Existence of positive solutions to some nonlinear equations on locally finite graphs

TL;DR

This paper proves the existence of positive solutions for certain nonlinear equations on locally finite graphs using variational methods, extending results known from Euclidean spaces and manifolds.

Contribution

It establishes new existence results for nonlinear equations on graphs by applying the mountain-pass theorem, a novel approach in this context.

Findings

Positive solutions exist under specific conditions on h and f.

The results extend to perturbed equations with additional terms.

Methodology adapts variational techniques to graph settings.

Abstract

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely $\Delta u+hu=f(x,u)$, $x\in V$. In particular, we prove that if $h$ and $f$ satisfy certain assumptions, then the above mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation $\Delta u+hu=f(x,u)+\epsilon g$. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.