Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in $${mathbb{R}^n}$$

In this paper we consider systems of coupled Schrödinger equations which appear in nonlinear optics. The problem has been considered mostly in the one-dimensional case. Here we make a rigorous study of the existence of least energy standing waves (solitons) in higher dimensions. We give: conditions on the parameters of the system under which it possesses a solution with least energy among all multi-component solutions; conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie.