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This work deals with existence of solutions for the class of quasilinearelliptic problems with cylindrical singularities and multiple criticalnonlinearities 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 $u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\})$ isproved with the help of the mountain pass theorem. We also prove a regularityresult, that is, using Moser's iteration scheme we show that $u \inL_{\operatorname{loc}}^{\infty}(\Omega)$ for domains $\Omega \subset\mathbb{R}^{N-k}\times\mathbb{R}^{k}\backslash \{ |y|=0 \}$ not necessarilybounded. Finally we show that if $ u \in\mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\}) $ is a weak solution tothe related problem \begin{align*} -\operatorname{div}\left[\frac{|\nablau|^{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 $u \inL_{bp^*(a,b)/q, \operatorname{loc}}^{q} (\mathbb{R}^N\backslash\{|y|=0\}) \capL_{\mathrm{loc}}^{\infty}(\mathbb{R}^{N-k}\times \mathbb{R}^{k} \backslash \{|y| = 0\})$. This nonexistence of nontrivial solution is proved by using aPohozaev-type identity.

Quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities: existence, regularity, nonexistence