A weighted fractional problem involving a singular nonlinearity and a $L^1$ data

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is bounded and contains the origin, $\alpha \in [0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N}<p<\mathrm{\infty }$, $sp<N$, $g\in L^1(\mathrm{\Omega })$, $f\in L^q(\mathrm{\Omega })$ 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 $$(-\mathrm{\Delta }{)}_{p,\alpha }^{s}u(x)=\text{P. V.}{\int }_{{\mathbb{R}}^N}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{p-2}(u(x)-u(y))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{N+ps}}\frac{dy}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }y{\mathrm{\mid }}^{\alpha }},\mathrm{\forall }x\in {\mathbb{R}}^N\mathrm{.}$$