Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant
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| Format: | Preprint |
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2024
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| _version_ | 1866910662139052032 |
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| author | Hou, Yongjun |
| author_facet | Hou, Yongjun |
| contents | We study positive properties of the quasilinear elliptic equation
$$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{ in } Ω,$$ where the function $\mathcal{A}(x,ξ)$ is induced by a family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points in the domain $Ω\subseteq\mathbb{R}^{n}$, and $V$ belongs to a certain local Morrey space. We first establish two-sided estimates for Bregman distances of $|ξ|^{p}_{s,a}$ ($1<s<\infty$), where $a=(a_{1},a_{2},\ldots,a_{n})$ and $a_{1},a_{2},\ldots,a_{n}$ are certain functions with positive local lower and upper bounds in $Ω$. These estimates lead to a Maz'ya-type characterization for Hardy-weights of the corresponding functionals. Then we prove three types of sufficient conditions for the attainment of the Hardy constant in a certain space $\widetilde{W}^{1,p}_{0}(Ω)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant Hou, Yongjun Analysis of PDEs Functional Analysis 35J20 (Primary) 35B09, 35J62, 35J70 (Secondary) We study positive properties of the quasilinear elliptic equation $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{ in } Ω,$$ where the function $\mathcal{A}(x,ξ)$ is induced by a family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points in the domain $Ω\subseteq\mathbb{R}^{n}$, and $V$ belongs to a certain local Morrey space. We first establish two-sided estimates for Bregman distances of $|ξ|^{p}_{s,a}$ ($1<s<\infty$), where $a=(a_{1},a_{2},\ldots,a_{n})$ and $a_{1},a_{2},\ldots,a_{n}$ are certain functions with positive local lower and upper bounds in $Ω$. These estimates lead to a Maz'ya-type characterization for Hardy-weights of the corresponding functionals. Then we prove three types of sufficient conditions for the attainment of the Hardy constant in a certain space $\widetilde{W}^{1,p}_{0}(Ω)$. |
| title | Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant |
| topic | Analysis of PDEs Functional Analysis 35J20 (Primary) 35B09, 35J62, 35J70 (Secondary) |
| url | https://arxiv.org/abs/2405.18159 |