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Multiple Positive Solutions for Fractional Schrödinger–Poisson System with Doubly Critical Exponents

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Abstract

In this article, we study the following fractional Schrödinger–Poisson system

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s} u+ V(x)u- \phi |u|^{2^{*}_{s}-3}u= |u|^{2^{*}_{s}-2}u+ \lambda f(x), &{} x\in {\mathbb {R}}^3, \\ (-\Delta )^{s} \phi = |u|^{2^{*}_{s}-1}, &{} x\in {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$

where \(s\in (0,1)\), \(2^{*}_{s}=\frac{6}{3-2s}\), \(\lambda >0\) is a real parameter, f and V satisfy some suitable hypothesis. Via applying the variational methods and mountain pass theorem, we prove that there exists \( \lambda ^{*}>0 \) such that the system has at least two positive solutions for any \(\lambda \in (0,\lambda ^{*})\) which supplements and generalizes the previous results in the literature.

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Acknowledgements

The authors express their gratitude to the reviewers for careful reading and helpful suggestions which led to an improvement of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, 637009, Sichuan, People’s Republic of China

    Wei Jiang & Jia-Feng Liao

  2. College of Mathematics Education, China West Normal University, Nanchong, 637009, Sichuan, People’s Republic of China

    Jia-Feng Liao

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  1. Wei Jiang
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Jiang Wei writes the first manuscript. Liao Jia-Feng completes this manuscipt.

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Correspondence to Jia-Feng Liao.

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Supported by the Natural Science Foundation of Sichuan (2022NSFSC1816) and Fundamental Research Funds of China West Normal University (18B015).

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Jiang, W., Liao, JF. Multiple Positive Solutions for Fractional Schrödinger–Poisson System with Doubly Critical Exponents. Qual. Theory Dyn. Syst. 22, 25 (2023). https://doi.org/10.1007/s12346-022-00726-3

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