Tr}\varphi)\subset\Omega$.

\[t:asym-ho\]

1.  Let $U\in\mathcal D$, $U\not\equiv0$, be a nontrivial weak solution to for some $h\in C^1(\Omega)$, with $\Omega$ being an open bounded set in ${{\mathbb R}}^N$ such that $0\in\Omega$. Then there exists $\ell\in{{\mathbb N}}$ such that $$\lambda^{-\ell}U(\lambda z)\to
    |z|^{\ell}
    \sum_{m=1}^{M_\ell}\alpha_{\ell,m}Y_{\ell,m}\Big(\frac
    z{|z|}\Big), \quad
    \lambda^{-\ell}\Delta U(\lambda z)\to
    |z|^{\ell}
    \sum_{m=1}^{M_\ell}\alpha'_{\ell,m}Y_{\ell,m}\Big(\frac
    z{|z|}\Big),$$ strongly in $H^1(B_1^+)$, where $\sum_{m=1}^{M_\ell}((\alpha_{\ell,m})^2+(\alpha'_{\ell,m})^2)\neq0$ and $\alpha_{\ell,m},\alpha'_{\ell,m}$ are given in – with $V=\Delta U$.

2.  If $U\in \mathcal D$ is a weak solution to such that $$U(z)=o(|z|^n)\quad\text{as }|z|\to 0\text{ for all $n\in
      {{\mathbb N}}$},$$ then $U\equiv 0$ in $B_R^+$.

As mentioned above, a motivation for the study of higher order equations of type and consequently of systems comes from the interest in higher order fractional laplacians and their characterization as a Dirichlet-to-Neumann map in the spirit of [@CS].

Let us consider the fractional laplacian $(-\Delta)^{3/2}$ defined as $$\widehat{(-\Delta)^{\frac32} u}(\xi)=|\xi|^{3}\widehat u(\xi) \, .$$ We also consider the space $\mathcal D^{1/2,2}({{\mathbb R}}^N)$ given by the completion of $C^\infty_c({{\mathbb R}}^N)$ with respect to the scalar product $$(u,v)_{\mathcal D^{1/2,2}({{\mathbb R}}^N)}:=\int_{{{\mathbb R}}^N} |\xi|\, \widehat
u(\xi)\overline{\widehat v(\xi)} \, d\xi.$$

\[t:frac\] For $N>3$, let $\Omega\subseteq{{\mathbb R}}^N$ be open, $a\in C^1(\Omega)$, and $u\in \mathcal D^{3/2,2}({{\mathbb R}}^N)$ be a weak solution to the problem $$\label{eq:48}
(-\Delta)^{3/2}u=a\,u,\quad\text{in }\Omega,$$ i.e. $$(u,\varphi)_{\mathcal
  D^{3/2,2}({{\mathbb 