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

TL;DR

This paper investigates the existence, regularity, and nonexistence of solutions for a class of quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities, employing variational methods, regularity techniques, and Pohozaev identities.

Contribution

It establishes new existence results for positive solutions using the mountain pass theorem, proves regularity via Moser iteration, and demonstrates nonexistence under certain conditions with a Pohozaev-type identity.

Findings

Existence of positive weak solutions proved.

Solutions are shown to be locally bounded.

Nonexistence results established for certain parameter ranges.

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.