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Sequential and continuum bifurcations in degenerate elliptic equations

Authors: R. E. Beardmore and R. Laister
Journal: Proc. Amer. Math. Soc. 132 (2004), 165-174
MSC (1991): Primary 34A09, 34B60, 35B32, 35J60, 35J70
Published electronically: May 7, 2003
MathSciNet review: 2021259
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Abstract: We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

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Additional Information

R. E. Beardmore
Affiliation: Department of Mathematics, Imperial College, South Kensington, London, SW7 2AZ, United Kingdom

R. Laister
Affiliation: Department of Mathematics, University of the West of England, Frenchay Campus, Bristol, United Kingdom

Keywords: Degenerate elliptic equations, sequential and continuum bifurcations, differential-algebraic equations, degenerate diffusion
Received by editor(s): May 13, 2002
Received by editor(s) in revised form: August 21, 2002
Published electronically: May 7, 2003
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2003 American Mathematical Society