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Bôcher-type theorem on n-dimensional manifolds with conical metric


Authors: Jiayu Li and Fangshu Wan
Journal: Proc. Amer. Math. Soc. 147 (2019), 4527-4538
MSC (2010): Primary 58J05; Secondary 35J15
DOI: https://doi.org/10.1090/proc/14554
Published electronically: June 27, 2019
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Abstract: We generalize the Bôcher-type theorem and give a sharp characterization of the behavior at the isolated singularities of a solution bounded on one side for the equation $ \Delta _g u =0$ on singular manifolds with conical metrics. Furthermore, we also obtain a Liouville-type result which demonstrates that the fundamental solution is the unique nontrivial solution of $ \operatorname {div}(\vert x\vert^\theta \nabla u)=0$ in $ \mathbb{R}^n\setminus \{0\}$ that is bounded on one side in both a neighborhood of the origin as well as at infinity.


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Additional Information

Jiayu Li
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, AMSS CAS, Beijing 100190, People’s Republic of China
Email: jiayuli@@ustc.edu.cn

Fangshu Wan
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China
Email: wfangshu@@mail.ustc.edu.cn

DOI: https://doi.org/10.1090/proc/14554
Keywords: Laplace's equation, B\^ocher-type theorem, Liouville-type theorem, singular manifolds, conical metric
Received by editor(s): October 4, 2018
Received by editor(s) in revised form: January 20, 2019
Published electronically: June 27, 2019
Additional Notes: The work was supported by NSFC, No. 11721101.
Communicated by: Guofang Wei
Article copyright: © Copyright 2019 American Mathematical Society