A free boundary problem for quasilinear elliptic equations in exterior domains
type: Item
instance of: scholarly article
title: A free boundary problem for quasilinear elliptic equations in exterior domains
zbMATH Open document ID: 1005.35091
author: S. Carl
author: S. Heikkilä
published in: Differential and Integral Equations
publication date: 2000-05-29T00:00:00Z
review text: This paper studies the quasilinear elliptic equations of the type \(\operatorname {div} a(x,u,\nabla u) =f(x,u,\nabla u)\) in an exterior domain in \(\mathbb{R}^n\). The fact that the function \(f\) is allowed to be discontinuous converts the problem into one of the free-boundary type. The main result of the paper is that, assuming an ordered pair of appropriately defined upper and lower solutions, there exist solutions of the problem within the interval specified by the upper and lower solution. Moreover, the provided solution is an extremal one. The proof of the main result is based on the abstract fixed point theorem. Since no decay of the solution is specified as \(|x |\to \infty \), suitable local Sobolev spaces are the solution spaces. An example with the \(p\)-Laplacian is given at the end of the paper. The paper is clearly written and concise. The result is interesting.
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 1453641
zbMATH Keywords: quasilinear elliptic equations
zbMATH Keywords: upper and lower solutions
zbMATH Keywords: fixed point theorem
zbMATH Keywords: local Sobolev spaces
zbMATH Keywords: \(p\)-Laplacian
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1453641