On extremal solutions of an elliptic boundary value problem involving discontinuous nonlinearities
type: Item
instance of: scholarly article
title: On extremal solutions of an elliptic boundary value problem involving discontinuous nonlinearities
zbMATH Open document ID: 0785.35034
author: S. Carl
author: S. Heikkilä
published in: Differential and Integral Equations
publication date: 1992-10-04T00:00:00Z
review text: Let \(\Omega\subset\mathbb{R}^ n\) be a bounded domain with a boundary \(\partial\Omega\) of class \(C^ 1\). We consider the following boundary value problem \[ -Lu=F(u,u)\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega, \tag{1.1} \] where \(F\) is the superposition operator associated with a function \(f:\Omega\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) by \(F(u,v)(x)=f(x,u(x),v(x))\), and \(L\) is assumed to be a uniformly elliptic linear operator of the form \[ Lu={\partial\over\partial x_ i}\left(a_{ij}{\partial u\over\partial x_ j}\right)-b_ i{\partial u\over\partial x_ i}, \] with coefficients \(a_{ij}\), \(b_ i\in L^ \infty(\Omega)\), \(i,j=1,\dots,n\). The aim of this paper is to show the existence of the greatest and the smallest solutions of the BVP (1.1) lying between upper and lower solutions.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35R05
zbMATH DE Number: 68016
zbMATH Keywords: extremal solutions between upper and lower solutions
zbMATH Keywords: discontinuous nonlinearities
zbMATH Keywords: semilinear elliptic equation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 68016