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Non-degeneracy for the critical Lane-Emden system


Authors: Rupert L. Frank, Seunghyeok Kim and Angela Pistoia
Journal: Proc. Amer. Math. Soc. 149 (2021), 265-278
MSC (2010): Primary 35J47, 35B40
DOI: https://doi.org/10.1090/proc/15217
Published electronically: October 16, 2020
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Abstract: We prove the non-degeneracy for the critical Lane-Emden system

$\displaystyle -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0$$\displaystyle \quad \text {in } \mathbb{R}^N$    

for all $ N \ge 3$ and $ p,q > 0$ such that $ \frac {1}{p+1} + \frac {1}{q+1} = \frac {N-2}{N}$. We show that all solutions to the linearized system around a ground state must arise from the symmetries of the critical Lane-Emden system provided that they belong to the corresponding energy space or they tend to zero at infinity.

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Rupert L. Frank
Affiliation: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstrasse 39, 80333 München, Germany; and Department of Mathematics 253-37, Caltech, Pasadena, California 91125
Email: r.frank@lmu.de, rlfrank@caltech.edu

Seunghyeok Kim
Affiliation: Department of Mathematics and Research Institute for Natural Sciences, College of Natural Sciences, Hanyang University, 222 Wangsimni-ro Seongdong-gu, Seoul 04763, Republic of Korea
Email: shkim0401@hanyang.ac.kr, shkim0401@gmail.com

Angela Pistoia
Affiliation: Dipartimento SBAI, “Sapienza” Università di Roma, via Antonio Scarpa 16, 00161 Roma, Italy
Email: angela.pistoia@uniroma1.it

DOI: https://doi.org/10.1090/proc/15217
Keywords: Lane--Emden system, critical hyperbola, non-degenerate solution.
Received by editor(s): September 22, 2019
Received by editor(s) in revised form: May 17, 2020
Published electronically: October 16, 2020
Additional Notes: The first author was partially supported by US National Science Foundation grant DMS-1363432.
The second author was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2017R1C1B5076384).
The third author was partially supported by Fondi di Ateneo “Sapienza” Università di Roma (Italy).
Communicated by: Ryan Hynd
Article copyright: © Copyright 2020 by the authors