Non-degeneracy for the critical Lane–Emden system

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.