Robin problems for the \(p\)-Laplacian with gradient dependence
type: Item
instance of: scholarly article
title: Robin problems for the \(p\)-Laplacian with gradient dependence
zbMATH Open document ID: 1432.35070
author: Genni Fragnelli
author: Dimitri Mugnai
author: Nikolaos S. Papageorgiou
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2019-03-27T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^N\) be a bounded smooth domain. The paper is concerned with the following \(p\)-Laplacian type Robin boundary problem: \[ -\operatorname{div}(|Du|^{p-2}Du(z)) = f(z, u(z),Du(z)) \text{ in } \Omega, \ \ \frac{\partial u}{{\partial n}_p} + \beta(z)|u|^{p-2}u = 0 \text{ on } \partial \Omega,\tag{P} \] where \(p\in (0,\infty)\), \(f(z, x, y)\) is a Carathéodory function, \(\frac{\partial u}{{\partial n}_p}\) denotes the conormal derivative and \(\beta(z)\geq 0\) is Hölder continuous on \(\partial \Omega\). Under some further conditions on \(f(z, x, y)\), such as \(f(z, x, y) \leq \Theta (z) |x|^{p-2}x\) uniformly for a.e. \(z\in \Omega\) and all \( y\in \mathbb{R}^N\), as \(|x| \rightarrow \infty\), if \( \Theta(z)\) is bounded by certain constant, the authors prove the existence of a smooth solution for problem (P) by the Leray-Schauder alternative principle.
reviewed by: Huansong Zhou
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J92
zbMATH DE Number: 7043507
zbMATH Keywords: $p$-Laplacian equation
zbMATH Keywords: Robin boundary condition
zbMATH Keywords: Leray-Schauder alternative principle
MaRDI profile type: Publication
full work available at URL: dcdss.2019020
OpenAlex ID: W2887153843
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DOI: 10.3934/DCDSS.2019020
description: scientific article; zbMATH DE number 7043507