In this work, we study a system of Schrödinger 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 of 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.

中文翻译：

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具有一般二次型非线性的薛定谔方程组

在这项工作中，我们研究了一个涉及具有二次增长的非线性的薛定谔方程系统。我们建立了关于有限时间内全局存在与爆炸的二分法的明确标准。这样的标准是根据与相应椭圆系统相关的基态解给出的，而这些基态解又是通过应用变分方法获得的。通过使用浓度紧致性方法，我们还研究了基态的非线性稳定性/不稳定性。