respect to the equator $t=0$[^2]. Therefore the eigenvalues of the problem $$\begin{aligned}
\label{eq:sph_eig}
  \begin{cases}
    -\Delta_{{\mathbb S}^{N}}\psi=\lambda\,\psi, &\text{in }{\mathbb S}^{N}_+,\\[5pt]
\nabla_{{\mathbb
    S}^{N}}\psi\cdot {\mathbf
  e}=0,&\text{on }\partial {\mathbb S}^{N}_+,
  \end{cases}\end{aligned}$$ with $${\mathbb S}^{N}_+=\{(\theta_1,\theta_2,\dots, \theta_{N+1})\in
  {\mathbb S}^{N}:\theta_{N+1}>0\},\quad {\mathbf e}=(0,0,\dots,0,1),$$ are given by the sequence $\{\lambda_\ell:\ell=0,1,2,\dots\}$; for every $\ell$, $\lambda_\ell$ has finite multiplicity $M_\ell$ as an eigenvalue of . For every $\ell {\geqslant}0$, let $\{Y_{\ell ,m}\}_{m=1,2,\dots ,M_{\ell }}$ be a $L^{2}(\mathbb{S}^{N}_+)$-orthonormal basis of the eigenspace of associated to $\lambda _{\ell }$ with $Y_{\ell ,m}$ being spherical harmonics of degree $\ell $.

We note that, if $\Psi$ is an eigenfunction of , then $\Psi\not\equiv0$ on $\partial {\mathbb
  S}^{N}_+={\mathbb S}^{N-1}$; indeed, by unique continuation, $\Psi$ and $\nabla_{{\mathbb
    S}^{N}}\Psi\cdot {\mathbf  e}$ can not both vanish on $\partial {\mathbb
  S}^{N}_+$. In particular $Y_{\ell,m}\not\equiv0$ on $\partial {\mathbb
  S}^{N}_+={\mathbb S}^{N-1}$ for all $\ell\in{{\mathbb N}}$ and $1{\leqslant}m{\leqslant}M_\ell$.

\[t:asym\] Let $(U,V)\in H^1(B_R^+)\times H^1(B_R^+)$ be a weak solution to such that $(U,V)\not=(0,0)$. Then there exists $\ell\in{{\mathbb N}}$ such that $$\lambda^{-\ell}U(\lambda z)\to \widehat U(z),\quad
\lambda^{-\ell}V(\lambda z)\to \widehat V(z),$$ as $\lambda\to0^+$ strongly in $H^1(B_1^+)$, where $$\widehat U(z)=|z|^{\ell}
\sum_{m=1}^{M_\ell}\alpha_{\ell,m}Y_{\ell,m}\Big(\frac
z{|z|}\Big),\quad
 \widehat V(z)=|z|^{\ell}
\sum_{m=1}^{M_\ell}\alpha'_{\ell,m}Y_{\ell,m}\Big(\frac
z{|z|}\Big),$$ $$\begin{aligned}
 \label{eq:45} \alpha_{\ell,m}&=R^{-\ell}\int_{{\mathbb S}^{N}_+}U(R\,\theta)
  Y_{\ell,m}(\theta)\,dS-
\frac{R^{-N-2\ell+1}}{N+2\ell-1}\int_0^R t^{N+\ell}\left(\int_{{\mathbb S}^{N}_+}V(t\,\theta)
    Y_{\ell,m}(\theta)\,dS\right)\,dt\\
\notag&\qquad\qquad
+\int_0^R\frac{t^{-\ell+1}}{2\ell+N-1}\left(\int_{{\mathbb S}^{N}_+}V(t\,\theta)
    Y_{\ell,m}(\theta)\,dS\right)\,dt,\\
 \label{eq:46} \alpha'_{\ell,m}&=R^{-\ell}\int_{{\mathbb S}^{N}_+}V(R\,\theta)
  Y_{\ell,m}(\theta)\,dS\\
\notag&\qquad\qquad-
\frac{R^{-N-2\ell+1}}{N+2\ell-1}\int_0^R t^{N+\ell-1}\left(
\int_{{\mathbb
      S}^{N-1}}h(t\theta')U(t\theta',0) Y_{\ell,m}(\theta',0) 