Nonexistence results for semilinear elliptic equations on weighted graphs

TL;DR

This paper establishes nonexistence results for nonnegative solutions to semilinear elliptic inequalities on infinite weighted graphs under certain geometric and potential conditions, demonstrating the optimality of these conditions.

Contribution

It provides new nonexistence theorems for semilinear elliptic inequalities on graphs, with conditions proven to be optimal.

Findings

No nonnegative nontrivial solutions under specified conditions

Conditions are proven to be optimal

Results extend understanding of elliptic equations on discrete structures

Abstract

We study semilinear elliptic inequalities with a potential on infinite graphs. Given a distance on the graph, we assume an upper bound on its Laplacian, and a growth condition on a suitable weighted volume of balls. Under such hypotheses, we prove that the problem does not admit any nonnegative nontrivial solution. We also show that our conditions are optimal.