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

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

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