Calculus of Variations and Geometric Measure Theory

A. Pinamonti - G. Stefani

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

created by stefani on 19 Jun 2021
modified on 23 Feb 2022


Accepted Paper

Inserted: 19 jun 2021
Last Updated: 23 feb 2022

Journal: Proc. Amer. Math. Soc.
Year: 2021


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.

Keywords: semi-linear equations on graphs, variational method, Yamabe-type equation, Kazdan-Warner-type equation, Brezis-Strauss Theorem