Stable standing waves for a class of nonlinear Schrödinger-Poisson equations

Zeitschrift fur Angewandte Mathematik und Physik

Volumn 62, Issue 2, 2011, Pages 267-280

  Bellazzini, Jacopo ^a   Siciliano, Gaetano ^b  

^a UNIVERSITY OF PISA   (Italy)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

Orbital stability; Schr dinger Poisson equations; Standing waves

Indexed keywords

BIHARMONIC OPERATORS; CONSTRAINED MINIMIZATION PROBLEM; DINGER EQUATION; EXISTENCE AND STABILITY; MINIMIZING SEQUENCE; ORBITAL STABILITY; SMALL CHARGES; STANDING WAVE; STANDING WAVES;

ELASTIC WAVES; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; OPTIMIZATION; WAVES;

POISSON EQUATION;

EID: 79953711999     PISSN: 00442275     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00033-010-0092-1     Document Type: Article

References (21)

1. Azzollini A., Pomponio A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345(1), 90-108 (2008).
2. Bellazzini, J., Bonanno, C.: Nonlinear Schrödinger equations with strongly singular potentials. Proc. Roy. Soc. Edimburgh sec A (in press), arXiv: 0903. 3301.
3. Bellazzini, J., Visciglia, N.: On the orbital stability for a class of nonautonmous NLS. Indiana Univ. Math. J. 59(2), (2010) online preprint.
4. Benci V., Fortunato D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283-293 (1998).
5. Brezis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88(3), 486-490 (1983).
6. Cazenave, T.: Semilinear Schrödinger equation, Courant. In: Lecture Notes in Mathematics, vol. 10, New York: New York University, Courant Institute of Mathematical Sciences.
7. Cazenave T., Lions P. L.: Orbital stability of standing waves for some non linear Schrödinger equations. Commun. Math. Phys. 85, 549-561 (1982).
8. Chen G., Zhang J., Wei Y.: Stability of standing waves for nonlinear defocusing fourth-order dispersive Schrödinger equation with unbounded potentials. Math. Nachr. 281(4), 517-524 (2008).
9. Coclite G. M.: A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun. Appl. Anal 7, 417-423 (2003).
10. D'Aprile T., Mugnai D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Royal Soc. Edinb. 134 A, 893-906 (2004).
11. d'Avenia P.: Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear. Stud. 2, 177-192 (2002).
12. Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160-197 (1987).
13. Ianni I., Le Coz S.: Orbital stability of standing waves of a semiclassical nonlinear Schrö dinger-Poisson equation. Adv. Differ. Equ. 14(7-8), 717-748 (2009).
14. Kikuchi H.: On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal. 67, 1445-1456 (2007).
15. Kikuchi H.: Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7(3), 403-437 (2007).
16. Lions, P. L.: The concentration-compactness principle in the calculus of variation. The locally compact case, part I and II. Ann. Inst. H. Poincar Anal. Non Linaire 1, 109-145 and 223-283 (1984).
17. Pisani L., Siciliano G.: Neumann condition in the Schrödinger-Maxwell system. Topol. Methods Nonlinear Anal. 29, 251-264 (2007).
18. Ruiz D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655-674 (2006).
19. Sanchez O., Soler J.: Long time dynamics of the Schrödinger-Poisson-Slater system. J. Stat. Phys. 114, 179-204 (2004).
20. 3. Discrete Contin. Dyn. Syst. 18, 809-816 (2007).
21. Zhao L., Zhao F.: On the existence of solutions for the Schrödinger-Poisson equations. J. Math. Anal. Appl 346, 155-169 (2008).