Optimal existence conditions for \(\varphi\)-Laplacian equations with upper and lower solutions in the reversed order
type: Item
instance of: scholarly article
title: Optimal existence conditions for \(\varphi\)-Laplacian equations with upper and lower solutions in the reversed order
zbMATH Open document ID: 0999.34011
author: Alberto Cabada
author: Patrick Habets
author: Rodrigo L. Pouso
published in: Journal of Differential Equations
publication date: 2002-11-25T00:00:00Z
review text: Consider the scalar \(\varphi\)-Laplacian problem \[ -\varphi' (u'(t))' = f(t,u(t)),\quad u'(a) = 0 = u'(b),\tag \(*\) \] under the following conditions: (i) \(\varphi: \mathbb{R}\to \mathbb{R}\) is an increasing homeomorphism such that \(\varphi (0) = 0\), (ii) for every compact interval \([k_1,k_2]\) there exists \(K>0\) such that for all \(u,v \in [k_1,k_2]\) holds \((\varphi (u)-\varphi (v))(u-v) \geq K(u-v)^2\), and (iii) \(f: I \times \mathbb{R}\to \mathbb{R}\) is an \(L^1\)-Carathéodory function. The authors prove an anti-maximum comparision principle for the operator \(-(\varphi(u'(t))'-Mu(t))\) and use it to establish the existence of extremal solutions to \((*)\) via the monotone method in the presence of upper and lower solutions in the reversed order.
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1530277
zbMATH Keywords: Neumann problem
zbMATH Keywords: \(\varphi\)-Laplacian equation
zbMATH Keywords: anti-maximum principle
zbMATH Keywords: monotone method
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3803
OpenAlex ID: W2009156549
cites work: Q4040886
cites work: An introduction to nonlinear boundary value problems
cites work: Existence results for nonlinear problems with separated boundary conditions
cites work: Monotone method for the Neumann problem with lower and upper solutions in the reverse order
cites work: Existence result for the problem (φ(u′))′ = f(t, u, u′) with periodic and Neumann boundary conditions
cites work: Existence results for the problem (φ(u′))′=f(t,u,u′) with nonlinear boundary conditions
cites work: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions
cites work: Monotone iterative methods for boundary value problems
cites work: Pairs of positive solutions for the one-dimensional p-Laplacian
cites work: Q4303531
cites work: Q4340284
cites work: A generalized periodic boundary value problem for the one-dimensional p-Laplacian
cites work: Q3807603
cites work: Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian
cites work: Q4000235
cites work: Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions
cites work: Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem
DOI: 10.1006/JDEQ.2000.3803
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description: scientific article; zbMATH DE number 1530277