Table of Contents
Geometry
Volume 2014, Article ID 582367, 9 pages
http://dx.doi.org/10.1155/2014/582367
Research Article

Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere

Department of Mathematics and Computer Science, University of Monastir, I.P.E.I.M., Avenue Ibn Al Jazzar, 5019 Monastir, Tunisia

Received 3 June 2013; Accepted 27 January 2014; Published 20 March 2014

Academic Editor: Constantin Udriste

Copyright © 2014 Ridha Yacoub. 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. Ben Ayed, K. El Mehdi, and M. Ould Ahmedou, “The scalar curvature problem on the four dimensional half sphere,” Calculus of Variations and Partial Differential Equations, vol. 22, no. 4, pp. 465–482, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Ben Ayed and H. Chtioui, “Topological tools in prescribing the scalar curvature on the half sphere,” Advanced Nonlinear Studies, vol. 4, no. 2, pp. 121–148, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Ben Ayed and H. Chtioui, “On the prescribed scalar curvature problem on the three-dimensional half sphere,” Pacific Journal of Mathematics, vol. 221, no. 2, pp. 201–226, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Ben Ayed and M. O. Ahmedou, “On the prescribed scalar curvature on 3-half spheres: multiplicity results and Morse inequalities at infinity,” Discrete and Continuous Dynamical Systems A, vol. 23, no. 3, pp. 655–683, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. P. Cherrier, “Problèmes de Neumann non linéaires sur les variétés riemanniennes,” Journal of Functional Analysis, vol. 57, no. 2, pp. 154–207, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. Djadli, A. Malchiodi, and M. Ould Ahmedou, “Prescribing scalar and boundary mean curvature on the three dimensional half sphere,” The Journal of Geometric Analysis, vol. 13, no. 2, pp. 255–289, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. F. Escobar, “Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary,” Annals of Mathematics, vol. 136, no. 1, pp. 1–50, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. F. Escobar, “Conformal metrics with prescribed mean curvature on the boundary,” Calculus of Variations and Partial Differential Equations, vol. 4, no. 6, pp. 559–592, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Z.-C. Han and Y. Li, “The Yamabe problem on manifolds with boundary: existence and compactness results,” Duke Mathematical Journal, vol. 99, no. 3, pp. 489–542, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z.-C. Han and Y. Li, “The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature,” Communications in Analysis and Geometry, vol. 8, no. 4, pp. 809–869, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Li, “The Nirenberg problem in a domain with boundary,” Topological Methods in Nonlinear Analysis, vol. 6, no. 2, pp. 309–329, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Yacoub, “Topological arguments in prescribing the scalar curvature under minimal boundary mean curvature condition on S+n,” Differential and Integral Equations, vol. 21, no. 5-6, pp. 459–476, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. Bianchi and X.-B. Pan, “Yamabe equations on half-spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 37, no. 2, pp. 161–186, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. Bahri, Critical Point at Infinity in Some Variational Problems, vol. 182 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1989. View at MathSciNet
  15. R. Schoen and D. Zhang, “Prescribed scalar curvature on the n-sphere,” Calculus of Variations and Partial Differential Equations, vol. 4, no. 1, pp. 1–25, 1996. View at Google Scholar · View at MathSciNet
  16. H. Brezis and J.-M. Coron, “Convergence of solutions of H-systems or how to blow bubbles,” Archive for Rational Mechanics and Analysis, vol. 89, no. 1, pp. 21–56, 1985. View at Google Scholar
  17. P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The limit case. II,” Revista Matemática Iberoamericana, vol. 1, no. 2, pp. 45–121, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. Struwe, “A global compactness result for elliptic boundary value problems involving limiting nonlinearities,” Mathematische Zeitschrift, vol. 187, no. 4, pp. 511–517, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Bahri, “An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension,” Duke Mathematical Journal, vol. 81, no. 2, pp. 323–466, 1996, A celebration of John F. Nash, Jr. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Bahri and J.-M. Coron, “The scalar-curvature problem on the standard three-dimensional sphere,” Journal of Functional Analysis, vol. 95, no. 1, pp. 106–172, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. O. Rey, “The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent,” Journal of Functional Analysis, vol. 89, no. 1, pp. 1–52, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Bahri and P. H. Rabinowitz, “Periodic solutions of Hamiltonian systems of three-body type,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 8, no. 6, pp. 561–649, 1991. View at Google Scholar