which requires differentiability of the potential $h$.

Let $N>3$, $R>0$, and $(U,V)\in H^1(B_R^+)\times H^1(B_R^+)$ be a weak solution to the system $$\label{eq:system}
  \begin{cases}
\Delta U=V,&\text{in }B_R^+,\\
\Delta V=0,&\text{in }B_R^+,\\
\frac{\partial U}{\partial \nu}= 0, &\text{in } B_R',\\
\frac{\partial V}{\partial \nu}=hu, &\text{in } B_R',
  \end{cases}$$ where $u=U(\cdot,0)$ (trace of $U$ on $B_R'$) and $h\in
C^1(B_R')$. We also denote $v=V(\cdot,0)$ (trace of $V$ on $B_R'$). By a weak solution to the system we mean a couple $(U,V)\in H^1(B_R^+)\times H^1(B_R^+)$ such that, for every $\varphi\in  H^1(B_R^+)$ having zero trace on $S_R^+$, $$\begin{cases}
\int_{B_R^+}\nabla U(z)\cdot\nabla
\varphi(z)\,dz=-\int_{B_R^+}V(z)\varphi(z)\,dz,\\
\int_{B_R^+}\nabla V(z)\cdot\nabla
\varphi(z)\,dz=\int_{B_R'}h(x)u(x)\mathop{\rm Tr}\varphi (x)\,dx,
\end{cases}$$ where $\mathop{\rm Tr}\varphi$ is the trace of $\varphi$ on $B_R'$.

Our first result is an asymptotic expansion of nontrivial solutions to ; more precisely we prove that blow-up profiles can be described as combinations of spherical harmonics symmetric with respect to the equator $t=0$.

Let $-\Delta _{\mathbb{S}^{N}}$ denote the Laplace Beltrami operator on the $N$-dimensional unit sphere $\mathbb{S}^{N}$. It is well known that the eigenvalues of $-\Delta _{\mathbb{S}^{N}}$ are given by $$\lambda _{\ell }=(N-1+\ell )\ell ,\quad \ell =0,1,2,\dots .$$ For every $\ell\in{{\mathbb N}}$, it is easy to verify that there exists a spherical harmonic on $\mathbb{S}^{N}$ of degree $\ell$ which is symmetric with 