LAUSR

Abstract In this paper, we study the following p ( x ) -curl systems: { ∇ × ( | ∇ × u | p ( x ) − 2 ∇ × u ) + a ( x ) | u | p ( x ) − 2 u = λ f ( x , u ) + μ g ( x , u ) , ∇ ⋅ u = 0 ,  in  Ω , | ∇ × u | p ( x ) − 2 ∇ × u × n = 0 , u ⋅ n = 0 ,  on  ∂ Ω , where Ω ⊂ R 3 is a bounded simply connected domain with a C 1 , 1 -boundary, denoted by ∂Ω, p : Ω ‾ → ( 1 , + ∞ ) is a continuous function, a ∈ L ∞ ( Ω ) , f , g : Ω × R 3 → R 3 are Caratheodory 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 Repovs (2018) [9] , and Ge and Lu (2019) [22] .