A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918494513135616 |
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| author | Gu, Qingsong Hao, Lu Huang, Xueping Sun, Yuhua |
| author_facet | Gu, Qingsong Hao, Lu Huang, Xueping Sun, Yuhua |
| contents | We prove a nonexistence result for nonnegative solutions of the quasi-linear
elliptic inequality
\[
-Δ_p u\ge u^σ
\]
on infinite locally finite connected weighted graphs, where $1<p<\infty$ and
$σ>p-1$. Under the non-$p$-parabolic setting, we show that every
nonnegative solution is identically zero, provided the weighted ball volumes
$W_n=μ(B(o,n))$ satisfy
\[
\sum_{n=1}^{\infty}
\frac{n^{\frac{pσ}{p-1}-1}}
{W_n^{\frac{σ-p+1}{p-1}}}
=\infty .
\]
This criterion recovers the known sharp pointwise critical volume-growth
threshold and is strictly more flexible, since it allows irregular growth and
does not require uniform upper bounds at every large radius. The proof adapts
the finite-network current method to the $p$-Laplace setting, combining a path
decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds
across metric cuts, and the global $p$-Green function furnished by
non-$p$-parabolicity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10446 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs Gu, Qingsong Hao, Lu Huang, Xueping Sun, Yuhua Analysis of PDEs Primary 35J92, 35R02, Secondary 31C20 We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -Δ_p u\ge u^σ \] on infinite locally finite connected weighted graphs, where $1<p<\infty$ and $σ>p-1$. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the weighted ball volumes $W_n=μ(B(o,n))$ satisfy \[ \sum_{n=1}^{\infty} \frac{n^{\frac{pσ}{p-1}-1}} {W_n^{\frac{σ-p+1}{p-1}}} =\infty . \] This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius. The proof adapts the finite-network current method to the $p$-Laplace setting, combining a path decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds across metric cuts, and the global $p$-Green function furnished by non-$p$-parabolicity. |
| title | A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs |
| topic | Analysis of PDEs Primary 35J92, 35R02, Secondary 31C20 |
| url | https://arxiv.org/abs/2605.10446 |