On a PDE involving the variable exponent operator with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: On a PDE involving the variable exponent operator with nonlinear boundary conditions
zbMATH Open document ID: 1323.35043
published in: Mediterranean Journal of Mathematics
publication date: 2015-09-03T00:00:00Z
review text: This paper is concerned with the study of the following boundary value problem \[ \begin{cases} -\mathcal{A}_{p(x)}u=\lambda f(x)|u|^{q(x)-2}u &\text{ in }\Omega,\\ \alpha(x,u){\langle A\nabla u, \nabla u \rangle}^{\frac{p(x)-2}{2}}\langle A\nabla u, \vec{n} \rangle=g(x)|u|^{r(x)-2}u\quad &\text{ on }\partial\Omega, \end{cases}\leqno{(1)} \] where \(\Omega \subset \mathbb R^N\) is a bounded domain with \(N \geq 2.\) The differential operator \({\mathcal{A}}_{p(x)}\) is defined by \[ {\mathcal{A}}_{p(x)}u=\operatorname{div} \left(\alpha(x,u){\langle A\nabla u, \nabla u \rangle}^{\frac{p(x)-2}{2}} A\nabla u \right), \] where \(A: \overline \Omega \rightarrow \mathbb R^{N^2}\) is a symmetric function matrix with \(a_{ij}\in L^\infty(\Omega)\cap C^1(\overline\Omega)\) and satisfies \[ \langle A\xi, \xi \rangle =\sum_{i, j=1}^Na_{ij}(x)\xi_i\xi_j \geq |\xi|^2 \quad \forall x\in \overline\Omega, \quad\xi\in \mathbb R^N. \] Under some conditions imposed on \(p(.)\), \(q(.)\), \(r(.),\) the weight functions \(f\) and \(g,\) and based on the Nehari manifold, the authors establish the existence and multiplicity of positive solutions for problem (1).
reviewed by: Said El Manouni
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 6478177
zbMATH Keywords: Nehari manifold
zbMATH Keywords: variable exponent operator
zbMATH Keywords: nonlinear boundary conditions
author: S. H. Rasouli
MaRDI profile type: Publication
full work available at URL: s00009-014-0424-z
OpenAlex ID: W1978920663
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DOI: 10.1007/S00009-014-0424-Z
description: scientific article; zbMATH DE number 6478177