Normalized solutions to Schrödinger systems with potentials (2406.13204v1)
Abstract: In this paper, we study the normalized solutions of the Schr\"{o}dinger system with trapping potentials \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1+V_1(x)u_1-\lambda_1 u_1=\mu_1 u_13+\beta u_1u_2{2}+\kappa u_2~\hbox{in}~ \mathbb{R}3,\ -\Delta u_2+V_2(x)u_2-\lambda_2 u_2=\mu_2 u_23+\beta u_12u_2+\kappa u_1~\hbox{in}~ \mathbb{R}3, u_1\in H1(\mathbb{R}3), u_2\in H1(\mathbb{R}3),\nonumber \end{cases} \end{equation} under the constraint \begin{equation} \int_{\mathbb{R}3} u_12=a_12,~\int_{\mathbb{R}3} u_22=a_22\nonumber, \end{equation} where $\mu_1,\mu_2,a_1,a_2,\beta>0$, $\kappa\in\mathbb{R}$, $V_1(x)$ and $V_2(x)$ are trapping potentials, and $\lambda_1,\lambda_2$ are lagrangian multipliers, this is a typical $L2$-supercritical case in $\mathbb{R}3$. We obtain the existence of solutions to this system by minimax theory on the manifold for $\kappa=0$ and $\kappa\neq 0$ respectively.