Runge property and multiplicity of solutions for elliptic equations in \({\mathbb{R}}^ N\) with a monotone nonlinearity
type: Item
instance of: scholarly article
title: Runge property and multiplicity of solutions for elliptic equations in \({\mathbb{R}}^ N\) with a monotone nonlinearity
zbMATH Open document ID: 0608.35020
DOI: 10.1007/BF01171705
author: J. M. Morel
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155107
review text: The author proves, that the solutions of \[ -\Delta u+g(u)=f\in L^ 1_{loc}({\mathbb{R}}^ n), \] where g is nondecreasing and Lipschitzcontinuous have the Runge property; that is, if \(\Omega\subset \Omega '\) are two simply connected domains, any solution u in \(\Omega\) can be uniformly approximated in compact subsets of \(\Omega\) by solutions in \(\Omega\) '. The proof relies on a homotopy method via the linearization \(-\Delta v+g'(u)v=0\), for which the Runge property is known to hold by works of \textit{P. D. Lax} [Commun. Pure Appl. Math. 9, 747-766 (1956; Zbl 0072.330)] and \textit{E. Heinz} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1, 1-12 (1955; Zbl 0067.075)]. As a consequence, \(-\Delta u+| u|^{\alpha -1}u=f\) with \(0<\alpha <1\) has infinitely many solutions in \(D'({\mathbb{R}}^ n)\).
reviewed by: Michael Wiegner
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 3983711
zbMATH Keywords: monotone nonlinearity
zbMATH Keywords: semilinear
zbMATH Keywords: multiplicity results
zbMATH Keywords: Runge property
zbMATH Keywords: homotopy method
zbMATH Keywords: linearization
zbMATH Keywords: infinitely many solutions
MaRDI profile type: Publication
cites work: Q4073824
cites work: Resolution of a semilinear equation in L1
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Integrals of subharmonic functions on manifolds of nonnegative curvature
cites work: Q3227818
cites work: A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations
cites work: On the behavior of the solutions of the differential equation ΔU = F(x,u) in the neighborhood of a point
cites work: Q3341251
OpenAlex ID: W2045309877
description: scientific article; zbMATH DE number 3983711