## Non-degeneracy for the critical Lane–Emden system

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- by Rupert L. Frank, Seunghyeok Kim and Angela Pistoia
- Proc. Amer. Math. Soc.
**149**(2021), 265-278 - DOI: https://doi.org/10.1090/proc/15217
- Published electronically: October 16, 2020

## Abstract:

We prove the non-degeneracy for the critical Lane–Emden system \begin{equation*} -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0 \quad \text {in } \mathbb {R}^N \end{equation*} 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.## References

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## Bibliographic Information

**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
- MR Author ID: 728268
- ORCID: 0000-0001-7973-4688
- 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
- MR Author ID: 997742
- ORCID: 0000-0003-2936-5858
- Email: shkim0401@hanyang.ac.kr, shkim0401@gmail.com
**Angela Pistoia**- Affiliation: Dipartimento SBAI, “Sapienza” Università di Roma, via Antonio Scarpa 16, 00161 Roma, Italy
- MR Author ID: 333401
- Email: angela.pistoia@uniroma1.it
- 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
- © Copyright 2020 by the authors
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 265-278 - MSC (2010): Primary 35J47, 35B40
- DOI: https://doi.org/10.1090/proc/15217
- MathSciNet review: 4172603