Nonlinear elliptic partial difference equations on graphs

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Abstract

This article furthers the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions u: V → R to the semilinear elliptic partial 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) Δu + 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-Calerkin algorithms for our discrete nonlinear problem, we compute and describe many solutions. Letting f = f(λ, u), we construct bifurcation diagrams and relate the results to the developed theory.

Original languageEnglish (US)
Pages (from-to)91-107
Number of pages17
JournalExperimental Mathematics
Volume15
Issue number1
StatePublished - 2006

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Keywords

  • Bifurcation
  • GNGA
  • Graphs
  • Mountain pass
  • Sign-changing solution
  • Superlinear
  • Symmetry
  • Variational method

ASJC Scopus subject areas

  • Mathematics(all)

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