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Asymptotic Behavior and Classification of Solutions to Hartree Type Equations with Exponential Nonlinearity

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Abstract

In this paper, we are mainly concerned with the following Hartree type equations with exponential nonlinearity

$$\begin{aligned} (-\Delta )u(x)=\left( \frac{1}{|x|^{\sigma }}*e^{pu}\right) e^{pu(x)}, \,\,\,\,\,\,\,\,\,\,\,\, \text {in} \,\,\, {\mathbb {R}}^{2} \end{aligned}$$

where \(p\in (0,+\infty )\), u may change sign. We first prove the equivalence between the PDEs and the corresponding integral equations, further to get the exact asymptotic behavior of solutions to the above PDEs equation. Finally, we classify all classical solutions to the integral equations via the method of moving spheres in integral form. Consequently, we obtain the classification results of classical solutions for the PDEs.

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Acknowledgements

The authors are grateful to the anonymous referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Yuxia Guo

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Shaolong Peng

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  1. Yuxia Guo
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  2. Shaolong Peng
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Correspondence to Shaolong Peng.

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Yuxia Guo was supported by NSFC (Nos. 12031015, 12271283).

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Guo, Y., Peng, S. Asymptotic Behavior and Classification of Solutions to Hartree Type Equations with Exponential Nonlinearity. J Geom Anal 34, 23 (2024). https://doi.org/10.1007/s12220-023-01470-z

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