Nonhomogeneous boundary value problems for some nonlinear equations with singular \(\phi\)-Laplacian
type: Item
instance of: scholarly article
title: Nonhomogeneous boundary value problems for some nonlinear equations with singular \(\phi\)-Laplacian
zbMATH Open document ID: 1170.34014
author: C. Bereanu
author: Jean Mawhin
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-14T00:00:00Z
review text: The paper studies existence and multiplicity results for the quasilinear equation \[ (\phi(u'))'=f(t,u,u') \] under nonhomogeneous Dirichlet conditions \[ u(0)=A,\qquad u(T)=B \] and Neumann-Steklov boundary conditions \[ \phi(u'(0)) = g_0(u(0)),\qquad \phi(u'(T)) = g_T(u(T)). \] It is assumed that \(\phi:(-a,a)\to \mathbb{R}\) is an increasing homeomorphism such that \(\phi(0)=0\), and \(f\), \(g_0\) and \(g_T\) are continuous functions. Using the well-known Leray-Schauder approach, it is proved that the problem has a solution for any right-hand term \(f\), provided that the Dirichlet boundary data satisfies the (sharp) condition \(|B-A|<aT\). Moreover, the Neumann-Steklov problem is solved under suitable sign conditions of Villari's type. Furthermore, the method of upper and lower solutions is extended to the case of nonhomogeneous Neumann conditions, without restrictions of Nagumo type. The combination of this method with Leray-Schauder degree allows the author to prove an Ambrosetti-Prodi type multiplicity result.
reviewed by: Pablo Amster
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5543851
zbMATH Keywords: Phi-Laplacian
zbMATH Keywords: nonhomogeneous Dirichlet problem
zbMATH Keywords: nonlinear Neumann-Steklov problem
zbMATH Keywords: Leray-Schauder degree
zbMATH Keywords: Ambrosetti-Prodi problem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.04.025
OpenAlex ID: W2051518027
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DOI: 10.1016/J.JMAA.2008.04.025
description: scientific article; zbMATH DE number 5543851