February  2022, 21(2): 639-667. doi: 10.3934/cpaa.2021192

Synchronized and ground-state solutions to a coupled Schrödinger system

1. 

Department of Mathematics, Kabul Polytechnic University, Kabul, Afghanistan

2. 

School of Mathematics and Statistics, Central China Normal University, Luo-Yu Road 152, Wuhan, 430079, China

3. 

School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Luo-Yu Road 152, Wuhan, 430079, China

* Corresponding author

Received  June 2021 Revised  October 2021 Published  February 2022 Early access  November 2021

Fund Project: The second author is supported by NSFC-11971191 and No. KJ02072020-0319

In this paper, we study the following coupled nonlinear Schrödinger system of the form
$ \left\{\begin{array}{l} -\Delta u_i-\kappa_iu_i = g_i(u_i)+\lambda\partial_iF(\vec{u}), \\ \vec{u} = (u_1,u_2,\cdots,u_m), u_i\in D_0^{1,2}(\Omega), \end{array}\right. $
for
$ m = 2,3 $
, where
$ \Omega\subset \mathbb{R}^N $
is a bounded domain or
$ \mathbb{R}^N $
,
$ N\geq 3 $
,
$ F(t_1,t_2\cdots,t_m)\in C^1(\mathbb{R}^m,\mathbb{R}) $
,
$ \kappa_i\in\mathbb{R} $
,
$ g_i\in C(\mathbb{R}) \ (i = 1,2,\cdots,m) $
and
$ \lambda>0 $
is large enough. In this work we mainly focus on the existence of fully nontrivial ground-state solutions and synchronized ground-state solutions under certain conditions.
Citation: Mohammad Ali Husaini, Chuangye Liu. Synchronized and ground-state solutions to a coupled Schrödinger system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 639-667. doi: 10.3934/cpaa.2021192
References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664. 

[2]

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.

[3] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Univ. Press, 2006.  doi: 10.1017/CBO9780511618260.
[4]

T. BartschN. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.

[5]

T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19 (2006), 200-207. 

[6]

T. BartschZ. 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.

[7]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[8]

Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.

[9]

M. Clapp and J. Faya, Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 3265-3289.  doi: 10.3934/dcds.2019135.

[10]

M. Clapp and A. Szulkin, Solutions to indefinite weakly coupled cooperative ellipitic systems, preprint, arXiv: 2003.12343v1.

[11]

N. DancerJ. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Ist. H. Poincaré Anal. Non Linéaire., 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.

[12]

W. Y. Ding, On a conformally invariant elliptic equation on $\mathbb{R}^n$, Commun. Math. Phys., 107 (1986), 331-335. 

[13]

B. EsryC. GreeneJ. Burke and J. Bohn, Hartree-Fock theory for double condesates, Phys. Rev. Lett., 78 (1997), 3594-3597. 

[14]

D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condesates: from theory to experiments, J. Phys. A, 43 (2010), 213001.  doi: 10.1088/1751-8113/43/21/213001.

[15]

F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differ. Equ., 2 (1997), 555-572. 

[16]

Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Reports, 298 (1998), 81-197.  doi: 10.1007/3-540-46629-0_8.

[17]

T. C. 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.

[18]

S. PengQ. Wang and Z. Q. Wang, On coupled nonlinear Schrödinger systems with mixed couplings, Trans. Amer. Math. Soc., 371 (2019), 7559-7583.  doi: 10.1090/tran/7383.

[19]

A. SzulkinT. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differ. Integral Equ., 22 (2009), 913-926. 

[20]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[21]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[22]

M. Willem, Analyse Harmonique Réelle, Hermann, Paris, 1995.

show all references

References:
[1]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664. 

[2]

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.

[3] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Univ. Press, 2006.  doi: 10.1017/CBO9780511618260.
[4]

T. BartschN. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.

[5]

T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19 (2006), 200-207. 

[6]

T. BartschZ. 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.

[7]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[8]

Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.

[9]

M. Clapp and J. Faya, Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 3265-3289.  doi: 10.3934/dcds.2019135.

[10]

M. Clapp and A. Szulkin, Solutions to indefinite weakly coupled cooperative ellipitic systems, preprint, arXiv: 2003.12343v1.

[11]

N. DancerJ. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Ist. H. Poincaré Anal. Non Linéaire., 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.

[12]

W. Y. Ding, On a conformally invariant elliptic equation on $\mathbb{R}^n$, Commun. Math. Phys., 107 (1986), 331-335. 

[13]

B. EsryC. GreeneJ. Burke and J. Bohn, Hartree-Fock theory for double condesates, Phys. Rev. Lett., 78 (1997), 3594-3597. 

[14]

D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condesates: from theory to experiments, J. Phys. A, 43 (2010), 213001.  doi: 10.1088/1751-8113/43/21/213001.

[15]

F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differ. Equ., 2 (1997), 555-572. 

[16]

Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Reports, 298 (1998), 81-197.  doi: 10.1007/3-540-46629-0_8.

[17]

T. C. 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.

[18]

S. PengQ. Wang and Z. Q. Wang, On coupled nonlinear Schrödinger systems with mixed couplings, Trans. Amer. Math. Soc., 371 (2019), 7559-7583.  doi: 10.1090/tran/7383.

[19]

A. SzulkinT. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differ. Integral Equ., 22 (2009), 913-926. 

[20]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[21]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[22]

M. Willem, Analyse Harmonique Réelle, Hermann, Paris, 1995.

[1]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

[2]

Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure and Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103

[3]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048

[4]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025

[5]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[6]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[7]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104

[8]

Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235

[9]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283

[10]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179

[11]

Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259

[12]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015

[13]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025

[14]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292

[15]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

[16]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131

[17]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329

[18]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257

[19]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[20]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (245)
  • HTML views (173)
  • Cited by (0)

Other articles
by authors

[Back to Top]