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abstract: |
    The purpose of this paper is to study the weak solutions of the fractional elliptic problem $$\label{000}
    \arraycolsep=1pt
    \begin{array}{lll}
     (-\Delta)^\alpha   u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad  &{\rm in}\quad\ \ \bar\Omega,\\[3mm]
     \phantom{(-\Delta)^\alpha   +\epsilon g(u)}
     u=0\quad &{\rm in}\quad\ \ \bar\Omega^c,
     \end{array}$$ where $k>0$, $\epsilon=1$ or $-1$, $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $\Omega$ is a bounded $C^2$ open set in ${\mathbb{R}}^N$ with $N\ge 2$, $\nu$ is a bounded Radon measure supported in $\partial\Omega$ and $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ is defined in the distribution sense, i.e. $$\langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\zeta\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\zeta(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \qquad \forall\zeta\in C^\alpha({\mathbb{R}}^N),$$ here $\vec{n}_x$ denotes the unit inward normal vector at $x\in\partial\Omega$.

    In this paper, we prove that (\[000\]) with $\epsilon=1$ admits a unique weak solution when $g$ is a continuous nondecreasing function satisfying $$\int_1^\infty (g(s)-g(-s))s^{-1-\frac{N+\alpha}{N-\alpha}}ds<+\infty.$$ Our interest then is to analyse the properties of weak solution when $\nu=\delta_{x_0}$ with $x_0\in\partial\Omega$, including the asymptotic behavior near $x_0$ and the limit of weak solutions as $k\to+\infty$. Furthermore, we show the optimality of the critical value $\frac{N+\alpha}{N-\alpha}$ in a certain sense, by proving the non-existence of weak solutions when $g(s)=s^{\frac{N+\alpha}{N-\alpha}}$.

    The final part of this article is devoted to the study of existence for positive weak solutions to (\[000\]) when 