constant curvature). However for the $SU(3)$ Toda system, the blowup point $p$ is a critical point of both $\log h_1$ and $\log h_2$, i.e. $p$ satisfies (see [@lwz]) $$\nabla h_1(p)=\nabla h_2(p)=0.$$ Second, for the $SU(3)$ Toda system, the convergence rate of $\rho_i^k-\rho_i$ is estimated to be $$\rho_i^k-\rho_i=(e^{\tilde c_i}b_i+o(1))\epsilon_k^2\log \epsilon_k^{-1},$$ where $b_i$ is the term in (\[12apr23e10\]). Nevertheless our result in (\[12apr23e10\]) is again a combination of the $b_i$s. The comparison of the results reflects some major difference between the Toda system and our Liouville system:

1.  The dimension of kernel space of the linearized operator at an entire solution is $8$ for $SU(3)$ Toda system, and is $3$ for our Liouville system.

2.  The set $\Gamma:=\{(\rho_1,...,\rho_n);\quad
    \rho_i=\int_{\mathbb R^2}e^{u_i},\quad (u_1,..,u_n) \mbox{ is an entire solution}\,\, \}$ is only a point for $SU(3)$ Toda system, while for the Liouville system it is a $(n-1)$ dimensional manifold.

As far as the blowup analysis is concerned, our Liouville system has disadvantages in both respects, as the kernel space is too small and $\Gamma$ is too large. For a sequence of bubbling solutions, it is extremely difficult to pin-point suitable approximating solutions from $\Gamma$, because at the beginning, the local energy of bubbling solutions could 