The Fredholm alternative for the one-dimensional \(p\)-Laplacian
type: Item
instance of: scholarly article
title: The Fredholm alternative for the one-dimensional \(p\)-Laplacian
zbMATH Open document ID: 1086.34019
author: Xiaojing Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-01-12T00:00:00Z
review text: This paper deals with a Dirichlet BVP of the form \[ \phi_p(u')'+\lambda_1 \phi_p(u)=f(u)+h(t), \quad u(0)=0=u(T), \] where \(\phi_p\) denotes the one-dimensional \(p\)-Laplacian, \(f\) is continuous and bounded, the limit \(\lim_{u \to +\infty} f(u)\) exists and \(h \in L^{\infty}(0,T)\). The following result is proved: Assume that \(H(t):=f(+\infty)+h(t) \not\equiv 0\). Suppose that \(\int_0^T H(t) \sin({{\pi_p t}\over{T}})dt =0,\) where \(\pi_p=2\pi/(p\sin(\pi/p)),\) and \(\lim_{u \to +\infty} | x| ^{p-1}[f(x)-f(+\infty)]=0\). Then the given problem has at least one solution. Moreover, if \(p\neq 2\) then the set of all possible solutions is bounded in \(C^1 [0,T]\). The proof is performed in the framework of the Leray-Scheuder continuation method; a generalized polar coordinates transformation is used. This result is strictly related to the work by \textit{R. F. Manásevich} and \textit{R. Takáč} [Proc. Lond. Math. Soc., III. Ser. 84, No. 2, 324--342 (2002; Zbl 1024.34010)].
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 2126902
zbMATH Keywords: one-dimensional \(p\)-Laplacian
zbMATH Keywords: Fredholm alternative
zbMATH Keywords: Leray-Schauder method
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.077
OpenAlex ID: W2018856843
cites work: On the Fredholm alternative for the $p$-Laplacian
cites work: Generic Fredholm alternative-type results for the one dimensional \(p\)-Laplacian.
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cites work: A homotopic deformation along \(p\) of a Leray-Schauder degree result and existence for \((| u'| ^{p-2}u')'+f(t,u)=0\), \(u(0)=u(T)=0\), \(p>1\)
cites work: On the Fredholm Alternative for the p -Laplacian in One Dimension
DOI: 10.1016/J.JMAA.2004.03.077
description: scientific article; zbMATH DE number 2126902