  Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite
measure $\mu$ has a positive lower bound. Motivated by wide interest in the
current literature, in this paper we study the existence of classical solutions
for a class of elliptic equations involving the $\mu$-Laplacian operator on the
graph $\mathscr G$, whose analytic expression is given by \begin{equation*}
\Delta_{\mu} u(x) := \frac{1}{\mu (x)} \sum_{y\sim x} w(x,y) (u(y)-u(x)),\
\hbox{for all} \ x\in V, \end{equation*} where $w : V\times V \rightarrow
[0,+\infty)$ is a weight symmetric function and the sum on the right-hand side
of the above expression is taken on the neighbors vertices $x,y\in V$, that is
$x\sim y$ whenever $w(x,y) > 0$. More precisely, by exploiting direct
variational methods, we study problems whose simple prototype has the following
form $$-\Delta_{\mu} u(x)=\lambda f(x,u(x)),\ x \in \mathop D\limits^ \circ , \
\hbox{and} \ u|_{\partial D}=0,$$ where $D$ is a bounded domain of $V$ such
that $\mathop D\limits^ \circ\neq \emptyset$ and $\partial D\neq \emptyset$,
the nonlinear term $f : D \times \mathbb{R} \rightarrow \mathbb{R}$ satisfy
suitable structure conditions and $\lambda$ is a positive real parameter. By
applying a critical point result coming out from a classical Pucci-Serrin
theorem in addition to a local minimum result for differentiable functionals
due to Ricceri, we are able to prove the existence of at least two solutions
for the treated problems. We emphasize the crucial role played by the famous
Ambrosetti-Rabinowitz growth condition along the proof of the main theorem and
its consequences. Our results improve the general results obtained by
Grigor'yan, Lin, and Yang (J. Differential Equations 261(9) (2016), 4924-4943).
