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Ground states of a nonlinear Schrödinger system with nonconstant potentials

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Abstract

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 Schrödinger 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.

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Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

    HaiDong Liu

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    ZhaoLi Liu

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  1. HaiDong Liu
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  2. ZhaoLi Liu
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Liu, H., Liu, Z. Ground states of a nonlinear Schrödinger system with nonconstant potentials. Sci. China Math. 58, 257–278 (2015). https://doi.org/10.1007/s11425-014-4914-z

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