Normalized solutions to Schr\"{o}dinger systems with potentials

TL;DR

This paper investigates the existence of normalized solutions to a coupled Schrödinger system with trapping potentials in three dimensions, employing minimax methods for both cases when the coupling parameter is zero and non-zero.

Contribution

It introduces a novel application of minimax theory to find normalized solutions for Schrödinger systems with potentials, covering both uncoupled and coupled scenarios.

Findings

Existence of solutions for the Schrödinger system with potentials.

Application of minimax methods on the constraint manifold.

Results valid for both zero and non-zero coupling parameters.

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_1^3+\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_2^3+\beta u_1^2u_2+\kappa u_1~\hbox{in}~ \mathbb{R}^3, u_1\in H^1(\mathbb{R}^3), u_2\in H^1(\mathbb{R}^3),\nonumber \end{cases} \end{equation} under the constraint \begin{equation} \int_{\mathbb{R}^3} u_1^2=a_1^2,~\int_{\mathbb{R}^3} u_2^2=a_2^2\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 $L^2$-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.