Normalized grounded states for a coupled nonlinear schr\"{o}dinger system on $\mathbb{R}^3$

TL;DR

This paper proves the existence of positive, radially symmetric ground states for a coupled nonlinear Schrödinger system with mass constraints in the super-critical case, and discusses their instability.

Contribution

It establishes conditions for the existence of normalized ground states in a coupled Schrödinger system with multiple components and super-critical nonlinearities.

Findings

Existence of ground states when coupling parameter exceeds a threshold

Ground states are positive and radially symmetric

The standing wave solutions are orbitally unstable

Abstract

We investigate the existence of normalized ground states to the system of coupled Schr\"odinger equations: \begin{equation}\label{eq:0.1} \begin{cases} -\Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{p_1-2}u_1 + \beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3}, -\Delta u_2 + \lambda_2 u_2 = \mu_2|u_2|^{p_2-2}u_2 + \beta r_2|u_1|^{r_1}|u_2|^{r_2-2}u_2 & \text{ in } \mathbb{R}^3, \end{cases} \end{equation} subject to the constraints $\mathcal{S}_{a_1} \times \mathcal{S}_{a_2} = \{(u_1 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_1^2 dx = a_1^2\} \times \{(u_2 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_2^2 dx = a_2^2\}$, where $\mu_1, \mu_2 > 0$, $r_1, r_2 > 1$, and $\beta \geq 0$. Our focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3} < p_1, p_2, r_1 + r_2 < 2^* = 6.$$ We demonstrate that there exists a $\tilde{\beta} \geq 0$ such that equation (\ref{eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when $\beta > \tilde{\beta}$. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable.