On boundary value problems for degenerate quasilinear elliptic equations and inequalities
type: Item
instance of: scholarly article
title: On boundary value problems for degenerate quasilinear elliptic equations and inequalities
zbMATH Open document ID: 0912.35069
author: Vy Khoi Le
author: Klaus Schmitt
published in: Journal of Differential Equations
publication date: 1998-11-19T00:00:00Z
review text: The boundary-value problem for a quasilinear elliptic equation \[ \text{div }A(x, \nabla u)= f,\quad x\in\Omega,\quad u= 0,\quad x\in\partial\Omega \] is considered. Solutions are sought for in weighted Sobolev spaces as extremum points of the variational inequality \[ L(u,\nu- u)\geq (f,\nu- u),\quad \forall\nu\in K,\quad u\in K. \] Using the fixed point theory and similar techniques, the authors study a nonlinear perturbed problem, where \(f\) depends not only on \(x\) but also on \(u\), the growth of \(f\) with respect to \(u\) being restricted. The existence and continuity of the solutions is proved under the restriction \[ | F(x,u)|\leq a(x)+ b| u|^\sigma,\quad f(x,u)= \int_\Omega F(x,u(x))\nu(x) dx. \] For the quasilinear degenerate case it is also proved that the existence of subsolutions and supersolutions follows from the existence of extremal solutions of the variational inequality.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 1206648
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: fixed point theory
zbMATH Keywords: existence
zbMATH Keywords: continuity
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3384
OpenAlex ID: W2051890064
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DOI: 10.1006/JDEQ.1997.3384
description: scientific article; zbMATH DE number 1206648