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

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 $u\in {\mathcal{D}}_a^{1,p}({\mathbb{R}}^N\mathrm{\backslash }\{\mathrm{\mid }y\mathrm{\mid }=0\})$ 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 $u \in L_{\operatorname{loc}}^{\infty}(\Omega)$fordomains$\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 $wheneither$ 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_{$\mathrm{l}\mathrm{o}\mathrm{c}$}^{\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.