Ground states of Nonlinear Schr\"{o}dinger System with Mixed Couplings

TL;DR

This paper systematically studies the existence of ground states in multi-component nonlinear Schrödinger systems with mixed attractive and repulsive couplings, providing new insights into complex interactions beyond the well-studied two-component case.

Contribution

It offers the first comprehensive analysis of ground state existence for systems with three or more components and mixed couplings, introducing block decomposition methods.

Findings

Nonexistence of ground states in repulsive-mixed cases.

Necessary conditions for ground state existence in total-mixed cases.

Estimates for Morse index related to interaction forces.

Abstract

We consider the following $k$-coupled nonlinear Schr\"odinger system: \begin{align*} \begin{cases} &-\Delta u_j + \lambda_j u_j = \mu_j u_j^3 + \sum_{i=1, i\not=j}^k \beta_{i,j} u_i^2 u_j \quad {\rm in}\ \mathbb{R}^N,\\ &u_j>0 \quad {\rm in}\ \mathbb{R}^N, \quad u_j(x) \to 0 \quad \text{as }|x|\to +\infty, \quad j=1,2,\cdots,k, \end{cases} \end{align*} where $N\leq 3$, $k\geq3$, $\lambda_j,\mu_j>0$ are constants and $\beta_{i,j}=\beta_{j,i}\not=0$ are parameters. There have been intensive studies for the above system when $k=2$ or the system is purely attractive ($ \beta_{i,j}>0, \forall i \not = j$) or purely repulsive ($\beta_{i,j}<0, \forall i\not = j $); however very few results are available for $k\geq 3$ when the system admits {\bf mixed couplings}, i.e., there exist $(i_1,j_1)$ and $(i_2,j_2)$ such that $\beta_{i_1,j_1}\beta_{i_2,j_2}<0$. In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the system admits mixed couplings. We first divide this system into {\bf repulsive-mixed} and {\bf total-mixed} cases. In the first case we prove nonexistence of ground states. In the second case we give an necessary condition for the existence of ground states and also provide estimates for the Morse index. The key idea is the {\bf block decomposition} of the system ({\bf optimal block decompositions, eventual block decompositions}), and the measure of total interaction forces between different {\bf blocks}. Finally the assumptions on the existence of ground states are shown to be {\bf optimal} in some special cases.