Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908707294543872 |
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| author | Chen, Lu Wu, Tian Yan, Jin Yang, Yabo |
| author_facet | Chen, Lu Wu, Tian Yan, Jin Yang, Yabo |
| contents | In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation}\label{0.1} -Δ^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation} without the finite volume constraint for $n \geq 3$ and $p_n(Λ) < p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent, $-Δ^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator and $Λ= λ\max\limits_{\substack{ξ\in \mathbb{R}^n\\1 \leq i, j \leq n}}\left\{\frac{|ξ|^{2}(\nabla^{2}_{ij}H^{p}(ξ))} {p(p-1)H^{p}(ξ)}\right\}$. By employing a novel approach based on invariant tensors technique, and using a Kato-type inequality, we prove that the positive solutions of \eqref{0.1} can be classified for $p_n(Λ) \leq p < n$, where $p_n(Λ)$ depends explicitly on $Λ$. This result removes the finite volume assumption on the classification of critical anisotropic $p$-Laplace equation which was obtained by Ciraolo-Figalli-Roncoroni in the literature \cite{CFR}. In particular, this results capture the precise dependence of critical exponents $p$ on both $n$ and $Λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_11628 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint Chen, Lu Wu, Tian Yan, Jin Yang, Yabo Analysis of PDEs 35J92, 35B33, 35B06 In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation}\label{0.1} -Δ^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation} without the finite volume constraint for $n \geq 3$ and $p_n(Λ) < p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent, $-Δ^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator and $Λ= λ\max\limits_{\substack{ξ\in \mathbb{R}^n\\1 \leq i, j \leq n}}\left\{\frac{|ξ|^{2}(\nabla^{2}_{ij}H^{p}(ξ))} {p(p-1)H^{p}(ξ)}\right\}$. By employing a novel approach based on invariant tensors technique, and using a Kato-type inequality, we prove that the positive solutions of \eqref{0.1} can be classified for $p_n(Λ) \leq p < n$, where $p_n(Λ)$ depends explicitly on $Λ$. This result removes the finite volume assumption on the classification of critical anisotropic $p$-Laplace equation which was obtained by Ciraolo-Figalli-Roncoroni in the literature \cite{CFR}. In particular, this results capture the precise dependence of critical exponents $p$ on both $n$ and $Λ$. |
| title | Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint |
| topic | Analysis of PDEs 35J92, 35B33, 35B06 |
| url | https://arxiv.org/abs/2404.11628 |