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

We consider a class of $p$-Laplace quasilinear Schrödinger Equations

$$\{\begin{array}{c}-{\mathrm{\Delta }}_{p}u-\frac{p}{2^{p-1}}u{\mathrm{\Delta }}_p(u^2)=\lambda u^{-\gamma }+u^q\text{in}\mathrm{\Omega },\\ u>0\text{in}\mathrm{\Omega },u=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with regular boundary, $1<p<\mathrm{\infty }$, $0<\gamma <1$, $2p-1<q\le 2\cdot p^{\ast }-1$ for $p\le N$, $2p-1<q<\mathrm{\infty }$ for $p>N$, where $p^{\ast }=\frac{Np}{N-p}$ if $1<p<N$, $p^{\ast }\in (p,\mathrm{\infty })$ is arbitrarily large if $p=N$, and $p^{\ast }=\mathrm{\infty }$ 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.