Global existence for a coupled system of Schrödinger equations with power-type nonlinearities

Abstract

In this manuscript, we consider the Cauchy problem for a Schrödinger system with power-type nonlinearities\documentclass[12pt]{minimal}\begin{document}$\left\lbrace \begin{array}{l}i\frac{\partial }{\partial t}u_j+ \triangle u_j + \sum _{k=1}^m a_{jk} |u_k|^{p}|u_j|^{p-2} u_j=0,\\u_j(x,0) = \psi _{j0}(x), \end{array} \right.$\end{document}i∂∂tuj+▵uj+∑k=1majk|uk|p|uj|p−2uj=0,uj(x,0)=ψj0(x),where \documentclass[12pt]{minimal}\begin{document}$u_j: \mathbb {R}^N\times \mathbb {R} \rightarrow \mathbb {C}$\end{document}uj:RN×R→C, \documentclass[12pt]{minimal}\begin{document}$\psi _{j0}:\mathbb {R}^N \rightarrow \mathbb {C}$\end{document}ψj0:RN→C for j = 1, 2, …, m and ajk = akj are positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.