Nonlinear Elliptic Partial Difference Equations on Graphs

Abstract

This article furthers the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions {\small $u:V\to\R$} to the semilinear elliptic partial difference equation {\small $-Lu + f(u) = 0$} on a graph {\small $G=(V,E)$}, where {\small $L$} is the (negative) Laplacian on the graph {\small $G$}. We extend techniques used to prove existence theorems and derive numerical algorithms for the partial differential equation (PDE) {\small $\Delta u + f(u) = 0$}. In particular, we prove the existence of sign-changing solutions and solutions with symmetry in the superlinear case. Developing variants of the mountain pass, modified mountain pass, and gradient Newton-Galerkin algorithms for our discrete nonlinear problem, we compute and describe many solutions. Letting {\small $f=f(\lambda,u)$}, we construct bifurcation diagrams and relate the results to the developed theory.