Indefinite Eigenvalue Problems for p-Laplacian Operators with Potential Terms on Networks

Abstract

We address some forward and inverse problems involving indefinite eigenvalues for discrete $p$-Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues of $p$-Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discrete $p$-Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discrete $p$-Laplacian operators with potential terms involving the smallest indefinite eigenvalues.