Weak positive solutions to singular quasilinear elliptic equation

In this paper, we study the existence of multiple solutions for the singular problem { a ⁢ ( x , u , ∇ ⁡ u ) - div ⁢ ( b ⁢ ( x , u , ∇ ⁡ u ) ) = u - α + λ ⁢ c ⁢ ( x , u ) in  ⁢ Ω , u > 0 in  ⁢ Ω , u = 0 on  ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle{}a(x,u,\nabla u)-{\rm div}(b(x,u,\nabla u% ))&\displaystyle=u^{-\alpha}+\lambda c(x,u)&&\displaystyle\phantom{}\text{in }% \Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }{\mathbb{R}}% ^{n}\setminus\Omega,\end{aligned}\right. where Ω ⊂