Superlinear elliptic inequalities on weighted graphs

Abstract

Let (V, µ) 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 ∆u + u σ ≤ 0 in V, where ∆ is the standard graph Laplacian on V and σ > 0. For σ ∈ (0, 1], the inequality admits no nontrivial positive solution. For σ > 1, assuming condition (p0) on (V, µ), we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that isfor some o ∈ V and all large enough n. For any ε > 0, we can construct an example on a homogeneous tree TN with µ(o, n) ≈ n 2σ σ−1 (ln n) 1 σ−1 +ε , and a solution to the inequality on (TN , µ) to illustrate the sharpness of 2σ σ−1 and 1 σ−1 .

Sharp Liouville type results for semilinear elliptic inequalities involving gradient terms on weighted graphs

“…Recently, Gu, Sun, and Huang in [10] proved that, for p > 1, if condition (p 0 ) is satisfied, and if…”