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# Existence and Multiplicity of Solutions for p(x)-Curl Systems Without the Ambrosetti–Rabinowitz Condition

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## Abstract

In this paper, we study the p(x)-curl systems:
\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}
where $$\Omega \subset \mathbb {R}^{3}$$ is a bounded simply connected domain with a $$C^{1,1}$$ boundary denoted by $$\partial \Omega$$ , $$p:\overline{\Omega }\rightarrow (1,+\infty )$$ is a continuous function, $$a\in L^{\infty }(\Omega )$$, and $$\mathbf {f}:\overline{\Omega }\times \mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$$ is a Carath$$\mathrm{{\acute{e}}}$$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.

## Keywords

p(x)-curl systems variable exponent mountain pass theorem critical point

## Mathematics Subject Classification

Primary 35G30 35J35 Secondary 35P30 58E05

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## Copyright information

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Ge Bin
• 1
• Lu Jian-Fang
• 1
1. 1.Department of Applied MathematicsHarbin Engineering UniversityHarbinPeople’s Republic of China