Classification of positive solutions to the Hénon-Sobolev critical systems
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914271108005888 |
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| author | Zhou, Yuxuan Zou, Wenming |
| author_facet | Zhou, Yuxuan Zou, Wenming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01784 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of positive solutions to the Hénon-Sobolev critical systems Zhou, Yuxuan Zou, Wenming Analysis of PDEs Functional Analysis 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. |
| title | Classification of positive solutions to the Hénon-Sobolev critical systems |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2312.01784 |