LAUSR

AbstractIn this paper, we consider the ground-states of the following M-coupled system: $$\left\{ {\begin{array}{*{20}{c}} { - \Delta {u_i} = \sum\limits_{j = 1}^M {{k_{ij}}\frac{{2{q_{ij}}}}{{2*}}{{\left| {{u_j}} \right|}^{{p_{ij}}}}{{\left| {{u_i}} \right|}^{{q_{ij}} - {2_{{u_i}}}}},x \in {\mathbb{R}^N},} } \\ {{u_i} \in {D^{1,2}}\left( {{\mathbb{R}^N}} \right),i = 1,2, \ldots ,M,} \end{array}} \right.$${−Δui=∑j=1Mkij2qij2*|uj|pij|ui|qij−2ui,x∈ℝN,ui∈D1,2(ℝN),i=1,2,…,M, where $$p_{ij} + q_{ij} = 2*: = \frac{{2N}} {{N - 2}}(N \geqslant 3)$$pij+qij=2∗:=2NN−2(N⩾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.