Classification of positive solutions of Hardy-Sobolev equation without the finite volume constraints
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909957216010240 |
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| author | Chen, Lu Yang, Yabo |
| author_facet | Chen, Lu Yang, Yabo |
| contents | In this paper, we are concerned with the critical Hardy-Sobolev equation \begin{equation*} -Δ_{p}u = \frac{u^{p^{*}_s-1}}{|x|^{s}}, \ \ x\in \mathbb{R}^n \end{equation*} where $p^{*}_s = \frac{(n-s)p}{n-p}$ denotes the critical Hardy-Sobolev exponent. We classify the positive solutions of this equation for $0 < s < \frac{p-1}{p}$ and $\frac{(2s+n+1)+\sqrt{(2s+n+1)^2-12s}}{6} \leq p < n$ without finite volume constraints, which extends Ou's result in \cite{9} in the literature. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_16017 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of positive solutions of Hardy-Sobolev equation without the finite volume constraints Chen, Lu Yang, Yabo Analysis of PDEs In this paper, we are concerned with the critical Hardy-Sobolev equation \begin{equation*} -Δ_{p}u = \frac{u^{p^{*}_s-1}}{|x|^{s}}, \ \ x\in \mathbb{R}^n \end{equation*} where $p^{*}_s = \frac{(n-s)p}{n-p}$ denotes the critical Hardy-Sobolev exponent. We classify the positive solutions of this equation for $0 < s < \frac{p-1}{p}$ and $\frac{(2s+n+1)+\sqrt{(2s+n+1)^2-12s}}{6} \leq p < n$ without finite volume constraints, which extends Ou's result in \cite{9} in the literature. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality. |
| title | Classification of positive solutions of Hardy-Sobolev equation without the finite volume constraints |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.16017 |