Non-degeneracy for the critical Lane–Emden system

Source abstract

We prove the non-degeneracy for the critical Lane–Emden system − Δ U = V p , − Δ V = U q , U , V > 0 in R N $$\begin{equation*} -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0 \quad \text {in } \mathbb {R}^N \end{equation*}$$ for all N ≥ 3 N \ge 3 and p , q > 0 p,q > 0 such that 1 p + 1 + 1 q + 1 = N − 2 N \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.