Abstract
This paper deals with the existence and uniqueness of a positive continuous
solution to the following singular semilinear fractional Dirichlet problem:
\begin{equation*} \left( -\Delta _{\mid D}\right) ^{\frac{\alpha
}{2}}u=a_{1}(x)u^{\sigma _{1}}+a_{2}(x)u^{\sigma _{2}}\text{ in }D,\text{
}\underset{x\rightarrow z\in \partial D}{\lim }\left( \delta (x)\right) ^{2-\alpha
}u(x)=0, \end{equation*} where $0 < \alpha < 2,$ $\sigma _{1},\sigma _{2}\in
(-1,1),$ $D$ is a bounded $C^{1,1}$-domain in $\mathbb{R}^{n},$ $n\geq 2,$ and
$\delta (x)$ denotes the Euclidian distance from $x$ to the boundary of $D$. The
nonnegative weight functions $a_{1}$ and $a_{2}$ are in $C_{loc}^{\gamma }(D),$ $ 0
< \gamma < 1,$ satisfying some appropriate assumptions related to Karamata
regular variation theory. We also give the global behavior of such a solution.