Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

Abstract

This paper is focused on following time‐harmonic Maxwell equation: ∇×false(μ−1false(xfalse)∇×ufalse)−ω2εfalse(xfalse)u=ffalse(x,ufalse),2emin.4emnormalΩ,ν×u=0,2em2emon.4em∂normalΩ, where normalΩ⊂R3 is a bounded Lipschitz domain, ν:∂normalΩ→R3 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 false|ufalse|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor ε∈R3×3, 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.