On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions
type: Item
instance of: scholarly article
title: On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions
zbMATH Open document ID: 0790.35038
DOI: 10.1007/BF01459750
published in: Mathematische Annalen
publication date: 1994-06-26T00:00:00Z
full work available at URL: https://eudml.org/doc/165185
review text: We use an implicit probabilistic representation and Schauder's fixed point theorem to obtain sufficient conditions for the existence of positive continuous solutions to the following semilinear elliptic equation with Dirichlet boundary conditions: \[ {\textstyle {1\over 2}} \Delta u+F(x,u) =-g(x) \qquad \text{for } x\in D, \] \[ u(x) =\varphi(x) \qquad \text{for } x\in \partial D, \qquad \lim_{{\textstyle {{| x|\to\infty}\atop {x\in D}}}} u(x)= \alpha \quad \text{when \(D\) is unbounded}, \] where \(D\) is a regular domain in \(\mathbb{R}^ d\) (\(d\geq 3\)), \(\varphi\) is a nonnegative continuous function on \(\partial D\), \(\alpha\) is a nonnegative constant such that when \(\partial D\) is unbounded, \(\lim_{| x|\to\infty,x\in\partial D} \varphi(x) =\alpha\), \(g\) is a nonnegative Green-tight function on \(D\), and \(F\) is a real-valued Borel measurable function defined on \(D\times (0,b)\) for some \(b\in(0,\infty]\) such that \(F(x,\cdot)\) is continuous on \((0,b)\) for each \(x\in D\) and \(-U(x)u\leq F(x,u)\leq V(x)f(u)\) for all \((x,u)\in D\times (0,b)\), where \(U\) and \(V\) are nonnegative Green-tight functions on \(D\) and \(f\) is a nonnegative Borel measurable function defined on \((0,b)\). Special cases of equations of this type arise in connection with superprocesses and problems in geometry concerning conformal deformation of metrics.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 562495
zbMATH Keywords: implicit probabilistic representation
zbMATH Keywords: Schauder's fixed point theorem
zbMATH Keywords: existence of positive continuous solutions
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: conformal deformation of metrics
author: Zhen-Qing Chen
author: R. J. Williams
author: Z. Zhao
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 562495