which is weaker than Radon measure. In the proof of Theorem \[teo 4\], we make use of the self-similar property in the half space.

The last goal of this paper is to consider the fractional elliptic problem (\[eq 1.1\]) with source nonlinearity, that is, $\epsilon=-1$. In the last decades, semilinear elliptic problems with source nonlinearity and measure data $$\label{eq01}
\arraycolsep=1pt
\begin{array}{lll}
 -\Delta   u=g(u)+k\nu\quad  &{\rm in}\quad\ \ \Omega,\\[3mm]
 \phantom{-\Delta }
 u=\mu\quad &{\rm on}\quad\ \ \partial\Omega,
 \end{array}$$ have attracted numerous interests. There are three basic methods to obtain weak solutions. The first one is to iterate $$u_{n+1}=\mathbb{G}_1[g(u_n)]+k\mathbb{G}_1[\nu],\quad \forall n\in{\mathbb{N}}$$ and look for a function $v$ satisfying $$v\ge \mathbb{G}_1[g(v)]+k\mathbb{G}_1[\nu].$$ When $g$ is a pure power source, the existence results could be found in the references [@BC; @BV; @BY; @KV; @V]. The second method is to apply duality argument to derive weak solution when the mapping $r\mapsto g(r)$ is nondecreasing, convex and continuous, see Baras-Pierre [@BP]. These two methods are very difficult to deal with for a general source nonlinearity. Recently, Chen-Felmer-Véron in [@CFV] introduced a new method to solve problem (\[eq01\]) when $g$ is a general nonlinearity, where the authors employed Schauder’s fixed point theorem to obtain the uniform bound and then to 