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

J. Korean Math. Soc. Published online May 31, 2019

Shoudong Man and Guoqing Zhang
Tianjin University of Finance and Economics, University of Shanghai for Science and Technology

Abstract : Let $G=(V, E)$ be a locally finite graph with vertex set $V$ and edge set $E$, $\Omega\subset V$ 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