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

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Abstract

In this paper, we study the following nonlinear Schrödinger-Poisson type equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u+K(x)\phi u=f(u)&{}\text {in}\ {\mathbb {R}}^3,\\ -\varepsilon ^2\Delta \phi =K(x)u^2&{}\text {in}\ {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$

where \(\varepsilon >0\) is a small parameter, \(V: {\mathbb {R}}^3\rightarrow {\mathbb {R}}\) is a continuous potential and \(K: {\mathbb {R}}^3\rightarrow {\mathbb {R}}\) 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 \(\varepsilon >0\) small. Moreover, we summarize some open problems for the Schrödinger-Poisson system.

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

We would like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments.

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Authors and Affiliations

  1. Mathematical Institute, Georg-August-University of Göttingen, 37073, Göttingen, Germany

    Zhipeng Yang

  2. University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Yuanyang Yu

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  1. Zhipeng Yang
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  2. Yuanyang Yu
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Correspondence to Yuanyang Yu.

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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

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Yang, Z., Yu, Y. Existence and concentration of solution for Schrödinger-Poisson system with local potential. Partial Differ. Equ. Appl. 2, 47 (2021). https://doi.org/10.1007/s42985-021-00105-8

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