  This work deals with existence of solutions for the class of quasilinear
elliptic problems 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)}} = \frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} +
\frac{u^{p^*(a,c)-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in
\mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} The existence of a positive,
weak solution $u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\})$ is
proved with the help of the mountain pass theorem. We also prove a regularity
result, that is, using Moser's iteration scheme we show that $u \in
L_{\operatorname{loc}}^{\infty}(\Omega)$ for domains $\Omega \subset
\mathbb{R}^{N-k}\times\mathbb{R}^{k}\backslash \{ |y|=0 \}$ not necessarily
bounded. Finally we show that if $ u \in
\mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\}) $ is a weak solution to
the related problem \begin{align*} -\operatorname{div}\left[\frac{|\nabla
u|^{p-2}}{|y|^{ap}}\nabla u\right] -\mu\,\frac{|u|^{p-2}u}{|y|^{p(a+1)}} =
\frac{|u|^{q-2}u}{|y|^{bp^*(a,b)}} + \frac{|u|^{p^*(a,c)-2}u}{|y|^{cp^*(a,c)}},
\qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k, \end{align*} then $ u
\equiv 0 $ when either $ 1 < q < p^*(a,b) $, or $ q > p^*(a,b) $ and $u \in
L_{bp^*(a,b)/q, \operatorname{loc}}^{q} (\mathbb{R}^N\backslash\{|y|=0\}) \cap
L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{N-k}\times \mathbb{R}^{k} \backslash \{
|y| = 0\})$. This nonexistence of nontrivial solution is proved by using a
Pohozaev-type identity.
