Existence of solutions for systems arising in electromagnetism

In this paper, we study the following p(x)-curl systems: {del x (vertical bar del x u vertical bar(p(x)-2)del x u) + a(x)vertical bar u vertical bar(p(x)-2)u = lambda f(x, u) + mu g(x, u), del . u = 0, in Omega, vertical bar del x u vertical bar(p(x)-2)del x u x n = 0, u . n = 0, on partial derivative Omega, where Omega subset of R-3 is a bounded simply connected domain with a C-1,C-1-boundary, denoted by partial derivative Omega, p : <(Omega) over bar> -> (1, +infinity) is a continuous function, a is an element of L-infinity (Omega), f,g : Omega x R-3 -> R-3 are Caratheodory functions, and lambda,mu 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]. (C) 2020 Elsevier Inc. All rights reserved.