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On elliptic equations with Stein–Weiss type convolution parts

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Abstract

The aim of this paper is to study the critical elliptic equations with Stein–Weiss type convolution parts

$$\begin{aligned} \displaystyle -\Delta u =\frac{1}{|x|^{\alpha }}\left( \int _{\mathbb {R}^{N}}\frac{|u(y)|^{2_{\alpha , \mu }^{*}}}{|x-y|^{\mu }|y|^{\alpha }}dy\right) |u|^{2_{\alpha , \mu }^{*}-2}u,\quad x\in \mathbb {R}^{N}, \end{aligned}$$

where the critical exponent is due to the weighted Hardy–Littlewood–Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, the subcritical case is also considered, the existence, symmetry, regularity of the positive solutions are obtained.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her useful comments and suggestions which help to improve the presentation of the paper greatly.

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  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, People’s Republic of China

    Lele Du & Minbo Yang

  2. Department of Mathematics and Physics, Henan University of Urban Construction, Pingdingshan, 467044, Henan, People’s Republic of China

    Fashun Gao

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  1. Lele Du
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Minbo Yang is the corresponding author who was partially supported by NSFC (11971436, 12011530199) and ZJNSF (LZ22A010001, LD19A010001) and Fashun Gao was partially supported by NSFC (11901155, 12011530199)

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Du, L., Gao, F. & Yang, M. On elliptic equations with Stein–Weiss type convolution parts. Math. Z. 301, 2185–2225 (2022). https://doi.org/10.1007/s00209-022-02973-1

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