N(r)\right)\quad\text{for a.e. }r\in(0,\tilde r)\cap\Sigma.$$ Since the above inequality is obviously true in $(0,\tilde
r)\setminus\Sigma$ (provided $\tilde r$ is sufficiently small), we deduce that $$\label{eq:25}
\mathcal N'(r){\geqslant}-\tilde C\left(
1+\mathcal N(r)\right)\quad\text{for a.e. }r\in(0,\tilde r).$$ Integrating the above inequality in $(r,\tilde r)$ we obtain that $$\mathcal N(r)+1{\leqslant}e^{\tilde C \tilde r} (\mathcal N(\tilde r)+1) \quad\text{for all }r\in(0,\tilde r).$$ The above estimate together with Corollary \[c:limpos\] yield . Furthermore implies that $$\left(e^{\tilde C r}(1+\mathcal N(r))\right)'{\geqslant}0 \quad\text{a.e. in
}(0,\tilde r),$$ hence the function $r\mapsto e^{\tilde C r}(1+\mathcal N(r))$ admits a limit as $r\to0^+$. Therefore also the limit $\gamma:=\lim_{r\to 0^+}\mathcal N(r)$ exists; furthermore $\gamma$ is finite in view of and $\gamma\geq0$ in view of .

A first consequence of the previous monotonicity argument is the following estimate of the function $H$.

\[l:uppb\] Letting $\gamma$ be as in Lemma \[l:limitN\], we have that $$\label{eq:24}
H(r)=O(r^{2\gamma})\quad \text{as $r\to0^+$}.$$ Furthermore, for any $\sigma>0$ there exist $K(\sigma)>0$ depending on $\sigma$ such that $$\label{2ndest}
H(r){\geqslant}K(\sigma)\,
  r^{2\gamma+\sigma} \quad \text{for all } r\in (0, r_0).$$

See the proof of [@FF Lemma 3.16].

Blow-up analysis {#sec:blow-up-analysis}
================

\[l:blowup\] Let $(U,V)\in H^1(B_R^+)\times H^1(B_R^+)$ be a weak solution to such that $(U,V)\not=(0,0)$, let $\mathcal N$ be defined in , and let $\gamma:=\lim_{r\rightarrow 0^+} {\mathcal
    N}(r)$ be as 