Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Solutions to a class of Schrödinger equations


Author: Yanheng Ding
Journal: Proc. Amer. Math. Soc. 130 (2002), 689-696
MSC (1991): Primary 35Q55; Secondary 58E55
DOI: https://doi.org/10.1090/S0002-9939-01-06225-6
Published electronically: July 25, 2001
MathSciNet review: 1866021
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We establish existence and multiplicity of solutions to a class of nonlinear Schrödinger equations with, e.g., ``atomic'' Hamiltonians, via critical point theory.


References [Enhancements On Off] (What's this?)

  • [1] S. Alama and Y. Y. Li, On ``Multibump'' bound states for certain semilinear elliptic equations, Indiana J. Math. 44 (1992), 983-1026. MR 94d:35044
  • [2] S. Alama and G. Tarantello, On the solvability of a semilinear elliptic equation via an associated eigenvalue problem, Math. Z. 221 (1996), 467-493. MR 97d:35067
  • [3] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal. 140 (1997), 285-300. MR 98k:35172
  • [4] T. Bartsch and Y. H. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann. 313 (1999), 15-37. MR 99m:35218
  • [5] R. Benguria, H. Brézis and E. H. Lieb, The Thomas-Fermi-von Weizäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981), 167-180. MR 83m:81114
  • [6] V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^{N}$, Comm. Pure Appl. Math. 45 (1992), 1217-1269. MR 93k:35087
  • [7] M. A. Del Pino and P. Felmer, Multiple solutions for a semilinear elliptic equation, Trans. Amer. Math. Soc. 347 (1995), 4839-4853. MR 96c:35062
  • [8] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Diff. Eq. 3 (1998), 441-472. CMP 2000:11
  • [9] S.J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), 6-32. MR 96a:58045
  • [10] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), 33-97. MR 88e:35170
  • [11] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C. B. M. S. 65, Amer. Math. Soc., Providence, R. I., 1986. MR 87j:58024
  • [12] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, ZAMP 43 (1992), 270-291. MR 93h:35194
  • [13] M. Reed and B. Simon, Methods of Mathematical Physics, vol. I-IV, Academic Press, 1978. MR 58:12429a; MR 58:12429b; MR 80m:81085; MR 58:12429c
  • [14] C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Diff. Eq. 21 (1996), 1431-1449. MR 98i:35034
  • [15] M. Willem, Minimax Methods, Birkhäuser, Boston, 1996.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35Q55, 58E55

Retrieve articles in all journals with MSC (1991): 35Q55, 58E55


Additional Information

Yanheng Ding
Affiliation: Morningside Center of Mathematics and Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100080 Beijing, People’s Republic of China
Email: dingyh@math03.math.ac.cn

DOI: https://doi.org/10.1090/S0002-9939-01-06225-6
Keywords: Schr\"{o}dinger equations, multiple solutions, critical point theory
Received by editor(s): August 15, 2000
Published electronically: July 25, 2001
Additional Notes: This research was supported by the Special Funds for Major State Basic Research Projects of China.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society