Global multiplicity of solutions for a singular $ p $-Laplacian quasilinear Schrödinger equation

We consider a class of $ p $-Laplace quasilinear Schrödinger Equations$ \left \{ \begin{array}{c} -\Delta_p u-\frac{p}{2^{p-1}}u\Delta_p (u^2) = \lambda u^{-\gamma}+u^q\; \mbox{in}\; \Omega, \\ u > 0 \; \; \mbox{in} \; \; \Omega, \; \; \; u = 0 \; \; \mbox{on} \; \; \partial \Omega , \end{array}\right. $where $ \Omega\subset\mathbb{R}^N $ is a bounded domain with regular boundary, $ 1 < p < \infty $, $ 0 < \gamma < 1 $, $ 2p - 1 < q\leq2\cdot p^*-1 $ for $ p\leq N $, $ 2p-1 < q < \infty $ for $ p > N $, where $ p^* = \frac{Np}{N-p} $ if $ 1 < p < N $, $ p^*\in(p, \infty) $ is arbitrarily large if $ p = N $, and $ p^* = \infty $ if $ p > N $. We establish global existence and multiplicity of positive solutions via a new strong comparison principle and a regularity result for weak solutions.