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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On Dancer’s conjecture for stable solutions with sign-changing nonlinearity
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by Yong Liu, Kelei Wang, Juncheng Wei and Ke Wu;
Proc. Amer. Math. Soc. 152 (2024), 3485-3497
DOI: https://doi.org/10.1090/proc/16881
Published electronically: June 20, 2024

Abstract:

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in $\mathbb {R}^{n}$ with sign-changing nonlinearity $f$. Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of $f$ at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer.
References
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Bibliographic Information
  • Yong Liu
  • Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
  • ORCID: 0000-0002-5967-4558
  • Email: yliumath@ustc.edu.cn
  • Kelei Wang
  • Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
  • MR Author ID: 866773
  • ORCID: 0000-0002-2815-0495
  • Email: wangkelei@whu.edu.cn
  • Juncheng Wei
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia 1Z2, Canada; \normalfont and Department of Mathematics, Chinese University of Hong Kong, Shatin, NT, Hong Kong
  • MR Author ID: 339847
  • ORCID: 0000-0001-5262-477X
  • Email: jcwei@math.ubc.ca
  • Ke Wu
  • Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
  • Email: wukemail@whu.edu.cn
  • Received by editor(s): December 9, 2023
  • Received by editor(s) in revised form: January 31, 2024
  • Published electronically: June 20, 2024
  • Additional Notes: The first author was supported by the National Key R&D Program of China 2022YFA1005400 and NSFC 11971026, NSFC 12141105. The second author was supported by National Key R&D Program of China (No. 2022YFA1005602) and NSFC (No. 12131017 and No. 12221001). The third author was supported by NSERC of Canada.
  • Communicated by: Ryan Hynd
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3485-3497
  • MSC (2020): Primary 35B53, 35J61
  • DOI: https://doi.org/10.1090/proc/16881
  • MathSciNet review: 4767278