The <i>p</i>-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.