Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

Abstract

We find solutions $E:Ω\to\mathbb{R}^3$ of the problem \begin{eqnarray*} \left\{ \begin{aligned} &\nabla\times(μ(x)^{-1}\nabla\times E) - ω^2ε(x) E = \partial_E F(x,E) &&\quad \text{in }Ω\\%\newline &ν\times E = 0 &&\quad \text{on }\partialΩ \end{aligned} \right. \end{eqnarray*} on a bounded Lipschitz domain $Ω\subset\mathbb{R}^3$ with exterior normal $ν:\partialΩ\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$. The equation describes the propagation of the time-harmonic electric field $\Re\{E(x)e^{iωt}\}$ in an anisotropic material with a magnetic permeability tensor $μ(x)\in\mathbb{R}^{3\times3}$ and a permittivity tensor $ε(x)\in\mathbb{R}^{3\times3}$. 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 $x\inΩ$, and that $μ,ε\in L^{\infty}(Ω,\mathbb{R}^{3\times 3})$. The nonlinearity $F:Ω\times\mathbb{R}^3\to\mathbb{R}$ 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 $Γ(x)\in GL(3)$ invertible for every $x\inΩ$ and $Γ,Γ^{-1}\in L^\infty(Ω, \mathbb{R}^{3\times 3})$. 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.