Singular elliptic problems with unbalanced growth and critical exponent

In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left{\begin{array}{rllll} -\Delta_{p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1}, \ u>0, \ $\text{ in }$ \Om \ u&=0 \quad $\text{ on }$ \pa\Om, \end{array} \right. \end{equation*} where $\text{Om}$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary, $1< q< p<r \leq p^{}$,where$p^{}=\ds \frac{np}{n-p}$,$0<\de< 1$,$n> p$and$\la,, \ba>0$ are parameters. We prove existence, multiplicity and regularity of weak solutions of $(P_{\text{la}})$ for suitable range of $\text{la}$. We also prove the global existence result for problem $(P_{\text{la}})$.