any $k\geq k_0$, the system (1.1) has a vector solution $(u_{1,k},\,u_{2,k})$ of the form

\[ (u_{1,k},u_{2,k})=(U_{1,r_k},U_{2,r_k})+(w_{1,k},w_{2,k}), \]

where $(w_{1,k},\,w_{2,k})\in (H_s\cap H^{1}(\mathbb {R}^{3}))^{2}$, $r_k\in [r_0klnk,\,r_1klnk]$, and as $k\rightarrow +\infty$,

\[ \|(w_{1,k},w_{2,k})\|_{H^{1}\times H^{1}}\rightarrow 0, \]

where $\|(w_{1,k},\,w_{2,k})\|_{H^{1}\times H^{1}}=(\|w_{1,k}\|_{H^{1}}+\|w_{2,k}\|_{H^{1}} )^{1/2}$.

Inspired by [Reference Guo, Musso, Peng and Yan10], we are aimed to prove the non-degeneracy property of the solution $(u_{1,k},\,u_{2,k})$ to (1.1) constructed in Theorem A, in the sense that the linearized operator $L_k$ defined by

\begin{align*} & \big\langle L_k (\psi_1,\psi_2),(\phi_1,\phi_2)\big\rangle_\lambda :=\displaystyle\sum^{2}_{i=1}\displaystyle\int_{\mathbb{R}^{3}} [\nabla \psi_i\nabla\phi_i + V_i(x)\psi_i\phi_i -3 \mu_i u_{i,k}^{2}\psi_i\phi_i]\\ & \quad-\int_{\mathbb{R}^{3}} [2\beta u_{1,k} u_{2,k} \psi_2\phi_1+\beta u_{2,k}^{2}\psi_1\phi_1 ]- \int_{\mathbb{R}^{3}} [2\beta u_{1,k}u_{2,k}\psi_1\phi_2+\beta u_{1,k}^{2}\psi_2\phi_2] \end{align*}

has trivial kernel in $(H_s\cap H^{1}(\mathbb {R}^{3}))^{2}$. We note that the non-degenerate result plays an essential role to construct solutions when Lyapunov Schmidt reduction is used in the systems. For the non-degeneracy and the locally uniqueness results for Schrödinger equations or systems, one can refer to [Reference Guo, Peng and Yan6, Reference Guo, Musso, Peng and Yan10, Reference Luo, Peng and Yan13, Reference Noussair and Yan14], etc.

To this aim, we need to impose the conditions for the potential functions $V_1$ and $V_2$:

($H$): There are constants $m_i>2$, $a_i>0,\,b_i>0$ and $\theta >0$ such that as $r\rightarrow +\infty$

\begin{align*} V_1(r)& =1+\displaystyle\frac{a_1}{r^{m_1}}+\frac{b_1}{r^{m_1+1}}+O\Big(\frac 1{r^{m_1+2}}\Big),\\ V'_1(r)& ={-}\frac{a_1m_1}{r^{m_1+1}}-\frac{b_1(m_1+1)}{r^{m_1+2}}+O\Big(\frac 1{r^{m_1+3}}\Big);\\ V_2(r)& =1+\displaystyle\frac{a_2}{r^{m_2}}+\frac{b_2}{r^{m_2+1}}+O\Big(\frac 1{r^{m_2+2}}\Big),\\ V'_2(r)& ={-}\frac{a_2m_2}{r^{m_2+1}}-\frac{b_2(m_2+1)}{r^{m_2+2}}+O\Big(\frac 1{r^{m_2+3}}\Big). \end{align*}

The main result of this paper is the following.

Theorem 1.1 Assume that ($H$) holds with $V_i(x),\, i=1,\,2$. Let $(\xi _{1},\,\xi _{2})\in (H_s\cap H^{1}(\mathbb {R}^{3}))^{2}$ be a solution of $L_k(\xi _{1},\,\xi _{2})=0$. Then $(\xi _{1},\,\xi _{2})=0$.

Remark 1.2 Compared with the study of the existence of the concentrated solution, the difficulties brought by the coupling terms of the system are more complex in the study of the non-degeneracy result. We have to deal with the delicate estimates for the interspecies interaction between the components of the system (1.1), which never appear in the study of single equations. We adopted some new observations and our method is inspired by the characterization of the Fermat points in the famous Fermat problem. For more details, one can see the arguments in §3.

The paper is organized as follows. Precise point-wise estimates are needed in the proof of the non-degeneracy result, so we first revisit the existence problem for (1.1) in § 2. In § 3, we establish some local Pohozaev identities and prove the non-degenerate result.

2. Revisit the existence problem

In this section, we give some preliminary results. We revisit the existence problem for (1.1) and give a different proof of Theorem A under the assumption $(H)$, so that we can obtain extra point-wise estimates for the proof non-degeneracy result.

Set

\[ Z_{i,j}=\frac{\partial Q_{i,x^{j}}}{\partial r},\ i=1,2, \]

where $x^{j}$ is defined by

\[ x^{j}=\left(r\cos\displaystyle\frac{2(j-1)\pi}{k},r\sin\frac{2(j-1)\pi}{k},0\right),\quad j=1,\ldots,k. \]

Note that for $k\gg 1$,

\[ |x^{1}-x^{2}|=2r\sin\displaystyle\frac\pi k\thicksim \frac{2r\pi}k. \]

We define

\begin{align*} \tilde H_s & =\left\{\vphantom{\left(\theta+\frac{2\pi j}k\right)}u:u\ is\ even\ in\ y_2,y_3,\ u(y',y'')=u(y',|y''|),\right.\\ & \quad \left.u(r\cos\theta,r\sin\theta,y'')=u\left(r\cos\left(\theta+\displaystyle\frac{2\pi j}k\right),r\sin\left(\theta+\frac{2\pi j}k\right),y''\right) \right\}. \end{align*}

Set

\[ \tilde E_k=\bigg\{(u,v)\in (\tilde H_s\cap C(R^{3}))^{2},\quad \sum_{j=1}^{k}\int_{\mathbb{R}^{3}}W_{x^{j}}^{2}(Z_{1,j}u+Z_{2,j}v){\rm d}y=0\bigg\}, \]

with $W_{x^{j}}(x)=W(x-x^{j}).$ Set

\[ S_k=\left[\left(\frac{m}{2\pi}-\delta\right)klnk,\left(\frac{m}{2\pi}+\delta\right)klnk\right], \]

where $m=\min \{m_1,\,m_2\}$, $\delta >0$ is a small constant, and we always assume $r\in S_k$.

Now we define the weighted $L^{\infty }$-norm as

\[ \|u\|_*=\sup_{y\in\mathbb{R}^{3}}\frac{|u(y)|}{\sum_{j=1}^{k}e^{-\tau|y-x^{j}|}}. \]

Here $\tau$ is a small parameter which will be determined later.

Let

\[ \mathbb{L}(\varphi_{1},\varphi_{2})=\begin{pmatrix} -\Delta\varphi_{1}+V_1(y)\varphi_{1}-3\mu_1U_{1,r}^{2}\varphi_{1} -2\beta U_{1,r}U_{2,r}\varphi_{2}-\beta U_{2,r}^{2}\varphi_{1}\nonumber \\ -\Delta\varphi_{2}+V_2(y)\varphi_{2}-3\mu_2U_{2,r}^{2}\varphi_{2} -2\beta U_{1,r}U_{2,r}\varphi_{1}-\beta U_{1,r}^{2}\varphi_{2} \end{pmatrix}^{\top}. \]

For $(f_1,\,f_2)\in (\tilde H_s\cap C(\mathbb {R}^{3}))^{2}$, we consider the following problem

(2.1)\begin{equation} \mathbb{L}(\varphi_{1},\varphi_{2})=(f_1,f_2)+b_k\left(\displaystyle\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{1,j},\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{2,j}\right) \end{equation}

with $(\varphi _{1},\,\varphi _{2})\in \tilde E_k$ and $b_k\in \mathbb {R}$.

Lemma 2.1 Assume $(\varphi _{1,k},\,\varphi _{2,k})$ and $b_k$ solve (2.1). If $\|(f_{1,k},\,f_{2,k})\|_*\rightarrow 0,$ as $k\rightarrow +\infty,$ then $\|(\varphi _{1,k},\,\varphi _{2,k})\|_*\rightarrow 0$.

Proof. By contradiction, we suppose there are $k_n\rightarrow +\infty,\, r_{k_n}\in S_{k_n}$ and $(\varphi _{1,k_n},\,\varphi _{2,k_n})$ and $b_{k_n}$ solve (2.1), such that $\|(f_{1,k_n},\,f_{2,k_n})\|_*\rightarrow 0$ and $\|(\varphi _{1,k_n},\,\varphi _{2,k_n})\|_*=1$. For simplicity, we drop the subscript $k_n$.

Using the Green's function $G_i(y,\,x)$ of $-\Delta +V_i(|y|)$ with $i=1,\,2$, we have

\begin{align*} & \varphi_1(x)=\displaystyle\int_{\mathbb{R}^{3}}G_1(y,x)\bigg(3\mu_1U_{1,r}^{2}\varphi_{1} +2\beta U_{1,r}U_{2,r}\varphi_{2}+\beta U_{2,r}^{2}\varphi_{1}+f_1+b_k\displaystyle\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{1,j}\bigg){\rm d}y,\\ & \varphi_2(x)=\displaystyle\int_{\mathbb{R}^{3}}G_2(y,x)\bigg(3\mu_2U_{2,r}^{2}\varphi_{2} +2\beta U_{1,r}U_{2,r}\varphi_{1}+\beta U_{1,r}^{2}\varphi_{2}+f_2+b_k\displaystyle\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{2,j}\bigg){\rm d}y. \end{align*}

First, we could prove that

\begin{align*} \left|\displaystyle\int_{\mathbb{R}^{3}}G_i(y,x)f_i \ {\rm d}y \right| & \leq C\|f_i\|_*\displaystyle\sum_{j=1}^{k}\int_{\mathbb{R}^{3}}G_i(y,x)e^{-\tau|y-x^{j}|}{\rm d}y\\ & \leq C \|f_i\|_*\sum_{j=1}^{k}e^{-\tau|x-x^{j}|},\quad i=1,2. \end{align*}

For $\tau >0$ small,

\begin{align*} & \left|\displaystyle\int_{\mathbb{R}^{3}}G_1(y,x)(3\mu_1U_{1,r}^{2}\varphi_{1} +2\beta U_{1,r}U_{2,r}\varphi_{2}+\beta U_{2,r}^{2}\varphi_{1}){\rm d}y\right|\\ & \quad\leq C\|(\varphi_{1},\varphi_{2})\|_*\int_{\mathbb{R}^{3}}G_1(y,x)\left(\displaystyle\sum_{j=1}^{k}e^{-|y-x^{j}|}\right)^{2}\sum_{j=1}^{k}e^{-\tau|y-x^{j}|}{\rm d}y\\ & \quad\leq C\|\left(\varphi_{1},\varphi_{2}\right)\|_*\sum_{j=1}^{k}e^{{-}2\tau|x-x^{j}|}, \end{align*}

and similarly,

\[ \left|\displaystyle\int_{\mathbb{R}^{3}}G_2(y,x)(3\mu_2U_{2,r}^{2}\varphi_{2} +2\beta U_{1,r}U_{2,r}\varphi_{1}+\beta U_{1,r}^{2}\varphi_{2}){\rm d}y\right|\leq C\|(\varphi_{1},\varphi_{2})\|_*\displaystyle\sum_{j=1}^{k}e^{{-}2\tau|x-x^{j}|}. \]

Moreover,

\[ \left|\displaystyle\int_{\mathbb{R}^{3}}G_i(y,x)\displaystyle\sum_{j=1}^{k}Q_{i,x^{j}}^{2}Z_{i,j}\right|\leq C\sum_{j=1}^{k}e^{-|x-x^{j}|},\quad i=1,2. \]

On the other hand, from

\[ b_k\displaystyle\int_{\mathbb{R}^{3}}\displaystyle\sum_{i=1}^{2}\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{i,j}Z_{i,1}=\langle\mathbb{L}(\varphi_1,\varphi_2)-(f_1,f_2),(Z_{1,1},Z_{2,1})\rangle, \]

we can show that $b_k\rightarrow 0$. Therefore,

(2.2)\begin{align} & |\varphi_1|\left(\displaystyle\sum_{j=1}^{k}e^{-\tau|x-x^{j}|}\right)^{{-}1}+|\varphi_2|\left(\sum_{j=1}^{k}e^{-\tau|x-x^{j}|}\right)^{{-}1}\nonumber\\ & \quad\leq o(1)+C\|(\varphi_{1},\varphi_{2})\|_*\displaystyle\sum_{j=1}^{k}e^{{-}2\tau|x-x^{j}|}\left(\sum_{j=1}^{k}e^{-\tau|x-x^{j}|}\right)^{{-}1}. \end{align}

Since $(\varphi _1,\,\varphi _2)\in \tilde E_k$, using the non-degeneracy of the linear operator $\mathbb {L}$ (see [Reference Peng and Wang15]), we could prove that $(\varphi _{1,k},\,\varphi _{2,k})\rightarrow (0,\,0)$ uniformly in $B_R(x^{j})$, for any $R>0$. Hence, combining (2.2), we find that $\|(\varphi _{1,k},\,\varphi _{2,k})\|_*\rightarrow 0$. This is a contradiction.

Now, we solve the following nonlinear problem. For $(\varphi _1,\,\varphi _2)\in \tilde E_k$ and $b_k\in \mathbb {R}$,

(2.3)\begin{equation} \mathbb{L}(\varphi_1,\varphi_2)=l_k+R(\varphi_1,\varphi_2)+ b_k\left(\displaystyle\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{1,j},\sum_{j=1}^{k}W_{x^{j}}^{2}Z_{2,j}\right), \end{equation}

where $l_k=(l_{1,k},\,l_{2,k})$ with

\begin{align*} & l_{1,k}=\displaystyle\sum_{j=1}^{k}(V_1(|x|)-1)Q_{1,x^{j}}+\mu_1\bigg(U_{1,r}^{3}-\sum_{j=1}^{k}Q_{1,x^{j}}^{3}\bigg)+ \beta\bigg(U_{2,r}^{2}U_{1,r}-\sum_{j=1}^{k}Q_{2,x^{j}}^{2}Q_{1,x^{j}}\bigg),\\ & l_{2,k}=\sum_{j=1}^{k}(V_2(|x|)-1)Q_{2,x^{j}}+\mu_2\bigg(U_{2,r}^{3}-\sum_{j=1}^{k}Q_{2,x^{j}}^{3}\bigg)+ \beta\bigg(U_{1,r}^{2}U_{2,r}-\sum_{j=1}^{k}Q_{1,x^{j}}^{2}Q_{2,x^{j}}\bigg), \end{align*}

and $R(\varphi _1,\,\varphi _2)=(R_{1}(\varphi _1,\,\varphi _2),\,R_{2}(\varphi _1,\,\varphi _2))$ with

\begin{align*} & R_{1}(\varphi_1,\varphi_2)=\mu_1(\varphi_1^{3}+3U_{1,r}\varphi_1^{2})+ \beta(\varphi_1\varphi_2^{2}+U_{1,r}\varphi_2^{2}+2U_{2,r}\varphi_1\varphi_2),\\ & R_{2}(\varphi_1,\varphi_2)=\mu_2(\varphi_2^{3}+3U_{2,r}\varphi_2^{2})+ \beta(\varphi_2\varphi_1^{2}+U_{2,r}\varphi_1^{2}+2U_{1,r}\varphi_1\varphi_2). \end{align*}

For the existence result, we first estimate $l_k$ just as in [Reference Guo, Musso, Peng and Yan10].

Lemma 2.2 We have

\[ \|l_{1,k}\|_*\leq\displaystyle\frac C{r^{m_1}}+Ce^{-|x^{1}-x^{2}|},\quad \|l_{2,k}\|_*\leq\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}. \]

Applying Lemma 2.1, we use the contraction mapping theorem to prove that there exists an integer $k_0>0$, such that for any integer $k\geq k_0$, (2.3) has a solution $(\varphi _1,\,\varphi _2)\in \tilde E_k$ satisfying

(2.4)\begin{equation} \|(\varphi_1,\varphi_2)\|_*\leq\displaystyle\frac C{r^{m_1}}+\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}. \end{equation}

Let

\begin{align*} \mathbb{R}(\varphi_1,\varphi_2)& =\displaystyle\int_{\mathbb{R}^{3}}\mu_1\left(\displaystyle\frac{\varphi_1^{4}}4+U_{1,r}\varphi_1^{3}\right)+\mu_2\left(\frac{\varphi_2^{4}}4+U_{2,r}\varphi_2^{3}\right)\\& \quad + \frac{\beta}{2}(\varphi_1^{2}\varphi_2^{2}+2U_{1,r}\varphi_1\varphi_2^{2}+2U_{2,r}\varphi_1^{2}\varphi_2).\\ I(u,v)& =\displaystyle\int_{\mathbb{R}^{3}}\displaystyle\frac 12(|\nabla u|^{2}+V_1(x)u^{2}+|\nabla v|^{2}+V_2(x)v^{2})-\frac 14(\mu_1u^{4}+\mu_2v^{4}+2\beta u^{2}v^{2}). \end{align*}

We give the following expansion of energy $I(U_{1,r}+\varphi _1,\,U_{2,r}+\varphi _2)$ by careful calculation.

Lemma 2.3

\begin{align*} I(U_{1,r}+\varphi_1,U_{2,r}+\varphi_2)& =k\displaystyle\frac{\mu_1+\mu_2-2\beta}{4(\mu_1\mu_2-\beta^{2})}\displaystyle\int_{R^{3}}W^{4} +\left(\frac{ka_1\gamma_1^{2}}{2r^{m_1}}+\frac{ka_2\gamma_2^{2}}{2r^{m_2}}\right)\int_{\mathbb{R}^{3}}W^{2}{\rm d}x \\ & \quad-k(D+\beta H)W(|x^{1}-x^{2}|)+O\left(\displaystyle\frac 1{k^{m_1-1+\sigma}}+\frac 1{k^{m_2-1+\sigma}}\right). \end{align*}

where $D,\,H$ are positive constants independent of $k$, and for $\beta >-\sqrt {\mu _1\mu _2}$ it holds that $D+\beta H>0$.

Proof. The proof is standard and we only sketch the main steps.

First, by expansion,

\begin{align*} I(U_{1,r}+\varphi_1,U_{2,r}+\varphi_2) & =I(U_{1,r},U_{2,r})+l_k(\varphi_1,\varphi_2)+\displaystyle\frac 12\langle \mathbb{L}(\varphi_1,\varphi_2),(\varphi_1,\varphi_2)\rangle+\mathbb{R}(\varphi_1,\varphi_2)\\ & =I(U_{1,r},U_{2,r})+ke^{\epsilon|x^{1}-x^{2}|}O(\|l_k\|_*\|(\varphi_1,\varphi_2)\|_*+\|(\varphi_1,\varphi_2)\|_*^{2})\\ & =I(U_{1,r},U_{2,r})+O\left(\displaystyle\frac 1{k^{m_1-1+\sigma}}+\frac 1{k^{m_2-1+\sigma}}\right). \end{align*}

Then we just need to estimate the main term $I(U_{1,r},\,U_{2,r})$ following [Reference Peng and Wang15] to have that

\begin{align*} I(U_{1,r},U_{2,r})& =k\displaystyle\frac{\mu_1+\mu_2-2\beta}{4(\mu_1\mu_2-\beta^{2})}\displaystyle\int_{R^{3}}W^{4}\\ & \quad +k\left(\left(\frac{a_1\gamma_1^{2}}{2r^{m_1}}+\frac{a_2\gamma_2^{2}}{2r^{m_2}}\right) \int_{\mathbb{R}^{3}}W^{2} +O\left(\frac 1{r^{m_1}k^{\sigma}}+\frac 1{r^{m_2}k^{\sigma}}\right)\right)\\ & \quad-k\int_{\Omega_1}\left(\frac{\mu_1\gamma_1}2Q_{1,x^{1}}^{3} +\frac{\mu_2\gamma_2}2Q_{2,x^{1}}^{3} +\frac{\beta\gamma_1}2Q_{1,x^{1}}Q_{2,x^{1}}^{2} +\frac{\beta\gamma_2}2Q_{2,x^{1}}Q_{1,x^{1}}^{2}\right)\\ & \quad\times\displaystyle\sum_{i=2}^{k}\frac{e^{-|x^{1}-x^{i}|}}{|x^{1}-x^{i}|} +kO\left(\frac 1{r^{m_1}k^{\sigma}}+\frac 1{r^{m_1}k^{\sigma}}+e^{-\frac{3\pi r}k}\right)\\ & =k\frac{\mu_1+\mu_2-2\beta}{4(\mu_1\mu_2-\beta^{2})}\int_{R^{3}}W^{4}\\ & \quad +k\left(\left(\frac{a_1\gamma_1^{2}}{2r^{m_1}} +\frac{a_2\gamma_2^{2}}{2r^{m_2}}\right)\int_{\mathbb{R}^{3}}W^{2} +O\left(\frac 1{r^{m_1}k^{\sigma}}+\frac 1{r^{m_2}k^{\sigma}}\right)\right)\\ & \quad-k(D+\beta H)W(|x^{1}-x^{2}|)+kO\left(\frac 1{r^{m_1}k^{\sigma}}+\frac 1{r^{m_1}k^{\sigma}}+e^{-\frac{3\pi r}k}\right), \end{align*}

where $D,\,H>0$ are constants independent of $k$. For $\beta >-\sqrt {\mu _1\mu _2}$, it is easy to get $D+\beta H>0$.

At the end of this section, we claim that

(2.5)\begin{equation} \displaystyle\sum_{i=1}^{2}\displaystyle\frac{b_i(m_i+1)\gamma_i^{2}}{2r^{m_i+1}}\displaystyle\int_{\mathbb{R}^{3}}W^{2}\sim W(|x^{1}-x^{2}|)\frac 1k, \end{equation}

As a matter of fact, we know from the equation that

\begin{align*} \begin{cases} \displaystyle\int_{\mathbb{R}^{3}}(-\Delta(U_{1,r}+\varphi_1)+V_1(|y|)(U_{1,r}+\varphi_1)-\mu_1(U_{1,r}+\varphi_1)^{3}\\ \displaystyle\quad-\beta(U_{1,r}+\varphi_1)(U_{2,r}+\varphi_2)^{2})\displaystyle\frac{\partial Q_{1,x^{1}}}{\partial y_1}=0 \\\displaystyle\int_{\mathbb{R}^{3}}(-\Delta(U_{2,r}+\varphi_2)+V_2(|y|)(U_{2,r}+\varphi_2) -\mu_2(U_{2,r}+\varphi_2)^{3}\\ \displaystyle\quad-\beta(U_{2,r}+\varphi_2)(U_{1,r}+\varphi_1)^{2})\frac{\partial Q_{2,x^{1}}}{\partial y_1}=0, \end{cases} \end{align*}

which gives then

\[ \begin{cases} \displaystyle\int_{\mathbb{R}^{3}}((V_1(|y|)-1)(U_{1,r}+\varphi_1) -\mu_1\bigg((U_{1,r}+\varphi_1)^{3}-\displaystyle\displaystyle\sum_{j=1}^{k}Q^{3}_{1,x^{j}}\bigg)\\ \quad -\beta\bigg((U_{1,r}+\varphi_1)(U_{2,r}+\varphi_2)^{2}-\displaystyle\sum_{j=1}^{k}Q^{2}_{2,x^{j}}Q_{1,x^{j}})\bigg)\displaystyle\displaystyle\frac{\partial Q_{1,x^{1}}}{\partial y_1}=0 \\ \displaystyle\int_{\mathbb{R}^{3}}\bigg((V_2(|y|)-1)(U_{2,r}+\varphi_2) -\mu_2\bigg((U_{2,r}+\varphi_2)^{3}-\sum_{j=1}^{k}Q^{3}_{2,x^{j}}\bigg)\\ \quad -\beta\bigg((U_{2,r}+\varphi_2)(U_{1,r}+\varphi_1)^{2}-\displaystyle\sum_{j=1}^{k}Q^{2}_{1,x^{j}}Q_{2,x^{j}}\bigg)\bigg) \displaystyle\frac{\partial Q_{2,x^{1}}}{\partial y_1}=0. \end{cases} \]

We check that

\begin{align*} & \displaystyle\int_{\mathbb{R}^{3}} (V_i(|y|)-1)(U_{i,r}+\varphi_i)\displaystyle\frac{\partial Q_{i,x^{1}}}{\partial y_1}= \int_{\mathbb{R}^{3}} (V_i(|y|)-1)Q_{i,x^{1}}\frac{\partial Q_{i,x^{1}}}{\partial y_1} \\ & \qquad+\displaystyle\sum_{j=2}^{k}\displaystyle\int_{\mathbb{R}^{3}} (V_i(|y|)-1)Q_{i,x^{j}}\displaystyle\frac{\partial Q_{i,x^{1}}}{\partial y_1} +\int_{\mathbb{R}^{3}} (V_i(|y|)-1)\varphi_i\frac{\partial Q_{i,x^{1}}}{\partial y_1}\\ & \quad={-}\frac 12\int_{\mathbb{R}^{3}} \frac{\partial V_i(y)}{\partial y_1}Q_{i,x^{1}}^{2}+O\left(\frac 1{r^{2m_i}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right)\\ & \quad=\frac{b_i(m_i+1)x^{1}_1}{2r^{m_i+2}}\int_{\mathbb{R}^{3}}Q_i^{2}+O\left(\frac 1{r^{m_i+2}}+\frac 1{r^{2m_i}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right). \end{align*}

Moreover,

\begin{align*} & \displaystyle\int_{\mathbb{R}^{3}} \bigg((U_{i,r}+\varphi_i)^{3}-\displaystyle\sum_{j=1}^{k}Q^{3}_{i,x^{j}}\bigg)\displaystyle\frac{\partial Q_{i,x^{1}}}{\partial y_1}\\ & \quad=3\gamma_i^{4}\displaystyle\int_{\mathbb{R}^{3}}W^{2}_{i,x^{1}} \displaystyle\sum_{j=2}^{k}W_{x^{j}}\displaystyle\frac{\partial W_{x^{1}}}{\partial y_1}+O\left(\frac 1{r^{2m_i}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right)\\ & \quad={-}\gamma_i^{4}\int_{\mathbb{R}^{3}}W^{3}_{x^{1}} \sum_{j=2}^{k}\frac{\partial W_{x^{j}}}{\partial y_1}+O\left(\frac 1{r^{2m_i}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right)\\ & \quad=(C_i+o(1))\gamma_i^{4}W(|x^{1}-x^{2}|)\left(\frac{(x^{1}-x^{2})_1}{|x^{1}-x^{2}|}+\frac{(x^{1}-x^{k})_1}{|x^{1}-x^{k}|}\right)\\ & \qquad+O\left(\frac 1{r^{2m_i}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right). \end{align*}

Similarly,

\begin{align*} & \displaystyle\int_{\mathbb{R}^{3}} \bigg((U_{1,r}+\varphi_1)(U_{2,r}+\varphi_2)^{2}-\displaystyle\sum_{j=1}^{k}Q_{1,x^{j}}Q^{2}_{2,x^{j}}\bigg)\displaystyle\frac{\partial Q_{1,x^{1}}}{\partial y_1}\\ & \displaystyle\quad=(C_3+o(1))\gamma_1\gamma_2^{2}W(|x^{1}-x^{2}|)(\displaystyle\frac{(x^{1}-x^{2})_1}{|x^{1}-x^{2}|}+\frac{(x^{1}-x^{k})_1}{|x^{1}-x^{k}|})\\ & \displaystyle\qquad+O\left(\frac 1{r^{2m_1}}+\frac 1{r^{2m_2}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right);\\ & \quad\displaystyle\int_{\mathbb{R}^{3}} \bigg((U_{2,r}+\varphi_1)(U_{1,r}+\varphi_2)^{2}-\displaystyle\sum_{j=1}^{k}Q_{2,x^{j}}Q^{2}_{1,x^{j}}\bigg)\frac{\partial Q_{2,x^{1}}}{\partial y_1}\\ & \displaystyle\quad=(C_4+o(1))\gamma_2\gamma_1^{2}W(|x^{1}-x^{2}|)\left(\frac{(x^{1}-x^{2})_1}{|x^{1}-x^{2}|}+\frac{(x^{1}-x^{k})_1}{|x^{1}-x^{k}|}\right)\\ & \displaystyle\qquad+O\left(\frac 1{r^{2m_1}}+\frac 1{r^{2m_2}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right). \end{align*}

Since

\[ (x^{1}-x^{2})_1=(x^{1}-x^{k})_1=r\left(1-\cos\frac{2\pi}k\right)=r\left(\frac{2\pi^{2}}{k^{2}}+O\left(\frac 1{k^{4}}\right)\right), \]

then we use the fact that

\[ e^{{-}2(1-\tau)|x^{1}-x^{2}|}\leq \frac C{r^{(2-\tau)m}}=o\left(\frac 1{r^{m+1}}\right) \]

to obtain

\begin{align*} \displaystyle\sum_{i=1}^{2}\displaystyle\frac{b_i(m_i+1)\gamma_i^{2}}{2r^{m_i+1}}\displaystyle\int_{\mathbb{R}^{3}}W^{2}+O\left(\frac 1{r^{m+2}}+\frac 1{r^{2m}}+e^{{-}2(1-\tau)|x^{1}-x^{2}|}\right)\\ =(C_1\gamma_1^{3}+C_2\gamma_2^{3}+C_3\gamma_1\gamma_2^{2}+C_4\gamma_2\gamma_1^{2}+o(1))W(|x^{1}-x^{2}|)\frac 1k, \end{align*}

implying (2.5).

3. Non-degeneracy of the solutions

Let $(u_1,\,u_2)$ solve (1.1), and $(\xi _1,\,\xi _2)$ solve

\[ \begin{cases} -\Delta \xi_1 +V_1(|y|) \xi_1=3 \mu_1 u_1^{2}\xi_1 +\beta \xi_1u_2^{2} +2\beta u_1u_2\xi_2 & ~\text{in }\mathbb{R}^{3},\\ -\Delta \xi_2+V_2(|y|) \xi_2= 3\mu_2 u_2^{2}\xi_2 +\beta u_1^{2}\xi_2 +2\beta u_1u_2\xi_1 & ~\text{in }\mathbb{R}^{3}. \end{cases} \]

Lemma 3.1 For $i=1,\,2,\,3$, any $\Omega \subset \mathbb {R}^{3},$ it holds that

\begin{align*} & -\displaystyle\int_{\partial\Omega}\left(\displaystyle\frac{\partial u_1}{\partial\nu}\frac{\partial \xi_1}{\partial y_i}+\frac{\partial \xi_1}{\partial\nu}\frac{\partial u_1}{\partial y_i} +\frac{\partial u_2}{\partial\nu}\frac{\partial \xi_2}{\partial y_i}+\frac{\partial \xi_2}{\partial\nu}\frac{\partial u_2}{\partial y_i}\right) +\int_{\partial\Omega}\langle\nabla u_1,\nabla\xi_1\rangle\nu_i+\langle\nabla u_2,\nabla\xi_2\rangle\nu_i\\ & \qquad+\int_{\partial\Omega}(V_1(|y|)u_1\xi_1+V_2(|y|)u_2\xi_2)\nu_i-\int_{\partial\Omega} (\mu_1 u_1^{3}\xi_1 +\mu_2 u_2^{3}\xi_2 +\beta u_1 \xi_1u_2^{2} +\beta \xi_2u_2u_1^{2})\nu_i\\ & \quad=\displaystyle\int_\Omega\left(\displaystyle\frac{\partial{V_1}}{\partial y_i}u_1\xi_1+\frac{\partial{V_2}}{\partial y_i}u_2\xi_2\right). \end{align*}

Proof. The proof is standard and can be followed by [Reference Guo, Musso, Peng and Yan10].

Proof of Theorem 1.1 We argue by contradiction. Suppose that there exist $k_n\rightarrow +\infty$, satisfying $(\xi _{1,n},\,\xi _{2,n})\in (H_s\cap H^{1}(\mathbb {R}^{3}))^{2},\, \|(\xi _{1,n},\,\xi _{2,n})\|_*=1$ and $L_{k_n}(\xi _{1,n},\,\xi _{2,n})=0$.

Let

\[ \tilde\xi_{i,n}(y)=\xi_{i,n}(y+x^{1}_{k_n}). \]

We need the following lemmas to show a contradiction.

Lemma 3.2 It holds that

\[ \tilde\xi_{i,n}\rightarrow b\frac{\partial Q_i}{\partial y_1} \]

uniformly in $C^{1}(B_R(0))$ for any $R>0,$ with a constant $b$.

Proof. By assumption, we get $|\tilde \xi _{i,n}|\leq C$, and we may assume that $\tilde \xi _{i,n}\rightarrow \xi _i$ in $C_{loc}(\mathbb {R}^{3})$. Then $(\xi _1,\,\xi _2)$ satisfies that

(3.1)\begin{equation} \begin{cases} \displaystyle-\Delta\xi_1+\xi_1=3\mu_1Q_1^{2}\xi_1+\beta Q_2^{2}\xi_1 +2\beta Q_1Q_2\xi_2\\ \displaystyle-\Delta\xi_2+\xi_2=3\mu_2Q_2^{2}\xi_2 +\beta Q_1^{2}\xi_2+2\beta Q_1Q_2\xi_1, \end{cases} \end{equation}

which gives

\[ \xi_i=\sum_{l=1}^{3}b^{l}\frac{\partial Q_i}{\partial y_l}. \]

Since $\xi _i$ is even in $y_l,\,l=2,\,3$, it holds that $b^{2}=b^{3}=0$, concluding the proof with ${b=b^{1}}$.

We decompose

\[ \xi_{i,n}(y)=b_n\sum_{j=1}^{k}\frac{\partial Q_{i,x^{j}}}{\partial r}+\xi_{i,n}^{*},\ \ i=1,2, \]

where

\[ (\xi_{1,n}^{*},\xi_{2,n}^{*})\in E_k=\bigg\{(u,v)\in( H_s\cap C(R^{3}))^{2},\ \sum_{j=1}^{k}\int_{\mathbb{R}^{3}}W_{x^{j}}^{2}(Z_{1,j}u+Z_{2,j}v){\rm d}y=0\bigg\}. \]

From Lemma 3.2, $b_n$ is bounded.

Lemma 3.3 We have

\[ \|(\xi_{1,n}^{*},\xi_{2,n}^{*})\|_*\leq\displaystyle\frac C{r^{m_1}}+\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}. \]

Proof. We know that $(\xi _{1,n}^{*},\,\xi _{2,n}^{*})$ satisfies the system

\begin{align*}\begin{cases} \displaystyle-\Delta\xi_{1,n}^{*}+V_1(y)\xi_{1,n}^{*}-3\mu_1u_1^{2}\xi_{1,n}^{*}-\beta u_2^{2}\xi_{1,n}^{*}-2\beta u_1u_2\xi_{2,n}^{*}\\\quad \displaystyle=b_k(V_1(|y|)-1)\displaystyle\sum_{j=1}^{k}\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r}+3\mu_1b_k\sum_{j=1}^{k}(Q_{1,x^{j}}^{2}-u_1^{2})\frac{\partial Q_{1,x^{j}}}{\partial r}\\ \displaystyle\qquad+b_k\beta\sum_{j=1}^{k}(Q_{2,x^{j}}^{2}-u_2^{2})\frac{\partial Q_{1,x^{j}}}{\partial r} +2b_k\beta\sum_{j=1}^{k}(Q_{1,x^{j}}Q_{2,x^{j}}-u_1u_2)\frac{\partial Q_{2,x^{j}}}{\partial r} \\ \displaystyle\displaystyle-\Delta\xi_{2,n}^{*}+V_2(y)\xi_{2,n}^{*}-3\mu_2u_2^{2}\xi_{2,n}^{*}-\beta u_1^{2}\xi_{2,n}^{*}-2\beta u_1u_2\xi_{1,n}^{*}\\ \displaystyle\quad =b_k(V_2(|y|)-1)\sum_{j=1}^{k}\frac{\partial Q_{1,x^{j}}}{\partial r}+3\mu_2b_k\sum_{j=1}^{k}(Q_{2,x^{j}}^{2}-u_2^{2})\frac{\partial Q_{2,x^{j}}}{\partial r}\\ \displaystyle\qquad+b_k\beta\sum_{j=1}^{k}(Q_{1,x^{j}}^{2}-u_1^{2})\frac{\partial Q_{2,x^{j}}}{\partial r} +2b_k\beta\sum_{j=1}^{k}(Q_{2,x^{j}}Q_{1,x^{j}}-u_1u_2)\frac{\partial Q_{1,x^{j}}}{\partial r}. \end{cases} \end{align*}

Then we know that

\[ L_k(\xi_{1,n}^{*},\xi_{2,n}^{*})=\begin{pmatrix} \displaystyle b_k(V_1(|y|)-1)\displaystyle\sum_{j=1}^{k}\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r}+3\mu_1b_k\sum_{j=1}^{k}(Q_{1,x^{j}}^{2}-u_1^{2})\frac{\partial Q_{1,x^{j}}}{\partial r}\\ \displaystyle+b_k\beta\displaystyle\sum_{j=1}^{k}(Q_{2,x^{j}}^{2}-u_2^{2})\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r} +2b_k\beta\sum_{j=1}^{k}(Q_{1,x^{j}}Q_{2,x^{j}}-u_1u_2)\frac{\partial Q_{2,x^{j}}}{\partial r} \\ \displaystyle b_k(V_2(|y|)-1)\sum_{j=1}^{k}\frac{\partial Q_{1,x^{j}}}{\partial r}+3\mu_2b_k\sum_{j=1}^{k}(Q_{2,x^{j}}^{2}-u_1^{2})\frac{\partial Q_{2,x^{j}}}{\partial r}\\ \displaystyle +b_k\beta\sum_{j=1}^{k}(Q_{1,x^{j}}^{2}-u_1^{2})\frac{\partial Q_{2,x^{j}}}{\partial r} +2b_k\beta\sum_{j=1}^{k}(Q_{2,x^{j}}Q_{1,x^{j}}-u_1u_2)\frac{\partial Q_{1,x^{j}}}{\partial r}\end{pmatrix}^{\top}.\]

Similar to Lemma 3.3, we could prove that

\[ \|L_k(\xi_{1,n}^{*},\xi_{2,n}^{*})\|_*\leq\displaystyle\frac C{r^{m_1}}+\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}. \]

Moreover, since $(\xi _{1,n}^{*},\,\xi _{2,n}^{*})\in \tilde E_k$, it is standard to prove that

\[ \|L_k(\xi_{1,n}^{*},\xi_{2,n}^{*})\|_*\geq\rho\|(\xi_{1,n}^{*},\xi_{2,n}^{*})\|_*, \]

which gives the conclusion.

Lemma 3.4 It holds that

\[ \tilde\xi_{i,n}\rightarrow 0,\ i=1,2 \]

uniformly in $C^{1}(B_R(0))$ for any $R>0$.

Proof. We apply Lemma 3.1 with $\Omega =B_{\frac 12|x^{1}-x^{2}|}(x^{1})$ and drop the subscript $n$. Define the following bilinear form

\begin{align*} \mathcal{L}(\vec{u},\vec\xi,\Omega)& ={-}\displaystyle\int_{\partial\Omega}\left(\displaystyle\frac{\partial u_1}{\partial\nu}\frac{\partial \xi_1}{\partial y_i}+\frac{\partial \xi_1}{\partial\nu}\frac{\partial u_1}{\partial y_i} +\frac{\partial u_2}{\partial\nu}\frac{\partial \xi_2}{\partial y_i}+\frac{\partial \xi_2}{\partial\nu}\frac{\partial u_2}{\partial y_i}\right)\\ & \quad+\int_{\partial\Omega}\langle\nabla u_1,\nabla\xi_1\rangle\nu_i+\langle\nabla u_2,\nabla\xi_2\rangle\nu_i +\int_{\partial\Omega}(u_1\xi_1+u_2\xi_2)\nu_i, \end{align*}

where $\vec u=(u_1,\,u_2),\,\vec \xi =(\xi _1,\,\xi _2)$, and $\nu$ is the outward unit normal of $\partial \Omega$. Then Lemma 3.1 implies that

(3.2)\begin{equation} \begin{aligned} & \mathcal{L}(\vec{u},\vec\xi,\Omega)+\displaystyle\int_{\partial\Omega}\left((V_1(|y|)-1)u_1\xi_1+(V_2(|y|)-1)u_2\xi_2\right)\nu_i\\ & \quad-\displaystyle\int_{\partial\Omega} (\mu_1 u_1^{3}\xi_1 +\mu_2 u_2^{3}\xi_2 +\beta u_1 \xi_1u_2^{2} +\beta \xi_2u_2u_1^{2})\nu_i =\int_\Omega\left(\displaystyle\frac{\partial{V_1}}{\partial y_i}u_1\xi_1+\frac{\partial{V_2}}{\partial y_i}u_2\xi_2\right). \end{aligned} \end{equation}

Now we rewrite $\mathcal {L}(\vec {u},\,\vec \xi,\,\Omega )$ as

\begin{align*} \mathcal{L}(\vec{u},\vec\xi,\Omega)& =\displaystyle\int_\Omega(-\Delta u_1+u_1-\mu_1u_1^{3}-\beta u_1u_2^{2})\displaystyle\frac{\partial \xi_1}{\partial y_1} +\int_\Omega(-\Delta u_2+u_2-\mu_2u_2^{3}-\beta u_2u_1^{2})\frac{\partial \xi_2}{\partial y_1}\nonumber\\ & \quad+\displaystyle\int_\Omega(-\Delta \xi_1+\xi_1-3\mu_1u_1^{2}\xi_1-2\beta u_1u_2\xi_2-\beta u_2^{2}\xi_1)\displaystyle\frac{\partial u_1}{\partial y_1} \nonumber\\ & \quad +\int_\Omega(-\Delta \xi_2+\xi_2-3\mu_2u_2^{2}\xi_2-2\beta u_1u_2\xi_1-\beta u_1^{2}\xi_2)\frac{\partial u_2}{\partial y_1}\nonumber\\ & \quad+\int_{\partial\Omega}(\mu_1u_1^{3}\xi_1+\mu_2u_2^{3}\xi_2+\beta u_1\xi_1u_2^{2}+\beta u_2\xi_2u_1^{2})\nu_1. \end{align*}

Recall that

\[ u_i=\sum_{j=1}^{k}Q_{i,x^{j}}+\varphi_1,\quad \xi_{i,n}(y)=b_n\sum_{j=1}^{k}\frac{\partial Q_{i,x^{j}}}{\partial r}+\xi_{i,n}^{*},\quad i=1,2. \]

We set

\[ I_{i,j,k,l}=\mathcal{L}\left((Q_{1,x^{i}},Q_{2,x^{j}}),\left(\frac{\partial Q_{1,x^{k}}}{\partial r},\frac{\partial Q_{2,x^{l}}}{\partial r}\right),\Omega\right). \]

Then

\begin{align*} I_{j,j,j,j}& =\displaystyle\int_{\partial\Omega}\left(\mu_1Q_{1,x^{j}}^{3}\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r}+\mu_2Q_{2,x^{j}}^{3}\frac{\partial Q_{2,x^{j}}}{\partial r}\right.\\ & \left.\quad+\beta Q_{1,x^{j}}\frac{\partial Q_{1,x^{j}}}{\partial r}Q_{2,x^{j}}^{2}+\beta Q_{2,x^{j}}\frac{\partial Q_{2,x^{j}}}{\partial r}Q_{1,x^{j}}^{2}\right)\nu_1\\ & =O(e^{-(2-\sigma)|x^{1}-x^{2}|}). \end{align*}

For general $I_{i,j,k,l}$, we observe first

(3.3)\begin{align} I_{i,j,k,l} & =\displaystyle\int_\Omega(-\Delta Q_{1,x^{i}}+Q_{1,x^{i}}-\mu_1Q_{1,x^{i}}^{3}-\beta Q_{1,x^{i}}Q_{2,x^{j}}^{2})\displaystyle\frac\partial{\partial y_1}\frac{\partial Q_{1,x^{k}}}{\partial r}\nonumber\\ & \quad+\displaystyle\int_\Omega(-\Delta Q_{2,x^{j}}+Q_{2,x^{j}}-\mu_2Q_{2,x^{j}}^{3}-\beta Q_{2,x^{j}}Q_{1,x^{j}}^{2})\displaystyle\frac\partial{\partial y_1}\frac{\partial Q_{2,x^{l}}}{\partial r}\nonumber\\ & \quad+\int_\Omega\left(-\Delta\frac{\partial Q_{1,x^{k}}}{\partial r}+\frac{\partial Q_{1,x^{k}}}{\partial r}-3\mu_1Q_{1,x^{i}}^{2}\frac{\partial Q_{1,x^{k}}}{\partial r}\right.\nonumber\\ & \left.\quad -\,2\beta Q_{1,x^{i}}Q_{2,x^{j}}\frac{\partial Q_{2,x^{l}}}{\partial r}-\beta Q_{2,x^{j}}^{2}\frac{\partial Q_{1,x^{k}}}{\partial r}\right)\frac{\partial Q_{1,x^{i}}}{\partial y_1}\nonumber\\ & \quad+\int_\Omega\left(-\Delta\frac{\partial Q_{2,x^{l}}}{\partial r}+\frac{\partial Q_{2,x^{l}}}{\partial r}-3\mu_2Q_{2,x^{j}}^{2}\frac{\partial Q_{2,x^{l}}}{\partial r}\right.\nonumber\\ & \left.\quad -\,2\beta Q_{1,x^{i}}Q_{2,x^{j}}\frac{\partial Q_{1,x^{k}}}{\partial r}-\beta Q_{1,x^{i}}^{2}\frac{\partial Q_{2,x^{l}}}{\partial r}\right) \frac{\partial Q_{2,x^{j}}}{\partial y_1}+ O(e^{-(2-\sigma)|x^{1}-x^{2}|})\nonumber\\ & =\int_\Omega \beta Q_{1,x^{i}}(Q_{2,x^{i}}^{2}-Q_{2,x^{j}}^{2})\frac\partial{\partial y_1}\frac{\partial Q_{1,x^{k}}}{\partial r} +\beta Q_{2,x^{j}}(Q_{1,x^{j}}^{2}-Q_{1,x^{i}}^{2})\frac\partial{\partial y_1}\frac{\partial Q_{2,x^{l}}}{\partial r}\nonumber\\ & \quad+\int_\Omega 3\mu_1(Q_{1,x^{k}}^{2}-Q_{1,x^{i}}^{2})\frac{\partial Q_{1,x^{k}}}{\partial r}\frac{\partial Q_{1,x^{i}}}{\partial y_1} +3\mu_2(Q_{2,x^{l}}^{2}-Q_{2,x^{j}}^{2})\frac{\partial Q_{2,x^{l}}}{\partial r}\frac{\partial Q_{2,x^{j}}}{\partial y_1}\nonumber\\ & \quad+2\beta \left(Q_{1,x^{k}}Q_{2,x^{k}}\frac{\partial Q_{2,x^{k}}}{\partial r}- Q_{1,x^{i}}Q_{2,x^{j}}\frac{\partial Q_{2,x^{l}}}{\partial r})\frac{\partial Q_{1,x^{i}}}{\partial y_1}\right.\nonumber\\ & \left.\quad+\,2\beta (Q_{1,x^{l}}Q_{2,x^{l}}\frac{\partial Q_{1,x^{l}}}{\partial r}- Q_{1,x^{i}}Q_{2,x^{j}}\frac{\partial Q_{1,x^{k}}}{\partial r}\right)\frac{\partial Q_{2,x^{j}}}{\partial y_1}\nonumber\\ & \quad+\beta (Q_{2,x^{k}}^{2}-Q_{2,x^{j}}^{2})\frac{\partial Q_{1,x^{k}}}{\partial r}\frac{\partial Q_{1,x^{i}}}{\partial y_1} +\beta (Q_{1,x^{l}}^{2}-Q_{1,x^{i}}^{2})\frac{\partial Q_{2,x^{l}}}{\partial r}\frac{\partial Q_{2,x^{j}}}{\partial y_1}\nonumber\\ & \quad+ O(e^{-(2-\sigma)|x^{1}-x^{2}|}). \end{align}

Remark 3.5 This general term does not appear in the study of a single equation [Reference Guo, Musso, Peng and Yan10]. Compared with the study of the existence of the concentrated solution, the difficulties brought by the coupling terms of the system are more evident and more complex in the study of the non-degeneracy result.

In order to estimate (3.3), we first refer the following result.

Lemma 3.6 (c.f. [Reference Ambrosetti, Colorado and Ruiz1])

Given $u_1,\,u_2,\,u_3: \mathbb {R}^{3}\rightarrow \mathbb {R}$ three positive continuous radial functions such that

\[ u_1(x)+u_2(x)+u_3(x)\leq C(1+|x|)^{a}e^{{-}b|x|} \]

for some $a\in \mathbb {R},\,b>0$. Take $p_1,\,p_2,\,p_3$ three fixed points in $\mathbb {R}^{3}$. Let $\lambda \rightarrow +\infty$, and denote $\xi _i=\lambda p_i$, $i=1,\,2,\,3$. Then,

\[ \displaystyle\int_{\mathbb{R}^{3}}u_{1}(x-\xi_1)u_{2}(x-\xi_2)u_{3}(x-\xi_3){\rm d}x\leq Ce^{{-}bqr}, \]

where $C=C(q)$ is some positive constant and $q$ is any fixed quantity satisfying

\[ q< q_0:=\min_{p\in\mathbb{R}^{3}}\{|p-p_1|+|p-p_2|+|p-p_3|\}. \]

Remark 3.7 From the well-known Fermat problem, we know that $q_0$ is obtained as the Fermat point, which is just $p_2$ when the angle between $\overrightarrow {p_2p_1}$ and $\overrightarrow {p_2p_3}$ is not smaller than $\frac 23\pi$. In this case, we in fact show that $q_0=|p_1-p_2|+|p_3-p_2|$.

In view of Remark 3.7, we know from Lemma 3.6 that when $i,\,j,\,k$ do not equal to each other, terms in (3.3) can be estimated very well, since we note that the angle between $\overrightarrow {x_{1,2}x_{1,1}}$ and $\overrightarrow {x_{1,2}x_{1,3}}$ is definitely larger than $\frac 23\pi$ as $k\rightarrow +\infty$. For example, we estimate

\begin{align*} & \displaystyle\int_\Omega \beta Q_{1,x^{i}}Q_{2,x^{j}}^{2}\displaystyle\frac\partial{\partial y_1}\frac{\partial Q_{1,x^{k}}}{\partial r} +\beta Q_{2,x^{j}}Q_{1,x^{i}}^{2}\frac\partial{\partial y_1}\frac{\partial Q_{2,x^{k}}}{\partial r}\\ & \quad -2\beta Q_{1,x^{i}}Q_{2,x^{j}}\displaystyle\frac{\partial Q_{2,x^{k}}}{\partial r}\frac{\partial Q_{1,x^{i}}}{\partial y_1}-2\beta Q_{1,x^{i}}Q_{2,x^{j}}\frac{\partial Q_{1,x^{k}}}{\partial r}\frac{\partial Q_{2,x^{j}}}{\partial y_1}= O(e^{-(2-\sigma)|x^{1}-x^{2}|}). \end{align*}

By analysing various cases on $i,\,j,\,k,\,l$, we can obtain that

\begin{align*} & \mathcal{L} \bigg(\bigg(\displaystyle\sum_{i=1}^{k}Q_{1,x^{i}},\sum_{i=1}^{k}Q_{2,x^{i}}\bigg), \bigg(\sum_{j=1}^{k}\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r},\sum_{j=1}^{k}\frac{\partial Q_{2,x^{j}}}{\partial r}\bigg),\Omega\bigg)\\ & \quad={-}C(\mu_1\gamma_1^{4}+\mu_2\gamma_2^{4}+2\beta\gamma_1^{2}\gamma_2^{2}) \left(\displaystyle\int_\Omega 3 W_{x^{1}}^{2}\frac{\partial W_{x^{1}} }{\partial y_1}\left(\frac{\partial W_{x^{2}} }{\partial y_1}+\frac{\partial W_{x^{k}} }{\partial y_1}+\frac{\partial W_{x^{2}} }{\partial r}+\frac{\partial W_{x^{k}} }{\partial r}\right)\right)\\ & \qquad+ O(e^{-(2-\sigma)|x^{1}-x^{2}|})\\ & \quad=C (\mu_1\gamma_1^{4}+\mu_2\gamma_2^{4}+2\beta\gamma_1^{2}\gamma_2^{2})\int_\Omega W_{x^{1}}^{3}\frac{\partial}{\partial y_1}\left(\frac{\partial W_{x^{2}}}{\partial y_1}+\frac{\partial W_{x^{k}}}{\partial y_1}+\frac{\partial W_{x^{2}}}{\partial r}+\frac{\partial W_{x^{k}}}{\partial r}\right)\\ & \qquad+ O(e^{-(2-\sigma)|x^{1}-x^{2}|})\\ & \quad=(B+o(1))\frac{\partial}{\partial y_1}\left(\frac{\partial W_{x^{2}}}{\partial y_1}+\frac{\partial W_{x^{k}}}{\partial y_1}+\frac{\partial W_{x^{2}}}{\partial r}+\frac{\partial W_{x^{k}}}{\partial r}\right)\Big|_{y=x_1}+ O(e^{-(2-\sigma)|x^{1}-x^{2}|}), \end{align*}

where $C,\,B>0$ are some constants. Following the arguments used in [Reference Guo, Musso, Peng and Yan10], for some $C>0$, there holds then

\[ \displaystyle\frac{\partial}{\partial y_1}\left(\frac{\partial W_{x^{2}}}{\partial y_1}+\frac{\partial W_{x^{k}}}{\partial y_1}+\frac{\partial W_{x^{2}}}{\partial r}+\frac{\partial W_{x^{k}}}{\partial r}\right)\Big|_{y=x_1} =\frac{W(|x^{1}-x^{2}|)}{k^{2}}(C+o(1)) \]

and therefore

(3.4)\begin{align} & \mathcal{L}\left(\left(\displaystyle\sum_{i=1}^{k}Q_{1,x^{i}},\sum_{i=1}^{k}Q_{2,x^{i}}\right), \left(\sum_{j=1}^{k}\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r},\sum_{j=1}^{k}\frac{\partial Q_{2,x^{j}}}{\partial r}\right),\Omega\right)\nonumber\\& \quad =(C+o(1))\frac{W(|x^{1}-x^{2}|)}{k^{2}}+ O(e^{-(2-\sigma)|x^{1}-x^{2}|}). \end{align}

In view of (2.4) and Lemma 3.3, we use $L^{p}$-estimates for the elliptic equations to prove that

\[ \|(\nabla\varphi_1,\nabla\varphi_2)\|_*,\|(\nabla\xi_1^{*},\nabla\xi_2^{*})\|_*\leq\frac C{r^{m_1}}+\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}, \quad \forall y\in\partial\Omega. \]

Therefore,

(3.5)\begin{equation} \begin{aligned} & \mathcal{L}\left(\vec\varphi,\left(\displaystyle\sum_{j=1}^{k}\displaystyle\frac{\partial Q_{1,x^{j}}}{\partial r},\sum_{j=1}^{k}\frac{\partial Q_{2,x^{j}}}{\partial r}\right),\Omega\right) =O\left(\frac C{r^{m_1}}+\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}\right)e^{-(1-\sigma)|x^{1}-x^{2}|},\\ & \mathcal{L}\left(\left(\displaystyle\sum_{i=1}^{k}Q_{1,x^{i}},\sum_{i=1}^{k}Q_{2,x^{i}}\right),\vec\xi^{*},\Omega\right) =O\left(\displaystyle\frac C{r^{m_1}}+\frac C{r^{m_2}}+Ce^{-|x^{1}-x^{2}|}\right)e^{-(1-\sigma)|x^{1}-x^{2}|}. \end{aligned} \end{equation}

Next, we check that

(3.6)\begin{equation} \displaystyle\int_{\partial\Omega}(V_i(y)-1)u_i\xi_i\nu_1=O\bigg(\displaystyle\frac 1{r^{m_i}}e^{-(1-\sigma)|x^{1}-x^{2}|}\bigg), \end{equation}

and

(3.7)\begin{equation} \displaystyle\int_{\partial\Omega} (\mu_1 u_1^{3}\xi_1 +\mu_2 u_2^{3}\xi_2 +\beta u_1 \xi_1u_2^{2} +\beta \xi_2u_2u_1^{2})\nu_1 =O( e^{-(2-\sigma)|x^{1}-x^{2}|}). \end{equation}

Combining (3.2), (3.4), (3.5), (3.6) and (3.7), we obtain that

(3.8)\begin{align} & \displaystyle\int_\Omega\left(\displaystyle\frac{\partial{V_1}}{\partial y_1}u_1\xi_1+\frac{\partial{V_2}}{\partial y_1}u_2\xi_2\right). \\ & \quad=b(C'+o(1))\frac{W(|x^{1}-x^{2}|)}{k^{2}}+O\left(\left(\frac 1{r^{m_1}}+ \frac 1{r^{m_2}}\right)e^{-(1-\sigma)|x^{1}-x^{2}|}+e^{-(2-\sigma)|x^{1}- x^{2}|}\right) \nonumber \end{align}

Using (2.4) and Lemma 3.3, we find

(3.9)\begin{equation} \begin{aligned} & \displaystyle\int_{ \Omega} u_1\xi_1 V_1'(|y|) |y|^{{-}1} y^{1}+u_2\xi_2 V_2'(|y|) |y|^{{-}1} y^{1}\\ & \quad=b\int_{ \mathbb{R}^{3}} Q_{1,x^{1}} \displaystyle\frac{\partial Q_{1,x^{1}}}{\partial y^{1}} V_1'(|y|) |y|^{{-}1} y^{1} +Q_{2,x^{1}} \frac{\partial Q_{2,x^{1}}}{\partial y^{1}} V_2'(|y|) |y|^{{-}1} y^{1}\\ & \qquad+\displaystyle\frac 1{|x^{1}|^{m+1}} O\left(\frac{1}{ |x^{1}|^{m}}+ e^{- |x^{2}-x^{1}|}\right). \end{aligned} \end{equation}

For any $h>1$, it holds

\[ \frac{ y^{1}+ x_{1, 1}} {|y+x^{1}|^{h}} = \frac{ x_{1, 1}} {|x^{1}|^{h}}- (h-1) \frac{ y^{1}} {|x^{1}|^{h}}+O\left(\frac 1{|x^{1}|^{h+1}}\right). \]

Thus

\[ \int_{\mathbb{R}^{3}}W \frac{\partial W}{\partial y^{1}} \frac{ y^{1}+ x_{1, 1}} {|y+x^{1}|^{h}}={-} (h-1)\int_{\mathbb{R}^{3}}W \frac{\partial W}{\partial y^{1}} \frac{ y^{1}} {|x^{1}|^{h}} +O\left(\frac 1{|x^{1}|^{h+1}}\right). \]

Therefore, by (V),

(3.10)\begin{equation} \begin{aligned} & \displaystyle\int_{ \mathbb{R}^{3}} Q_{1,x^{1}} \displaystyle\frac{\partial Q_{1,x^{1}}}{\partial y^{1}} V_1'(|y|) |y|^{{-}1} y^{1}\\ & \quad=\int_{ \mathbb{R}^{3} } Q_1 \frac{\partial Q_1}{\partial y^{1}} \left( -\frac{ \alpha a_1 }{ |y+ x^{1}|^{m_1+1}} -\frac{ (m_1+1) b_1 } {|y+ x^{1}|^{m_1+2}} + O\left(\frac{ 1 }{ |y+ x^{1}|^{m_1+3}}\right)\right)\frac{ y^{1}+ x_{1,1}}{|y+ x^{1}| }\\ & \quad=\frac{\alpha(\alpha+1)a_1}{|r_{k}|^{m_1+2}} \left(\int_{\mathbb{R}^{3}} Q_1Q_1'(|y|) (y^{1})^{2} +o(1)\right). \end{aligned} \end{equation}

Similarly,

(3.11)\begin{equation} \displaystyle\int_{ \mathbb{R}^{3}} Q_{2,x^{1}} \displaystyle\frac{\partial Q_{2,x^{1}}}{\partial y^{1}} V_2'(|y|) |y|^{{-}1} y^{1} =\frac{\alpha(\alpha+1)a_2}{|r_{k}|^{m_2+2}} \left(\int_{\mathbb{R}^{3}} Q_2Q_2'(|y|) (y^{1})^{2} +o(1)\right). \end{equation}

Combining (3.8), (3.10) and (3.11), we obtain

(3.12)\begin{equation} \begin{aligned} & b\left( \displaystyle\frac{m_1(m_1+1)a_1}{r_{k}^{m_1+2}}+ \frac{m_2(m_2+1)a_2}{r_{k}^{m_2+2}}\right) \left( \displaystyle\int_{\mathbb{R}^{3}}(Q_1 Q_1'(|y|)+ Q_2Q_2'(|y|))(y^{1})^{2} +o(1)\right)\\ & \quad+\displaystyle\frac 1{|x^{1}|^{m+1}} O\left(\frac{1}{ |x^{1}|^{m}}+ e^{- |x^{2}-x^{1}|}\right)\\ & = b ( B'+o(1)) \frac{ W(|x^{1}- x^{2}|)}{k^{2}} +O\left(\frac 1{r_k^{m}} e^{-(1-\sigma)|x^{1}-x^{2}|}+e^{-(2 -\sigma)|x^{2}-x^{1}|}\right). \end{aligned} \end{equation}

Since $m>2$, we see that

\[ \frac 1{r_k^{m}} e^{-(1-\sigma)|x^{1}-x^{2}|}+e^{-(2-\sigma)|x^{2}-x^{1}|}=o\left(\frac{ W(|x^{1}- x^{2}|)}{k^{2}}\right). \]

On the other hand, by (2.5), we have

\[ \displaystyle\frac{1 }{r_k^{m+1}} \sim W(|x^{2}-x^{1}|)\frac 1{k}. \]

Hence since $r_k\sim k\ln k$,

\[ \frac{ W(|x^{1}- x^{k}|)}{k^{2}}>{>}\frac{1 }{r_k^{m_1+2}}+\frac{1 }{r_k^{m_2+2}}, \]

i.e. $\frac {1 }{r_k^{m_1+2}}+\frac {1 }{r_k^{m_2+2}}$ is the high order infinitely small quantity compared with $\frac {W(|x^{1}- x^{k}|)}{k^{2}}$. Thus, (3.12) gives $b:=b_{k_n}\to 0$.

Proof of Theorem 1.1 Let $G_i(y,\, x)$ be the Green's function of $-\Delta u + V_i(|y|) u,\, i=1,\,2$ in $\mathbb {R}^{3}$. Then

\[ 0< G_i(z, y) \le C e^{-|z-y|},\quad i=1,2 \]

as $|z-y|\to +\infty$. We have

\[ \begin{cases} \displaystyle\xi_{1,k} (y)= \displaystyle\int_{\mathbb{R}^{3}} G_1(z, y)( 3\mu_1u_{1,k}^{2}\xi_{1,k}(z)+2\beta u_{1,k}u_{2,k} \xi_{2,k}(z) +\beta u_{2,k}^{2} \xi_{1,k}(z))\,dz,\\ \displaystyle\xi_{2,k} (y)= \displaystyle\int_{\mathbb{R}^{3}} G_2(z, y)( 3\mu_2u_{2,k}^{2}\xi_{2,k}(z)+2\beta u_{1,k}u_{2,k} \xi_{1,k}(z) +\beta u_{1,k}^{2} \xi_{2,k}(z))\,dz. \end{cases} \]

Now we estimate

\begin{align*} & \left|\int_{\mathbb{R}^{3}} G(z, y) u_k^{p-1}(z) \xi_k(z)\,dz\right|\le C\|\xi _k\|_{*} \int_{\mathbb{R}^{3}} G(z, y) u_k^{p-1}(z)\sum_{j=1}^{k} e^{-\tau|z-x_j|}\,dz \\ & \quad\le C\|\xi _k\|_{*} \int_{\mathbb{R}^{3}} G(z, y)\bigl( W_r^{p-1}(z)+ |\omega|^{p-1}\bigr)\sum_{j=1}^{k} e^{-\tau|z-x_j|}\,dz\\ & \quad\le C\|\xi _k\|_{*} \int_{\mathbb{R}^{3}} G(z, y) \left(\sum_{j=1}^{k} e^{- (p-1-\sigma+\tau)|z-x_j|}+\|\omega\|_*^{p-1} \sum_{j=1}^{k} e^{-(p\tau-\sigma) |z-x_j|}\right) \\ & \quad\le C\|\xi_k\|_{*} \sum_{j=1}^{k} e^{-(p\tau-\sigma)|y-x_j|}, \end{align*}

where $\sigma \in (0,\, \tau )$ is any fixed small constant. So we obtain

\[ \frac{ |\xi_k(y)|}{ \sum_{j=1}^{k} e^{-\tau |y-x_j|}}\le C\|\xi_k\|_{*}\frac{ \sum_{j=1}^{k} e^{-(p\tau-\sigma) |y-x_j|}}{ \sum_{j=1}^{k} e^{-\tau |y-x_j|}}. \]

Since $\xi _k\to 0$ in $B_R(x_{j, k})$ and $\|\xi _k\|_*=1$, we know that $\frac { |\xi _k(y)|}{ \sum \nolimits _{j=1}^{k} e^{-\tau |y-x_j|}}$ attains its maximum in $\mathbb {R}^{3} \setminus \cup _{j=1}^{k} B_R(x_{j, k})$. Thus

\[ \|\xi_k\|_* \le o(1)\|\xi_k\|_* . \]

So $\|\xi _k\|_*\to 0$ as $k\to +\infty$. This is a contradiction to $\|\xi _k\|_*=1$.

Acknowledgements

We are very grateful to the anonymous referees for the valuable comments and suggestions, which helped us to improve our manuscript. This work is supported by NSFC (No.11771469,12171014).

References

Ambrosetti, A., Colorado, E. and Ruiz, D., Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calculus Variat. 30 (2007), 85–112.CrossRefGoogle Scholar
Bao, W. and Cai, Y., Mathmatical theory and numerical methods for Bose-Einstein condensation, Kinetic Relat. Models 6 (2013), 1–135.CrossRefGoogle Scholar
Guo, Y. and Seiringer, R., On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141–156.CrossRefGoogle Scholar
Guo, Y., Zeng, X. and Zhou, H., Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 809–828.Google Scholar
Guo, Y., Zeng, X. and Zhou, H., Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions, Discrete Contin. Dyn. Syst. 37 (2017), 3749–3786.CrossRefGoogle Scholar
Guo, Y. X., Peng, S. and Yan, S., Local uniqueness and periodicity induced by concentration, Proc. Lond. Math. Soc. 114 (2017), 1005–1043.CrossRefGoogle Scholar
Guo, Y., Lin, C. and Wei, J., Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal. 49 (2017), 3671–3715.CrossRefGoogle Scholar
Guo, Y., Li, S., Wei, J. and Zeng, X., Ground states of two-component attractive Bose-Einstein condensates I: Existence and uniqueness, J. Funct. Anal. 276 (2019), 183–230.CrossRefGoogle Scholar
Guo, Y., Li, S., Wei, J. and Zeng, X., Ground states of two-component attractive Bose-Einstein condensates II: Semi-trivial limit behavior, Trans. Amer. Math. Soc. 371 (2019), 6903–6948.CrossRefGoogle Scholar
Guo, Y., Musso, M., Peng, S. and Yan, S., Non-degeneracy and existence of new solutions for the Schrödinger equations, preprint (2021).CrossRefGoogle Scholar
Lin, T. and Wei, J., Ground state of N coupled nonlinear Schrödinger equations in $\mathbb {R}^{N}$, $N\leq 3$., Comm. Math. Phys. 255 (2005), 629–653.CrossRefGoogle Scholar
Long, W. and Peng, S., Segregated vector solutions for a class of Bose-Einstein systems, J. Differ. Equ. 257 (2014), 207–230.CrossRefGoogle Scholar
Luo, P., Peng, S. and Yan, S., Excited states on Bose-Einstein condensates with attractive interactions.Google Scholar
Noussair, E. and Yan, S., On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc. 62 (2000), 213–227.CrossRefGoogle Scholar
Peng, S. and Wang, Z., Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal. 208 (2013), 305–339.CrossRefGoogle Scholar
Royo-Letelier, J., Segregation and symmetry breaking of strongly coupled two component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differ. Equ. 49 (2014), 103–124.CrossRefGoogle Scholar