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: $$-\mathrm{div}(|x{|}^{-2a}\nabla u)=|x{|}^{-bp}|u{|}^{p-2}u+\nu \alpha |x{|}^{-bp}|u{|}^{\alpha -2}|v{|}^{\beta }u\text{in }{\mathbb{R}}^n,$$ $$-\mathrm{div}(|x{|}^{-2a}\nabla v)=|x{|}^{-bp}|v{|}^{p-2}v+\nu \beta |x{|}^{-bp}|u{|}^{\alpha }|v{|}^{\beta -2}v\text{in }{\mathbb{R}}^n,$$ $$u,v\in D_a^{1,2}({\mathbb{R}}^n),$$ where $n\ge 3,-\infty <a<\frac{n-2}{2},a\le b<a+1,p=\frac{2n}{n-2+2(b-a)},\nu >0$ 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 $a\ge 0$, 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.

23 pages