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Journal of Function Spaces and Applications
Volume 2013, Article ID 510943, 8 pages
http://dx.doi.org/10.1155/2013/510943
Research Article

Multiplicity and Bifurcation of Solutions for a Class of Asymptotically Linear Elliptic Problems on the Unit Ball

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 13 December 2012; Revised 25 January 2013; Accepted 6 February 2013

Academic Editor: Ti J. Xiao

Copyright © 2013 Benlong Xu. 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. H. Amann, “Nonlinear eigenvalue problems having precisely two solutions,” Mathematische Zeitschrift, vol. 150, no. 1, pp. 27–37, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Ambrosetti and P. Hess, “Positive solutions of asymptotically linear elliptic eigenvalue problems,” Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 411–422, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. G. Crandall and P. H. Rabinowitz, “Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems,” Archive for Rational Mechanics and Analysis, vol. 58, no. 3, pp. 207–218, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. P. Mironescu and V. D. Rădulescu, “The study of a bifurcation problem associated to an asymptotically linear function,” Nonlinear Analysis. Theory, Methods & Applications, vol. 26, no. 4, pp. 857–875, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C.-H. Hsu and Y.-W. Shih, “Solutions of semilinear elliptic equations with asymptotic linear nonlinearity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 50, no. 2, pp. 275–283, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Duo, J. Shi, and Y. Wang, “Structure of the solution set for a class of semilinear elliptic equations with asymptotic linear nonlinearity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2369–2378, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. Xu, “Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 783–790, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. C. Evans, Partial Differential Equations, vol. 19, American Mathematical Society, Providence, RI, USA, 1998. View at MathSciNet
  9. M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34, Springer, Berlin, Germany, 2nd edition, 1996. View at MathSciNet
  10. H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620–709, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W.-M. Ni, “Recent progress in semilinear elliptic equations,” Publications of the Research Institute for Mathematical Sciences, vol. 679, pp. 1–39, 1989. View at Google Scholar
  12. P. Korman, Y. Li, and T. Ouyang, “An exact multiplicity result for a class of semilinear equations,” Communications in Partial Differential Equations, vol. 22, no. 3-4, pp. 661–684, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Ouyang and J. Shi, “Exact multiplicity of positive solutions for a class of semilinear problems,” Journal of Differential Equations, vol. 146, no. 1, pp. 121–156, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Du and Y. Lou, “Proof of a conjecture for the perturbed Gelfand equation from combustion theory,” Journal of Differential Equations, vol. 173, no. 2, pp. 213–230, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry and related properties via the maximum principle,” Communications in Mathematical Physics, vol. 68, no. 3, pp. 209–243, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. S. Lin and W. Ni, “A conterexample to the nodal domain conjecture and related semilinear equation,” Proceedings of the American Mathematical Society, vol. 102, pp. 271–277, 1998. View at Google Scholar
  17. P. Korman, “Solution curves for semilinear equations on a ball,” Proceedings of the American Mathematical Society, vol. 125, no. 7, pp. 1997–2005, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. T. Ouyang and J. Shi, “Exact multiplicity of positive solutions for a class of semilinear problem—II,” Journal of Differential Equations, vol. 158, no. 1, pp. 94–151, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. View at MathSciNet
  20. M. G. Crandall and P. H. Rabinowitz, “Bifurcation, perturbation of simple eigenvalues and linearized stability,” Archive for Rational Mechanics and Analysis, vol. 52, pp. 161–180, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet