    - 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. 2019 Vol. 56, No. 4, 857-867 https://doi.org/10.4134/JKMS.j180456Published online July 1, 2019 Shoudong Man, Guoqing Zhang Tianjin University of Finance and Economics; University of Shanghai for Science and Technology Abstract : 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 Downloads: Full-text PDF   Full-text HTML