Combined effects in nonlinear singular fractional Dirichlet problem in bounded domains

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.