\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 05, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/05\hfil Linearly coupled Schr\"odinger systems] {Ground states of linearly coupled \\ Schr\"odinger systems} \author[H. Liu \hfil EJDE-2017/05\hfilneg] {Haidong Liu} \address{Haidong Liu \newline College of Mathematics, Physics and Information Engineering \\ Jiaxing University, Zhejiang 314001, China} \email{liuhaidong@mail.zjxu.edu.cn} \thanks{Submitted January 11, 2016. Published January 5, 2017.} \subjclass[2010]{35B40, 35J47, 35J50} \keywords{Linearly coupled Schrodinger system; ground states; \hfill\break\indent asymptotic behavior} \begin{abstract} This article concerns the standing waves of a linearly coupled Schr\"odinger system which arises from nonlinear optics and condensed matter physics. The coefficients of the system are spatially dependent and have a mixed behavior: they are periodic in some directions and tend to positive constants in other directions. Under suitable assumptions, we prove that the system has a positive ground state. In addition, when the $L^\infty$-norm of the coupling coefficient tends to zero, the asymptotic behavior of the ground states is also obtained. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction and statement of main results} Nonlinear Schr\"odinger systems of the form \begin{equation}\label{1.1} \begin{gathered} -i\frac{\partial}{\partial t}\Psi_1=\Delta \Psi_1-V_1(x)\Psi_1 +\mu_1|\Psi_1|^2\Psi_1+\beta|\Psi_2|^2\Psi_1+\gamma \Psi_2\\ -i\frac{\partial}{\partial t}\Psi_2=\Delta \Psi_2-V_2(x)\Psi_2 +\mu_2|\Psi_2|^2\Psi_2+\beta|\Psi_1|^2\Psi_2+\gamma \Psi_1 \\ x\in\mathbb{R}^N,\; t>0,\\ \Psi_j=\Psi_j(x,t)\in \mathbb C,\quad t>0,\; j=1,2 \end{gathered} \end{equation} model several interesting phenomena in physics. Physically, $\Psi_j$ are two components of a quantum system, $\mu_j$ and $\beta$ are the intraspecies and interspecies scattering lengths, $\gamma$ is the Rabi frequency related to the external electric field. The sign of the scattering length $\beta$ determines whether the interaction is repulsive or attractive. We refer to \cite{AA, GGR, Hioe1, JW, Ti} and references therein for more information on the physical background of \eqref{1.1}. To study standing waves of the system \eqref{1.1}, we set $\Psi_j(x,t)= e^{i\lambda t} u_j(x)$ for $j=1, 2$. Then \eqref{1.1} is reduced to the following elliptic system \begin{equation}\label{1.2} \begin{gathered} -\Delta u_1+(V_1(x)+\lambda) u_1=\mu_1u_1^3+\beta u_1u_2^2+\gamma u_2 \quad \text{in } \mathbb{R}^N,\\ -\Delta u_2+(V_2(x)+\lambda)u_2=\mu_2u_2^3+\beta u_1^2u_2+\gamma u_1 \quad \text{in } \mathbb{R}^N,\\ u_j(x)\to 0\quad \text{as }|x|\to\infty,\; j=1,2. \end{gathered} \end{equation} In the presence of only nonlinearly coupling terms (i.e., $\gamma=0$), \eqref{1.2} has been studied extensively in recent years for the existence, multiplicity and asymptotic behavior of nontrivial solutions. We make no attempt here to give a complete survey of all related results and only refer the reader to \cite{ACo, BDW, Bh, LW1, LL, LW2, SW, Si, WW1} and references therein. However, In the presence of only linearly coupling terms (i.e., $\beta=0$), \eqref{1.2} has not been much studied and we are only aware of a few papers in this direction (\cite{Am, ACeR, ACoR, CZ1, CZ2, LP}). On the other hand, the single elliptic equation \begin{equation}\label{1.3} -\Delta u+ V(x)u =\mu(x)|u|^{p-2}u, \end{equation} has been deeply studied in the literature. Among other solutions, ground states are physically and mathematically of particular interest. We refer the reader to \cite{AFM, CRV, CM, CR, DN, Ga, LWZ, Lio, MR, RR, Will, Zh} for related results. Here we only mention that \eqref{1.3} has a positive ground state if $02$ and $2^*=+\infty$ for $N=2$, and the coefficients $V_j, \mu_j, \gamma$ are continuous functions on $\mathbb{R}^N$. For system \eqref{1.4}, a nontrivial solution is a solution $(u_1, u_2)$ with $(u_1, u_2)\neq(0,0)$. A nontrivial solution of \eqref{1.4} will be called a ground state if it has the least energy among all nontrivial solutions. A positive ground state means a ground state with each component being positive. We remark that each component of a nontrivial solution of \eqref{1.4} must be nonzero. Write $N=k+l$ with $1\leq k\leq N-1$ and $x=(x', x'')\in \mathbb{R}^k\times \mathbb{R}^l=\mathbb{R}^N$. We study \eqref{1.4} under the following assumptions. \begin{itemize} \item[(H1)] The five functions $V_j, \mu_j, \gamma$ are $\tau_i$-periodic in $x_i$, $\tau_i>0$, $i=1,\dots,k$. \item[(H2)] $00$, $i=1,\dots,k$. \item[(H2')] $00$ such that, for $\vec{u}\in{\mathcal N}$, $\|\vec{u}\|\geq\delta$. \end{lemma} \begin{proof} For $\vec{u}\in \mathcal{N}$, we use the H\"older inequality and Sobolev inequality to deduce \begin{align*} \Big(1-\big|\gamma(V_1V_2)^{-1/2}\big|_\infty\Big)\|\vec{u}\|^2 &\leq\|\vec{u}\|^2-2\int_{\mathbb{R}^N}\gamma u_1u_2 \\ &=\int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right) \leq C\|\vec{u}\|^p, \end{align*} which implies the desired result. \end{proof} Note that for $\vec{u}\in\mathcal{N}$, $$ I(\vec{u})=\big(\frac{1}{2}-\frac{1}{p}\big) \Big(\|\vec{u}\|^2-2\int_{\mathbb{R}^N}\gamma u_1u_2\Big) \geq\big(\frac{1}{2}-\frac{1}{p}\big) \Big(1-\big|\gamma(V_1V_2)^{-1/2}\big|_\infty\Big)\|\vec{u}\|^2. $$ Therefore, as a consequence of Lemma \ref{lem2.1}, we have \begin{lemma}\label{lem2.2} $c=\inf_{\vec{u}\in\mathcal{N}}I(\vec{u})>0$. \end{lemma} \begin{lemma}\label{lem2.3} If $c$ is achieved, then \eqref{1.4} has a positive ground state. \end{lemma} \begin{proof} We first claim that any minimizer of $c$ is a ground state of \eqref{1.4}. Indeed, if $\vec{u}=(u_1, u_2)\in\mathcal{N}$ is a minimizer of $c$, then there exists $\lambda\in\mathbb{R}$ such that $$ I'(\vec{u})=\lambda J'(\vec{u}). $$ From $\langle I'(\vec{u}),\vec{u}\rangle=0$ and \begin{align*} \langle J'(\vec{u}),\vec{u}\rangle &=2\Big(\|\vec{u}\|^2-2\int_{\mathbb{R}^N}\gamma u_1u_2\Big) -p\int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right)\\ &=-(p-2)(\|\vec{u}\|^2-2\int_{\mathbb{R}^N}\gamma u_1u_2)\\ &\leq -(p-2)\Big(1-\big|\gamma(V_1V_2)^{-1/2}\big|_\infty\Big)\|\vec{u}\|^2\\ &\leq -(p-2)\Big(1-\big|\gamma(V_1V_2)^{-1/2}\big|_\infty\Big)\delta^2 <0 \end{align*} it follows that $\lambda=0$. Then $I'(\vec{u})=0$ and so $\vec{u}$ is a ground state of \eqref{1.4}. Next we prove that \eqref{1.4} has a positive ground state. Assume $\vec{u}=(u_1, u_2)\in\mathcal{N}$ is a minimizer of $c$ and let $t>0$ be such that $(t|u_1|, t|u_2|)\in\mathcal{N}$. Then $$ t^{p-2}=\frac{\|\vec{u}\|^2-2\int_{\mathbb{R}^N}\gamma |u_1||u_2|} {\int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right)} \leq \frac{\|\vec{u}\|^2-2\int_{\mathbb{R}^N}\gamma u_1u_2} {\int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right)}=1, $$ which implies \begin{align*} c\leq I(t|u_1|, t|u_2|)&=\big(\frac12-\frac{1}{p}\big)t^p \int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right)\\ &\leq\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right)\\ &=I(\vec{u})=c. \end{align*} This means $t=1$ and $(|u_1|, |u_2|)$ is also a minimizer of $c$. From the above claim, $(|u_1|, |u_2|)$ is a ground state of \eqref{1.4}. Note that none of $|u_1|$ and $|u_2|$ can be identically zero. Using the strong maximum principle, we have $|u_1|>0$ and $|u_2|>0$. Therefore $(|u_1|, |u_2|)$ is a positive ground state of \eqref{1.4}. The proof is complete. \end{proof} \section{Proof of Theorem \ref{thm1.1}} In this section we prove the existence of ground states of \eqref{1.4}. From Lemma \ref{lem2.3}, it suffices to prove that $c$ is achieved. For this purpose, we need to compare the value of $c$ with $$ c_{\infty}=\inf_{\vec{u}\in \mathcal{N}_\infty} I_\infty(\vec{u}), $$ where the functional $I_\infty : \mathcal{H}\to\mathbb{R}$ is defined by \begin{align*} I_\infty(\vec{u}) &=\frac{1}{2}\int_{\mathbb{R}^N}\big(|\nabla u_1|^2 +V_{1\infty}u_1^2+|\nabla u_2|^2+V_{2\infty}u_2^2\big)\\ &\quad -\frac{1}{p}\int_{\mathbb{R}^N} \left(\mu_{1\infty}|u_1|^p+\mu_{2\infty}|u_2|^p\right) -\int_{\mathbb{R}^N} \gamma_\infty u_1u_2, \end{align*} and $$ \mathcal{N}_\infty=\left\{\vec{u}\in \mathcal{H}\setminus\{(0,0)\}: \langle I_\infty'(\vec{u}),\vec{u}\rangle=0\right\} $$ is the Nehari manifold associated with $I_\infty$. \begin{lemma}\label{lem3.1} Assume that {\rm (H1)--(H3)} are satisfied and suppose in addition that at least one of the five functions $V_j, \mu_j, \gamma$ is not a constant, then $c0$ be such that $t \vec{u}_\infty\in\mathcal{N}$. Then \begin{align*} c_{\infty} & =I_\infty(\vec{u}_\infty)\geq I_\infty(t \vec{u}_\infty)\\ & =I(t\vec{u}_\infty)+\frac{t^2}{2}\int_{\mathbb{R}^N} \left[\left(V_{1\infty}-V_1\right)u_{1\infty}^2+ \left(V_{2\infty}-V_2\right)u_{2\infty}^2\right]\\ &\quad +\frac{t^p}{p}\int_{\mathbb{R}^N}\left[\left(\mu_1-\mu_{1\infty} \right)|u_{1\infty}|^p+\left(\mu_2-\mu_{2\infty}\right)|u_{2\infty}|^p\right] \\ &\quad +t^2\int_{\mathbb{R}^N}\left(\gamma-\gamma_{\infty}\right)u_{1\infty}u_{2\infty}\\ &>I(t \vec{u}_\infty)\geq c. \end{align*} The proof is complete. \end{proof} Now we are in a position to prove the first main result. \begin{proof}[Proof of Theorem \ref{thm1.1}] By Lemma \ref{lem2.3}, it suffices to prove that the infimum $$c=\inf_{\vec{u}\in\mathcal{N}}I(\vec{u})$$ is achieved. If the five functions $V_j, \mu_j, \gamma$ are all constants, then the result has been proved in \cite[Lemma 3.2]{ACeR}. In what follows we always assume that at least one of the five functions $V_j, \mu_j, \gamma$ is not a constant. By Ekeland's variational principle, there exists $\{\vec{u}_m\}\subset\mathcal{N}$ with $\vec{u}_m=(u_{m1},u_{m2})$ such that $$ I(\vec{u}_m)\to c\quad \text{and}\quad (I|_{\mathcal{N}})'(\vec{u}_m)\to 0\quad \text{as } m\to\infty. $$ Then there exists a sequence $\{\lambda_m\}$ of real numbers such that $$ I'(\vec{u}_m)-\lambda_m J'(\vec{u}_m)\to 0\quad \text{as } m\to\infty. $$ From (H3) and $$ I(\vec{u}_m)=\big(\frac12-\frac{1}{p}\big) \Big(\|\vec{u}_m\|^2-2\int_{\mathbb{R}^N}\gamma u_{m1}u_{m2}\Big)=c+o(1), $$ we see that $\{\vec{u}_m\}$ is bounded in $\mathcal{H}$. Then \begin{align*} o(1) &=\langle I'(\vec{u}_m)-\lambda_m J'(\vec{u}_m), \vec{u}_m\rangle\\ &=\lambda_m(p-2)\Big(\|\vec{u}_m\|^2 -2\int_{\mathbb{R}^N}\gamma u_{m1}u_{m2}\Big)\\ &=\lambda_m(2pc+o(1)), \end{align*} which implies that $\lambda_m=o(1)$ and so $I'(\vec{u}_m)\to 0$ as $m\to\infty$. Since $\{\vec{u}_m\}\subset\mathcal{H}$ is bounded, up to a subsequence, it can be assumed that \begin{gather*} (u_{m1},u_{m2})\rightharpoonup (u_1, u_2)\quad \text{in } \mathcal{H},\\ (u_{m1},u_{m2})\to (u_1, u_2) \quad \text{in } L^p_{\rm loc}(\mathbb{R}^N)\times L^p_{\rm loc}(\mathbb{R}^N),\\ (u_{m1},u_{m2})\to (u_1, u_2) \quad \text{a.e. in }\mathbb{R}^N. \end{gather*} Then $\vec{u}=(u_1, u_2)$ is a critical point of $I$. We have the following two cases. \smallskip \noindent\textbf{Case 1.} $\vec{u}\neq(0,0)$. In this case, $\vec{u}\in\mathcal{N}$ and \begin{align*} c\leq I(\vec{u}) &=\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_1|u_1|^p+\mu_2|u_2|^p\right)\\[1mm] &\leq\lim_{m\to\infty}\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right)\\[1mm] &=\lim_{m\to\infty} I(\vec{u}_m)=c. \end{align*} Then $c$ is achieved by $\vec{u}$. \smallskip \noindent\textbf{Case 2.} $\vec{u}=(0,0)$. In this case, we decompose $\mathbb{R}^N$ into $N$-dimensional intervals $\{Q_j\}_{j\in\mathbb N\cup\{0\}}$ with each of them having sides of size $(\tau_1,\dots,\tau_k,1,\dots,1)$ and chosen in such a way that $0$ is the center of $Q_0$. For each $m$, we set $$ d_m=\sup_{j\in\mathbb N\cup\{0\}}\Big[\int_{Q_j} \left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right)\Big]^{1/p}. $$ Then there exists $\eta>0$ such that \begin{equation}\label{3.1} d_m\geq \eta>0 \end{equation} for all $m$. Indeed, using the Sobolev embeddings $H^1(Q_j)\hookrightarrow L^p(Q_j)$ and the boundedness of $\{\vec{u}_m\}\subset\mathcal{H}$ leads to \begin{align*} \frac{2p}{p-2}(c+o(1)) &=\int_{\mathbb{R}^N}\left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right)\\ &=\sum_{j=0}^\infty\int_{Q_j}\left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right)\\ &\leq d_m^{p-2}\sum_{j=0}^\infty\Big[\int_{Q_j}\left(\mu_1|u_{m1}|^p +\mu_2|u_{m2}|^p\right)\Big]^{2/p}\\ &\leq C d_m^{p-2}\|\vec{u}_m\|^2\\ &\leq C d_m^{p-2}, \end{align*} which implies \eqref{3.1}. From $$ \sum_{j=0}^\infty\int_{Q_j} \left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right)\leq C, $$ we see that $d_m$ is achieved. Let $y_m\in\mathbb{R}^N$ be the center of the interval $Q_m^*$ satisfying $$ d_m=\Big[\int_{Q_m^*}\left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right) \Big]^{1/p}. $$ It follows from \eqref{3.1} and $(u_{m1},u_{m2})\rightharpoonup(0,0)$ in $\mathcal{H}$ that $\{y_m\}$ is unbounded. Without loss of generality, we assume $|y_m|\to\infty$ as $m\to\infty$. Set $\tilde u_{mj}=u_{mj}(\cdot+ y_m)$ for $j=1, 2$ and assume that \begin{gather*} (\tilde u_{m1},\tilde u_{m2}) \rightharpoonup (\tilde u_{1},\tilde u_{2}) \quad \text{in}\ \mathcal{H},\\ (\tilde u_{m1},\tilde u_{m2}) \to (\tilde u_{1},\tilde u_{2}) \quad \text{in}\ L^p_{\rm loc}(\mathbb{R}^N)\times L^p_{\rm loc}(\mathbb{R}^N),\\ (\tilde u_{m1},\tilde u_{m2}) \to (\tilde u_{1},\tilde u_{2}) \quad \text{a.e. in } \mathbb{R}^N. \end{gather*} Then $(\tilde u_{1},\tilde u_{2})\neq (0,0)$ as showed by \begin{align*} 0<\eta &\leq\varliminf_{m\to\infty}\Big[\int_{ Q_m^*}\left(\mu_1|u_{m1}|^p+\mu_2|u_{m2} |^p\right)\Big]^{1/p}\\ &\leq C\varliminf_{m\to\infty}\Big[\int_{ Q_m^*}\left(|u_{m1}|^p+|u_{m2} |^p\right)\Big]^{1/p}\\ &=C\lim_{m\to\infty}\Big[\int_{Q_0}\left(|\tilde u_{m1}|^p+|\tilde u_{m2} |^p\right)\Big]^{1/p}\\ &=C\Big[\int_{Q_0}\left(|\tilde u_1|^p+|\tilde u_2|^p\right)\Big]^{1/p}. \end{align*} Next we claim that $\{|y_m''|\}$ is bounded. Suppose by contradiction that $|y_m''|\to\infty$ as $m\to\infty$. Then, using (H2), we deduce from $I'(u_{m1},u_{m2})\to 0$ that $$ I'_\infty(\tilde u_{1},\tilde u_{2})=0. $$ Therefore, since $(\tilde u_{1},\tilde u_{2})\neq (0,0)$, $(\tilde u_{1},\tilde u_{2})\in\mathcal{N}_\infty$ and we have \begin{align*} c_\infty\leq I_\infty(\tilde u_{1},\tilde u_{2}) &=\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_{1\infty}|\tilde u_1|^p+\mu_{2\infty}|\tilde u_2|^p\right)\\[1mm] &\leq\varliminf_{m\to\infty}\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_{1\infty}|\tilde u_{m1}|^p+\mu_{2\infty}|\tilde u_{m2}|^p\right)\\[1mm] &\leq\lim_{m\to\infty} I(\vec{u}_m)=c, \end{align*} a contradiction to Lemma \ref{lem3.1}. Decompose $\mathbb{R}^N$ into bigger $N$-dimensional intervals $\{\hat Q_j\}_{j\in\mathbb N\cup\{0\}}$ and assume that $Q_m^*\subset \hat Q_m^*$ with $(y_m',0)$ being the center of $\hat Q_m^*$. Setting $\hat u_{mj}=u_{mj}(\cdot+(y_m',0))$ for $j=1,2$, we see from the assumption (H1) that $$ I(\hat u_{m1},\hat u_{m2})\to c\quad \text{and}\quad I'(\hat u_{m1},\hat u_{m2})\to0\quad \text{as } m\to\infty. $$ Since $\{(\hat u_{m1},\hat u_{m2})\}$ is bounded in $\mathcal{H}$, we may assume that \begin{gather*} (\hat u_{m1},\hat u_{m2}) \rightharpoonup (\hat u_{1},\hat u_{2}) \quad \text{in } \mathcal{H},\\ (\hat u_{m1},\hat u_{m2}) \to (\hat u_{1},\hat u_{2}) \quad \text{in}\ L^p_{\rm loc}(\mathbb{R}^N)\times L^p_{\rm loc}(\mathbb{R}^N),\\ (\hat u_{m1},\hat u_{m2}) \to (\hat u_{1},\hat u_{2}) \quad \text{a.e. in } \mathbb{R}^N. \end{gather*} Then $(\hat u_{1},\hat u_{2})$ is a critical point of $I$. Furthermore, \begin{align*} 0<\eta &\leq\varliminf_{m\to\infty}\Big[\int_{ Q_m^*} \left(\mu_1|u_{m1}|^p+\mu_2|u_{m2}|^p\right)\Big]^{1/p}\\ &\leq C\varliminf_{m\to\infty}\Big[\int_{\hat Q_m^*} \left(|u_{m1}|^p+|u_{m2}|^p\right)\Big]^{1/p}\\ &=C\lim_{m\to\infty}\Big[\int_{\hat Q_0} \left(|\hat u_{m1}|^p+|\hat u_{m2}|^p\right)\Big]^{1/p}\\ &=C\Big[\int_{\hat Q_0}\left(|\hat u_1|^p+|\hat u_2|^p\right)\Big]^{1/p}, \end{align*} which implies $(\hat u_{1},\hat u_{2})\neq (0,0)$. Using the same arguments as in Case 1, we see that $c$ is achieved by $(\hat u_{1},\hat u_{2})$. In both cases we reach the conclusion. The proof is complete. \end{proof} \section{Proof of Theorem \ref{thm1.3}} In this section, we describe the asymptotic behavior of the ground states of \eqref{1.4} when the $L^\infty$-norm of the coupling coefficient tends to zero. To underline the dependence on the coupling coefficient, for \eqref{1.4} with $\gamma=\gamma_n$, the corresponding functional and the Nehari manifold will be denoted by $I_{\gamma_n}$ and $\mathcal{N}_{\gamma_n}$ respectively. Then we see from Theorem \ref{thm1.1} that, for $n$ sufficiently large, the infimum $$ c_{\gamma_n}=\inf_{\vec{u}\in\mathcal{N}_{\gamma_n}}I_{\gamma_n}(\vec{u}) $$ is achieved at $(u_{n1}, u_{n2})$. We also use the functional $I_0: {\mathcal{H}}\to\mathbb{R}$ defined by $$ I_0(\vec{u})=\frac{1}{2}\|\vec{u}\|^2-\frac{1}{p}\int_{\mathbb{R}^N} \left(\mu_1|u_1|^p+\mu_2|u_2|^p\right) $$ and its corresponding Nehari manifold $$ \mathcal{N}_0=\left\{\vec{u}\in \mathcal{H}\setminus\{(0,0)\}: \langle I'_0(\vec{u}),\vec{u}\rangle=0\right\}. $$ Define $c_0=\inf_{\vec{u}\in\mathcal{N}_0}I_0(\vec{u})$, then we have $$ c_0=\min_{j\in\{1,2\}}\Phi_j(w_j), $$ where $$ \Phi_j(u)=\frac{1}{2}\|u\|_j^2-\frac{1}{p}\int_{\mathbb{R}^N}\mu_j|u|^p $$ and $w_j$ is the positive ground state of the single elliptic equation $$ -\Delta u+V_ju=\mu_j|u|^{p-2}u\quad\text{in } \mathbb{R}^N. $$ \begin{lemma}\label{lem4.1} $c_{\gamma_n}\to c_0$ as $n\to\infty$. \end{lemma} \begin{proof} Since $(w_1,0)$ and $(0,w_2)$ are contained in $\mathcal{N}_{\gamma_n}$, we have \begin{equation}\label{4.1} c_{\gamma_n}\leq \min\{I_{\gamma_n}(w_1,0), I_{\gamma_n}(0,w_2)\} =\min_{j\in\{1,2\}}\Phi_j(w_j)=c_0. \end{equation} Then it is easy to see that $\{(u_{n1}, u_{n2})\}$ is bounded in $\mathcal{H}$. We define a sequence $\{t_n\}$ of positive numbers by $$ t_n^{p-2}=\frac{\|(u_{n1}, u_{n2})\|^2}{\int_{\mathbb{R}^N} \left(\mu_1|u_{n1}|^p+\mu_2|u_{n2}|^p\right)}. $$ Then $(t_nu_{n1}, t_nu_{n2})\in \mathcal{N}_0$. We claim that $t_n=1+o(1)$ as $n\to\infty$. Indeed, since $(u_{n1}, u_{n2})$ is a positive ground state of \eqref{1.4} with $\gamma=\gamma_n$, we have \begin{align*} &\Big(1-\big|\gamma_n(V_1V_2)^{-1/2}\big|_\infty\Big)\|(u_{n1}, u_{n2})\|^2\\ &\leq\|(u_{n1}, u_{n2})\|^2-2\int_{\mathbb{R}^N}\gamma_n u_{n1}u_{n2}\\ &=\int_{\mathbb{R}^N}\left(\mu_1|u_{n1}|^p+\mu_2|u_{n2}|^p\right) \leq C\|(u_{n1}, u_{n2})\|^p. \end{align*} Then, since $|\gamma_n|_\infty\to 0$, we see that $\|(u_{n1}, u_{n2})\|^2$ and $\int_{\mathbb{R}^N} \left(\mu_1|u_{n1}|^p+\mu_2|u_{n2}|^p\right)$ have a positive lower bound. From this, it can be seen that $t_n=1+o(1)$ as $n\to\infty$. Therefore, \begin{align*} c_0\leq I_0(t_nu_{n1}, t_nu_{n2}) &=\big(\frac12-\frac{1}{p}\big)t_n^p \int_{\mathbb{R}^N}\left(\mu_1|u_{n1}|^p+\mu_2|u_{n2}|^p\right)\\ &=\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_1|u_{n1}|^p+\mu_2|u_{n2}|^p\right)+o(1)\\ &=c_{\gamma_n}+o(1), \end{align*} which combined with \eqref{4.1} completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.3}] We use an argument of contradiction and, up to a subsequence, suppose that \begin{equation}\label{4.2} \|u_{nj}\|_j\geq \alpha>0 \end{equation} for $j=1,2$. Define two sequences $\{s_n\}$ and $\{t_n\}$ of positive numbers by $$ s_n^{p-2}=\frac{\|u_{n1}\|_1^2}{\int_{\mathbb{R}^N}\mu_1|u_{n1}|^p},\quad t_n^{p-2}=\frac{\|u_{n2}\|_2^2}{\int_{\mathbb{R}^N}\mu_2|u_{n2}|^p}. $$ Since $(u_{n1}, u_{n2})$ is a positive ground state of \eqref{1.4} with $\gamma=\gamma_n$ and $|\gamma_n|_\infty\to0$, we have \begin{gather*} \|u_{n1}\|_1^2=\int_{\mathbb{R}^N}\mu_1|u_{n1}|^p +\int_{\mathbb{R}^N}\gamma_n u_{n1}u_{n2} =\int_{\mathbb{R}^N}\mu_1|u_{n1}|^p+o(1), \\ \|u_{n2}\|_2^2=\int_{\mathbb{R}^N}\mu_2|u_{n2}|^p +\int_{\mathbb{R}^N}\gamma_n u_{n1}u_{n2} =\int_{\mathbb{R}^N}\mu_2|u_{n2}|^p+o(1), \end{gather*} which combined with \eqref{4.2} implies that $s_n=1+o(1)$ and $t_n=1+o(1)$ as $n\to\infty$. Then we have \begin{align*} 2c_0 &\leq \Phi_1(s_nu_{n1})+\Phi_2(t_nu_{n2})\\ &=\big(\frac12-\frac{1}{p}\big)s_n^p\int_{\mathbb{R}^N}\mu_1|u_{n1}|^p +\big(\frac12-\frac{1}{p}\big)t_n^p\int_{\mathbb{R}^N}\mu_2|u_{n2}|^p\\ &=\big(\frac12-\frac{1}{p}\big) \int_{\mathbb{R}^N}\left(\mu_1|u_{n1}|^p+\mu_2|u_{n2}|^p\right)+o(1)\\ &=c_{\gamma_n}+o(1), \end{align*} which contradicts the result in Lemma \ref{lem4.1}. This contradiction implies that either $u_{n1}\to 0$ in $H^1(\mathbb{R}^N)$ or $u_{n2}\to 0$ in $H^1(\mathbb{R}^N)$. \end{proof} \subsection*{Acknowledgments} This research was supported by the ZJNSF (LQ15A010011). \begin{thebibliography}{99} \bibitem{AA} N. Akhmediev, A. Ankiewicz; \emph{Partially coherent solitons on a finite background}, Phys. Rev. Lett., 82 (1999), 2661--2664. \bibitem{Am} A. Ambrosetti; \emph{Remarks on some systems of nonlinear Schr\"odinger equations}, J. Fixed Point Theory Appl., 4 (2008), 35--46. \bibitem{ACeR} A. Ambrosetti, G. Cerami, D. 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