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:

$$ \mathcal L_{p,\omega}u=-\Delta _{p,\omega} u + q |u|^{p-2} u = f $$

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.