it is easy to see that for $t=T-\epsilon$ with $\epsilon>0$ small, we also have $$\int_0^re^{u_i}\psi_i^t(\sigma)
\sigma^3d\sigma>0,\quad \mbox{for all } r>0,$$ a contradiction to the definition of $T$. Lemma \[1129lem1\] is proved. $\Box $

[**Proof of Theorem \[uniqlin\]:**]{} We first prove the third statement. The following function plays an important role: Let $f=(f_1,..,f_n)=(u_1',..,u_n')$. Then $$\label{85e1}
-\Delta f_i=\sum_ja_{ij}e^{u_j}f_j-\frac 1{r^2}f_i,\quad i=1,..,n.$$

Let $\phi^k=(\phi_1^k,..,\phi_n^k)$ be defined as $$\phi_i^k(r)=\frac 1{2\pi}\int_0^{2\pi}\phi_i(r\cos \theta,r\sin \theta)\cos k\theta d\theta,\quad i\in I.$$ Then $\phi^k$ satisfies $$\label{85e2}
-\Delta \phi_i^k=\sum_ja_{ij}e^{u_j}\phi_j^k-\frac{k^2}{r^2}\phi_i^k,\quad i\in I, \quad k=2,..$$ Clearly $\phi_i^k(r)=o(r)$ near $0$ and $\phi_i^k(r)=O(r^{2-\epsilon_0})$ at $\infty$. We claim that $$\label{11mar29e1}
\phi_i^k(r)\equiv 0,\quad \forall k\ge 2,\quad \mbox{provided that }\,\, \phi_i^k(r)=O(r^{k-1+\tau}), \,\, r>1,\,\,  k\ge 2.$$ Note that the growth condition in (\[11mar29e1\]) is weaker than what is assumed in the assumption in Theorem \[uniqlin\].

The argument below also applies if $\phi$ is projected on $\sin k\theta$. First we show that $\phi_i^k=o(r^{-1})$ as $r\to \infty$. Indeed, using $\phi_j^k(x)\le C|x|^{k-1+\tau}$ we write $\sum_ja_{ij}e^{u_j}\phi^k_j$ as $O(r^{k-1+\tau-2-\delta_0})$ (for some $\delta_0>0$). Let $g(t)=\phi_i^k(e^t)$, then from (\[85e2\]) $g(t)$ satisfies $$g''(t)-k^2g(t)=h(t),\quad t\in \mathbb R$$ where $$\label{11mar29e2}
h(t)=O(e^{(k-1+\tau-\delta_0)t})\quad  t>0.$$ Let $g_1(t)=e^{kt}$ and $g_2(t)=e^{-kt}$ be two fundamental solutions of the homogeneous equation, a general solution $g(t)$ is of the form $$g(t)=c_1g_1(t)+c_2g_2(t)-\frac{g_1(t)}{2k}\int_0^tg_2(s)h(s)ds+\frac {g_2(t)}{2k}\int_0^tg_1(s)h(s)ds$$ where $c_1,c_2$ are constants. Using (\[11mar29e2\]) 