# 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 and Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040
##### 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. [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. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82. [4] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. [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. [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. [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. [8] J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys., 28 (1987), 13-24. [9] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. [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. [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. [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. [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. [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. [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. [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. [17] Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. [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. [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. [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. [21] Ch. Rüegg, Bose-Einstein condensate of the triplet ststes in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.  doi: 10.1038/nature01617. [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. [23] H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. [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. [25] E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718. [26] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. [27] Z. Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math., 122 (2014), 69-85. [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. [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. [30] T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158. [31] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [32] M. Willem, Principles d'analyse fonctionnelle, Cassini, Pairs, 2007.

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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. [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. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82. [4] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. [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. [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. [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. [8] J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys., 28 (1987), 13-24. [9] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. [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. [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. [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. [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. [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. [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. [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. [17] Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. [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. [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. [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. [21] Ch. Rüegg, Bose-Einstein condensate of the triplet ststes in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.  doi: 10.1038/nature01617. [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. [23] H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. [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. [25] E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718. [26] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. [27] Z. Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math., 122 (2014), 69-85. [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. [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. [30] T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158. [31] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [32] M. Willem, Principles d'analyse fonctionnelle, Cassini, Pairs, 2007.
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