Poisson Equations with Nonlinear Source Terms on Networks

Abstract

C. Berenstein and the first author introduced an elliptic operator ${∆}_{\omega }$ and an $\omega$-harmonic function on graphs, with its physical interpretation as a diffusion equation on graphs, which models electric networks. They also proved the solvability of the problems such as the Dirichlet and Neumann boundary value problems for the Laplace equation and the Poisson equation on networks. In this paper, we consider the Poisson equation with nonlinear source term on networks and show the existence and the uniqueness of a solution with suitable source term.