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Finite Morse index solutions of the Allen-Cahn equation with energy bound
Abstract
We study bounded entire solutions of the Allen-Cahn equation in dimension three with quadratic energy growth. We prove that such a solution of finite Morse index I has at most 2I ends. Furthermore, we show that Morse index one solutions have two ends and are axially symmetric, which generalizes an earlier result by Florit-Simon.
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BibTeX
@misc{liu2026finite,
title = {{Finite Morse index solutions of the Allen-Cahn equation with energy bound}},
author = {Yong Liu and Kelei Wang and Juncheng Wei and Ke Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15688},
eprint = {2609.15688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15688},
}
RIS
TY - UNPB
TI - Finite Morse index solutions of the Allen-Cahn equation with energy bound
AU - Yong Liu
AU - Kelei Wang
AU - Juncheng Wei
AU - Ke Wu
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15688
UR - https://arxiv.org/abs/2609.15688
AB - We study bounded entire solutions of the Allen-Cahn equation in dimension three with quadratic energy growth. We prove that such a solution of finite Morse index I has at most 2I ends. Furthermore, we show that Morse index one solutions have two ends and are axially symmetric, which generalizes an earlier result by Florit-Simon.
ER -