Existence of solutions for systems arising in electromagnetism

Abstract

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 Repov\v{s} (Complex Var. Elliptic Equ., 2018), and Ge and Lu (Mediterr. J. Math., 2019).