Multiplicity and uniqueness of positive solutions for a superlinear-singular $(p,q)$-Laplacian equation on locally finite graphs

We investigate the multiplicity and uniqueness of positive solutions for the superlinear singular $(p,q)$-Laplacian equation \begin{eqnarray*} \begin{cases} -Δ_p u-Δ_q u+a(x)u^{p-1}+b(x)u^{q-1}=f(x)u^{-γ}+λg(x)u^α, ;;;;\hfill \mbox{in};; V,\ u>0,;;u\in W_a^{1,p}(V) \cap W_b^{1,q}(V), \end{cases} \end{eqnarray*} on a weighted locally finite graph $G=(V,E)$, where $0<\gamma <1<q\le p<\alpha +1$, $\lambda$ is a parameter, the potential functions $a(x)$ and $b(x)$ satisfy some suitable conditions, $f>0,g\ge 0$, $f\in L^1(V)\cap L^{\frac{p}{p-1+\gamma }}(V)\cap L^{\frac{q}{q-1+\gamma }}(V)$ and $g\in L^1(V)\cap L^{\mathrm{\infty }}(V)$. By making use of the method of Nehari manifold and the Ekeland’s variational principle, we prove that there exist two positive solutions for $\lambda$ belonging to some precise interval. Besides, we also investigate the existence and uniqueness of positive solution for $\lambda <0$. We overcome some difficulties which are caused by: $(i)$ the singular term; $(ii)$ the definition of gradient $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ on graph which is different from that on ${\mathbb{R}}^N$; $(iii)$ the lack of compactness of Sobolev embedding.