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

TL;DR

This paper investigates the non-degeneracy of ground state solutions to the critical Lane-Emden system using a perturbative approach, focusing on parameter regimes near specific critical points.

Contribution

It introduces a perturbative method to establish non-degeneracy of solutions near particular critical parameter values in the Lane-Emden system.

Findings

Solutions are non-degenerate near (1, (n+4)/(n-4)) for n≥5.

Solutions are non-degenerate near ((n+2)/(n-2), (n+2)/(n-2)) for n≥3.

The approach applies to the critical hyperbola in the Lane-Emden system.

Abstract

We consider ground state solutions of the critical Lane-Emden system \[\begin{cases} -\Delta u = v^p &\text{in } \mathbb{R}^n,\\ -\Delta v = u^q &\text{in } \mathbb{R}^n,\\ u,v >0\ &\text{in } \mathbb{R}^n, \end{cases}\] 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 $(1,{n+4\over n-4})$ (if $n\ge5$) or $(p,q)$ is close to $({n+2\over n-2},{n+2\over n-2})$ (if $n\ge3$).