of $e^{\tilde U_i}$ for $\rho=Q$ is $O(|x|^{-4})$. The difference on the decay rate for $Q$ and $\rho\neq Q$ will have great effects on bubbling analysis later.

Let $u^k=(u_1^k,...,u_n^k)$ be a sequence of blow up solutions to (\[mainsys\]) with $\rho=\rho^k$ such that $\rho^k\to \rho\in \Gamma_1$. The point $Q$ defined above is of particular importance, the readers will see that in our main theorems below, the asymptotic behavior of blowup solutions, the nature of $\Lambda_I(\rho^k)$ and the location of blowup point are all significantly different depending on $\rho=Q$ or not.

Let $p$ be the blowup point of $u^k$ and $$\label{12apr23e4}
 M_k=\max_{B(p,\delta)} \bigg ( u_1^k(x)-\log \int_M h_1e^{u_1^k}dV_g \bigg ),$$ $$\label{12apr23e5}
 \epsilon_k=e^{-\frac 12 M_k}.$$ Since $\rho^k\to \Gamma_1$, there is only one blowup point $p$. It is easy to see that $u^k$ fully blows up at $p$ (see Lemma \[ful\]): $$\label{12apr23e6}
 \max_{B(p, \delta_0)}(u_i^k(x)-\log \int_M h_i e^{u_i^k}dV_g)=M_k+O(1), \quad i\in I.$$

Our first result is on the location of the blowup point $p$. Let $p_k\to p$ be where the maximum of $\{u_1^k,..,u_n^k\}$ is attained, then we have

\[locations\] Let $\rho^k\to \rho\in \Gamma_1$ and all $\rho_i^k-\rho_i$ have the same sign.

1.  If $\rho\neq Q$, then $$\label{11july13e7}
    \sum_{i\in I} \bigg (\nabla (\log h_i)(p_k)+2\pi m\nabla_1 \gamma(p_k,p_k)\bigg )
    \rho_{i} =O(\epsilon_k^{m-2}).$$

2.  