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Multiplicity of semiclassical states for Schrödinger–Poisson systems with critical frequency

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Abstract

We are concerned with the Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{array}{lll} -\,\epsilon ^2\Delta u+V(x)u+K(x)\phi u=u^5,&{}\quad x\in {\mathbb {R}}^3,\\ -\,\Delta \phi =K(x)u^2, &{}\quad x\in {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$

where \(\epsilon >0\) is a parameter, \(V\in L^{\frac{3}{2}}({\mathbb {R}}^3)\) and \(K\in L^{2}({\mathbb {R}}^3)\) are nonnegative functions and V is assumed to be zero in some region of \({\mathbb {R}}^3\), which means it is of the critical frequency case. By virtue of a global compactness lemma and Lusternik–Schnirelman theory, we show the multiplicity of high energy semiclassical states.

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Acknowledgements

The authors would like to express their sincere gratitude to the referees for careful reading of the manuscript and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, China

    Hui Zhang

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Junxiang Xu & Fubao Zhang

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  1. Hui Zhang
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  2. Junxiang Xu
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Correspondence to Hui Zhang.

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The work was supported by the National Natural Science Foundation of China (Nos. 11601204,11671077 and 11571140), and Qing Lan Project.

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Zhang, H., Xu, J. & Zhang, F. Multiplicity of semiclassical states for Schrödinger–Poisson systems with critical frequency. Z. Angew. Math. Phys. 71, 5 (2020). https://doi.org/10.1007/s00033-019-1226-8

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  • DOI: https://doi.org/10.1007/s00033-019-1226-8

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