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Existence and concentration of solution for Schrödinger-Poisson system with local potential

Author

Listed:
  • Zhipeng Yang

    (Georg-August-University of Göttingen)

  • Yuanyang Yu

    (University of Chinese Academy of Sciences)

Abstract

In this paper, we study the following nonlinear Schrödinger-Poisson type equation $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u+K(x)\phi u=f(u)&{}\text {in}\ {\mathbb {R}}^3,\\ -\varepsilon ^2\Delta \phi =K(x)u^2&{}\text {in}\ {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$ - ε 2 Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) in R 3 , - ε 2 Δ ϕ = K ( x ) u 2 in R 3 , where $$\varepsilon >0$$ ε > 0 is a small parameter, $$V: {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ V : R 3 → R is a continuous potential and $$K: {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ K : R 3 → R is used to describe the electron charge. Under suitable assumptions on V(x), K(x) and f, we prove existence and concentration properties of ground state solutions for $$\varepsilon >0$$ ε > 0 small. Moreover, we summarize some open problems for the Schrödinger-Poisson system.

Suggested Citation

  • Zhipeng Yang & Yuanyang Yu, 2021. "Existence and concentration of solution for Schrödinger-Poisson system with local potential," Partial Differential Equations and Applications, Springer, vol. 2(4), pages 1-22, August.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:4:d:10.1007_s42985-021-00105-8
    DOI: 10.1007/s42985-021-00105-8
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    References listed on IDEAS

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    1. Zhipeng Yang & Yuanyang Yu & Fukun Zhao, 2019. "The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system," Mathematische Nachrichten, Wiley Blackwell, vol. 292(8), pages 1837-1868, August.
    2. Tao Xie & Lu Xiao & Jun Wang, 2015. "Existence of Multiple Positive Solutions for Choquard Equation with Perturbation," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-10, September.
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