dS'$$ for a.e. $r\in (0,R)$ and in the distributional sense. Therefore $D\in W^{1,1}_{{\rm loc}}(0,R)$ and, replacing , (\[I(r)\]), and (\[I’(r)\]) into $D'(r)=r^{-N}[-(N-1)I(r)+rI'(r)]$, we obtain (\[D’F\]).

In view of Lemma \[l:Hpos\], the function $$\label{eq:7}
\mathcal N:(0,r_0)\to{{\mathbb R}},\quad
\mathcal N(r)=\frac{D(r)}{H(r)}$$ is well defined. As a consequence of estimate we obtain the following corollary.

\[c:limpos\] Let $(U,V)\in H^1(B_R^+)\times H^1(B_R^+)$ be as in Lemma \[l:Hpos\] and let $D,H,\mathcal N$ be defined in , , and respectively. For every $\e>0$ there exists $r_\e>0$ such that $$\mathcal N(r)+\e{\geqslant}0\quad\text{for all }0<r<r_\e,$$ i.e. $$\label{eq:liminf}
\liminf_{r\to0^+}\mathcal N(r)\geq0.$$

\[mono\] The function ${\mathcal N}$ defined in belongs to $W^{1,1}_{{\rm
       loc}}(0, r_0)$ and $$\begin{aligned}
\label{formulona}
{\mathcal N}'(r)=\nu_1(r)+\nu_2(r)\end{aligned}$$ in a distributional sense and for a.e. $r\in (0,r_0)$, where $$\begin{aligned}
\nu_1(r)=\frac{2r\Big[
    \left(\int_{S_r^+}
  \left(\left|\frac{\partial U}{\partial \nu}\right|^2+
\left|\frac{\partial V}{\partial \nu}\right|^2\right) dS\right) \cdot
    \left(\int_{S_r^+}
  (U^2+V^2)\,dS\right)-\left(
\int_{S_r^+}
  \left(U\frac{\partial U}{\partial \nu}+V\frac{\partial V}{\partial \nu}\right)\, dS\right)^{\!2} \Big]}
{\left(
\int_{S_r^+}
  (U^2+V^2)\,dS\right)^2}\end{aligned}$$ and $$\begin{aligned}
\label{eq:16}
  \nu_2(r)= &\, \frac{r\int_{S_r^+}  UV\,dS-2\int_{B_r^+} V\,\nabla
              U\cdot z\,dz-(N-1)\int_{B_r^+} UV\,dz}{\int_{S_r^+}
              (U^2+V^2)\,dS}\\
\notag&\quad+ \frac{(N-1)\int_{ B_r'}huv\, dx-r\int_{\partial B_r'}huv\,dS'+2\int_{B_r'}hu\,x\cdot
  \nabla_x v\,dx} {\int_{S_r^+}
    (U^2+V^2)\,dS}.\end{aligned}$$

It follows directly from the definition of $\mathcal N$ and Lemmas \[l:hprime\] and \[l:dprime\].

We now estimate the term $\nu_2$ in . This is the most delicate point in the development of the monotonicity argument for system , due to 