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 & 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.   Google Scholar

[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.  Google Scholar

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

show all references

References:
[1]

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

[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.  Google Scholar

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

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