Ground states of a nonlinear Schrödinger system with nonconstant potentials

In the case where either the potentials Vj, µj and β are periodic or Vj are well-shaped and µj and β are anti-well-shaped, existence of a positive ground state of the Schrödinger system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\{ \begin{gathered} - \Delta u_1 + V_1 (x)u_1 = \mu _1 (x)u_1^3 + \beta (x)u_1 u_2^2 in\mathbb{R}^N , \hfill \\ - \Delta u_2 + V_2 (x)u_2 = \beta (x)u_1^2 u_2 + u_2 (x)u_2^3 in\mathbb{R}^N , \hfill \\ u_j \in H^1 (\mathbb{R}^N ),j = 1,2, \hfill \\ \end{gathered} \right.$\end{document} where N = 1, 2, 3, is proved provided that β is either small or large in terms of Vj and µj. The system with constant coefficients has been studied extensively in the last ten years, and the nonconstant coefficients case has seldom been studied. It turns out that new technical machineries in the setting of variational methods are needed in dealing with the nonconstant coefficients case.