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

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.