Let (V,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(V,\mu )$$\end{document} 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 Δu+uσ≤0inV,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta u+u^{\sigma }\le 0\quad \text {in}\,\,V, \end{aligned}$$\end{document}where Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} is the standard graph Laplacian on V and σ>0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma >0$$\end{document}. For σ∈(0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \in (0,1]$$\end{document}, the inequality admits no nontrivial positive solution. For σ>1\documentclass[12pt]{minimal}
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\begin{document}$$\sigma >1$$\end{document}, assuming condition (p0\documentclass[12pt]{minimal}
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\begin{document}$$p_0$$\end{document}) on (V,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(V,\mu )$$\end{document}, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is μ(B(o,n))≲n2σσ-1(lnn)1σ-1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mu (B(o,n))\lesssim n^{\frac{2\sigma }{\sigma -1}}(\ln n)^{\frac{1}{\sigma -1}} \end{aligned}$$\end{document}for some o∈V\documentclass[12pt]{minimal}
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\begin{document}$$o\in V$$\end{document} and all large enough n. For any ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document}, we can construct an example on a homogeneous tree TN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}_N$$\end{document} with μ(B(o,n))≍n2σσ-1(lnn)1σ-1+ε\documentclass[12pt]{minimal}
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\begin{document}$$\mu (B(o,n))\asymp n^{\frac{2\sigma }{\sigma -1}}(\ln n)^{\frac{1}{\sigma -1}+\varepsilon }$$\end{document}, and a solution to the inequality on (TN,μ)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbb {T}}_N,\mu )$$\end{document} to illustrate the sharpness of 2σσ-1\documentclass[12pt]{minimal}
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\begin{document}$$\frac{2\sigma }{\sigma -1}$$\end{document} and 1σ-1\documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{\sigma -1}$$\end{document}.
