Existence of Standing Waves for Coupled Nonlinear Schrödinger Equations

Abstract

In this paper we study the existence of standing waves for coupled nonlinear Schrödinger equations. The interaction between equations plays an important role in our study. When the interaction is strong, the least energy solution is a solution whose both components are positive. When the interaction is weak, the least energy solution is a semitrivial solution, namely a solution of a form $(u_1,0)$ or $(0,u_2)$. Moreover, minimizing method on the Nehari type manifold with codimension 2 gives us a positive solution when the interaction is weak.