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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.10272 |
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| _version_ | 1866917955779952640 |
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| author | Li, Kui Liu, Mengyao Wu, Jianfeng |
| author_facet | Li, Kui Liu, Mengyao Wu, Jianfeng |
| contents | We study the weighted elliptic equation \begin{equation} -div(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u~~~\mbox{in}~\mathbb{R}^N ~~~~~~~~~~~~~~~~~~~~(0.1)\end{equation} with $N\geq 2$, which arises from the Caffarelli-Kohn-Nirenberg inequalities. Under the assumptions of finite energy and $a_1+a_2=N-2$, for nonnegative solutions we prove the equivalence between equation (0.1) with $a=a_1$ and equation (0.1) with $a=a_2$. Without finite energy assumptions, for $2\leq p<2^*$ we give the optimal parameter range in which nonnegative solutions of (0.1) in $\mathbf{L}^\infty_{Loc}(\mathbb{R}^N)$ must be radially symmetric, and give a complete classification for these solutions in this range. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10272 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry and classification of positive solutions of some weighted elliptic equations Li, Kui Liu, Mengyao Wu, Jianfeng Analysis of PDEs We study the weighted elliptic equation \begin{equation} -div(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u~~~\mbox{in}~\mathbb{R}^N ~~~~~~~~~~~~~~~~~~~~(0.1)\end{equation} with $N\geq 2$, which arises from the Caffarelli-Kohn-Nirenberg inequalities. Under the assumptions of finite energy and $a_1+a_2=N-2$, for nonnegative solutions we prove the equivalence between equation (0.1) with $a=a_1$ and equation (0.1) with $a=a_2$. Without finite energy assumptions, for $2\leq p<2^*$ we give the optimal parameter range in which nonnegative solutions of (0.1) in $\mathbf{L}^\infty_{Loc}(\mathbb{R}^N)$ must be radially symmetric, and give a complete classification for these solutions in this range. |
| title | Symmetry and classification of positive solutions of some weighted elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.10272 |