Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 564930, 14 pages
http://dx.doi.org/10.1155/2011/564930
Research Article

Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

Energy Edge Pty Ltd., P.O. Box 10755, Brisbane, QLD 4000, Australia

Received 30 December 2010; Revised 28 February 2011; Accepted 4 March 2011

Academic Editor: Marco Squassina

Copyright © 2011 Elliot Tonkes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan, New York, NY, USA, 1964.
  2. A. Ambrosetti and G. Prodi, “On the inversion of some differentiable mappings with singularities between Banach spaces,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 93, pp. 231–246, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. R. Böhme, “Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme,” Mathematische Zeitschrift, vol. 127, pp. 105–126, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Marino, “La biforcazione nel caso variazionale,” Conferenze del Seminario di Matematica dell'Università di Bari, no. 132, p. 14, 1977. View at Google Scholar
  5. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, New York, NY, USA, 1986.
  6. R. Chiappinelli, “An estimate on the eigenvalues in bifurcation for gradient mappings,” Glasgow Mathematical Journal, vol. 39, no. 2, pp. 211–216, 1997. View at Publisher · View at Google Scholar
  7. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, The Clarendon Press, New York, NY, USA, 1987.
  8. R. Chiappinelli, “A-priori bounds and asymptotics on the eigenvalues in bifurcation problems for perturbed self-adjoint operators,” Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 263–272, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. Chiappinelli, “Upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3772–3777, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. J. Chabrowski, P. Drábek, and E. Tonkes, “Asymptotic bifurcation results for quasilinear elliptic operators,” Glasgow Mathematical Journal, vol. 47, no. 1, pp. 55–67, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. D. A. Kandilakis, M. Magiropoulos, and N. Zographopoulos, “Existence and bifurcation results for fourth-order elliptic equations involving two critical Sobolev exponents,” Glasgow Mathematical Journal, vol. 51, no. 1, pp. 127–141, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  13. L. A. Lusternik, “On the conditions for extremals of functionals,” Matematicheskii Sbornik, vol. 41, no. 3, 1934. View at Google Scholar
  14. S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, NY, USA, 1982.
  15. J. Chabrowski, “On multiple solutions for the nonhomogeneous p-Laplacian with a critical Sobolev exponent,” Differential and Integral Equations, vol. 8, no. 4, pp. 705–716, 1995. View at Google Scholar
  16. H. Brézis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,” Communications on Pure and Applied Mathematics, vol. 36, no. 4, pp. 437–477, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet