Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems

In this paper we deal with the cubic Schr"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$,$u_1,\dots,u_n \geq 0$in$$\mathbb{R}$^N (N\leq 3)$,where$\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients and ${\beta }_{ii}\ge 0$ for every $i=1,\dots ,n$. We analyse the existence and nonexistence of nontrivial solutions in connection with the properties of the matrix $\beta$, and provide a complete characterization in dimensions $N=1,2$. Extensions to more general power-type nonlinearities are given.