Positive ground state solutions of the critical nonlinear Schrödinger system with the harmonic potentials for the cooperative case

Abstract

In this paper, we study the following two coupled nonlinear Schrödinger system with critical exponent: \begin{equation*} \begin{cases} -\Delta u+|x-x_{0}|^{2}u-\lambda_{1}u=\mu_{1}u^{2^{*}-1}+\beta u^{{2^{*}}/{2}-1}v^{{2^{*}}/{2}}, & x\in \mathbb{R}^{N}, \\ -\Delta v+|x-y_{0}|^{2}v-\lambda_{2}v=\mu_{2}v^{2^{*}-1}+\beta v^{{2^{*}}/{2}-1}u^{{2^{*}}/{2}}, & x\in \mathbb{R}^{N}, \\ u> 0,\quad v> 0,& x\in \mathbb{R}^N,\\ u(x)\to0,\quad v(x)\to0, &\text{as }|x|\to+\infty, \end{cases} \end{equation*} where $N\geq3$, $2^{*}={2N}/({N-2})$, $\mu_{1},\mu_{2},\beta> 0$, $x_{0},y_{0}\in \mathbb{R}^{N}$ and $\lambda_{1},\lambda_{2}\in(\lambda_{*},N)$ with $\lambda_{*}=1$ for $N=3$ and $\lambda_{*}=0$ for $N\geq4$. By the variational method, we show that the existence of a positive ground state solution for the above system under the cooperative case. Furthermore, we also partially reveal the influence mechanisms of $x_{0}$, $y_{0}$, $\lambda_{i}$ and $\beta$ on the existence of positive ground state solutions.