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Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities

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Abstract

In this paper, we first study a Schrödinger system with nonlocal coupling nonlinearities of Hartree type

$$\left\{\begin{array}{ll} -\varepsilon^{2}\Delta u +V_1(x)u = \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)u\,+\, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d} y \right)u,\\ -\varepsilon^{2} \Delta v +V_2(x)v = \left(\int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d}y \right)v \,+ \, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)v. \end{array}\right.$$

Using variational methods, we prove the existence of purely vector ground state solutions for the Schrödinger system if the parameter \({\varepsilon}\) is small enough. Secondly, we also establish some existence results for the coupled Schrödinger system with critical exponents.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Minbo Yang

  2. College of Mathematics, Jilin University, Jilin, 130012, People’s Republic of China

    Yuanhong Wei

  3. Institute of Mathematics, Academy of Mathematics and Systems Sciences Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Yanheng Ding

Authors
  1. Minbo Yang
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  2. Yuanhong Wei
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  3. Yanheng Ding
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Corresponding author

Correspondence to Minbo Yang.

Additional information

M. Yang is supported by ZJNSF (Y7080008) and NSFC (11101374, 11271331). Y. Ding is supported by NSFC (10831005).

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Yang, M., Wei, Y. & Ding, Y. Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities. Z. Angew. Math. Phys. 65, 41–68 (2014). https://doi.org/10.1007/s00033-013-0317-1

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  • DOI: https://doi.org/10.1007/s00033-013-0317-1

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