Ground states of time-harmonic semilinear Maxwell equations in R^3 with vanishing permittivity

Abstract

We investigate the existence of solutions $E:{\mathbb{R}}^3\to {\mathbb{R}}^3$ of the time-harmonic semilinear Maxwell equation $$\nabla \times (\nabla \times E)+V(x)E={\partial }_{E}F(x,E)\text{in}{\mathbb{R}}^3,$$ where $V:{\mathbb{R}}^3\to \mathbb{R}$, $V(x)\le 0$ a.e. on ${\mathbb{R}}^3$, $\nabla \times$ denotes the curl operator in ${\mathbb{R}}^3$ and $F:{\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R}$ is a nonlinear function in $E$. In particular we find a ground state solution provided that suitable growth conditions on $F$ are imposed and $L^{3/2}$-norm of $V$ is less than the best Sobolev constant. In applications $F$ is responsible for the nonlinear polarization and $V(x)=-\mu {\omega }^2\epsilon (x)$ where $\mu >0$ is the magnetic permeability, $\omega$ is the frequency of the time-harmonic electric field $\Re \{E(x)e^{i\omega t}\}$ and $\epsilon$ is the linear part of the permittivity in an inhomogeneous medium.

arXiv commentSome slight errors and misprints have been corrected in comparison to the published article, in particular Claim 2 of Lemma 5.2 has been modified, and an additional explanation in Claim 3 has been provided