Free access
Issue
ESAIM: COCV
Volume 15, Number 3, July-September 2009
Page(s) 653 - 675
DOI http://dx.doi.org/10.1051/cocv:2008055
Published online 20 August 2008
  1. A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140 (1997) 285–300. [CrossRef]
  2. A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159 (2001) 253–271. [CrossRef] [MathSciNet]
  3. G. Arioli and A. Szulkin, A semilinear Schrödinger equations in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170 (2003) 277–295. [CrossRef] [MathSciNet]
  4. S. Barile, S. Cingolani and S. Secchi, Single-peaks for a magnetic Schrödinger equation with critical growth. Adv. Diff. Equations 11 (2006) 1135–1166.
  5. T. Bartsch, E.N. Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Diff. Equations 11 (2006) 781–812.
  6. H. Berestycki and P.L. Lions, Nonlinear scalar field equation I. Arch. Ration. Mech. Anal. 82 (1983) 313–346.
  7. J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185 (2007) 185–200. [CrossRef] [MathSciNet]
  8. J. Byeon and L. Jeanjean, Erratum: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. DOI 10.1007/s00205-006-0019-3.
  9. J. Byeon and L. Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete Cont. Dyn. Systems 19 (2007) 255–269. [CrossRef]
  10. J. Byeon and Z.-Q. Wang, Standing waves with critical frequency for nonlinear Schrödinger equations. Arch. Rat. Mech. Anal. 165 (2002) 295–316. [CrossRef]
  11. J. Byeon and Z.-Q. Wang, Standing waves with critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 18 (2003) 207–219. [CrossRef]
  12. J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Comm. Partial Diff. Eq. 33 (2008) 1113–1136. [CrossRef]
  13. D. Cao and E.-S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations. J. Diff. Eq. 203 (2004) 292–312. [CrossRef] [MathSciNet]
  14. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes. AMS (2003).
  15. J. Chabrowski and A. Szulkin, On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Topol. Methods Nonlinear Anal. 25 (2005) 3–21. [MathSciNet]
  16. S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Diff. Eq. 188 (2003) 52–79. [CrossRef]
  17. S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Diff. Eq. 160 (2000) 118–138. [CrossRef]
  18. S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. Royal Soc. Edinburgh 128 (1998) 1249–1260.
  19. S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275 (2002) 108–130. [CrossRef] [MathSciNet]
  20. S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46 (2005) 1–19.
  21. M. Clapp, R. Iturriaga and A. Szulkin, Periodic solutions to a nonlinear Schrödinger equations with periodic magnetic field. Preprint.
  22. V. Coti-Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991) 693–727. [CrossRef] [MathSciNet]
  23. V. Coti-Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on Formula . Comm. Pure Appl. Math. 45 (1992) 1217–1269. [CrossRef] [MathSciNet]
  24. M. Del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996) 121–137. [CrossRef] [MathSciNet]
  25. M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149 (1997) 245–265. [CrossRef] [MathSciNet]
  26. M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 127–149. [CrossRef] [MathSciNet]
  27. M.J. Esteban and P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, in PDE and Calculus of Variations, in honor of E. De Giorgi, Birkhäuser (1990).
  28. A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986) 397–408. [CrossRef] [MathSciNet]
  29. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Grundlehren 224. Springer, Berlin, Heidelberg, New York and Tokyo (1983).
  30. C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations. Comm. Partial Diff. Eq. 21 (1996) 787–820. [CrossRef]
  31. H. Hajaiej and C.A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Advances Nonlinear Studies 4 (2004) 469–501.
  32. L. Jeanjean and K. Tanaka, A remark on least energy solutions in Formula . Proc. Amer. Math. Soc. 131 (2003) 2399–2408. [CrossRef] [MathSciNet]
  33. L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions. Advances Nonlinear Studies 3 (2003) 461–471.
  34. L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asympotically linear nonlinearities. Calc. Var. Partial Diff. Equ. 21 (2004) 287–318.
  35. K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 41 (2000) 763–778. [CrossRef] [MathSciNet]
  36. Y.Y. Li, On a singularly perturbed elliptic equation. Adv. Diff. Equations 2 (1997) 955–980.
  37. P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 223–283.
  38. Y.G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations. Comm. Partial Diff. Eq. 13 (1988) 1499–1519. [CrossRef]
  39. M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations. Springer-Verlag, New York, Berlin, Heidelberg and Tokyo (1984).
  40. P. Rabinowitz, On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992) 270–291. [CrossRef] [MathSciNet]
  41. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II. Academic press, New York (1972).
  42. S. Secchi and M. Squassina, On the location of spikes for the Schrödinger equations with electromagnetic field. Commun. Contemp. Math. 7 (2005) 251–268. [CrossRef] [MathSciNet]
  43. W. Strauss, Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977) 149–162. [CrossRef] [MathSciNet]
  44. M. Struwe, Variational Methods, Application to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag (1990).
  45. X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal. 28 (1997) 633–655. [CrossRef] [MathSciNet]