Ambrosetti-Prodi type result to a Neumann problem via a topological approach
type: Item
instance of: scholarly article
title: Ambrosetti-Prodi type result to a Neumann problem via a topological approach
zbMATH Open document ID: 1392.34017
author: Elisa Sovrano
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2017-11-21T00:00:00Z
review text: The author studies the Neumann problem \[ \begin{aligned} u''+a(x)g(u) &=\mu+p(x), \\ u'(0)\;\;=\;\;u'(T)&=0, \end{aligned} \] where \(p\) is a bounded, measurable real function, \(g\) is \(C^1\) in \(\mathbb R\) with \(-\infty<g'(-\infty)<0<g'(+\infty)<+\infty\), and \(a\in L^\infty(0,T)\) is a non-negative function with \(\int_0^Ta(x)\,dx>0\). The main result says that if the number \(\mu\) is sufficiently large, the problem has at least two solutions. One of the solutions, say \(\tilde u\), is obtained by truncation and shooting and by construction it is negative (assuming, without loss of generality, that \(g(0)=0\) and \(g'(s)<0\) if \(s<0\)). A second solution is also obtained by shooting between the initial conditions of \(\tilde u+\varepsilon\), with \(\varepsilon>0\) small, and initial conditions \((u(0),0)\) with \(u(0)\) positive and large. The article contains a history of this kind of problems and stresses that a significant feature is the fact that the result holds with weight functions \(a\) that may vanish in a set of positive measure.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6810423
zbMATH Keywords: Ambrosetti-Prodi problems
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: multiplicity results
zbMATH Keywords: Shooting method
MaRDI profile type: Publication
full work available at URL: dcdss.2018019
OpenAlex ID: W2765158811
cites work: A multiplicity result for a class of elliptic boundary value problems
cites work: Observations on global inversion theorems
cites work: On the inversion of some differentiable mappings with singularities between Banach spaces
cites work: Q4002249
cites work: Sharp existence results for a class of semilinear elliptic problems
cites work: Q4095271
cites work: On the ranges of certain weakly nonlinear elliptic partial differential equations
cites work: Two-point boundary value problems. Lower and upper solutions
cites work: A Multiplicity Result for Periodic Solutions of Forced Nonlinear Second order Ordinary Differential Equations
cites work: On a singular periodic Ambrosetti-Prodi problem
cites work: Boundary value problems with jumping nonlinearities
cites work: Remarks on some quasilinear elliptic equations
cites work: Q5184084
cites work: Q3795986
cites work: The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the \(p\)-Laplacian
cites work: Q4204461
cites work: Stability of a periodic problem of Ambrosetti-Prodi type
cites work: A Neumann problem of Ambrosetti-Prodi type
cites work: Q4300917
cites work: The Ambrosetti-Prodi periodic problem: Different routes to complex dynamics
DOI: 10.3934/DCDSS.2018019
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description: scientific article; zbMATH DE number 6810423