On the existence of positive solutions for a class of semilinear elliptic equations
type: Item
instance of: scholarly article
title: On the existence of positive solutions for a class of semilinear elliptic equations
zbMATH Open document ID: 1027.34043
DOI: 10.1016/S0362-546X(02)00163-3
author: Imed Bachar
author: Noureddine Zeddini
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The authors are concerned with the existence and uniqueness of large, i.e., unbounded, solutions to semi-linear elliptic equations. The first step is to establish conditions for the existence of positive solutions to the one-dimensional nonlinear boundary value problem \[ (A(x)y^{\prime }(x))^{\prime }=A(x)p(x)f(y(x)) \quad\text{for }0\leq x<\infty, \] where \(f\) is Lipschitz continuous. Then they apply their result to partial differential equations in the special case when the solution is radial, to prove the existence of a large solution, since in this case the semi-linear partial differential equation \(\Delta u=q(x)F(u)\) reduces to a nonlinear ordinary differential equation. Otherwise, when \(q\) is not radial but positive, an average function of \(q\) is used to prove the existence of a bounded positive solution to \(\Delta u=q(x)F(u)\) over \(\mathbb{R}^{n}\).
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 1880006
zbMATH Keywords: large solutions
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: nonlinear ordinary differential equation
reviewed by: Amin Boumenir
MaRDI profile type: Publication
cites work: From Brownian Motion to Schrödinger’s Equation
cites work: Q3490335
cites work: Large solutions of sublinear elliptic equations
cites work: Q4316851
description: scientific article; zbMATH DE number 1880006