Nonlinear time-harmonic Maxwell equations in domains

Abstract: The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\nabla\times\left(\mu(x)^{-1} \nabla\times u\right) - \omega^2\varepsilon(x)u = f(x,u)$$ for the field $u:\Omega\to\mathbb{R}^3$ in a domain $\Omega\subset\mathbb{R}^3$. Here $\varepsilon(x) \in \mathbb{R}^{3\times3}$ is the (linear) permittivity tensor of the material, and $\mu(x) \in \mathbb{R}^{3\times3}$ denotes the magnetic permeability tensor. The nonlinearity $f:\Omega\times\mathbb{R}^3\to\mathbb{R}^3$ comes from the nonlinear polarization. If $f=\nabla_uF$ is a gradient then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.