Uniqueness for elliptic problems with locally Lipschitz continuous dependence on the solution
type: Item
instance of: scholarly article
title: Uniqueness for elliptic problems with locally Lipschitz continuous dependence on the solution
zbMATH Open document ID: 1364.35093
author: Filomena Feo
author: O. Guibé
published in: Journal of Differential Equations
publication date: 2016-12-22T00:00:00Z
review text: The paper under review is concerned with the nonlinear elliptic problem \[ -\mathrm{div}(a(x,u,\nabla u)= f-\mathrm{div} \, g \quad \text{in} \quad \Omega,\quad u=0 \quad \text{on} \quad \partial\Omega, \] where \(\Omega \subset \mathbb{R}^{^N}\) open, bounded, \(f\in L^{1}(\Omega)\), \(g\in (L^{p}(\Omega))^{N} \quad (p>1)\). Besides the standard growth and strong monotonicity conditions, the authors impose the following local Lipschitz condition on the function \(a\): \[ \begin{cases} \forall \, k>0 \, \exists \, L_{k}>0 \quad \text{such that}\\ \mid a(x,s,\xi) - a(x,r,\xi)\mid \leq L_{k}\mid s-r\mid \, \mid\xi\mid^{p-1}\\ \forall \, s,r,\in \mathbb{R}: \mid s\mid, \mid r\mid \leq k, \quad \forall \, \xi \in \mathbb{R}^{N} \end{cases} \] \[ \begin{cases} \mid a(x,s,\xi)-a(x,s,\xi')\mid \leq b(s)\mid \xi-\xi'\mid \; (\mid \xi\mid + \mid\xi'\mid)^{p-2} \\ \forall \, s\in \mathbb{R}, \quad \forall \, \xi, \xi' \in \mathbb{R}^{N} \end{cases} \] for a.e. \(x\in \Omega\), where \(b:\mathbb{R} \longrightarrow ]0,+\infty[\) is a continuous function such that \(b(s)\geq\) const \(>0\) for all \(s\in \mathbb{R}\). Under an additional structural condition on \((a(x,s,\xi)\cdot \xi'\) and the assumption that one of the conditions (A1) \(\quad 1<p\leq 2\), \(\quad\) (A2) \(\quad p>2\) and \(\text{sign}(f-\text{div} g) =\) const in \(\Omega\) holds, the authors prove the existence and uniqueness of a renormalized solution of (1).
reviewed by: Joachim Naumann
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 6668261
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: locally Lipschitz continuous coefficients
zbMATH Keywords: renormalized solution
zbMATH Keywords: uniqueness result
MaRDI profile type: Publication
full work available at URL: j.jde.2016.10.029
OpenAlex ID: W2553857383
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DOI: 10.1016/J.JDE.2016.10.029
description: scientific article; zbMATH DE number 6668261