Non-degeneracy for the critical Lane–Emden system,Proceedings of the American Mathematical Society

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Non-degeneracy for the critical Lane–Emden system
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-10-16 , DOI: 10.1090/proc/15217
Rupert Frank , Seunghyeok Kim , Angela Pistoia

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

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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