Positive ground state solutions of the two coupled nonlinear Schrödinger system with the harmonic potential and the energy critical nonlinearities

In this paper, we consider the following two coupled nonlinear Schrödinger system:$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+|x|^{2}u-\lambda_{1}u = \mu_{1}|u|^{2^{*}-2}u+\beta |v|^{\frac{2^{*}}{2}}|u|^{\frac{2^{*}}{2}-2}u, \quad\text{in }\mathbb{R}^{N}, \\ -\Delta v+|x|^{2}v-\lambda_{2}v = \mu_{2}|v|^{2^{*}-2}v+\beta |u|^{\frac{2^{*}}{2}}|v|^{\frac{2^{*}}{2}-2}v, \quad \quad\text{in }\mathbb{R}^{N}, \\ u(x)\to0, \; v(x)\to0, \quad\text{as }|x|\to+\infty, \end{array} \right. \end{equation*} $where $ N\geq3 $, $ 2^{*} = \frac{2N}{N-2} $ is the critical Sobolev exponent, $ \lambda_1, \lambda_2, \mu_{1}, \mu_{2} $ are constants and $ \beta $ is the coupling parameter. By using the variational method, we construct a positive ground state solution of the above system under some suitable assumptions. Our results indicate that, unlike the previous studies on elliptic systems with energy critical nonlinearities but without the harmonic potential, where the physical dimension $ N = 3 $ is the special dimension for the existence of positive ground state solutions, the dimension $ N = 4 $ plays a special role in the existence of positive ground state solutions for the above coupled nonlinear Schrödinger system with both harmonic potential and energy critical nonlinearities.