in the Pohozaev identity for $\sigma$: $$4\sum_{i} \sigma_{i}=\sum_{i,j}a_{ij}\sigma_{i}\sigma_{j}.$$ Lemma \[globalcpt\] is established. $\Box$.

First order estimates {#blowup}
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Let $(h_1^k,...,h_n^k)$ be a family of positive, $C^3$ functions on $B_1$ with a uniform bound on their positivity and $C^3$ norm: $$\label{hikn}
\frac 1C\le h_i^k(x)\le C,\quad \|h_i^k\|_{C^3(B_1)}\le C, \quad x\in B_1, \quad i=1,..,n.$$ In the next three sections we consider a sequence of locally defined, fully blown-up solutions $u^k=(u_1^k,...,u_n^k)$ and we shall derive their precise asymptotic behavior near their singularity and the precise location of their singularity. Here we abuse the notation $u^k=(u_1^k,...,u_n^k)$ and it is independent of the one used in the introduction.

Specifically we assume that $u^k$ satisfies the following equation in $B_1$, the unit ball: $$\label{uik}
-\Delta u_i^k=\sum_{j=1}^na_{ij}h_j^ke^{u_j^k},\quad i=1,..,n,\quad x\in B_1,$$ with $0$ being the only blowup point in $B_1$: $$\max_Ku_i^k\le C(K), \quad \forall K\subset\subset \bar B_1\setminus \{0\}, \,\, \mbox{ and } \max_{B_1}u_i^k\to \infty,$$ with bounded oscillation on $\partial B_1$: $$\label{88e1}
|u_i^k(x)-u_i^k(y)|\le C,\forall x,y\in \partial B_1,\quad C\mbox{ independent of }k$$ and uniformly bounded energy: $$\label{89e1}
\int_{B_1}e^{u_i^k}\le C,\quad C \mbox{ is independent of } k.$$ Finally we assume that $u^k$ is a fully blown-up sequence, which means when re-scaled according its maximum, $\{u_k\}$ converges to a system of $n$ equations: 