Non-degeneracy for the critical Lane–Emden system

Frank, Rupert; Kim, Seunghyeok; Pistoia, Angela

doi:10.1090/proc/15217

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

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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.

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Proceedings of the American Mathematical Society

Non-degeneracy for the critical Lane–Emden system
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Abstract: