### 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 language | English (US) |
---|---|

Pages (from-to) | 91-107 |

Number of pages | 17 |

Journal | Experimental Mathematics |

Volume | 15 |

Issue number | 1 |

State | Published - 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Experimental Mathematics*,

*15*(1), 91-107.

**Nonlinear elliptic partial difference equations on graphs.** / Neuberger, John M.

Research output: Contribution to journal › Article

*Experimental Mathematics*, vol. 15, no. 1, pp. 91-107.

}

TY - JOUR

T1 - Nonlinear elliptic partial difference equations on graphs

AU - Neuberger, John M

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Bifurcation

KW - GNGA

KW - Graphs

KW - Mountain pass

KW - Sign-changing solution

KW - Superlinear

KW - Symmetry

KW - Variational method

UR - http://www.scopus.com/inward/record.url?scp=33745676148&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745676148&partnerID=8YFLogxK

M3 - Article

VL - 15

SP - 91

EP - 107

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 1

ER -