On a system of Schrödinger equations with general quadratic-type nonlinearities

Abstract

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 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. Contents 1. Introduction 1 2. Preliminaries 4 3. Local and global well-posedness in L 2 and H 1 9 3.1. Local existence of L 2 -solutions 10 3.2. Local existence of H 1 -solutions 12 3.3. Global solutions 13 4. Existence of ground state solutions 17 5. Global solutions versus blow-up 23 5.1. Virial Identities 23 5.2. Global existence in H 1 27 5.3. Blow-up results 28 6. Stability and instability of standing waves 32 6.1. Stability 32 6.2. Instability 45 Acknowledgement 49 References 49