Monotonicity and 1-Dimensional Symmetry for Solutions of an Elliptic System Arising in Bose–Einstein Condensation
Tóm tắt
We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system:
$$\left\{\begin{array}{ll} -\Delta u = -u \upsilon^2 &\quad {\rm in}\, \mathbb{R}^N\\
-\Delta \upsilon= -u^2 \upsilon &\quad {{\rm in}\, \mathbb{R}^N},\end{array}\right.$$
for every dimension
$${N \geqq 2}$$
. In particular, we prove a Gibbons-type conjecture proposed by Berestycki et al.
Tài liệu tham khảo
Ambrosetti A., Colorado E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67–82 (2007)
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)
Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2(2), 353–367 (2007)
Berestycki, H., Caffarelli, L., Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipschitz domains, Commun. Pure Appl. Math. 50(11), 1089–1111 (1997)
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)
Berestycki, H., Terracini, S., Wang, K., Wei, J.: On entire solutions of an elliptic system modelling phase-separation. Adv. Math. 243, 102–126 (2013)
Conti M., Terracini S., Verzini G.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195, 524–560 (2005)
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)
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)
Farina, A.: Symmetry for solutions of semilinear elliptic equations in \({\mathbb{R}^N}\) and related conjectures, Papers in memory of Ennio De Giorgi (Italian). Ricerche Mat. 48(suppl.), 129–154 (1999)
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”
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
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001
Lin, T.-C., Wei, J.: Ground state of Ncoupled nonlinear Schrödinger equations in \({\mathbb{R}^n, n \leqq 3}\). Commun. Math. Phys. 255(3), 629–653 (2005)
Liu Z., Wang, Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282(3), 721–731 (2008)
Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)
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)
Papp, S.B., Pino, J.M., Wieman, C.E.: Tunable miscibility in a dual-species Bose–Einstein condensate. Phys. Rev. Lett. 101, 040402 (2008)
Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}^N}\). Commun. Math. Phys. 271(1), 199–221 (2007)
Soave N., Zilio A.: Entire solutions with exponential growth for an elliptic system modelling phase separation. Nonlinearity 27(2), 305–342 (2014)
Tavares, H., Terracini, S.: Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. PDE 45, 273–317 (2012)
Terracini, S., Verzini, G.: Multipulses phases in k-mixtures of Bose–Einstein condensates. Arch. Ration. Mech. Anal. 194(3), 717–741 (2009)
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
Wei J., Weth T.: Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21(2), 305–317 (2008)
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)