  This work deals with the existence of at least two positive solutions for the
class of quasilinear elliptic equations with cylindrical singularities and
multiple critical nonlinearities that can be written in the form \begin{align*}
-\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right]
-\mu\,\frac{u^{p-1}}{|y|^{p(a+1)}} = h\,\frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}}
+\lambda g\,\frac{u^{q-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in
\mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} We consider $N \geqslant 3$,
$\lambda >0$, $p < k \leqslant N$, $1<p<N$, $0 \leqslant \mu <\bar{\mu} \equiv
\left\{[k-p(a+1)]/p\right\}^p$, $0 \leqslant a < (k-p)/p$, $a \leqslant b <
a+1$, $a \leqslant c < a+1$, $1\leqslant q <p$, $p^*(a,b)=Np/[N-p(a+1-b)]$, and
$p^*(a,c) \equiv Np/[N-p(a+1-c)]$; in particular, if $\mu = 0$ we can include
the cases $(k-p)/p \leqslant a < k(N-p)/Np$ and $a < b< c<k(N-p(a+1))/p(N-k) <
a+1$. We suppose that $g\in L_{\alpha}^{r}(\mathbb{R}^{N})$, where $r =
p^*(a,c)/[p^*(a,c)-q]$ and $\alpha = c(p^*(a,c)-q)$, is positive in a ball and
that it can change sign; we also suppose that $h \in L^\infty(\mathbb{R}^k)$
and that it has a finite, positive limit $h_0$ at the origin and at infinity.
To prove our results we use the Nehari manifold methods and we establish
sufficient conditions to overcome the lack of compactness.
