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Transactions of the American Mathematical Society

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Axially symmetric solutions of the Allen-Cahn equation with finite Morse index


Authors: Changfeng Gui, Kelei Wang and Jucheng Wei
Journal: Trans. Amer. Math. Soc. 373 (2020), 3649-3668
MSC (2010): Primary 35B53, 35J15, 35J20, 35J91, 53A05, 53A10
DOI: https://doi.org/10.1090/tran/8035
Published electronically: February 19, 2020
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Abstract: In this paper we study axially symmetric solutions of the Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between $ 4$ and $ 10$. In dimension $ 3$, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals $ 1$.


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Additional Information

Changfeng Gui
Affiliation: Department of Mathematics, University of Texas at San Antonio, San Antonio, Texas 78249; and School of Mathematics and Statistics, The Central South University, Changsha 410082, People’s Republic of China
Email: changfeng.gui@utsa.edu

Kelei Wang
Affiliation: School of Mathematics and Statistics & Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, People’s Republic of China
Email: wangkelei@whu.edu.cn

Jucheng Wei
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: jcwei@math.ubc.ca

DOI: https://doi.org/10.1090/tran/8035
Keywords: Allen-Cahn, stable or finite Morse index solutions, axially symmetric solutions
Received by editor(s): February 3, 2019
Received by editor(s) in revised form: February 28, 2019, and September 22, 2019
Published electronically: February 19, 2020
Additional Notes: The research of the first author was partially supported by NSF grant DMS-1601885.
The research of the second author was supported by NSFC. 11871381 and 11631011.
The research of the third author was partially supported by NSERC of Canada.
Article copyright: © Copyright 2020 American Mathematical Society