defined on Riemann surface.

Let $V^k=(V_1^k,..,V_n^k)$ be the radial solutions of $$\label{88e2}
\left\{\begin{array}{ll}-\Delta V_i^k=\sum_{j=1}^na_{ij}h_j^k(0)e^{V_j^k}\quad \mathbb R^2,\quad i\in I\\  \\
V_i^k(0)=v_i^k(0),\quad i\in I
\end{array}
\right.$$ where $v_i^k$ is defined in (\[817e1\]). It is easy to see that any radial solution $V$ of (\[88e2\]) exists for all $r>0$ and $e^{V_i}\in L^2(\mathbb R^2)$. The main result of this section is to prove that $V_i^k(y)+\phi_i^k(\epsilon_ky)$ is the first order approximation to $v_i^k(y)$.

\[thm2\] Let $A$, $u^k$, $h^k=(h_1^k,..,h_n^k)$ ,$\phi_i^k$ and $v^k$ be described as above. Then for any $\delta>0$, there exist $k_0(\delta)>1$ and $C$ independent of $k$ and $\delta$ such that for all $k\ge k_0$, $$\begin{aligned}
\label{11mar9e2}
&&|D^{\alpha}(v_i^k(y)-V_i^k(y)-\phi_i^k(\epsilon_ky))|\\
&\le &\left\{\begin{array}{ll}C\epsilon_k(1+|y|)^{3-m+\delta-|\alpha|},\quad m\le 3,\\
C\epsilon_k(1+|y|)^{\delta-l},\quad m>3,
\end{array}
\right.
\quad |y|<\epsilon_k^{-1}/2,\quad |\alpha|=0,1,2. \nonumber\end{aligned}$$

\[11def1\] $$\sigma_i^k=\frac 1{2\pi}\int_{\mathbb R^2} h_i^k(0)e^{V_i^k},\,\, m_i^k=\sum_{j=1}^n a_{ij}\sigma_j^k,\,\, m^k=\min \{m_1^k,..,m_n^k\}.$$ From Theorem \[thm2\] it is easy to see that $\lim_{k\to\infty}m_i^k=m_i$. Thus $m_i^k\ge 2+\delta_0$ for some $\delta_0>0$ independent of $k$.

To prove Theorem \[thm2\], we have $$\label{1020e1}
-\Delta (v_i^k(y)-\phi_i^k(\epsilon_ky))=\sum_ja_{ij}H_j^k(\epsilon_ky)e^{v_j^k(y)-\phi_j^k(\epsilon_ky)},\,\, \mbox{ in } \Omega_k\,\, \big( \mbox{ see (\ref{vikdef})} \big )$$ where $$\label{Hik}
H_i^k(\cdot)=h_i^k(\cdot)e^{\phi_i^k(\cdot)}.$$ Since $\phi_i^k(0)=0$ we have $H_i^k(0)=h_i^k(0)$. Also, the definition of $\phi_i^k$ implies that $v_i^k-\phi_i^k(\epsilon_k\cdot)$ is a constant on $\partial \Omega_k$.

To estimate the error term $w_i^k=v_i^k-\phi_i^k(\epsilon_k \cdot )-V_i^k$. We find $w_i^k$ satisfies $$\label{wik01}
\left\{\begin{array}{ll}
\Delta w_i^k(y)+\sum_ja_{ij}H_j^k(\epsilon_ky)e^{\xi_j^k}w_j^k=-\sum_ja_{ij}(H_j^k(\epsilon_ky)-H_j^k(0))e^{V_j^k}, 