Positive ground state solutions of the critical nonlinear Schrödinger system with the harmonic potentials for the cooperative case - Wu, Zhiguo; Wu, Yuanze (Topological Methods in Nonlinear Analysis, 2025)
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.