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On the number of concentrating solutions of a fractional Schrödinger–Poisson system with doubly critical growth

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Abstract

In this paper we study the existence, multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson system with doubly critical growth

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

where \(s \in (\frac{3}{4},1)\), \(\varepsilon \) is a positive parameter, \(2^*_s = \frac{6}{3-2s}\) is the fractional critical Sobolev exponent, \((-\Delta )^s\) is the fractional Laplacian operator, and f is a continuous nonlinearity with subcritical growth. With the help of Nehari manifold and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value for small values of the parameter \(\varepsilon \).

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Acknowledgements

The authors would like to thank the anonymous reviewers for their careful reading the manuscript and giving valuable comments. This work is supported by NSFC (11771468, 11971027, 12171497).

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  1. College of Science, Minzu University of China, Beijing, 100081, China

    Siqi Qu & Xiaoming He

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Qu, S., He, X. On the number of concentrating solutions of a fractional Schrödinger–Poisson system with doubly critical growth. Anal.Math.Phys. 12, 59 (2022). https://doi.org/10.1007/s13324-022-00675-9

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