Multiple solutions for p(x)${p(x)}$‐curl systems with nonlinear boundary condition

Bin Ge and Jian‐Fang Lu

Mathematische Nachrichten, 2022, vol. 295, issue 3, 512-535

Abstract: In this paper, we study the p(x)$p(x)$‐curl problem of the type: ∇×(|∇×u|p(x)−2∇×u)+a(x)|u|p(x)−2u=λf(x,u),inΩ,∇·u=0,inΩ,|∇×u|p(x)−2∇×u×n=0,u·n=0,on∂Ω,\begin{equation*} \hspace*{13pc}{\left\lbrace \def\eqcellsep{&}\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} =\lambda \mathbf {f}(x,\mathbf {u}), &{\rm in}\; \Omega , \\ [0.1cm] \nabla \cdot \mathbf {u}=0,\;&{\rm in}\; \Omega , \\ [0.1cm] |\nabla \times \mathbf {u}|^{p(x)-2}\nabla \times \mathbf {u} \times \mathbf {n}=0,\mathbf {u}\cdot \mathbf {n}=0, \; &\text{on} \; \partial \Omega , \end{array} \right.} \end{equation*}where Ω⊂R3$\Omega \subset \mathbb {R}^{3}$ is a bounded simply connected domain with a C1, 1 boundary denoted by ∂Ω$\partial \Omega$, λ>0$\lambda >0$, p(x):Ω¯→(1,+∞)$p(x):\overline{\Omega }\rightarrow (1,+\infty )$ is a continuous function, a(x)∈L∞(Ω)$a(x)\in L^{\infty }(\Omega )$, and f:Ω¯×R3→R3$\mathbf {f}:\overline{\Omega }\times \mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$ is a Carathéodory function. We obtain the existence and multiplicity of solutions for a class of p(x)$p(x)$‐curl systems in the absence of Ambrosetti–Rabinowitz condition under superlinear case. Besides, we also obtain infinitely many solutions under sublinear case.

Date: 2022
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