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Ground State Solutions for a Class of Choquard Equations Involving Doubly Critical Exponents

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Abstract

In this paper, we are concerned with the autonomous Choquard equation

$$ - \Delta u + u = ({I_\alpha } \ast {\left| u \right|^{{\alpha \over N} + 1}}){\left| u \right|^{{\alpha \over N} - 1}}u + {\left| u \right|^{{2^ \ast } - 2}}u + f(u)\,\,\,\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N},$$

where N ≥ 3, Iα denotes the Riesz potential of order α ∈ (0, N), the exponents \({\alpha \over N} + 1\) and \({2^ * } = {{2N} \over {N - 2}}\) are critical with respect to the Hardy-Littlewood-Sobolev inequality and Sobolev embedding, respectively. Based on the variational methods, by using the minimax principles and the Pohožaev manifold method, we prove the existence of ground state solution under some suitable assumptions on the perturbation f.

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Acknowledgments

The authors would like to thank the referees and editors for carefully reading this paper and making valuable suggestions which greatly improve the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Yong-yong Li, Gui-dong Li & Chun-lei Tang

  2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

    Yong-yong Li

Authors
  1. Yong-yong Li
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  2. Gui-dong Li
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Correspondence to Chun-lei Tang.

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This paper is supported by the National Natural Science Foundation of China (No. 11971393).

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Li, Yy., Li, Gd. & Tang, Cl. Ground State Solutions for a Class of Choquard Equations Involving Doubly Critical Exponents. Acta Math. Appl. Sin. Engl. Ser. 37, 820–840 (2021). https://doi.org/10.1007/s10255-021-1046-4

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