# American Institute of Mathematical Sciences

November  2019, 12(7): 2143-2161. doi: 10.3934/dcdss.2019138

## Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems

 1 School of Mathematical Science, Capital Normal University, Beijing 10048, China 2 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China 3 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

* Corresponding author: Rushun Tian and Zhi-Qiang Wang

Received  November 2017 Revised  April 2018 Published  December 2018

Fund Project: This paper is supported by Beijing Natural Science Foundation (1174013), National Natural Science Foundation of China (11601353, 11771302, 11771324, 11671026, 11831009).

In this paper, we study the following doubly coupled multicomponent system
 $\begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + \lambda_ju_j+ \sum_{k\neq j}\gamma_{jk}u_k = \mu_ju_j^3+ u_j\sum_{k\neq j}\beta_{jk}u_k^2,\\ u_j(x)\geq0\ \ \hbox{and}\ \ u_j\in H_0^1(\Omega), \end{array} \right. \end{equation*}$
where
 $\Omega\subset \mathbb{R} ^N$
and
 $N = 2,3$
;
 $\lambda_j, \gamma_{jk} = \gamma_{kj}, \mu_j, \beta_{jk} = \beta_{kj}$
are constants,
 $j, k = 1, 2, ..., n$
,
 $n\geq 2$
. We prove some existence and nonexistence results for positive solutions of this system. If the system is fully symmetric, i.e.
 $\lambda_j\equiv\lambda, \gamma_{jk}\equiv\gamma, \mu_j\equiv\mu, \beta_{jk}\equiv\beta$
, we study the multiplicity and bifurcation phenomena of positive solution.
Citation: Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138
##### References:
 [1] A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb{R} ^n$, J. Funct. Anal., 254 (2008), 2816-2845.  doi: 10.1016/j.jfa.2007.11.013. [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.  doi: 10.1016/j.crma.2006.01.024. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020. [4] T. Bartsch, Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13 (2013), 37-50.  doi: 10.1007/s11784-013-0109-4. [5] T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Vari. Part. Diff. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y. [6] T. Bartsch, R. Tian and Z.-Q. Wang, Bifurcations for a coupled Schr dinger system with multiple components,, Angew. Math. Phys., 66 (2015), 2109-2123.  doi: 10.1007/s00033-015-0498-x. [7] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. [8] T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6. [9] G. Dai, R. Tian and Z. Zhang, Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger systems., Preprint. [10] 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.  doi: 10.1016/j.anihpc.2010.01.009. [11] B. Deconinck, P. G. Kevrekidis, H. E. Nistazakis and D. J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 063605. [12] B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. [13] P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation thereoms: a Unified approach via complementing maps, Math. Ann., 263 (1983), 61-73.  doi: 10.1007/BF01457084. [14] K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17pp.  doi: 10.1063/1.4960046. [15] T. Lin and J. Wei, Ground state of $N$ Coupled Nonlinear Schrödinger equations in $\mathbb{R} ^n, n\leq3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x. [16] T. Lin and J. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Physics D: Nonlinear Phenomena, 220 (2006), 99-115.  doi: 10.1016/j.physd.2006.07.009. [17] Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731.  doi: 10.1007/s00220-008-0546-x. [18] Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109. [19] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002. [20] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian System, Spinger-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [21] M. Mitchell, Z. Chen, M. Shih and M. Segev, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493. [22] Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwischu, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. [23] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R} ^n$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x. [24] R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. [25] R. Tian and Z.-Q. Wang, Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discrete Contin. Dyn. Syst. - Series A, 33 (2013), 335-344.  doi: 10.3934/dcds.2013.33.335. [26] R. Tian and Z.-Q. Wang, Bifurcation results on positive solutions of an indefinite nonlinear elliptic system Ⅱ, Adv. Nonlinear Stud., 13 (2013), 245-262.  doi: 10.1515/ans-2013-0115. [27] R. 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.  doi: 10.1007/s11425-015-5028-y. [28] Z.-Q. Wang, A Zp index theory, Acta Mathematica Sinica, New Series, 6 (1990), 18-23.  doi: 10.1007/BF02108859. [29] J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.  doi: 10.4171/RLM/495.

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##### References:
 [1] A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb{R} ^n$, J. Funct. Anal., 254 (2008), 2816-2845.  doi: 10.1016/j.jfa.2007.11.013. [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.  doi: 10.1016/j.crma.2006.01.024. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020. [4] T. Bartsch, Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13 (2013), 37-50.  doi: 10.1007/s11784-013-0109-4. [5] T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Vari. Part. Diff. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y. [6] T. Bartsch, R. Tian and Z.-Q. Wang, Bifurcations for a coupled Schr dinger system with multiple components,, Angew. Math. Phys., 66 (2015), 2109-2123.  doi: 10.1007/s00033-015-0498-x. [7] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. [8] T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6. [9] G. Dai, R. Tian and Z. Zhang, Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger systems., Preprint. [10] 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.  doi: 10.1016/j.anihpc.2010.01.009. [11] B. Deconinck, P. G. Kevrekidis, H. E. Nistazakis and D. J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 063605. [12] B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. [13] P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation thereoms: a Unified approach via complementing maps, Math. Ann., 263 (1983), 61-73.  doi: 10.1007/BF01457084. [14] K. Li and Z. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 081504, 17pp.  doi: 10.1063/1.4960046. [15] T. Lin and J. Wei, Ground state of $N$ Coupled Nonlinear Schrödinger equations in $\mathbb{R} ^n, n\leq3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x. [16] T. Lin and J. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Physics D: Nonlinear Phenomena, 220 (2006), 99-115.  doi: 10.1016/j.physd.2006.07.009. [17] Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731.  doi: 10.1007/s00220-008-0546-x. [18] Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109. [19] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002. [20] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian System, Spinger-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [21] M. Mitchell, Z. Chen, M. Shih and M. Segev, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996), 490-493. [22] Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht and P. Vorderwischu, Bose-Einstein condensation of the triplet states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. [23] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R} ^n$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x. [24] R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. [25] R. Tian and Z.-Q. Wang, Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discrete Contin. Dyn. Syst. - Series A, 33 (2013), 335-344.  doi: 10.3934/dcds.2013.33.335. [26] R. Tian and Z.-Q. Wang, Bifurcation results on positive solutions of an indefinite nonlinear elliptic system Ⅱ, Adv. Nonlinear Stud., 13 (2013), 245-262.  doi: 10.1515/ans-2013-0115. [27] R. 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.  doi: 10.1007/s11425-015-5028-y. [28] Z.-Q. Wang, A Zp index theory, Acta Mathematica Sinica, New Series, 6 (1990), 18-23.  doi: 10.1007/BF02108859. [29] J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.  doi: 10.4171/RLM/495.
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