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2006 Nonlinear Elliptic Partial Difference Equations on Graphs
John M. Neuberger
Experiment. Math. 15(1): 91-107 (2006).


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.


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John M. Neuberger. "Nonlinear Elliptic Partial Difference Equations on Graphs." Experiment. Math. 15 (1) 91 - 107, 2006.


Published: 2006
First available in Project Euclid: 16 June 2006

zbMATH: 1160.35412
MathSciNet: MR2229389

Primary: 05C50 , 35A05 , 35A15 , 49M15 , 58J55 , 58J70 , 65K10 , 90C47

Keywords: bifurcation , GNGA , Graphs , mountain pass , sign-changing solution , superlinear , symmetry , variational method

Rights: Copyright © 2006 A K Peters, Ltd.


Vol.15 • No. 1 • 2006
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