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

DOI:
https://doi.org/10.1090/S0002-9939-03-06979-X

Published electronically:
May 7, 2003

MathSciNet review:
2021259

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**A. Ambrosetti, J. Garcia-Azorero, and I. Peral,*Quasilinear equations with a multiple bifurcation*, Differential and Integral Equations**10**(1997), no. 1, 37-50. MR**97i:35036****2.**D. Aronson and L.A. Peletier,*Large time behaviour of solutions of the porous medium equation in bounded domains*, J. Differential Equations**39**(1981), 378-412. MR**82g:35047****3.**H. Berestycki,*On some nonlinear Sturm-Louiville problems*, J. Differential Equations.**26**(1977), 375-390. 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), 1-25. MR**97k:34052****5.**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.**6.**D. Henry,*Geometrical theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, New York, 1981. MR**83j:35084****7.**T. Kato,*Perturbation theory for linear operators*, vol. Corrected 2nd Edition, Springer-Verlag, New York, 1980. MR**96a:47025****8.**M.K. Kwong and L. Zhang,*Uniqueness of the positive solution of in an annulus*, Differential Integral Equations**4**(1991), 583-596. MR**92b:35015****9.**R. Laister and R. E. Beardmore,*Transversality and separation of zeros in second order differential equations*, Proc. AMS., to appear.**10.**M. A. Lewis,*Spatial coupling of plant and herbivore dynamics: the contribution of herbivore dispersal to transient and persistant ``waves'' of damage*, Theoretical Population Biology**45**(1994), 277-312.**11.**P.L. Lions,*Structure of the set of steady-state solutions and asymptotic behaviour of semilinear heat equations*, J. Differential Equations**53**(1984), no. 3, 362-386. MR**86b:35092****12.**M. Protter and H. Weinberger,*Maximum principles in differential equations*, Prentice Hall, Englewood Cliffs, N.J. (1967). MR**36:2935****13.**P. Rabinowitz,*Some global results for nonlinear eigenvalue problems*, J. Funct. Anal.**7**(1971), 487-513. MR**46:745****14.**A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov,*Blow-up in quasilinear parabolic equations*, de Gruyter Expositions in Mathematics, 19, Walter de Gruyter, Berlin, 1995. MR**96b:35003****15.**C.A. Stuart,*Bifurcation for Dirichlet problems without eigenvalues*, Proc. London Math. Soc.**45**(1982), 169-192. MR**83k:58021****16.**S. C. Welsh,*A priori bounds and nodal properties for periodic solutions to a class of ordinary differential equations*, J. Math. Anal. Appns.**171**(1992), 395-406. MR**93m:34060****17.**G. T. Whyburn,*Topological analysis*, Princeton University Press, 1964. MR**29:2758**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34A09,
34B60,
35B32,
35J60,
35J70

Retrieve articles in all journals with MSC (1991): 34A09, 34B60, 35B32, 35J60, 35J70

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:
https://doi.org/10.1090/S0002-9939-03-06979-X

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