In this paper, we study the p(x)-curl systems: ∇×(|∇×u|p(x)-2∇×u)+a(x)|u|p(x)-2u=f(x,u),inΩ,∇·u=0,inΩ,|∇×u|p(x)-2∇×u×n=0,u·n=0,on∂Ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \nabla \times \big (|\nabla \times \mathbf {u} |^{p(x)-2}\nabla \times \mathbf {u}\big )+a(x)|\mathbf {u}|^{p(x)-2}\mathbf {u} =\mathbf {f}(x,\mathbf {u}),&{} \mathrm{in}\; \Omega ,\\ \nabla \cdot \mathbf {u}=0, &{}\mathrm{in}\; \Omega ,\\ |\nabla \times \mathbf {u}|^{p(x)-2}\nabla \times \mathbf {u} \times \mathbf {n}=0, \mathbf {u}\cdot \mathbf {n}=0,&{} \mathrm{on} \; \partial \Omega ,\\ \end{array} \right. \end{aligned}$$\end{document}where Ω⊂R3\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset \mathbb {R}^{3}$$\end{document} is a bounded simply connected domain with a C1,1\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,1}$$\end{document} boundary denoted by ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document} , p:Ω¯→(1,+∞)\documentclass[12pt]{minimal}
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\begin{document}$$p:\overline{\Omega }\rightarrow (1,+\infty )$$\end{document} is a continuous function, a∈L∞(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$a\in L^{\infty }(\Omega )$$\end{document}, and f:Ω¯×R3→R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {f}:\overline{\Omega }\times \mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$$\end{document} is a Carathe´\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{{\acute{e}}}$$\end{document}odory function. We use mountain pass theorem and symmetric mountain pass theorem to obtain the existence and multiplicity of solutions for a class of p(x)-curl systems in the absence of Ambrosetti–Rabinowitz condition.
