Classification of positive solutions to the critical Lane-Emden system

Abstract

We classify all positive classical solutions of the Lane--Emden system $$ -Δu=v^p \, \, \text{in } \, \, \mathbb{R}^N, \, \, -Δv=u^q \, \, \text{in} \, \, \mathbb{R}^N, $$ where $N\geq3$ and $p,q>0$ satisfy $$ \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$$Without assuming boundedness, decay at infinity, or finite energy, we prove that every such solution has the form $$ u(x)=λ^{\frac{N}{q+1}}U(λ(x-a)), \, \, v(x)=λ^{\frac{N}{p+1}}V(λ(x-a)), $$ for some $a\in \mathbb{R}^N$ and $λ>0$, where $(U,V)$ is the positive radial ground state centered at the origin and normalized by $U(0)=1$. Consequently, both components are radially symmetric about the same point and satisfy $$ \int_{\mathbb{R}^N}u^{q+1}\, dx = \int_{\mathbb{R}^N}v^{p+1}\,dx<\infty. $$ Our result extends the classical classification of positive solutions of the critical Sobolev equation by Caffarelli, Gidas and Spruck and Chen and Li to the critical Lane-Emden system.