## Abstract

In this paper, we consider the ground-states of the following *M*-coupled system:

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.

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## Acknowledgements

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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He, Q., Yang, J. Quantitative properties of ground-states to an *M*-coupled system with critical exponent in ℝ^{N}.
*Sci. China Math.* **61**, 709–726 (2018). https://doi.org/10.1007/s11425-016-0464-4

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DOI: https://doi.org/10.1007/s11425-016-0464-4