Sharp Criteria for the existence of positive solutions to Lane-Emden-type inequalities on weighted graphs
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866916060531261440 |
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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 study positive solutions of the superlinear Lane-Emden inequality
\(-Δu\ge σu^q\), \(q>1\), on infinite locally finite weighted
graphs and connected domains of such graphs. We first prove that solvability is
equivalent to the
pointwise test
\[
G_Ω(σg_Ω(o,\cdot)^q)(x)\le Cg_Ω(o,x)
\]
for every fixed pole \(o\inΩ\). We also prove sharp existence criteria
under \textnormal{(VD)},
\textnormal{(PI)}, and \textnormal{(P$_0$)}, and applications
giving the Serrin-type exponents on \(\mathbb Z^d\) and
orthant domains including half-spaces.
Our main result resolves the volume-growth conjecture for arbitrary weighted
graphs: if
\[
\sum_{n\ge1}\frac{n^{2q-1}}{μ(B(o,n))^{q-1}}=\infty,
\]
then every nonnegative solution of \(-Δu\ge u^q\) is identically zero.
The proof combines a flow decomposition with Hardy estimates along paths. For general positive $σ$, an
intrinsic-metric version is obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24932 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Criteria for the existence of positive solutions to Lane-Emden-type inequalities on weighted graphs Gu, Qingsong Hao, Lu Huang, Xueping Sun, Yuhua Analysis of PDEs We study positive solutions of the superlinear Lane-Emden inequality \(-Δu\ge σu^q\), \(q>1\), on infinite locally finite weighted graphs and connected domains of such graphs. We first prove that solvability is equivalent to the pointwise test \[ G_Ω(σg_Ω(o,\cdot)^q)(x)\le Cg_Ω(o,x) \] for every fixed pole \(o\inΩ\). We also prove sharp existence criteria under \textnormal{(VD)}, \textnormal{(PI)}, and \textnormal{(P$_0$)}, and applications giving the Serrin-type exponents on \(\mathbb Z^d\) and orthant domains including half-spaces. Our main result resolves the volume-growth conjecture for arbitrary weighted graphs: if \[ \sum_{n\ge1}\frac{n^{2q-1}}{μ(B(o,n))^{q-1}}=\infty, \] then every nonnegative solution of \(-Δu\ge u^q\) is identically zero. The proof combines a flow decomposition with Hardy estimates along paths. For general positive $σ$, an intrinsic-metric version is obtained. |
| title | Sharp Criteria for the existence of positive solutions to Lane-Emden-type inequalities on weighted graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.24932 |