Quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities: existence, regularity, nonexistence (1506.09152v1)

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.