Nonexistence results for semilinear elliptic equations on metric graphs

TL;DR

This paper proves that under certain conditions, semilinear elliptic equations with positive potential on metric graphs have only trivial solutions, using a modified distance function and volume growth conditions.

Contribution

It establishes new nonexistence results for solutions of semilinear elliptic equations on metric graphs, considering a special Laplacian related to vertices and edges.

Findings

Nonnegative solutions are trivial under volume growth conditions.

Sign-changing solutions are also shown to be trivial.

A modified distance function is key to the analysis.

Abstract

In this paper, we study the nonexistence of solutions to semilinear elliptic equations with a positive potential on metric graphs. In particular, the Laplacian under consideration is of a special type, related to both the vertices and edges of metric graphs. We construct a modified distance function, introduce appropriate test functions, and establish the nonexistence of global solutions under suitable volume growth conditions imposed on the potential. More precisely, the nonnegative solutions or sign-changing solutions to the equations are the trivial zero solutions.