We consider the following k-coupled nonlinear Schrödinger systems:$\{\begin{array}{cc}\hfill & -\mathrm{\Delta }{u}_j+{\lambda }_j{u}_j={\mu }_j{u}_j^3+\sum _{i=1,i\ne j}^k{\beta }_{i,j}{u}_i^2{u}_j\text{in }{\mathbb{R}}^N,\hfill \\ \hfill & u_j>0\text{in }{\mathbb{R}}^N,u_j(x)\to 0\text{as }|x|\to +\infty ,j=1,2,\cdots ,k,\hfill \end{array}$ where $N\le 3$, $k\ge 3$, ${\lambda }_j,{\mu }_j>0$ are constants and ${\beta }_{i,j}={\beta }_{j,i}\ne 0$ are parameters. There have been intensive studies for the above systems when $k=2$ or the systems are purely attractive (${\beta }_{i,j}>0,\forall i\ne j$) or purely repulsive (${\beta }_{i,j}<0,\forall i\ne j$); however very few results are available for $k\ge 3$ when the systems admit mixed couplings and the components are organized into groups, i.e., there exist $(i_1,j_1)$ and $(i_2,j_2)$ such that ${\beta }_{i_1,j_1}>0$ and ${\beta }_{i_2,j_2}<0$. In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the systems admit mixed couplings and the components are organized into groups. We first divide these systems into repulsive-mixed and total-mixed cases. In the first case we prove nonexistence of ground states. In the second case we give a necessary condition for the existence of ground states and also provide estimates for Morse index. The key idea is the block decomposition of the systems (optimal block decompositions, eventual block decompositions), and the measure of total interaction forces between different blocks. Finally the assumptions on the existence of ground states are shown to be optimal in some special cases.

我们考虑以下k耦合非线性Schrödinger系统：$\{\begin{array}{cc}\hfill & -\mathrm{\Delta }{ü}_{Ĵ}+{\lambda }_{Ĵ}{ü}_{Ĵ}={\mu }_{Ĵ}{ü}_{Ĵ}^3+\sum _{\mathrm{一世}=\mathrm{1个}，\mathrm{一世}\ne Ĵ}^{ķ}{\beta }_{\mathrm{一世}，Ĵ}{ü}_{\mathrm{一世}}^2{ü}_{Ĵ}\text{在 }{\mathbb{[R}}^{ñ}，\hfill \\ \hfill & {ü}_{Ĵ}>0\text{在 }{\mathbb{[R}}^{ñ}，{ü}_{Ĵ}（X）\to 0\text{如 }|X|\to +\infty ，Ĵ=\mathrm{1个}，2，\cdots ，ķ，\hfill \end{array}$ 哪里 $ñ\le 3$， $ķ\ge 3$， ${\lambda }_{Ĵ}，{\mu }_{Ĵ}>0$ 是常数， ${\beta }_{\mathrm{一世}，Ĵ}={\beta }_{Ĵ，\mathrm{一世}}\ne 0$是参数。对于上述系统，已经进行了深入研究。$ķ=2$ 或系统纯粹是吸引人的（${\beta }_{\mathrm{一世}，Ĵ}>0，\forall \mathrm{一世}\ne Ĵ$）或纯粹排斥（${\beta }_{\mathrm{一世}，Ĵ}<0，\forall \mathrm{一世}\ne Ĵ$）; 但是很少有结果可用于$ķ\ge 3$当系统允许混合联轴器并且组件被分组时，即存在$（{\mathrm{一世}}_{\mathrm{1个}}，{Ĵ}_{\mathrm{1个}}）$ 和 $（{\mathrm{一世}}_2，{Ĵ}_2）$ 这样 ${\beta }_{{\mathrm{一世}}_{\mathrm{1个}}，{Ĵ}_{\mathrm{1个}}}>0$ 和 ${\beta }_{{\mathrm{一世}}_2，{Ĵ}_2}<0$。在本文中，当系统允许混合耦合并将各组成部分分组时，我们对基态的存在进行了系统的（几乎）完整的研究。我们首先将这些系统分为排斥混合和完全混合两种情况。在第一种情况下，我们证明不存在基态。在第二种情况下，我们给出了基态存在的必要条件，并提供了莫尔斯指数的估计。关键思想是系统的块分解（最佳块分解，最终块分解），以及不同块之间总相互作用力的度量。最后，关于基态存在的假设被证明是在某些特殊情况下最佳。