Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 684679, 16 pages
http://dx.doi.org/10.1155/2014/684679
Research Article

Existence of Nonradial Solutions for Hénon Type Biharmonic Equation Involving Critical Sobolev Exponents

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 13 June 2014; Accepted 19 August 2014; Published 14 October 2014

Academic Editor: Antonio Suárez

Copyright © 2014 Yajing Zhang et al. 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.-Ch. Grunau, Polyharmonische Dirichletprobleme: Positivität, kritische Exponenten und kritische Dimensionen, Habilitationsschrift, Universität Bayreuth, 1996.
  2. F. Gazzola, H. Grunau, and M. Squassina, “Existence and nonexistence results for critical growth biharmonic elliptic equations,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 2, pp. 117–143, 2003. View at Publisher · View at Google Scholar
  3. F. Bernis, J. García Azorero, and I. Peral, “Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order,” Advances in Differential Equations, vol. 1, no. 2, pp. 219–240, 1996. View at Google Scholar · View at MathSciNet
  4. T. Bartsch, T. Weth, and M. Willem, “A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 3, pp. 253–268, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. A. Bahri and J. Coron, “On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain,” Communications on Pure and Applied Mathematics, vol. 41, no. 3, pp. 253–294, 1988. View at Publisher · View at Google Scholar
  6. E. Berchio, F. Gazzola, and T. Weth, “Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems,” Journal für die Reine und Angewandte Mathematik, vol. 620, pp. 165–183, 2008. View at Publisher · View at Google Scholar
  7. W. M. Ni, “A nonlinear Dirichlet problem on the unit ball and its applications,” Indiana University Mathematics Journal, vol. 31, no. 6, pp. 801–807, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Byeon and Z. Wang, “On the Hénon equation: asymptotic profile of ground states, I,” Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, vol. 23, no. 6, pp. 803–828, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Byeon and Z. Wang, “On the Hénon equation: asymptotic profile of ground states, II,” Journal of Differential Equations, vol. 216, no. 1, pp. 78–108, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. D. Cao and S. Peng, “The asymptotic behaviour of the ground state solutions for Hénon equation,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 1–17, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. Smets, M. Willem, and J. Su, “Non-radial ground states for the Hénon equation,” Communications in Contemporary Mathematics, vol. 4, no. 3, pp. 467–480, 2002. View at Publisher · View at Google Scholar
  12. E. Serra, “Non radial positive solutions for the Hénon equation with critical growth,” Calculus of Variations and Partial Differential Equations, vol. 23, no. 3, pp. 301–326, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Luckhaus, “Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order,” Journal für die Reine und Angewandte Mathematik, vol. 306, pp. 192–207, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solution of elliptic partial differential equations satisfying general boundary conditions I,” Communications on Pure and Applied Mathematics, vol. 12, pp. 623–727, 1959. View at Publisher · View at Google Scholar
  15. F. Gazzola, H. Grunau, and G. Sweers, Polyharmonic Boundary Value Problems, vol. 1991 of Lecture Notes in Mathematics, Springer, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. H. Grunau and G. Sweers, “Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,” Mathematische Annalen, vol. 307, no. 4, pp. 589–626, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. K. Tintarev and K.-H. Fieseler, Concentration Compactness, Functional-Analytic Grounds and Applications, Imperial College Press, London, UK, 2007. View at MathSciNet
  18. C. A. Swanson, “The best Sobolev constant,” Applicable Analysis, vol. 47, no. 4, pp. 227–239, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  19. H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983. View at Publisher · View at Google Scholar