In this article, we show the existence of a unique entropy solution to the following problem: \begin{ equation} \begin{ split} (-\Delta)_{ p,\alpha}^su&= f(x)h(u)+g(x) ~\text{ in}~\Omega,\\ u&>0~\text{ in}~\Omega,\\ u&= 0~\text{ in}~\mathbb{ R}^N\setminus\Omega,\nonumber \end{ split} \end{ equation} where the domain $\Omega\subset \mathbb{ R}^N$ is bounded and contains the origin, $ \alpha\in[0,\frac{ N-ps}{ 2})$, $s\in (0,1)$, $2-\frac{ s}{ N}<p<\infty$, $sp<N$, $g\in L^1(\Omega)$, $f\in L^q(\Omega)$ for $q>1$ and $h$ is a general singular function with singularity at 0. Further, the fractional $p$-Laplacian with weight $\alpha$ is given by $$(-\Delta)_{ p,\alpha}^su(x)=\text{ P. V.}\int_{ \mathbb{ R}^N}\frac{ |u(x)-u(y)|^{ p-2}(u(x)-u(y))}{ |x-y|^{ N+ps}}\frac{ dy}{ |x|^\alpha|y|^{ \alpha}},~\forall x\in \mathbb{ R}^N.$$
Categories: math.AP