Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation:∇ × ( μ − 1 ( x ) ∇ × u ) − ω 2 ε ( x ) u = f ( x , u ) ,in Ω ,ν × u = 0 ,on ∂ Ω ,where Ω ⊂ R 3 is a bounded Lipschitz domain, ν : ∂ Ω → R 3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as | u | → ∞ , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ ∈ R 3 × 3and permittivity tensor ε ∈ R 3 × 3 , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.