Journal of the
Korean Mathematical Society JKMS

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.

Keywords: indefinite weights, quasilinear elliptic equation on graphs, eigenvalue problem on graphs

MSC numbers: Primary 34B15, 35A15, 58E30

Supported by: This first author is supported by the National Natural Science Foundation of China (Grant No. 11601368).
This second author is supported by the National Natural Science Foundation of China (Grant No.11771291).