Sequential and continuum bifurcations in degenerate elliptic equations
Authors:
R. E. Beardmore and R. Laister
Journal:
Proc. Amer. Math. Soc. 132 (2004), 165174
MSC (1991):
Primary 34A09, 34B60, 35B32, 35J60, 35J70
Published electronically:
May 7, 2003
MathSciNet review:
2021259
Fulltext PDF Free Access
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Abstract: We examine the bifurcations to positive and signchanging 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 nonmonotone eigenvalue problems, degenerate semilinear elliptic equations, boundary value differentialalgebraic equations and fully nonlinear elliptic equations.
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 H. Berestycki, On some nonlinear SturmLouiville problems, J. Differential Equations. 26 (1977), 375390. MR 58:1358
 4.
 H. Berestycki and M.J. Esteban, Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations 134 (1997), 125. MR 97k:34052
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 K.P. Hadeler, Free boundary problems in biology, in Free Boundary Problems: Theory and Applications Vol.II, eds. A. Fasano and M. Primicerio, Pitman Advanced Publishing Program, Pitman, New York, 1983.
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 M.K. Kwong and L. Zhang, Uniqueness of the positive solution of in an annulus, Differential Integral Equations 4 (1991), 583596. MR 92b:35015
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 R. Laister and R. E. Beardmore, Transversality and separation of zeros in second order differential equations, Proc. AMS., to appear.
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 C.A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. 45 (1982), 169192. MR 83k:58021
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Additional Information
R. E. Beardmore
Affiliation:
Department of Mathematics, Imperial College, South Kensington, London, SW7 2AZ, United Kingdom
Email:
r.beardmore@ic.ac.uk
R. Laister
Affiliation:
Department of Mathematics, University of the West of England, Frenchay Campus, Bristol, United Kingdom
Email:
robert.laister@uwe.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299390306979X
PII:
S 00029939(03)06979X
Keywords:
Degenerate elliptic equations,
sequential and continuum bifurcations,
differentialalgebraic 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
