Stable solitary waves with prescribed L2-mass for the cubic Schrödinger system with trapping potentials

Discrete and Continuous Dynamical Systems- Series A

Volumn 35, Issue 12, 2015, Pages 6085-6112

  Noris, Benedetta ^a   Tavares, Hugo ^b   Verzini, Gianmaria ^c  

^a UNIVERSITÉ DE VERSAILLES SAINT QUENTIN EN YVELINES   (France)

^b INSTITUTO SUPERIOR TÉCNICO   (Portugal)

^c POLITECNICO DI MILANO   (Italy)

Author keywords

Ambrosetti Prodi type problem; Constrained critical points; Cooperative and competitive elliptic systems; Gross Pitaevskii systems; Orbital stability

Indexed keywords

EID: 84930268398     PISSN: 10780947     EISSN: 15535231     Source Type: Journal    
DOI: 10.3934/dcds.2015.35.6085     Document Type: Article

References (37)

1. A. Aftalion, B. Noris and C. Sourdis, Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation, Communications in Mathematical Physics, 336 (2015), 509-579.
2. S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model, J. Funct. Anal., 256 (2009), 1118-1136.
3. A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4), 93 (1972), 231-246.
4. A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2), 75 (2007), 67-82.
5. A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993.
6. T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
7. T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
8. H. Brezis, Semilinear equations in RN without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.
9. T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
10. S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of twodimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.
11. Z. Chen, C.-S. Lin and W. Zou, Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations, J. Differential Equations, 255 (2013), 4289-4311.
12. E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
13. DispersiveWiki project, URL http://wiki.math.toronto.edu/DispersiveWiki/index.php/.
14. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.
15. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348.
16. N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Adv. Nonlinear Stud., 14 (2014), 115-136.
17. O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.
18. n, n ≤ 3, Comm. Math. Phys., 255 (2005), 629-653.
19. Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
20. L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.
21. L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations, Adv. Nonlinear Stud., 10 (2010), 681-705.
22. N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system, Commun. Math. Sci., 9 (2011), 997-1012.
23. N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differential Equations, 16 (2011), 977-1000.
24. B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proc. Amer. Math. Soc., 138 (2010), 1681-1692.
25. B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245-1273.
26. 2-critical and supercritical NLS on bounded domains, arXiv:1307.3981.
27. B. Noris and G. Verzini, A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems, J. Differential Equations, 254 (2013), 1529-1547.
28. M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26 (1996), 933-939.
29. J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations, 49 (2014), 103-124.
30. J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations., Commun. Math. Phys., 91 (1983), 313-327.
31. n, Comm. Math. Phys., 271 (2007), 199-221.
32. N. Soave, On existence and phase separation of positive solutions to nonlinear elliptic systems modelling simultaneous cooperation and competition, arXiv:1310.8492.
33. H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740.
34. H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 279-300.
35. S. Terracini and G. Verzini, Multipulse phases in k-mixtures of Bose-Einstein condensates, Arch. Ration. Mech. Anal., 194 (2009), 717-741.
36. R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
37. W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989.