1.1\]) when $\nu$ is Dirac mass. The nonexistence of weak solution in the critical case is addressed in Section 5. Finally we give the proof of Theorem \[teo 3\] in Section 6.

General measure concentrated on boundary
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In this section, we first build the one-to-one connection between the Radon measure space $\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ and the bounded Radon measure space $\mathfrak{M}^b(\partial\Omega)$.

On the one hand, for any $\mu\in \mathfrak{M}^b(\partial\Omega)$, we denote by $\tilde \mu$ the measure generated by $\mu$ extending inside $\Omega$ by zero, that is, $$\tilde \mu(E):=\mu(E\cap\partial\Omega),\qquad \forall E\subset \bar\Omega\ {\rm Borel\ set},$$ then $\tilde\mu\in \mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$.

On the other hand, let $\tilde \mu\in \mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, we see that $$\tilde \mu(E)=\tilde \mu(E\cap\partial\Omega),\qquad \forall E\subset \bar\Omega\ {\rm Borel\ set}.$$ Denote by $\mu$ a Radon measure such that $\mu(F):=\tilde \mu(F),\ F\subset \partial\Omega\ {\rm Borel\ set}$. Then $\tilde \mu(E)=\mu(E\cap\partial\Omega)$ for any Borel set $E\subset\bar\Omega$ and $${\|\tilde \mu\|}_{\mathfrak{M}^b(\bar\Omega)}={\|\mu\|}_{\mathfrak{M}^b(\partial\Omega)}.$$

Now we make an approximation of $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ by a sequence Radon measure concentrated on one type of manifolds inside of $\Omega$. Indeed, we observe that there exists $\sigma_0>0$ small such that $$\Omega_t:=\{x\in \Omega,\ \rho_{\partial\Omega}(x)>t\}$$ is a $C^2$ domain in ${\mathbb{R}}^N$ for $t\in[0,\sigma_0]$ and for any $x\in\partial\Omega_t$, there exists a unique $x_\partial\in\partial\Omega$ such that $|x-x_\partial|=\rho_{\partial\Omega}(x)$. Conversely, for any 