Science China Mathematics

, Volume 61, Issue 4, pp 709–726 | Cite as

Quantitative properties of ground-states to an M-coupled system with critical exponent in ℝ N



In this paper, we consider the ground-states of the following M-coupled system:
$$\left\{ {\begin{array}{*{20}{c}} { - \Delta {u_i} = \sum\limits_{j = 1}^M {{k_{ij}}\frac{{2{q_{ij}}}}{{2*}}{{\left| {{u_j}} \right|}^{{p_{ij}}}}{{\left| {{u_i}} \right|}^{{q_{ij}} - {2_{{u_i}}}}},x \in {\mathbb{R}^N},} } \\ {{u_i} \in {D^{1,2}}\left( {{\mathbb{R}^N}} \right),i = 1,2, \ldots ,M,} \end{array}} \right.$$
where \(p_{ij} + q_{ij} = 2*: = \frac{{2N}} {{N - 2}}(N \geqslant 3)\). We prove the existence of ground-states to the M-coupled system. At the same time, we not only give out the characterization of the ground-states, but also study the number of the ground-states, containing the positive ground-states and the semi-trivial ground-states, which may be the first result studying the number of not only positive ground-states but also semi-trivial ground-states.


ground-states quantitative properties critical exponent 


35J20 35J60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by National Natural Science Foundation of China (Grant No. 11601194) and PhD Start-Up Funds of Jiangsu University of Science and Technology (Grant Nos. 1052931601 and 1052921513). The authors sincerely thank Professor S. Peng for helpful discussions and suggestions.


  1. 1.
    Akhmediev N, Ankiewicz A. Partially coherent solitons on a finite background. Phys Rev Lett, 1999, 82: 26–61CrossRefGoogle Scholar
  2. 2.
    Ambrosetti A, Colorado E. Bound and ground-states of coupled nonlinear Schrödinger equations. C R Math Acad Sci Paris, 2006, 342: 453–458MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen Z, Zou W. Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch Ration Mech Anal, 2012, 205: 515–551MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen Z, Zou W. An optimal constant for the existence of least energy solutions of a coupled Schrödinger system. Calc Var Partial Differential Equations, 2013, 48: 695–711MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Correia S. Characterization of ground-states for a system of M coupled semilinear Schrödinger equations and applications. J Differential Equations, 2016, 260: 3302–3326MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Correia S. Ground-states for systems of M coupled semilinear Schrödinger equations with attraction-repulsion effects: Characterization and perturbation results. Nonlinear Anal, 2016, 140: 112–129MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers II: Normal dispersion. Appl Phys Lett, 1973, 23: 171–172CrossRefGoogle Scholar
  9. 9.
    He Q, Peng S J, Peng Y F. Existence, non-degeneracy and proportion of positive solution for a fractional elliptic equations in ℝN. Adv Difference Equ, 2017, in pressGoogle Scholar
  10. 10.
    Lin T C, Wei J. Ground state of N coupled nonlinear Schrödinger equations in ℝn; n ≤ 3. Comm Math Phys, 2005, 255: 629–653MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lions P L. The concentration-compactness principle in the calculus of variations: The locally compact case. I. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 109–145MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lions P L. The concentration-compactness principle in the calculus of variations: The locally compact case. II. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 223–283MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ma L, Zhao L. Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application. J Differential Equations, 2008, 245: 2551–2565MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Struwe M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math Z, 1984, 187: 511–517MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Swanson C A. The best Sobolev constant. Appl Anal, 1992, 47: 227–239MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Talenti G. Best constant in Sobolev inequality. Ann Mat Pura Appl (4), 1976, 110: 353–372MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Wei J, Yao W. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Commun Pure Appl Anal, 2012, 11: 1003–1011MathSciNetMATHGoogle Scholar
  18. 18.
    Zakharov V E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys, 1968, 9: 190–194CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceGuangxi UniversityNanningChina
  2. 2.School of ScienceJiangsu University of Science and TechnologyZhenjiangChina

Personalised recommendations