  We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$
in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in
(\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha
\in(0,1)$. We establish a universal a priori estimate for positive solutions of
(E), as well as for their gradients. Then for $C^2$ bounded domain $\Omega$, we
prove the existence of positive solutions of (E) with prescribed boundary value
$\rho \nu$, where $\rho>0$ and $\nu$ is a positive Radon measure on $\partial
\Omega$ with total mass $1$, and discuss regularity property of the solutions.
When $f(u)=u^p$, we demonstrate that there exists a critical exponent
$p_s:=\frac{N+s}{N-s}$ in the following sense. If $p\geq p_s$, the problem does
not admit any positive solution with $\nu$ being a Dirac mass. If $p\in(1,p_s)$
there exits a threshold value $\rho^*>0$ such that for $\rho\in (0, \rho^*]$,
the problem admits a positive solution and for $\rho>\rho^*$, no positive
solution exists. We also show that, for $\rho>0$ small enough, the problem
admits at least two positive solutions.
