Non-degeneracy for the critical Lane-Emden system

Abstract

We prove the non-degeneracy for the critical Lane--Emden system $$-\Delta U=V^p,-\Delta V=U^q,U,V>0\text{in }{\mathbb{R}}^N$$ 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 decay to 0 uniformly as the point tends to infinity.

15 pages