Classification of positive solutions to the critical Lane-Emden system

We classify all positive classical solutions of the Lane–Emden system $$-\mathrm{\Delta }u=v^p\text{in }{\mathbb{R}}^N,-\mathrm{\Delta }v=u^q\text{in}{\mathbb{R}}^N,$$ where $N\ge 3$ and $p,q>0$ satisfy $$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}\mathrm{.}$$Without assuming boundedness, decay at infinity, or finite energy, we prove that every such solution has the form $$u(x)={\lambda }^{\frac{N}{q+1}}U(\lambda (x-a)),v(x)={\lambda }^{\frac{N}{p+1}}V(\lambda (x-a)),$$ for some $a\in {\mathbb{R}}^N$ and $\lambda >0$, 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 $${\int }_{{\mathbb{R}}^N}{u}^{q+1}dx={\int }_{{\mathbb{R}}^N}{v}^{p+1}dx<\mathrm{\infty }\mathrm{.}$$ 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.