Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in $\mathbb {R}^6$
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- by Xuemei Li, Xingdong Tang and Guixiang Xu PDF
- Proc. Amer. Math. Soc. 150 (2022), 5203-5216 Request permission
Abstract:
We prove that any positive solution for the critical nonlinear Hartree equation \begin{equation*} -\operatorname {\bigtriangleup }{u}\left (x\right ) -\int _{\mathbb {R}^6} \frac {\left |{u}\left (y\right )\right |^2 }{ \left |x-y\right |^4 }\mathrm {d}y \,{u}\left (x\right )=0,\quad \quad x\in \mathbb {R}^6, \end{equation*} is nondegenerate. Firstly, in terms of spherical harmonics, we show that the corresponding linear operator can be decomposed into a series of one dimensional linear operators. Secondly, by making use of the Perron-Frobenius property, we show that the kernel of each one dimensional linear operator is finite. Finally, we show that the kernel of the corresponding linear operator is the direct sum of the kernel of all one dimensional linear operators.References
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Additional Information
- Xuemei Li
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 10087, People’s Republic of China
- Email: xuemei_li@mail.bnu.edu.cn
- Xingdong Tang
- Affiliation: School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China
- MR Author ID: 901729
- Email: txd@nuist.edu.cn
- Guixiang Xu
- Affiliation: Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- MR Author ID: 715732
- ORCID: 0000-0001-7376-3760
- Email: guixiang@bnu.edu.cn
- Received by editor(s): December 27, 2021
- Published electronically: September 30, 2022
- Additional Notes: The second author was supported by NSFC (No. 12001284). The third author was supported by National Key Research and Development Program of China (No. 2020YFA0712900) and by NSFC (No. 11831004)
- Communicated by: Ariel Barton
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 5203-5216
- MSC (2020): Primary 35J91, 35B38
- DOI: https://doi.org/10.1090/proc/16088