Existence result for an elliptic equation on locally finite graphs with singularity

We study the following equation with singular nonlinearity on locally finite graphs $G=(V,E)$: $$\begin{aligned} \left \{ \begin{aligned} -\Delta u &=\frac{\lambda}{u^{\gamma}}+f(x,u) && \text{in }\Omega , \\ u &=0 \qquad &&\text{on }\partial \Omega , \end{aligned} \right . \end{aligned}$$ where Δ denotes the discrete Laplacian, V denotes the vertices set and E denotes the edges set, $0<\gamma <1$, $\Omega \subset V$ is a bounded domain and $\lambda >0$. By employing a local minimization approach within the variational framework, we establish the existence of a strictly positive solution to the above problem, when f has a subquadratic growth.