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Existence of solutions for systems arising in electromagnetismHamdani, Mohamed Karim ; Repovš, Dušan, 1954-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).Source: Journal of mathematical analysis and applications. - ISSN 0022-247X (Vol. 486, iss.2, June 2020, art. 123898 [18 str.])Type of material - article, component part ; adult, seriousPublish date - 2020Language - englishCOBISS.SI-ID - 18900057
source: Journal of mathematical analysis and applications. - ISSN 0022-247X (Vol. 486, iss.2, June 2020, art. 123898 [18 str.])
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Hamdani, Mohamed Karim | |
Repovš, Dušan, 1954- | 07083 |
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