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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 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).

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:0022-247X
DOI:10.1016/j.jmaa.2020.123898 This link opens in a new window
COBISS.SI-ID:18900057 This link opens in a new window
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Downloads:277
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