Existence of solutions for systems arising in electromagnetism

In this paper, we study the following \(p(x)\)-curl systems: \begin{eqnarray*} \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}=0,\; \mbox{ in } \Omega, \\ |\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u}\times \mathbf{n}=0,\quad \mathbf{u}\cdot \mathbf{n}=0, \mbox{ on } \partial\Omega, \end{cases} \end{eqnarray*} where \(\Omega \subset \mathbb{R}^{3}\) is a bounded simply connected domain with a \(C^{1,1}\)-boundary, denoted by \(\partial \Omega\), \(p:\overline{\Omega}\to (1, +\infty)\) is a continuous function, \(a \in L^\infty(\Omega)\), \(f,g : \Omega \times \mathbb{R}^{3}\to \mathbb{R}^{3}\) are Carath\'{e}odory 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 et al. (J. Math. Anal. Appl., 2017), Bahrouni and Repovš (Complex Var. Elliptic Equ., 2018), and Ge and Lu (Mediterr. J. Math., 2019).