Bulletin of the
Korean Mathematical Society BKMS

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