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Groundstates of the Schrödinger–Poisson–Slater equation with critical growth

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Abstract

In this paper, we study the existence of ground state solutions for the following Schrödinger–Poisson–Slater equation:

$$\begin{aligned} - \Delta u+(|x|^{\alpha -n}*|u|^2)u=\mu |u|^{p-2}u+ |u|^{2^*-2}u,~{\textrm{in}}~ {\mathbb {R}}^n, \end{aligned}$$

where \(n\geqslant 3\), \(\alpha \in (0,n)\). By combining the Nehari–Pohozaev method with compactness arguments, we first obtain a positive ground state solution for above equation. Then we establish several qualitative properties of the ground state solutions.

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Acknowledgements

C. Lei is supported by Science and Technology Foundation of Guizhou Province (No.ZK[2022]199)). B. Zhang was supported by National Natural Science Foundation of China (No. 11871199 and No. 12171152) and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province. The research of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within PNCDI III.

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Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Minzu University, Guiyang, 550025, People’s Republic of China

    Chunyu Lei

  2. Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059, Kraków, Poland

    Vicenţiu D. Rădulescu

  3. Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

  4. Brno University of Technology, Technická 3058/10, 61600, Brno, Czech Republic

    Vicenţiu D. Rădulescu

  5. School of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Vicenţiu D. Rădulescu & Binlin Zhang

  6. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Binlin Zhang

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  1. Chunyu Lei
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Lei, C., Rădulescu, V.D. & Zhang, B. Groundstates of the Schrödinger–Poisson–Slater equation with critical growth. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 128 (2023). https://doi.org/10.1007/s13398-023-01457-z

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  • DOI: https://doi.org/10.1007/s13398-023-01457-z

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