Existence of solutions for systems arising in electromagnetism

Mohamed Karim Hamdani (Author), Dušan Repovš (Author)

Abstract

In this paper, we study the following ▫$p(x)$▫-curl systems: ▫$$\begin{cases} \nabla \times (|\nabla \times \mathbf{u}|^{p(x)-2}\nabla \times \mathbf{u}) + a(x)|\mathbf{u}|^{p(x)-2}\mathbf{u} = \lambda f(x, \mathbf{u}) + \mu g(x, \mathbf{u}), \quad \nabla \cdot \mathbf{u} & \text{in} \; \Omega, \\ |\nabla \times \mathbf{u}|^{p(x)-2}\nabla \times \mathbf{u} \times \mathbf{n} = 0, \quad \mathbf{u} \cdot \mathbf{n} = 0 & \text{on} \; \partial\Omega, \end{cases}$$▫ where ▫$\Omega \subset \mathbb{R}^3$▫ is a bounded simply connected domain with a ▫$C^{1,1}$▫-boundary, denoted by ▫$\delta\Omega$▫, ▫$p \colon \overline{\Omega} \to (1, +\infty)$▫ is a continuous function, ▫$a \in L^\infty(\Omega$▫, ▫$f, g \colon \Omega \times \mathbb{R}^3 \to \mathbb{R}^3$▫ are Carathéodory functions, and ▫$\lambda, \mu$▫ are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang, Wang and Zhang (J. Math. Anal. Appl., 2016), Bahrouni and Repovš (Complex Var. Elliptic Equ., 2018), and Bin and Fang (Mediterr. J. Math., 2019).

Keywords

variable exponent;p(x)-curl system;Palais Smale compactness condition;Fountain theorem;Dual Fountain theorem;existence of solutions;multiplicity of solutions;electromagnetism;

Data

Language:	English
Year of publishing:	2020
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.956.2
COBISS:	18900057
ISSN:	0022-247X
Views:	578
Downloads:	378
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	art. 123898 [18 str.]
Volume:	ǂVol. ǂ486
Issue:	ǂiss.ǂ2
Chronology:	June 2020
DOI:	10.1016/j.jmaa.2020.123898
ID:	11404448