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(\mu(x){-1}\nabla\times E) - \omega2\epsilon(x) E = \partial_E F(x,E) &&\quad \text{in }\Omega\%\newline &\nu\times E = 0 &&\quad \text{on }\partial\Omega \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 $\mu(x)\in\mathbb{R}{3\times3}$ and a permittivity tensor $\epsilon(x)\in\mathbb{R}{3\times3}$. 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.
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