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

Abstract

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.