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Liouville Theorem Involving the Uniformly Nonlocal Operator

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Abstract

We prove that u is constant if u is a bounded solution of

$$\begin{aligned} A_{\alpha } u(x) = C_{n,\alpha } \text {P.V.} \int _{\mathbb {R}^n} \frac{a(x-y)(u(x)-u(y))}{|x-y|^{n+\alpha }} \mathrm{d}y=0, \;\; x \in \mathbb {R}^n, \end{aligned}$$

where the function \(a:\mathbb {R}^n \mapsto \mathbb {R}\) be uniformly bounded and radial decreasing. This result can be regarded as the generalization of usual Liouville theorem. To get the proof, we establish a maximum principle involving the nonlocal operator \( A_{\alpha }\) for antisymmetric functions on any half space.

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Acknowledgements

The authors would like to thank the anonymous referee for her/his useful comments and valuable suggestions which improved and clarified the paper. The work was supported by the National Natural Science Foundation of China (11871096 ).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, China

    Meng Qu

  2. School of Mathematics, Zhejiang University, Hangzhou, 310000, China

    Jiayan Wu

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  1. Meng Qu
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  2. Jiayan Wu
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Correspondence to Meng Qu.

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Communicated by Maria Alessandra Ragusa.

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Qu, M., Wu, J. Liouville Theorem Involving the Uniformly Nonlocal Operator. Bull. Malays. Math. Sci. Soc. 44, 1893–1903 (2021). https://doi.org/10.1007/s40840-020-01039-x

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  • DOI: https://doi.org/10.1007/s40840-020-01039-x

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