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

  • Ge BinEmail author
  • Lu Jian-Fang


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.


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

Mathematics Subject Classification

Primary 35G30 35J35 Secondary 35P30 58E05 



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Authors and Affiliations

  1. 1.Department of Applied MathematicsHarbin Engineering UniversityHarbinPeople’s Republic of China

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