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Solutions of a quasilinear Schrödinger–Poisson system with linearly bounded nonlinearities

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Abstract

In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+ K(x)\phi u=f(x,u),\quad &{}x\in {\mathbb {R}}^3,\\ -\Delta \phi -\varepsilon ^4\Delta _4\phi = K(x) u^2, &{}x\in {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$

where \(\varepsilon \) is a positive parameter and f is linearly bounded in u at infinity. Under suitable assumptions on V, K and f, we establish the existence and asymptotic behavior of ground state solutions to the system. We prove that they converge to the solutions of the classic Schrödinger–Poisson system associated as \(\varepsilon \) tends to zero.

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Acknowledgements

We are grateful to the referees for the insightful comments and valuable suggestions to our paper. This work is supported by NSFC (12171014, 12071266, 11701346) and Fundamental Research Program of Shanxi Province (202303021212001, 202203021221005, 202103021224013).

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Anran Li & Chongqing Wei

  2. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, People’s Republic of China

    Leiga Zhao

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  1. Anran Li
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  2. Chongqing Wei
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  3. Leiga Zhao
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Anran Li and Leiga Zhao prepare the manuscript. Chongqing Wei and Leiga Zhao wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Leiga Zhao.

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Li, A., Wei, C. & Zhao, L. Solutions of a quasilinear Schrödinger–Poisson system with linearly bounded nonlinearities. Nonlinear Differ. Equ. Appl. 31, 28 (2024). https://doi.org/10.1007/s00030-023-00912-5

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