Nonlinear singular problems with convection
type: Item
instance of: scholarly article
title: Nonlinear singular problems with convection
zbMATH Open document ID: 1471.35170
author: Nikolaos S. Papageorgiou
author: Andrea Scapellato
published in: Journal of Differential Equations
publication date: 2021-07-12T00:00:00Z
review text: The authors study the following nonlinear Dirichlet problem with singular and gradient dependent terms \begin{align*} -\Delta_pu=u^{-\eta}+f(z,u,\nabla u)\text{ in }\Omega,\\ u|_{\partial\Omega}=0, \, u>0, \, 1<p<N,\, 0<\eta, \end{align*} where \(\Omega\subseteq\mathbb{R}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\) and \(\Delta_pu:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) for all \(u\in W^{1,p}_0(\Omega)\) is the \(p\)-Laplacian differential operator. A main feature of the problem is that the singular term \(u\to u^{-\eta}\) leads to an energy functional, associated to the problem, which is not \(C^1\). By using the so-called frozen variable method and working on appropriate auxiliary problems, the authors establish the existence of a smallest element of the set of positive solutions to the main problem. Finally, as the given minimal solution map possesses certain regularities, the authors apply the Leray-Schauder alternative principle and conclude that the main problem admits a positive solution.
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J75
zbMATH DE Number: 7370842
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: singular and convection term
zbMATH Keywords: nonlinear regularity
zbMATH Keywords: minimal positive solution
zbMATH Keywords: fixed point theory
zbMATH Keywords: frozen variable method
MaRDI profile type: Publication
full work available at URL: j.jde.2021.06.001
OpenAlex ID: W3170050531
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DOI: 10.1016/J.JDE.2021.06.001
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description: scientific article; zbMATH DE number 7370842