Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

Abstract

We find solutions $E:\Omega \to {\mathbb{R}}^3$ 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 $\Omega \subset {\mathbb{R}}^3$ with exterior normal $\nu :\partial \Omega \to {\mathbb{R}}^3$. Here $\nabla \times$ denotes the curl operator in ${\mathbb{R}}^3$. The equation describes the propagation of the time-harmonic electric field $\Re \{E(x)e^{i\omega t}\}$ in an anisotropic material with a magnetic permeability tensor $\mu (x)\in {\mathbb{R}}^{3\times 3}$ and a permittivity tensor $\epsilon (x)\in {\mathbb{R}}^{3\times 3}$. The boundary conditions are those for $\Omega$ surrounded by a perfect conductor. It is required that $\mu (x)$ and $\epsilon (x)$ are symmetric and positive definite uniformly for $x\in \Omega$, and that $\mu ,\epsilon \in L^{\infty }(\Omega ,{\mathbb{R}}^{3\times 3})$. The nonlinearity $F:\Omega \times {\mathbb{R}}^3\to \mathbb{R}$ is superquadratic and subcritical in $E$, the model nonlinearity being of Kerr-type: $F(x,E)=|\Gamma (x)E{|}^p$ for some $2<p<6$ with $\Gamma (x)\in GL(3)$ invertible for every $x\in \Omega$ and $\Gamma ,{\Gamma }^{-1}\in L^{\infty }(\Omega ,{\mathbb{R}}^{3\times 3})$. 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.

to appear in J. Funct. Anal