Classification of positive solutions to the Hénon-Sobolev critical systems

TL;DR

The paper addresses the classification of positive solutions to the Hénon-Sobolev critical system in $\mathbb{R}^n$ with weights, under the regime $n\ge3$, $-\infty<a<\frac{n-2}{2}$ and $a\le b<a+1$, with $p=\frac{2n}{n-2+2(b-a)}$ and $\alpha+\beta=p$. The authors combine a generalized moving plane method, ODE reduction via radial symmetry, and a sharp vector-valued Caffarelli–Kohn–Nirenberg inequality to obtain a full classification: for $a\ge0$ any positive solution is synchronized with a scalar ground state of the decoupled equation, and for $a<0$, $b>a$ all nonnegative ground states are of the synchronized form $(sc_1W, sc_2W)$ with $W$ solving the scalar Hénon equation; the constants $c_1,c_2$ are characterized by a variational minimization. They further establish nondegeneracy criteria for synchronized solutions and extend the framework to general $k$-coupled systems, providing a comprehensive picture that links symmetry, asymptotics, and sharp inequalities. The results have implications for the structure of ground states and their stability in weighted critical systems and connect to Hardy–Sobolev-type inequalities and symmetry thresholds in related problems.

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 $n\geq 3,-\infty< a<\frac{n-2}{2},a\leq b<a+1,p=\frac{2n}{n-2+2(b-a)},ν>0$ 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.

Theorems & Definitions (26)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3: Radial symmetry
• Theorem 1.4: Asymptotic behavior
• Theorem 1.5: Modified inversion symmetry
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10