# American Institute of Mathematical Sciences

May  2018, 17(3): 787-806. doi: 10.3934/cpaa.2018040

## Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction

 HLM, CEMS, Academy of Mathematics and Systems Science, the Chinese, Academy of Sciences, Beijing 100190, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author

Received  June 2017 Revised  June 2017 Published  January 2018

Fund Project: Supported by the National Natural Science Foundation of China 11325107,11331010,11771428

We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system
 $\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$
where
 $N≤3, Ω\subseteq\mathbb{R}^N$
is a smooth domain. First we establish the symmetry of ground state solutions, that is, when
 $Ω$
is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that
 $Ω$
is a ball or the whole space
 $\mathbb{R}^N$
. Next we investigate the asymptotic behavior of positive ground state solution as
 $κ\to 0^-$
, which shows that the limiting profile is exactly a minimizer for
 $c_0$
(the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.
Citation: Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040
##### References:
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Nonlinear Sci., 19 (2009), 437-451. Google Scholar [7] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. Google Scholar [8] J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys., 28 (1987), 13-24. Google Scholar [9] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. Google Scholar [10] B. Deconinck, Linearly coupled Bose-Einstein condesates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706. doi: 10.1103/PhysRevA.70.063605. Google Scholar [11] G. W. Dai, R. S. Tian and Z. T. Zhang, Global bifurcation, priori bounds and uniqueness of positive solutions for coupled nonlinear Schrödinger systems, preprint.Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. Google Scholar [13] D. S. Hall, M. R. Matthews, J. R. Ensher and C. E. Wieman, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1539-1542. doi: 10.1103/PhysRevLett.81.1539. Google Scholar [14] N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480. Google Scholar [15] K. Li and Z. T. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 17 pp. Google Scholar [16] T. C. Lin and J. C. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n≤3$, Comm. Math. Phys., 255 (2005), 629-653. Google Scholar [17] Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. 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Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. Google Scholar [24] R. S. Tian and Z. T. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math., 58 (2015), 1607-1620. Google Scholar [25] E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718. Google Scholar [26] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. Google Scholar [27] Z. Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math., 122 (2014), 69-85. Google Scholar [28] J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), 279-293. Google Scholar [29] J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. Google Scholar [30] T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158. Google Scholar [31] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar [32] M. Willem, Principles d'analyse fonctionnelle, Cassini, Pairs, 2007. Google Scholar

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##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent soltions on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664. doi: 10.1103/PhysRevLett.82.2661. Google Scholar [2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. Google Scholar [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82. Google Scholar [4] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. Google Scholar [5] T. Bartsch, Z. Q. Wang and J. C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. Google Scholar [6] J. Belmonte-Beitia, V. M. Pérez-García and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451. Google Scholar [7] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. Google Scholar [8] J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys., 28 (1987), 13-24. Google Scholar [9] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. Google Scholar [10] B. Deconinck, Linearly coupled Bose-Einstein condesates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706. doi: 10.1103/PhysRevA.70.063605. Google Scholar [11] G. W. Dai, R. S. Tian and Z. T. Zhang, Global bifurcation, priori bounds and uniqueness of positive solutions for coupled nonlinear Schrödinger systems, preprint.Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. Google Scholar [13] D. S. Hall, M. R. Matthews, J. R. Ensher and C. E. Wieman, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1539-1542. doi: 10.1103/PhysRevLett.81.1539. Google Scholar [14] N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480. Google Scholar [15] K. Li and Z. T. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 17 pp. Google Scholar [16] T. C. Lin and J. C. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n≤3$, Comm. Math. Phys., 255 (2005), 629-653. Google Scholar [17] Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. Google Scholar [18] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. Google Scholar [19] C. J. Myatt, Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett., 78 (1997), 586-589. doi: 10.1103/PhysRevLett.78.586. Google Scholar [20] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. Google Scholar [21] Ch. Rüegg, Bose-Einstein condensate of the triplet ststes in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. doi: 10.1038/nature01617. Google Scholar [22] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Comm. Math. Phys., 271 (2007), 199-221. Google Scholar [23] H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. Google Scholar [24] R. S. Tian and Z. T. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math., 58 (2015), 1607-1620. Google Scholar [25] E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718. Google Scholar [26] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. Google Scholar [27] Z. Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math., 122 (2014), 69-85. Google Scholar [28] J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), 279-293. Google Scholar [29] J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. Google Scholar [30] T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158. Google Scholar [31] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar [32] M. Willem, Principles d'analyse fonctionnelle, Cassini, Pairs, 2007. Google Scholar
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