The $p$-Schrödinger Equations on Finite Networks

Abstract

We introduce the discrete $p$-Schrödinger operator ${\mathcal{L}}_{p,\omega }$ and solve the following $p$-Schrödinger equation:

on networks. To show the uniqueness of solutions of the $p$-Schrödinger equation, we first solve the eigenvalue problem for the $p$-Schrödinger operator and obtain some properties of the smallest eigenvalue and its corresponding eigenfunction of the $p$-Schrödinger operator.