Monotonicity and 1-Dimensional Symmetry for Solutions of an Elliptic System Arising in Bose-Einstein Condensation

Archive for Rational Mechanics and Analysis

Volumn 213, Issue 1, 2014, Pages 287-326

  Farina, Alberto ^a,b   Soave, Nicola ^a,c  

^a UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

^b UNIVERSITÉ LYON 1   (France)

^c UNIVERSITY OF MILANO BICOCCA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84899993644     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-014-0724-2     Document Type: Article

References (25)

1. Ambrosetti A., Colorado E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67-82 (2007).
2. Bartsch T., Dancer E. N., Wang Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37(3-4), 345-361 (2010).
3. Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2(2), 353-367 (2007).
4. Berestycki, H., Caffarelli, L., Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipschitz domains, Commun. Pure Appl. Math. 50(11), 1089-1111 (1997).
5. Berestycki, H., Lin, T. C., Wei, J., Zhao, C.: On phase-separation model: asymptotics and qualitative properties. Arch. Ration. Mech. Anal. 208, 163-200 (2013).
6. Berestycki, H., Terracini, S., Wang, K., Wei, J.: On entire solutions of an elliptic system modelling phase-separation. Adv. Math. 243, 102-126 (2013).
7. Conti M., Terracini S., Verzini G.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195, 524-560 (2005).
8. Dancer, E. N., Wang, K., Zhang, Z.: The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture. J. Funct. Anal. 262, 1087-1131 (2012).
9. Hall, D. S., Matthews, M. R., Ensher, J. R., Wieman, C. E., Cornell, E. A.: Dynamics of component separation in a binary mixture of Bose-Einstein condensates. Phys. Rev. Lett. 81, 1539-1542 (1998).
10. N and related conjectures, Papers in memory of Ennio De Giorgi (Italian). Ricerche Mat. 48(suppl.), 129-154 (1999).
11. Farina, A.: Some symmetry results for entire solutions of an elliptic system arising in phase separation (2012, preprint). http://arxiv. org/abs/1307. 5537. To appear on Discrete Contin. Dyn. Syst. A, special volume "Qualitative properties of solutions of nonlinear elliptic equations and systems".
12. Farina, A., Valdinoci, E.: The State of the Art of a Conjecture of De Giorgi and Related Problems. Recent Progress in Reaction-Diffusion Systems and Viscosity Solutions, pp. 74-96. World Scientific Publishers, Hackensack, 2009.
13. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.
14. nRn, n≦3. Commun. Math. Phys. 255(3), 629-653 (2005).
15. Liu Z., Wang, Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282(3), 721-731 (2008).
16. Maia, L. A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743-767 (2006).
17. Noris B., Tavares H., Terracini S., Terracini S., Terracini S.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63(3), 267-302 (2010).
18. Papp, S. B., Pino, J. M., Wieman, C. E.: Tunable miscibility in a dual-species Bose-Einstein condensate. Phys. Rev. Lett. 101, 040402 (2008).
19. N. Commun. Math. Phys. 271(1), 199-221 (2007).
20. Soave N., Zilio A.: Entire solutions with exponential growth for an elliptic system modelling phase separation. Nonlinearity 27(2), 305-342 (2014).
21. Tavares, H., Terracini, S.: Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. PDE45, 273-317 (2012).
22. Terracini, S., Verzini, G.: Multipulses phases in k-mixtures of Bose-Einstein condensates. Arch. Ration. Mech. Anal. 194(3), 717-741 (2009).
23. Wang, K.: On the De Giorgi type conjecture for an elliptic system modeling phase separation. Commun. Partial Differ. Equ. 39(4), 696-739 (2014). doi: 10. 1080/03605302. 2013. 856916.
24. Wei J., Weth T.: Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21(2), 305-317 (2008).
25. Wei, J., Weth, T.: Radial solutions and phase seperation in a system of two coupled Schrödinger equations. Arch. Ration. Mech. Anal. 190, 83-106 (2008).