The number of positive solutions for n$n$‐coupled elliptic systems

We study the number of positive solutions to the n$n$‐coupled elliptic system −Δui=μiui2∗−1+∑j=1,j≠inβijuipij−1ujqij,ui∈D1,2(RN),i=1,2,…,n,$$\begin{align*} -\Delta u_i&=\mu _iu_i^{2^*-1}+\sum _{j=1,\,j\ne i}^n\beta _{ij}u_i^{p_{ij}-1}u_j^{q_{ij}},\ u_i\in \mathcal {D}^{1,2}(\mathbb {R}^N),\\ &\quad i=1,2,\ldots ,n, \end{align*}$$where N⩾3$N\geqslant 3$, n⩾2$n\geqslant 2$, μi>0$\mu _i>0$, βij>0$\beta _{ij}>0$, pij