Semi-linear elliptic inequalities on weighted graphs

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 delta u + u(sigma) <= 0 in V, where delta is the standard graph Laplacian on V and sigma > 0. For sigma is an element of (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 mu(B(o, n)) ? n (2 sigma/sigma-1) (ln n) (1/sigma-1) for some o is an element of V and all large enough n. For any epsilon > 0, we can construct an example on a homogeneous tree T-N with mu(B(o, n)) sic n (2 sigma/sigma-1) (ln n) (1/sigma-1+epsilon), and a solution to the inequality on (T-N, mu) to illustrate the sharpness of 2 sigma/sigma-1 and 1/sigma-1.