A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

In this paper we consider the following critical nonlocal problem $$\{\begin{array}{ll}M\left({\displaystyle {\iint }_{{\mathbb{R}}^{2N}}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{N+2s}}dxdy}\right)(-\mathrm{\Delta }{)}^{s}u={\displaystyle \frac{\lambda }{u^{\gamma }}+u^{2_s^{\ast }-1}}& \text{quadin}\mathrm{\Omega },\\ u>0& \text{quadin}\mathrm{\Omega },\\ u=0& \text{quadin}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is an open bounded subset of ${\mathbb{R}}^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2_s^{\ast }=2N\mathrm{/}(N-2s)$ is the fractional critical Sobolev exponent, $\lambda >0$ is a real parameter, exponent $\gamma \in (0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\mathrm{\Delta }{)}^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.