Abstract:We study positive solutions of the superlinear Lane-Emden inequality
\(-\Delta u\ge \sigma u^q\), \(q>1\), on infinite locally finite weighted
graphs and connected domains of such graphs. We first prove that solvability is
equivalent to the
pointwise test
\[
G_\Omega(\sigma g_\Omega(o,\cdot)^q)(x)\le Cg_\Omega(o,x)
\]
for every fixed pole \(o\in\Omega\). 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
\[
\sum_{n\ge1}\frac{n^{2q-1}}{\mu(B(o,n))^{q-1}}=\infty,
\]
then every nonnegative solution of \(-\Delta u\ge u^q\) is identically zero.
The proof combines a flow decomposition with Hardy estimates along paths. For general positive $\sigma$, an
intrinsic-metric version is obtained.
From: Yuhua Sun [view email]
[v1]
Mon, 27 Apr 2026 19:16:53 UTC (38 KB)
[v2]
Thu, 28 May 2026 14:02:34 UTC (42 KB)
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