On coupled systems of Schrödinger equations

Abstract

In this paper we study the following system of nonlinear Schrödinger equations: $$ \begin{cases}-\Delta u +u = f(x,u)+\lambda v, & x\in \mathbb R^N,\\ -\Delta v +v =g(x,v)+\lambda u, & x\in \mathbb R^N.\end{cases} $$ Under some assumptions on $f$ and $g$, we obtain the existence of positive ground and bound states of the coupled system for $\lambda \in (0, 1)$. More importantly, we will give more precise descriptions of the limit behavior and energy estimates of the bound states as $\lambda$ changes.