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

  • Qihan He
  • Jing Yang


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 


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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.


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© 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

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