  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.
