A singular elliptic problem involving fractional <i>p</i>-Laplacian and a discontinuous critical nonlinearity

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with singualrity and discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob} (-\Delta)_p^su&=\mu g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega u&>0,~\text{in}~\Omega, u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega, \end{split} \end{align} where $\Omega\subset\mathbb{R}^N$ bounded domain Lipschitz boundary, $s\in (0,1)$, $20$, $\alpha\geq 0$ real, $H$ Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ $a>0$ $p_s^*=\frac{Np}{N-sp}$ fractional Sobolev exponent. Under suitable assumptions on function $g$, solution problem. Furthermore, show that $\alpha\rightarrow0^+$, sequence $\eqref{main_prob}$ for each such $\alpha$ converges $\alpha=0$.