Existence and uniqueness theorems for some semi-linear equations on locally finite graphs

TL;DR

This paper investigates semi-linear equations involving the $(m,p)$-Laplacian on locally finite graphs, establishing existence and uniqueness of solutions and applying these results to Yamabe-type and Kazdan-Warner-type equations.

Contribution

It extends existence results for semi-linear equations on graphs to all $m$ and $p$, and proves uniqueness for the case $m=1$, with applications to geometric equations.

Findings

Proved existence of weak solutions for all $m$ and $p$ using variational methods.

Established uniqueness for the case $m=1$ similar to Brezis-Strauss Theorem.

Applied results to Yamabe-type and Kazdan-Warner-type equations on graphs.

Abstract

We study some semi-linear equations for the $(m,p)$-Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all $m\in\mathbb{N}$ and $p\in(1,+\infty)$ via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When $m=1$, we also establish a uniqueness result in the spirit of the Brezis-Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan-Warner-type equations on locally finite weighted graphs.