Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

Proceedings of the London Mathematical Society

Volumn 107, Issue 2, 2013, Pages 303-339

  Bellazzini, Jacopo ^a   Jeanjean, Louis ^b   Luo, Tingjian ^b  

^a UNIVERSITY OF SASSARI   (Italy)

^b UNIVERSITÉ DE FRANCHE COMTÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84880299253     PISSN: 00246115     EISSN: 1460244X     Source Type: Journal    
DOI: 10.1112/plms/pds072     Document Type: Article

References (31)

1. A. Ambrosetti, A. Malchiodi and D. Ruiz, 'Bound states of nonlinear Schr̈odinger equations with potential vanishing at infinity', J. Anal. Math. 98 (2006) 317-348. (Pubitemid 44369827)
2. A. Azzollini, A. Pomponio and P. d'Avenia, 'On the Schr̈odinger- Maxwell equations under the effect of a general nonlinear term', Ann. Inst. H. Poincaŕe Anal. Non Lińeaire 27 (2010) 779-791.
3. C. Bardos, F. Golse, A. D. Gottlieb and N. Mauser, 'Mean field dynamics of fermions and the time-dependent Hartree-Fock equation', J. Math. Pures Appl. (9) 82 (2003) 665-683. (Pubitemid 36910332)
4. J. Bellazzini and G. Siciliano, 'Stable standing waves for a class of nonlinear Schr̈odinger-Poisson equations', Z. Angew. Math. Phys. 62 (2011) 267-280.
5. J. Bellazzini and G. Siciliano, 'Scaling properties of functionals and existence of constrained minimizers', J. Funct. Anal. 261 (2011) 2486-2507.
6. H. Berestycki and T. Cazenave, 'Instabilit́e des ́etats stationnaires dans les ́equations de Schr̈odinger et de Klein-Gordon non lińeaire', C. R. Acad. Sci. Paris Śer. I Math. 293 (1981) 489-492.
7. H. Berestycki and P. L. Lions, 'Nonlinear scalar field equations I', Arch. Ration. Mech. Anal. 82 (1983) 313-346.
8. H. Berestycki and P. L. Lions, 'Nonlinear scalar field equations II, existence of infinitely many solutions', Arch. Ration. Mech. Anal. 82 (1983) 347-375. (Pubitemid 13555202)
9. H. Brezis and E. Lieb, 'A relation between pointwise convergence of functions and convergence of functionals', Proc. Amer. Math. Soc. 88 (1983) 486-490.
10. H. Brezis and A. C. Ponce, 'Kato's inequality when Δu is a measure (English, French summary)', C. R. Math. Acad. Sci. Paris. 338 (2004) 599-604. (Pubitemid 38457758)
11. I. Catto and P. L. Lions, 'Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I. A necessary and sufficient condition for the stability of general molecular systems', Comm. Partial Differential Equations 17 (1992) 1051-1110.
12. T. Cazenave, Semilinear Schr̈odinger equations, Courant Lecture Notes in Mathematics 10 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003) xiv+323 pp. ISBN: 0-8218-3399-5.
13. T. Cazenave and P. L. Lions, 'Orbital stability of standing waves for some non linear Schr̈odinger equations', Commun. Math. Phys. 85 (1982) 549-561.
14. T. D'Aprile and D. Mugnai, 'Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations', Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 893-906. (Pubitemid 39604178)
15. R. Fukuizumi and T. Ozawa, 'Exponential decay of the solutions to nonlinear elliptic equations with potential', Z. Angew. Math. Phys. 56 (2005) 1000-1011. (Pubitemid 41605301)
16. V. Georgiev, F. Prinari and N. Visciglia, 'On the radiality of constrained minimizers to the Schr̈odinger-Poisson-Slater energy', Ann. Inst. H. Poincaŕe Anal. Non Lińeaire 29 (2012) 369-376.
17. H. Hajaiej and C. A. Stuart, 'On the variational approach to the stability of standing waves for the nonlinear Schr̈odinger equation', Adv. Nonlinear Stud. 4 (2004) 469-501. (Pubitemid 40019600)
18. I. Ianni and D. Ruiz, 'Ground and bound states for a static Schr̈odinger-Poisson-Slater problem', Comm. Comp. Math., to appear. doi no: 10.1142/S0219199712500034.
19. L. Jeanjean, 'Existence of solutions with prescribed norm for semilinear elliptic equations', Nonlinear Anal. Theory T. M.& A. 28 (1997) 1633-1659. (Pubitemid 127410939)
20. L. Jeanjean and M. Squassina, 'An approach to minimization under a constraint: the added mass technique', Calc. Var. Partial Differential Equations 41 (2011) 511-534.
21. M. Kaminaga and M. Ohta, 'Stability of standing waves for nonlinear Schr̈odinger equation with attractive delta potential and repulsive nonlinearity', Saitama Math. J. 26 (2009) 39-48.
22. H. Kikuchi, 'Existence and stability of standing waves for Schr̈odinger-Poisson-Slater equation', Adv. Nonlinear Stud. 7 (2007) 403-437.
23. H. Kikuchi, 'Existence and orbital stability of the standing waves for nonlinear Schr̈odinger equations via the variational method', Doctoral Thesis, 2008.
24. E. H. Lieb and B. Simon, 'The Thomas-Fermi theory of atoms, molecules, and solids', Adv. Math. 23 (1977) 22-116.
25. P. L. Lions, 'The concentration-compactness principle in the Calculus of Variation. The locally compact case, part I and II', Ann. Inst. H. Poincaŕe Anal. Non Lińeaire 1 (1984) 109-145 and 223-283.
26. P. L. Lions, 'Solutions of Hartree-Fock equations for Coulomb systems', Comm. Math. Phys. 109 (1987) 33-97.
27. N. J. Mauser, 'The Schr̈odinger-Poisson-Xα equation', Appl. Math. Lett. 14 (2001) 759-763.
28. D. Ruiz, 'The Schr̈odinger-Poisson equation under the effect of a nonlinear local term', J. Funct. Anal. 237 (2006) 655-674. (Pubitemid 43792923)
29. D. Ruiz, 'On the Schr̈odinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases', Arch. Ration. Mech. Anal. 198 (2010) 349-368.
30. O. Sanchez and J. Soler, 'Long-time dynamics of the Schr̈odinger- Poisson-Slater system', J. Statist. Phys. 114 (2004) 179-204.
31. L. Zhao and F. Zhao, 'On the existence of solutions for the Schr̈odinger-Poisson equations', J. Math. Anal. Appl. 346 (2008) 155-169.