Nonlinear Elliptic Partial Difference Equations on Graphs

This article initiates the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions u : V → R to the semilinear elliptic difference equation −Lu + f(u) = 0 on a graph G = (V,E), where L is the (negative) Laplacian on the graph G. We extend techniques used to prove existence theorems and derive numerical algorithms for the partial differential equation (PDE) ∆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 to our discrete nonlinear problems, we compute and describe many solutions. Letting f = f(λ, u), we construct bifurcation diagrams and relate the results to the developed theory.