Positive solutions for nonlinear Choquard equation with singular nonlinearity

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^_{\mu}}}{|x-y|^{\mu}}$\mathrm{d}$y \right)|u|^{2^{\mu}-2}u, \quad u>0 ; $\text{in}$; \Om,\quad u = 0 ; \mbox{on}; \partial\Om, \end{equation*} where $\text{Om}$ is a bounded domain in $\text{mb}{R}^n$ with smooth boundary $\mathrm{\partial }\text{Om}$, $n > 2,; \la >0,; 0 < q < 1, ; 0<\mu<n$and$2^*\mu=\frac{2n-\mu}{n-2}$. Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if $\text{la}$ is less than some positive constant. We also study the regularity of these weak solutions.