Existence of solutions for systems arising in electromagnetism

In this paper, we study the following ▫$p(x)$▫-curl systems: ▫$$begin{cases} nabla times (|nabla times mathbf{u}|^{p(x)-2}nabla times mathbf{u}) + a(x)|mathbf{u}|^{p(x)-2}mathbf{u} = lambda f(x, mathbf{u}) + mu g(x, mathbf{u}), quad nabla cdot mathbf{u} & text{in}Omega, \ |nabla times mathbf{u}|^{p(x)-2}nabla times mathbf{u} times mathbf{n} = 0, quad mathbf{u} cdot mathbf{n} = 0 & text{on}partialOmega, end{cases}$$▫ where ▫$Omega subset mathbb{R}^3$▫ is a bounded simply connected domain with a ▫$C^{1,1}$▫-boundary, denoted by ▫$deltaOmega$▫, ▫$p colon overline{Omega} to (1, +infty)$▫ 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)