文摘

In the case where either the potentials V j , μ j and β are periodic or V j are well-shaped and μ j and β are anti-well-shaped, existence of a positive ground state of the Schr?dinger system $\left\{ \begin{gathered} - \Delta u_1 + V_1 (x)u_1 = \mu _1 (x)u_1^3 + \beta (x)u_1 u_2^2 in\mathbb{R}^N , \hfill \\ - \Delta u_2 + V_2 (x)u_2 = \beta (x)u_1^2 u_2 + u_2 (x)u_2^3 in\mathbb{R}^N , \hfill \\ u_j \in H^1 (\mathbb{R}^N ),j = 1,2, \hfill \\ \end{gathered} \right.$ where N = 1, 2, 3, is proved provided that β is either small or large in terms of V j and μ j . 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.