A perturbative approach to non-degeneracy of the Lane-Emden system

Abstract

We consider ground state solutions of the critical Lane-Emden system \[\begin{cases} -Δu = v^p &\text{in } \mathbb{R}^n,\\ -Δv = u^q &\text{in } \mathbb{R}^n,\\ u,v >0\ &\text{in } \mathbb{R}^n, \end{cases}\] where $n\ge 3$ and $p,q>0$ and $(p,q)$ belongs to the critical hyperbola $\frac{1}{p+1}+\frac{1}{q+1}=\frac{n-2}{n}.$ We prove that they are non-degenerate when either $(p,q)$ is close to $(1,\frac{n+4}{n-4})$ (if $n\ge 5$) or $(p,q)$ is close to $(\frac{n+2}{n-2},\frac{n+2}{n-2})$ (if $n\ge 3$).

15 pages, Comments are welcome