Classification of positive solutions to the Hénon-Sobolev critical systems

Abstract

In this paper, we investigate positive solutions to the following Hénon-Sobolev critical system: $$ -\mathrm{div}(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u+να|x|^{-bp}|u|^{α-2}|v|^βu\quad\text{in }\mathbb{R}^n,$$ $$ -\mathrm{div}(|x|^{-2a}\nabla v)=|x|^{-bp}|v|^{p-2}v+νβ|x|^{-bp}|u|^α|v|^{β-2}v\quad\text{in }\mathbb{R}^n,$$ $$u,v\in D_a^{1,2}(\mathbb{R}^n),$$ where $n\geq 3,-\infty< a<\frac{n-2}{2},a\leq b 0$ and $α>1,β>1$ satisfying $α+β=p$. 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 Hénon 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 Hénon-Sobolev critical systems.