  Let $G=(V,E)$ be a finite connected weighted graph, and assume
$1\leq\alpha\leq p\leq q$. In this paper, we consider the following $p$-th
Yamabe type equation $$-\Delta_pu+hu^{q-1}=\lambda fu^{\alpha-1}.$$ on $G$,
where $\Delta_p$ is the $p$-th discrete graph Laplacian, $h\leq0$ and $f>0$ are
real functions defined on all vertices of $G$. Instead of the approach in
[Ge3], we adopt a new approach, and prove that the above equation always has a
positive solution $u>0$ for some constant $\lambda\in\mathbb{R}$. In
particular, when $q=p$ our result generalizes the main theorem in [Ge3] from
the case of $\alpha\geq p>1$ to the case of $1\leq\alpha\leq p$. It's
interesting that our new approach can also work in the case of $\alpha\geq
p>1$.
