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Exact Number of Peak Solutions for Nonlinear Schrödinger-Poisson Systems

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Abstract

This paper is concerned with the number of multi-peak solutions for the Schrödinger-Poisson system

$$ \left \{ \textstyle\begin{array}{lll} -\varepsilon ^{2} \Delta u + V(x) u + \Phi (x) u = |u|^{p-1} u, && \text{ in } \mathbb{R}^{3}, \\ - \Delta \Phi = u^{2}, && \text{ in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $$

where \(\varepsilon \) is a parameter, \(p \in (1, 5)\) and \(V (x)\) is the potential function. We obtain the number of peak solutions for the system when the solutions concentrate at the critical points of \(V(x)\) through Pohozaev identities and the blow-up analysis.

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Acknowledgements

L. Chen acknowledges support from Jiangxi Provincial Natural Science Foundation (20212ACB201003). H.-S. Ding acknowledges support from Two Thousand Talents Program of Jiangxi Province (jxsq2019201001) and NSFC. Q. He acknowledges support from NSFC (12061012).

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Authors and Affiliations

  1. School of Mathematics and Statistics, and Jiangxi Provincial Center for Applied Mathematics, Jiangxi Normal University, Nanchang, Jiangxi, 330022, P.R. China

    Lin Chen & Hui-Sheng Ding

  2. College of Mathematics and Information Sciences, Guangxi Center for Mathematical Research, Guangxi University, Guangxi, 530003, P.R. China

    Qihan He

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  1. Lin Chen
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Correspondence to Hui-Sheng Ding.

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Chen, L., Ding, HS. & He, Q. Exact Number of Peak Solutions for Nonlinear Schrödinger-Poisson Systems. Acta Appl Math 185, 8 (2023). https://doi.org/10.1007/s10440-023-00579-1

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