American Institute of Mathematical Sciences

March  2011, 10(2): 785-802. doi: 10.3934/cpaa.2011.10.785

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

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

Received  January 2010 Revised  July 2010 Published  December 2010

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.
Citation: Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure & Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785
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References:
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