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

TL;DR

This paper investigates the conditions under which entire solutions exist or do not exist for a class of non-cooperative cubic elliptic systems, providing a complete characterization in low dimensions and extending to general nonlinearities.

Contribution

It offers a comprehensive analysis of existence and nonexistence of solutions for cubic Schrödinger systems based on matrix properties, including new results in low dimensions and generalizations.

Findings

Complete characterization in dimensions N=1,2

Conditions linking matrix properties to solution existence

Extensions to general power-type nonlinearities

Abstract

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 $\beta_{ii}\geq 0$ for every $i=1,\ldots,n$. We analyse the existence and nonexistence of nontrivial solutions in connection with the properties of the matrix $\beta$, and provide a complete characterization in dimensions $N=1,2$. Extensions to more general power-type nonlinearities are given.