Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 651871, 20 pages
http://dx.doi.org/10.1155/2009/651871
Research Article

Existence of Infinitely Many Distinct Solutions to the Quasirelativistic Hartree-Fock Equations

1Department of Mathematics, Uppsala University, 751 06 Uppsala, Sweden
2School of Mathematical Sciences, Dublin Institute of Technology, Dublin 8, Ireland

Received 18 March 2009; Revised 8 July 2009; Accepted 10 August 2009

Academic Editor: Irena Lasiecka

Copyright © 2009 M. Enstedt and M. Melgaard. 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. P.-L. Lions, “Solutions of Hartree-Fock equations for Coulomb systems,” Communications in Mathematical Physics, vol. 109, no. 1, pp. 33–97, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. T. Bartsch, Z.-Q. Wang, and M. Willem, “The Dirichlet problem for superlinear elliptic equations,” in Stationary Partial Differential Equations. Vol. II, Handbook of Differential Equations, pp. 1–55, Elsevier/North-Holland, Amsterdam, The Netherlands, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  3. J. P. Solovej, “Proof of the ionization conjecture in a reduced Hartree-Fock model,” Inventiones Mathematicae, vol. 104, no. 2, pp. 291–311, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Dall'Acqua, T. Ø. Sørensen, and E. Stockmeyer, “Hartree-Fock theory for pseudorelativistic atoms,” Annales Henri Poincaré, vol. 9, no. 4, pp. 711–742, 2008. View at Google Scholar · View at MathSciNet
  5. E. H. Lieb, “Bound on the maximum negative ionization of atoms and molecules,” Phys. Rev. A, vol. 29, pp. 3018–3028, 1984. View at Google Scholar
  6. M. Enstedt and M. Melgaard, “Non-existence of a minimizer to the magnetic Hartree-Fock functional,” Positivity, vol. 12, pp. 653–666, 2008. View at Google Scholar
  7. J. P. Solovej, “The ionization conjecture in Hartree-Fock theory,” Ann. of Math (2), vol. 158, no. 2, pp. 509–576, 2003. View at Google Scholar
  8. M. Lewin, “Solutions to the multiconfiguration equations in quantum chemistry,” Arch Rat. Mech. Anal., vol. 171, pp. 83–114, 2004. View at Google Scholar
  9. C. Le Bris and P.-L. Lions, “From atoms to crystals: a mathematical journey,” Bull. Amer. Math. Soc. (N.S.), vol. 42, no. 3, pp. 291–363, 2005. View at Google Scholar
  10. G. Fang and N. Ghoussoub, “Second-order information on Palais-Smale sequences in the mountain pass theorem,” Manuscripta Mathematica, vol. 75, no. 1, pp. 81–95, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G. Fang and N. Ghoussoub, “Morse-type information on Palais-Smale sequences obtained by min-max principles,” Communications on Pure and Applied Mathematics, vol. 47, no. 12, pp. 1595–1653, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. E. N. Dancer, “Finite Morse index solutions of exponential problems,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 25, no. 1, pp. 173–179, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. G. de Figueiredo, P. N. Srikanth, and S. Santra, “Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball,” Communications in Contemporary Mathematics, vol. 7, no. 6, pp. 849–866, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. G. Flores, P. Padilla, and Y. Tonegawa, “Higher energy solutions in the theory of phase transitions: a variational approach,” Journal of Differential Equations, vol. 169, no. 1, pp. 190–207, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K. Tanaka, “Morse indices at critical points related to the symmetric mountain pass theorem and applications,” Communications in Partial Differential Equations, vol. 14, no. 1, pp. 99–128, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Cingolani and M. Lazzo, “Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,” Topological Methods in Nonlinear Analysis, vol. 10, no. 1, pp. 1–13, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. N. Ghoussoub and C. Yuan, “Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents,” Transactions of the American Mathematical Society, vol. 352, no. 12, pp. 5703–5743, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. C. Lazer and S. Solimini, “Nontrivial solutions of operator equations and Morse indices of critical points of min-max type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 12, no. 8, pp. 761–775, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. L. Jeanjean, “On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on N,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 129, no. 4, pp. 787–809, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. K.-H. Fieseler and K. Tintarev, “Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds,” The Journal of Geometric Analysis, vol. 13, no. 1, pp. 67–75, 2003. View at Google Scholar · View at MathSciNet
  21. K. Tintarev and K.-H. Fieseler, Concentration Compactness: Functional-Analytic Grounds and Applications, Imperial College Press, London, UK, 2007. View at MathSciNet
  22. E. A. Mazepa, “On the existence of entire solutions of a semilinear elliptic equation on noncompact Riemannian manifolds,” Matematicheskie Zametki, vol. 81, no. 1, pp. 153–156, 2007 (Russian), translation in: Mathematical Notes, vol. 81, no. 1-2, pp. 135–139, 2007. View at Google Scholar · View at MathSciNet
  23. K. Tanaka, “Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 17, no. 1, pp. 1–33, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Clarendon Press, Oxford University Press, New York, NY, USA, 1987. View at MathSciNet
  25. T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, Germany, 1995. View at MathSciNet
  26. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: An Introduction to Advanced Electronic Structure Theory, MacMillan, New York, NY, USA, 1982.
  27. M. Enstedt and M. Melgaard, “Existence of a solution to Hartree-Fock equations with decreasing magnetic fields,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 7, pp. 2125–2141, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. I. W. Herbst, “Spectral theory of the operator (p2+m2)1/2Ze2/r,” Communications in Mathematical Physics, vol. 53, no. 3, pp. 285–294, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J.-M. Barbaroux, W. Farkas, B. Helffer, and H. Siedentop, “On the Hartree-Fock equations of the electron-positron field,” Communications in Mathematical Physics, vol. 255, no. 1, pp. 131–159, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. M. Borwein and D. Preiss, “A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions,” Transactions of the American Mathematical Society, vol. 303, no. 2, pp. 517–527, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, vol. 107 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1993. View at MathSciNet
  32. P. H. Rabinowitz, “Variational methods for nonlinear eigenvalue problems,” in Eigenvalues of Non-Linear Problems (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1974), pp. 139–195, Edizioni Cremonese, Rome, Italy, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet