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Existence of solutions for systems arising in electromagnetism
Hamdani, Mohamed Karim
(
Author
),
Repovš, Dušan
(
Author
)
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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 nonexistence 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).
Language:
English
Keywords:
variable exponent
,
p(x)curl system
,
Palais Smale compactness condition
,
Fountain theorem
,
Dual Fountain theorem
,
existence of solutions
,
multiplicity of solutions
,
electromagnetism
Work type:
Article (dk_c)
Tipology:
1.01  Original Scientific Article
Organization:
PEF  Faculty of Education
Year:
2020
Number of pages:
art. 123898 [18 str.]
Numbering:
Vol. 486, iss.2
UDC:
517.956.2
ISSN on article:
0022247X
DOI:
10.1016/j.jmaa.2020.123898
COBISS.SIID:
18900057
Views:
325
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277
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