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: $\left\{\begin{aligned}\hfill -{{\Delta}}_{p}u-\beta {{\Delta}}_{q}u& =\lambda {u}^{-\delta }+{u}^{r-1},\quad u{ >}0,\;\;\text{in}\;{\Omega}\hfill \\ \hfill u& =0\quad \;\text{on}\;\partial {\Omega},\hfill \end{aligned}\right.$ where Ω is a bounded domain in ${\mathbb{R}}^{n}$ with smooth boundary, 1 < q < p < r ⩽ p*, where ${p}^{{\ast}}=\frac{np}{n-p}$, 0 < δ < 1, n > p and λ, β > 0 are parameters. We prove existence, multiplicity and regularity of weak solutions of (P_λ) for suitable range of λ. We also prove the global existence result for problem (P_λ).