UNIQUENESS OF GROUND STATES FOR PSEUDORELATIVISTIC HARTREE EQUATIONS

Analysis and PDE

Volumn 2, Issue 1, 2009, Pages

  Lenzmann, Enno ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Boson stars; Ground state; Pseudorelativistic hartree equation; Uniqueness

Indexed keywords

EID: 70449525913     PISSN: 21575045     EISSN: 1948206X     Source Type: Journal    
DOI: 10.2140/apde.2009.2.1     Document Type: Article

References (34)

1. [Abou Salem 2007] W. K. Abou Salem, “Solitary wave dynamics in time-dependent potentials”, 2007. arXiv 0707.0272
2. [Albert 1995] J. Albert, “Positivity properties and uniqueness of solitary wave solutions of the intermediate long-wave equa- tion”, pp. 11–20 in Evolution equations (Baton Rouge, LA, 1992), edited by G. Ferreyra et al., Lecture Notes in Pure and Appl. Math. 168, Dekker, New York, 1995. MR 95i:35240
3. [Amick and Toland 1991] C. J. Amick and J. F. Toland, “Uniqueness and related analytic properties for the Benjamin–Ono equation—a nonlinear Neumann problem in the plane”, Acta Math. 167:1-2 (1991), 107–126. MR 92i:35099 Zbl 0755.35108
4. [Bronski and Jerrard 2000] J. C. Bronski and R. L. Jerrard, “Soliton dynamics in a potential”, Math. Res. Lett. 7:2-3 (2000), 329–342. MR 2001g:37139 Zbl 0955.35067
5. [Carmona and Simon 1981] R. Carmona and B. Simon, “Pointwise bounds on eigenfunctions and wave packets in N -body quantum systems, V: Lower bounds and path integrals”, Comm. Math. Phys. 80:1 (1981), 59–98. MR 84m:81052a Zbl 0464. 35085
6. [Chang 2005] K.-C. Chang, Methods in nonlinear analysis, Springer Monographs in Mathematics, Springer, Berlin, 2005. MR 2007b:47169 Zbl 1081.47001
7. [Chang et al. 2007] S.-M. Chang, S. Gustafson, K. Nakanishi, and T.-P. Tsai, “Spectra of linearized operators for NLS solitary waves”, SIAM J. Math. Anal. 39:4 (2007), 1070–1111. MR 2008k:35400 Zbl 05306587
8. [Cho and Ozawa 2006] Y. Cho and T. Ozawa, “On the semirelativistic Hartree-type equation”, SIAM J. Math. Anal. 38:4 (2006), 1060–1074. MR 2007m:35249 Zbl 1122.35119
9. [Elgart and Schlein 2007] A. Elgart and B. Schlein, “Mean field dynamics of boson stars”, Comm. Pure Appl. Math. 60:4 (2007), 500–545. MR 2009a:85001 Zbl 1113.81032
10. [Esteban and Séré 2001] M. J. Esteban and E. Séré, “Nonrelativistic limit of the Dirac–Fock equations”, Ann. Henri Poincaré 2:5 (2001), 941–961. MR 2003d:81066 Zbl 1092.81520
11. [Fröhlich and Lenzmann 2007] J. Fröhlich and E. Lenzmann, “Blowup for nonlinear wave equations describing boson stars”, Comm. Pure Appl. Math. 60:11 (2007), 1691–1705. MR 2008m:85002 Zbl 1126.35064
12. [Fröhlich et al. 2002] J. Fröhlich, T.-P. Tsai, and H.-T. Yau, “On the point-particle (Newtonian) limit of the nonlinear Hartree equation”, Comm. Math. Phys. 225:2 (2002), 223–274. MR 2003e:81047 Zbl 1025.81015
13. [Fröhlich et al. 2004] J. Fröhlich, S. Gustafson, B. L. G. Jonsson, and I. M. Sigal, “Solitary wave dynamics in an external potential”, Comm. Math. Phys. 250:3 (2004), 613–642. MR 2005h:35320 Zbl 1075.35075
14. [Fröhlich et al. 2007a] J. Fröhlich, B. L. G. Jonsson, and E. Lenzmann, “Boson stars as solitary waves”, Comm. Math. Phys. 274:1 (2007), 1–30. MR 2008e:35183
15. [Fröhlich et al. 2007b] J. Fröhlich, B. L. G. Jonsson, and E. Lenzmann, “Effective dynamics for boson stars”, Nonlinearity 20:5 (2007), 1031–1075. MR 2312382 Zbl 1124.35084
16. [Harrison et al. 2003] R. Harrison, I. Moroz, and K. P. Tod, “A numerical study of the Schrödinger–Newton equations”, Nonlinearity 16:1 (2003), 101–122. MR 2004a:81082 Zbl 1040.81554
17. [Herbst 1977] I. W. Herbst, “Spectral theory of the operator ( p2 m2)1/2 Ze2/r ”, Comm. Math. Phys. 53:3 (1977), 285–294. MR 55 #9790 Zbl 0375.35047
18. [Holmer and Zworski 2008] J. Holmer and M. Zworski, “Soliton interaction with slowly varying potentials”, Int. Math. Res. Not. IMRN 10 (2008), Art. ID rnn026, 36. MR MR2429244 Zbl 1147.35084
19. [Jonsson et al. 2006] B. L. G. Jonsson, J. Fröhlich, S. Gustafson, and I. M. Sigal, “Long time motion of NLS solitary waves in a confining potential”, Ann. Henri Poincaré 7:4 (2006), 621–660. MR 2007f:35269 Zbl 1100.81019
20. [Krieger et al. 2008] J. Krieger, I. Martel, and P. Raphaël, “Two soliton solutions to the three dimensional gravitational Hartree equation”, Preprint, 2008.
21. [Kwong 1989] M. K. Kwong, “Uniqueness of positive solutions of ∆u u u p 0 in Rn ”, Arch. Rational Mech. Anal. 105:3 (1989), 243–266. MR 90d:35015 Zbl 0676.35032
22. [Lenzmann 2006] E. Lenzmann, Nonlinear dispersive equations describing boson stars, Ph.D. dissertation No. 16572, ETH Zürich, 2006.
23. [Lenzmann 2007] E. Lenzmann, “Well-posedness for semirelativistic Hartree equations of critical type”, Math. Phys. Anal. Geom. 10:1 (2007), 43–64. MR 2008i:35228
24. [Lieb 1977] E. H. Lieb, “Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation”, Studies in Appl. Math. 57:2 (1977), 93–105. MR 57 #11508 Zbl 0369.35022
25. [Lieb and Loss 2001] E. H. Lieb and M. Loss, Analysis, Second ed., Graduate Studies in Math. 14, Amer. Math. Soc., Provi- dence, RI, 2001. MR 2001i:00001 Zbl 0966.26002
26. [Lieb and Yau 1987] E. H. Lieb and H.-T. Yau, “The Chandrasekhar theory of stellar collapse as the limit of quantum mechan- ics”, Comm. Math. Phys. 112:1 (1987), 147–174. MR 89b:82014 Zbl 0641.35065
27. [McLeod 1993] K. McLeod, “Uniqueness of positive radial solutions of ∆u f (u) 0 in Rn. II”, Trans. Amer. Math. Soc. 339:2 (1993), 495–505. MR 94b:35105 Zbl 0804.35034
28. [McLeod and Serrin 1987] K. McLeod and J. Serrin, “Uniqueness of positive radial solutions of ∆u f (u) 0 in Rn ”, Arch. Rational Mech. Anal. 99:2 (1987), 115–145. MR 88c:35057 Zbl 0667.35023
29. [Peletier and Serrin 1983] L. A. Peletier and J. Serrin, “Uniqueness of positive solutions of semilinear equations in Rn ”, Arch. Rational Mech. Anal. 81:2 (1983), 181–197. MR 84b:35046 Zbl 0516.35031
30. [Reed and Simon 1978] M. Reed and B. Simon, Methods of modern mathematical physics, IV, Analysis of operators, Academic Press (Harcourt Brace Jovanovich), New York, 1978. MR 0493421 Zbl 0401.47001
31. [Reed and Simon 1980] M. Reed and B. Simon, Methods of modern mathematical physics, I, Functional analysis, Second ed., Academic Press (Harcourt Brace Jovanovich), New York, 1980. MR 85e:46002 Zbl 0459.46001
32. [Strauss 1977] W. A. Strauss, “Existence of solitary waves in higher dimensions”, Comm. Math. Phys. 55:2 (1977), 149–162. MR 56 #12616 Zbl 0356.35028
33. [Tod and Moroz 1999] P. Tod and I. M. Moroz, “An analytical approach to the Schrödinger–Newton equations”, Nonlinearity 12:2 (1999), 201–216. MR 2000j:81053 Zbl 0942.35077
34. [Weinstein 1985] M. I. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equations”, SIAM J. Math. Anal. 16:3 (1985), 472–491. MR 86i:35130 Zbl 0583.35028