A conjecture on multiple solutions of a nonlinear elliptic boundaryvalue problem: some numerical evidence

Lisa Hollman; P. J. McKenna

doi:10.3934/cpaa.2011.10.785

Article Contents

2011, Volume 10, Issue 2: 785-802. Doi: 10.3934/cpaa.2011.10.785

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A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence

Lisa Hollman^1, and
P. J. McKenna^2,

1.
Quinnipiac University, 275 Mt Carmel Avenue, Hamden, CT 06518
2.
Department of Mathematics, 196 Auditorium Road, Unit 3009, University of Connecticut, Storrs, CT 06269-3009

Received: December 31, 2009

Revised: June 30, 2010

Published: February 2011

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Abstract

We investigate a conjecture regarding the number of solutions of a second order elliptic boundary value problem with an asymmetric nonlinearity. This investigation makes use of several computer assisted techniques. First, we compute approximate solutions using Newton's Iteration for small $b$ and then use a continuation method to show that the number of solutions becomes large as $b$ increases.

Keywords:

Second order elliptic boundary value problem,
nonlinear equation,
computer assisted proof,
newton’s iteration,
continuation method.

Mathematics Subject Classification: Primary: 35B40, 35J25; Secondary: 35B06.

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References

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