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

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.