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

Abstract

We study some semi-linear equations for the ( m , p ) (m,p) -Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all m ∈ N m\in \mathbb {N} and p ∈ ( 1 , + ∞ ) 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 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.