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Symmetry of positive solutions for equations involving higher order fractional Laplacian


Authors: Yan Li and Ran Zhuo
Journal: Proc. Amer. Math. Soc. 144 (2016), 4303-4318
MSC (2010): Primary 35S05, 35B09, 35C15, 35J08
DOI: https://doi.org/10.1090/proc/13052
Published electronically: May 25, 2016
MathSciNet review: 3531181
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Abstract: In this paper, we consider problems associated with the higher order fractional Laplacian. Through the method of moving planes, we derive rotational symmetry of positive solutions and show their dependence on the $ x_n$ variable only. We also establish the equivalence between a semilinear higher order fractional Laplacian equation and its corresponding integral equation, so as to further deduce a Liouville type theorem and obtain a priori estimates for positive solutions.


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Additional Information

Yan Li
Affiliation: Department of Mathematical Sciences, Yeshiva University, New York, New York 10033
Email: yali3@mail.yu.edu

Ran Zhuo
Affiliation: Department of Mathematical Sciences, Yeshiva University, New York, New York 10033
Address at time of publication: Department of Mathematical Sciences, Huanghuai University, Zhumadian, Henan, People’s Republic of China, 463000
Email: zhuoran1986@126.com

DOI: https://doi.org/10.1090/proc/13052
Keywords: Higher order fractional Laplacian, fractional Laplacian, the method of moving planes, rotational symmetry, equivalence
Received by editor(s): November 3, 2015
Received by editor(s) in revised form: November 30, 2015
Published electronically: May 25, 2016
Additional Notes: Corresponding author for this article is Ran Zhuo
Dedicated: This paper is dedicated to our advisor, Professor Wenxiong Chen
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society