Synchronized and ground-state solutions to a coupled Schrödinger system

Mohammad Ali Husaini; Chuangye Liu

doi:10.3934/cpaa.2021192

Article Contents

2022, Volume 21, Issue 2: 639-667. Doi: 10.3934/cpaa.2021192

This issue Previous Article Blowing-up solutions for a supercritical elliptic equation Next Article Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

Synchronized and ground-state solutions to a coupled Schrödinger system

Mohammad Ali Husaini^1,2, and
Chuangye Liu^3, ,

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
^* Corresponding author

Received: May 31, 2021

Revised: September 30, 2021

Early access: November 2021

Published: February 2022

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35D30, 35J50; Secondary: 35B33.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[4]	T. Bartsch, N. 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. 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.
[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. Dancer, J. 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. Esry, C. Greene, J. 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. Peng, Q. 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. Szulkin, T. 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.