Existence of solutions for systems arising in electromagnetism

In this paper, we study the following p(x)-curl systems: {∇×(|∇×u|p(x)−2∇×u)+a(x)|u|p(x)−2u=λf(x,u)+μg(x,u),∇⋅u=0, in Ω,|∇×u|p(x)−2∇×u×n=0,u⋅n=0, on ∂Ω, where Ω⊂R3 is a bounded simply connected domain with a C1,1-boundary, denoted by ∂Ω, p:Ω‾→(1,+∞) is a continuous function, a∈L∞(Ω), f,g:Ω×R3→R3 are Carathéodory functions, and λ,μ are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang et al. (2017) [41], Bahrouni and Repovš (2018) [9], and Ge and Lu (2019) [22].