A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a 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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2<p<\frac{N}{s}$, $\gamma \in (0,1)$, $\lambda ,\mu >0$, $\alpha \ge 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$,$H(a)=1$if$a>0$and$p_s^=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent. Under suitable assumptions on the function $g$, we prove the existence of solution to the problem. Furthermore, we show that as $\alpha \to 0^{+}$, the sequence of solutions of $\text{eqref}mai{n}_{p}rob$ for each such $\alpha$ converges to a solution of the problem for which $\alpha =0$.