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Hopf’s lemma and radial symmetry for the Logarithmic Laplacian problem

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Abstract

In this paper, we prove Hopf’s lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf’s lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf’s lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf’s lemma can be applied to the study of the symmetry of the positive solution of the nonlinear Logarithmic Laplacian problem by the moving plane method.

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Acknowledgements

The first author thanks National Natural Science Foundation of China (No.12001344) and Natural Science Foundation of Shanxi Province, China (No.20210302123339) for the support.

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  1. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030031, Shanxi, China

    Lihong Zhang & Xiaofeng Nie

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  1. Lihong Zhang
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  2. Xiaofeng Nie
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Correspondence to Lihong Zhang.

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Zhang, L., Nie, X. Hopf’s lemma and radial symmetry for the Logarithmic Laplacian problem. Fract Calc Appl Anal (2024). https://doi.org/10.1007/s13540-024-00285-1

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  • DOI: https://doi.org/10.1007/s13540-024-00285-1

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