Classification of positive solutions to the H\'enon-Sobolev critical systems

In this paper, we investigate positive solutions to the following H\'enon-Sobolev critical system: $$-\mathrm{div}(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u+\nu\alpha|x|^{-bp}|u|^{\alpha-2}|v|^{\beta}u\quad\text{in }\mathbb{R}^n,$$ $$-\mathrm{div}(|x|^{-2a}\nabla v)=|x|^{-bp}|v|^{p-2}v+\nu\beta|x|^{-bp}|u|^{\alpha}|v|^{\beta-2}v\quad\text{in }\mathbb{R}^n,$$ $$u,v\in D_a^{1,2}(\mathbb{R}^n),$$ where $n\geq 3,-\infty< a<\frac{n-2}{2},a\leq b<a+1,p=\frac{2n}{n-2+2(b-a)},\nu>0$ and $\alpha>1,\beta>1$ satisfying $\alpha+\beta=p$. Our findings are divided into two parts, according to the sign of the parameter $a$. For $a\geq 0$, we demonstrate that any positive solution $(u,v)$ is synchronized, indicating that $u$ and $v$ are constant multiples of positive solutions to the decoupled H\'enon equation: \begin{equation*} -\mathrm{div}(|x|^{-2a}\nabla w)=|x|^{-bp}|w|^{p-2}w. \end{equation*} For $a<0$ and $b>a$, we characterize all nonnegative ground states. Additionally, we study the nondegeneracy of nonnegative synchronized solutions. This work also delves into some general $k$-coupled H\'enon-Sobolev critical systems.