the limit of these weak solutions (if it exists) is called a weak solution of the following fractional elliptic problem with measure concentrated on boundary $$\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha   u+g(u)=\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad  &{\rm in}\quad\ \ \bar\Omega,\\[3mm]
 \phantom{------\ }
 u=0\quad &{\rm in}\quad\ \ \bar\Omega^c.
 \end{array}$$ This will be our main focus in this paper.

Statement of our problem and main results
-----------------------------------------

Let $\alpha\in(0,1)$, $g:{\mathbb{R}}\to{\mathbb{R}}$ be a continuous function, $\Omega$ be a bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 2$ and denote by $\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ the bounded Radon measure in $\bar\Omega$ with the support in $\partial\Omega$. Our purpose in this article is to investigate the existence, non-existence and uniqueness of weak solutions to semilinear fractional elliptic problem $$\label{eq 1.1}
\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha   u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad  &{\rm in}\quad\ \ \bar\Omega,\\[3mm]
 \phantom{------\ \  }
 u=0\quad &{\rm in}\quad\ \ \bar\Omega^c,
 \end{array}$$ where $\epsilon=1$ or $-1$, $k>0$, $(-\Delta)^\alpha$ is the fractional Laplacian and denote $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ with $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ by $$\langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\xi\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \qquad \xi\in \mathbb{X}_\alpha,$$ with $\vec{n}_x$ being the unit inward normal vector at $x$. We call $g$ the absorption nonlinearity if $\epsilon=1$, otherwise it is called as source nonlinearity.

Before starting our main theorems we make precise the notion of weak solution used in this article.

\[weak 