\,dS'\right)\,dt\\
\notag&\qquad\qquad
+\int_0^R\frac{t^{-\ell}}{2\ell+N-1}\left(
\int_{{\mathbb
      S}^{N-1}}h(t\theta')U(t\theta',0)Y_{\ell,m}(\theta',0)\,dS'
\right)\,dt,\end{aligned}$$ and $$\sum_{m=1}^{M_\ell}((\alpha_{\ell,m})^2+(\alpha'_{\ell,m})^2)\neq0.$$

A first remarkable consequence of Theorem \[t:asym\] is the validity of a [*[strong unique continuation property]{}*]{} (from the boundary point $0$) for solutions to .

\[t:sun-ext\] Let $(U,V)\in H^1(B_R^+)\times H^1(B_R^+)$ be a weak solution to . If $$\label{eq:case2}
U(z)=o(|z|^n)\quad\text{as }|z|\to 0\text{ for all $n\in
  {{\mathbb N}}$}$$ then $U\equiv V\equiv 0$ in $B_R^+$.

We observe that in the case of a single equation a blow result as the one stated in Theorem \[t:asym\] directly yields the strong unique continuation: indeed, if the solution has a precise vanishing order it cannot vanish of any order. On the other hand, in the case of a system of type , the blow-up Theorem \[t:asym\] ensures that the couple of the limit profiles $(\widehat U, \widehat V)$ is not trivial, i.e. at least one of the two components $U,V$ has a precise vanishing order; hence some further analysis is needed to deduce strong unique continuation from Theorem \[t:asym\].

System is related to fourth order elliptic equations arising in Caffarelli-Silvestre type extensions for higher order fractional laplacians in the spirit of [@Y]. Let us define $\mathcal D$ as the completion of $$\label{eq:space}
\mathcal T:=\left\{U\in C^\infty_c(\overline{{{\mathbb R}}^{N+1}_+}):U_t\equiv 0 \ 