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{ equation*} \Delta u+u^{ \sigma}\leq0\quad \text{ in}\,\,V, \end{ equation*} 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 \textbf{ ($p_0$)} 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{ equation*} \mu(o,n)\lesssim n^{ \frac{ 2\sigma}{ \sigma-1}}(\ln n)^{ \frac{ 1}{ \sigma-1}} \end{ equation*} 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 $\mu(o,n)\approx n^{ \frac{ 2\sigma}{ \sigma-1}}(\ln n)^{ \frac{ 1}{ \sigma-1}+\varepsilon}$, and a solution to the inequality on $(\mathbb T_N,\mu)$ to illustrate the sharpness of $\frac{ 2\sigma}{ \sigma-1}$ and $\frac{ 1}{ \sigma-1}$.
Categories: math.AP