On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911284076740608 |
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| author | Hao, Lu Sun, Yuhua |
| author_facet | Hao, Lu Sun, Yuhua |
| contents | We study the equivalence between the $L^p$-parabolicity, the $L^q$-Liouville property of positive super-harmonic functions, and the existence of nonharmonic positive solutions to the following elliptic differential system
\begin{equation*}
\left\{
\begin{array}{lr}
-Δu\geq 0,
Δ(|Δu|^{p-2}Δu)\geq 0,
\end{array}
\right.
\end{equation*}
on weighted graphs, where $1\leq p< \infty$, and $(p, q)$ are Hölder conjugate exponent pair. Furthermore, by refining a new technique on estimate of heat kernel, we can establish two-sided estimates of Green function on graph, and find the sharp volume growth criteria for the $L^q$-Liouville property on a large class of graphs. As an application, many non-trivial interesting examples are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs Hao, Lu Sun, Yuhua Analysis of PDEs We study the equivalence between the $L^p$-parabolicity, the $L^q$-Liouville property of positive super-harmonic functions, and the existence of nonharmonic positive solutions to the following elliptic differential system \begin{equation*} \left\{ \begin{array}{lr} -Δu\geq 0, Δ(|Δu|^{p-2}Δu)\geq 0, \end{array} \right. \end{equation*} on weighted graphs, where $1\leq p< \infty$, and $(p, q)$ are Hölder conjugate exponent pair. Furthermore, by refining a new technique on estimate of heat kernel, we can establish two-sided estimates of Green function on graph, and find the sharp volume growth criteria for the $L^q$-Liouville property on a large class of graphs. As an application, many non-trivial interesting examples are presented. |
| title | On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.15420 |