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.

In this work we give a sharp criterion for the global well-posedness, in the energy space, for a system of nonlinear Schrödinger equations with quadratic interaction in dimension n = 5. The criterion is given in terms of the charge and energy of the ground states associated with the system, which are obtained by minimizing a Weinstein-type functional. The main result is then obtained in view of a sharp Gagliardo-Nirenberg-type inequality.

In this work, we show the existence of ground state solutions for an l-component system of non-linear Schrödinger equations with quadratic-type growth interactions in the energy-critical case. They are obtained analyzing a critical Sobolev-type inequality and using the concentration-compactness method. As an application, we prove the existence of blow-up solutions of the system without the mass-resonance condition in dimension six (and five), when the initial data is radial.

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