Semi-linear elliptic inequalities on weighted graphs

Abstract

Let $(V,\mu )$ be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality

$$\begin{aligned} \Delta u+u^{\sigma }\le 0\quad \text {in}\,\,V, \end{aligned}$$

where $\Delta$ is the standard graph Laplacian on V and $\sigma >0$. For $\sigma \in (0,1]$, the inequality admits no nontrivial positive solution. For $\sigma >1$, assuming condition ($p_0$) on $(V,\mu )$, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is

$$\begin{aligned} \mu (B(o,n))\lesssim n^{\frac{2\sigma }{\sigma -1}}(\ln n)^{\frac{1}{\sigma -1}} \end{aligned}$$

for some $o\in V$ and all large enough n. For any $\varepsilon >0$, we can construct an example on a homogeneous tree ${\mathbb {T}}_N$ with $\mu (B(o,n))\asymp n^{\frac{2\sigma }{\sigma -1}}(\ln n)^{\frac{1}{\sigma -1}+\varepsilon }$, and a solution to the inequality on $({\mathbb {T}}_N,\mu )$ to illustrate the sharpness of $\frac{2\sigma }{\sigma -1}$ and $\frac{1}{\sigma -1}$.