Title: The $p$-Schrödinger Equations on Finite Networks
Abstract: We introduce the discrete p-Schrödinger operator Lp,ω and solve the following p-Schrödinger equation: Lp,ωu = −Δ\_p,ωu + q\_|u|p\_−2\_u = f 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.