On a class of quasilinear elliptic equation with indefinite weights on graphs

Suppose that $G=(V, E)$ is a connected locally finite graph with the vertex set $V$ and the edge set $E$. Let $Omegasubset V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph $G$ $$ left { begin{array}{lcr} -Delta_{p}u= lambda K(x)|u|^{p-2}u+f(x,u), xinOmega^{circ}, u=0, xinpartial Omega, end{array} right. $$ where $Omega^{circ}$ and $partial Omega$ denote the interior and the boundary of $Omega$ respectively, $Delta_{p}$ is the discrete $p$-Laplacian, $K(x)$ is a given function which may change sign, $lambda$ is the eigenvalue parameter and $f(x,u)$ has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.