Existence of solutions for nonlocal elliptic systems involving \(p(x)\)-Laplace operator
type: Item
instance of: scholarly article
title: Existence of solutions for nonlocal elliptic systems involving \(p(x)\)-Laplace operator
zbMATH Open document ID: 1499.35170
author: S. Firouzi
author: Ghasem A. Afrouzi
published in: Periodica Mathematica Hungarica
publication date: 2022-02-22T00:00:00Z
review text: The authors of the paper study the existence and the multiplicity of solutions of the following nonlocal elliptic systems \[-M\left( \int_{\Omega}\frac{1}{p(x)}\left\vert \Delta u\right\vert ^{p(x)}\,dx\right) \Delta_{p(x)}u=a(x)\left\vert u\right\vert ^{p(x)-2}\text{in }\Omega, \] \[\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{\partial v}=\lambda f(x,u)\text{ in }\partial \Omega ,\] where \(\Omega \)\ is a bounded smooth domain of \(\mathbb{R}^{n}\), \(v\)\ is the outer unit normal to \(\partial \Omega ,\) \(\lambda \) is a positive scalar, \(\Delta_{p(x)}\)\ denotes the \(p(x)\)-Laplace operator with \(p\in C(\overline{\Omega}),\) \(f:\partial \Omega \times \mathbb{R}\rightarrow \mathbb{R}\)\ is a Carathéodory function, \(a\in L^{\infty}(\Omega)\)\ with \(\operatorname{essinf}_{\Omega}a>0\) and \(M\in C(\mathbb{R})\). The results proposed by the authors are obtained in Sobolev spaces with variable exponent under suitable hypothesis on the given functions \(p\), \(f\), \(a\) and \(M\).
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 7479506
zbMATH Keywords: \(p(x)\)-Laplace operator
zbMATH Keywords: variable exponent Sobolev spaces
zbMATH Keywords: nonlinear elliptic systems
zbMATH Keywords: variational methods
zbMATH Keywords: nonlocal condition
reviewed by: Chikh Bouzar
MaRDI profile type: Publication
full work available at URL: s10998-021-00396-9
OpenAlex ID: W3177579890
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DOI: 10.1007/S10998-021-00396-9