Science in China Series A: Mathematics

, Volume 51, Issue 10, pp 1753–1762 | Cite as

Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain



We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.


imperfect bifurcation exact multiplicity perturbed logistic equation 


35J60 35J55 35B32 35P30 58C25 47J15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Golubitsky M, Schaeffer D. A theory for imperfect bifurcation via singularity theory. Comm Pure Appl Math, 32(1): 21–98 (1979)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ikeda K, Murota K. Imperfect Bifurcation in Structures and Materials. Engineering Use of Group-Theoretic Bifurcation Theory. Applied Mathematical Sciences, Vol. 149. New York: Springer-Verlag, 2002Google Scholar
  3. 3.
    Reiss E L. Imperfect bifurcation. In: Rabinowitz P H, eds. Applications of Bifurcation Theory, Proc Advanced Sem, Univ Wisconsin, Madison, Wis, 1976. Publ Math Res Center Univ Wisconsin, No. 38. New York: Academic Press, 1977, 37–71Google Scholar
  4. 4.
    Liu P, Shi J P, Wang Y W. Imperfect transcritical and pitchfork bifurcations. J Funct Anal, 251(2): 573–600 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Shi J P. Persistence and bifurcation of degenerate solutions. J Funct Anal, 169(2): 494–531 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Crandall M G, Rabinowitz P H. Bifurcation from simple eigenvalues. J Funct Anal, 8: 321–340 (1971)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Crandall M G, Rabinowitz P H. Bifurcation perturbation of simple eigenvalues and linearized stability. Arch Ration Mech Anal, 52: 161–180 (1973)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Liu P, Wang Y W. The generalized saddle-node bifurcation of degenerate solution. Comment Math Prace Mat, 45(2): 145–150 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    Wang Y W. Theory and Applications of Generalized Inverses in Banach spaces (in Chinese). Beijing: Science Press, 2005Google Scholar
  10. 10.
    Wang Y W, Yin H C, Sun X M. Bifurcation problems of nonlinear operator equations from non-simple eigenvalues (in Chinese). Acta Math Appl Sin, 28(2): 236–242 (2005)MathSciNetGoogle Scholar
  11. 11.
    Shi J P. Multi-parameter bifurcation and applications. In: Brezis H, Chang K C, Li S J, Rabinowitz P, eds. ICM 2002 Satellite Conference on Nonlinear Functional Analysis: Topological Methods, Variational Methods and Their Applications. Singapore: World Scientific, 2003, 211–222CrossRefGoogle Scholar
  12. 12.
    Shi J P, Shivaji R. Global bifurcation for concave semipositon problems. In: Goldstein G R, Nagel R, Romanelli S, eds. Advances in Evolution Equations: Proceedings in Honor of Goldstein J A’s 60th birthday. New York-Basel: Marcel Dekker, Inc., 2003, 385–398Google Scholar
  13. 13.
    Dancer E N, Shi J P. Uniqueness and nonexistence of positive solutions to semipositone problems. Bull London Math Soc, 38(6): 1033–1044 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Oruganti S B, Shi J P, Shivaji R. Diffusive logistic equation with constant yield harvesting. I. Steady states. Trans Amer Math Soc, 354(9): 3601–3619 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Yuan-Yung Tseng Functional Analysis Research Center and School of Mathematical SciencesHarbin Normal UniversityHarbinChina
  2. 2.School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina
  3. 3.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations