Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

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We find solutions E:Ω→R3 of the problem{∇×(μ(x)−1∇×E)−ω2ε(x)E=∂EF(x,E)in Ων×E=0on ∂Ω on a bounded Lipschitz domain Ω⊂R3 with exterior normal ν:∂Ω→R3. Here ∇× denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field ℜ{E(x)eiωt} in an anisotropic material with a magnetic permeability tensor μ(x)∈R3×3 and a permittivity tensor ε(x)∈R3×3. The boundary conditions are those for Ω surrounded by a perfect conductor. It is required that μ(x) and ε(x) are symmetric and positive definite uniformly for x∈Ω, and that μ,ε∈L∞(Ω,R3×3). The nonlinearity F:Ω×R3→R is superquadratic and subcritical in E, the model nonlinearity being of Kerr-type: F(x,E)=|Γ(x)E|p for some 2<p<6 with Γ(x)∈GL(3) invertible for every x∈Ω and Γ,Γ−1∈L∞(Ω,R3×3). We prove the existence of a ground state solution and of bound states if F is even in E. Moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries.