One-dimensional nonlinear boundary value problems with variable exponent
type: Item
instance of: scholarly article
title: One-dimensional nonlinear boundary value problems with variable exponent
zbMATH Open document ID: 1414.34017
author: Gabriele Bonanno
author: Giuseppina D'Aguì
author: Angela Sciammetta
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2017-11-21T00:00:00Z
review text: The authors prove the existence of at least one non-zero solution of a nonlinear boundary value problem with variable exponent of the form \[ -(|u'(x)|^{p(x)-2}u'(x))'+a(x) |u(x)|^{p(x)-2} u(x)= \lambda f(x,u(x)), \quad u(0)=0=u(1), \] where $f$ is an $L^1$-Carathéodory function, $a\in L^{\infty}[0,1]$ with $\operatorname{essinf} a \geq 0, p \in C([0,1]$ with $\min p >1$ and $\lambda$ a positive real parameter. It is assumed a technical condition stated by means of an algebraic inequality for the primitive of $f$; no condition at zero or at infinity on the nonlinear term is required. The assumptions of the main result are satisfied e.g. when $p(x)=x^2, \lambda =1$ and $f(x,u)=x^4 u^2$. For the proof, it is applied an abstract result in Orlicz-Sobolev spaces due to the first author.
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 6810415
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: $p(x)$-Laplacian
MaRDI profile type: Publication
full work available at URL: dcdss.2018011
OpenAlex ID: W2765228071
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DOI: 10.3934/DCDSS.2018011
description: scientific article; zbMATH DE number 6810415