\\ \\
w_i^k(0)=0,\quad i\in I,\quad \nabla w_1^k(0)=O(\epsilon_k),
\end{array}
\right.$$ where $\xi_i^k$ is defined by $$\label{1211e1}
e^{\xi_i^k}=\int_0^1e^{tv_i^k+(1-t)V_i^k}dt.$$

Since both $v^k$ and $V^k$ converge to $v$, $w_i^k=o(1)$ over any compact subset of $\mathbb R^2$. The first estimate of $w_i^k$ is the following

\[1020lem1\] $$\label{820e1}
w_i^k(y)=o(1)\log (1+|y|)+O(1),\quad \mbox{ for } y\in \Omega_k.$$

[**Proof**]{}: By (\[11mar9e3\]) $$|v_i^k(y)-\bar v_i^k(|y|)|\le C, \quad \forall y\in \Omega_k$$ where $\bar v_i^k(r)$ is the average of $v_i^k$ on $\partial B_r$: $$\bar v_i^k(r)=\frac{1}{2\pi r}\int_{\partial B_r}v_i^k.$$ Thus we have $e^{v_i^k(y)}=O(r^{-2-\delta_0})$ and $e^{V_i^k(y)}=O(r^{-2-\delta_0})$ where $r=|y|$ and $\delta_0>0$. Then $$r(\bar w_i^k)'(r)=\frac 1{2\pi }\bigg (\int_{B_r}\sum_ja_{ij}H_j^k(\epsilon_k\cdot)e^{v_j^k}
-\int_{B_r}\sum_ja_{ij}h_j^k(0)e^{V_j^k}\bigg )$$

It is easy to use the decay rate of $e^{v_i^k}$, $e^{V_i^k}$ and the closeness between $v_i^k$ and $V_i^k$ to obtain $$r(\bar w_i^k)'(r)=o(1),\quad r\ge 1.$$ Hence $\bar w_i^k(r)=o(1)\log r$ and (\[820e1\]) follows from this easily. Lemma \[1020lem1\] is established. $\Box$

The following estimate is immediately implied by Lemma \[1020lem1\]: $$e^{\xi_i^k(y)}\le C(1+|y|)^{-m+o(1)}\quad \mbox{ for } y\in \Omega_k=B(0,\epsilon_k^{-1}).$$

Before we derive further estimate for $w_i^k$ we establish a useful estimate for the Green’s function on $\Omega_k$ with respect to the Dirichlet boundary condition:

\[greenk\] Let $G(y,\eta)$ be the Green’s function with respect to Dirichlet boundary condition on $\Omega_k$. For $y\in \Omega_k$, let $$\begin{aligned}
\Sigma_1&=&\{\eta \in \Omega_k;\quad |\eta |<|y|/2 \quad \}\\
\Sigma_2&=&\{\eta 