- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 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 MSC numbers : 34B15, 35A15, 58E30 Full-Text :