# On a system of Schr\"odinger equations with general quadratic-type nonlinearities

## TL;DR

This paper analyzes a system of Schrödinger equations with quadratic nonlinearities, establishing criteria for global existence or blow-up, and examining the stability of ground states using variational and concentration-compactness methods.

## Contribution

It provides sharp criteria for global existence versus blow-up and investigates ground state stability in quadratic nonlinear Schrödinger systems using variational techniques.

## Key findings

- Sharp criterion for global existence vs. blow-up.
- Ground state solutions characterized via variational methods.
- Stability and instability results for ground states.

## Abstract

In this work we study a system of Schr\"odinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms on the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.