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
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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