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 $\Omega\subset V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph $G$ $$ \left \{ \begin{array}{lcr} -\Delta_{p}u= \lambda K(x)|u|^{p-2}u+f(x,u), \ \ x\in\Omega^{\circ}, u=0, \ \ x\in\partial \Omega, \\ \end{array} \right. $$ 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.