Multiple solutions to discrete boundary value problems for the $p$-Laplacian with potential terms on finite graphs
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1517-1533
https://doi.org/10.4134/BKMS.2015.52.5.1517
Printed September 1, 2015
Soon-Yeong Chung and Jea-Hyun Park
Sogang University, Kunsan National University
Abstract : In this paper, we prove the existence of at least three non-trivial solutions to nonlinear discrete boundary value problems \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p,\omega} u(x) + V(x) |u(x)|^{q-2}u(x) = f(x, u(x)), & x \in S,\\ u(x) =0, & x \in \partial S, \end{array} \right. \end{equation*} involving the discrete $p$-Laplacian on simple, finite and connected graphs $\overline S (S \cup \partial S , E)$ with weight $\omega$, where $1< q< p< \infty$. The approach is based on a suitable combine of variational and truncations methods.
Keywords : $p$-Laplacian difference equation, discrete $p$-Laplacian, discrete boundary value problems
MSC numbers : Primary 34B45, 39A12, 39A20
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