Normalized grounded states for a coupled nonlinear schrödinger system on ${\mathbb{R}}^3$

Abstract

We investigate the existence of normalized ground states to the system of coupled Schrödinger equations: \begin{equation}\label{eq:0.1} \begin{cases} -Δu_1 + λ_1 u_1 = μ_1 |u_1|^{p_1-2}u_1 + βr_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3}, -Δu_2 + λ_2 u_2 = μ_2|u_2|^{p_2-2}u_2 + βr_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 $\hat{I}{¼}_1,\hat{I}{¼}_2>0$, $r_1,r_2>1$, and ${\hat{I}}^2\ge 0$. Our focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3}<p_1,p_2,r_1+r_2<2^{\ast }=6.$$ We demonstrate that there exists a ${\tilde{\hat{I}}}^2\ge 0$ such that equation (\ref{eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when ${\hat{I}}^2>{\tilde{\hat{I}}}^2$. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable.

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