Existence of solutions for systems arising in electromagnetism

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

Povzetek

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).

Ključne besede

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

Podatki

Jezik:	Angleški jezik
Leto izida:	2020
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UL FMF - Fakulteta za matematiko in fiziko
UDK:	517.956.2
COBISS:	18900057
ISSN:	0022-247X
Št. ogledov:	578
Št. prenosov:	378
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Vrsta dela (COBISS):	Članek v reviji
Strani:	art. 123898 [18 str.]
Letnik:	ǂVol. ǂ486
Zvezek:	ǂiss.ǂ2
Čas izdaje:	June 2020
DOI:	10.1016/j.jmaa.2020.123898
ID:	11404448