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

TL;DR

This paper proves the existence of ground state solutions for time-harmonic semilinear Maxwell equations in three-dimensional space with vanishing permittivity, under certain growth and norm conditions on the nonlinear and linear parts.

Contribution

It establishes the existence of ground state solutions for a class of Maxwell equations with inhomogeneous media and vanishing permittivity, extending previous results to more general nonlinearities and conditions.

Findings

Existence of ground state solutions under growth conditions on F.

Solutions exist when the L^{3/2}-norm of V is below a critical threshold.

Application to nonlinear polarization in inhomogeneous media.

Abstract

We investigate the existence of solutions $E:\mathbb{R}^3\to\mathbb{R}^3$ of the time-harmonic semilinear Maxwell equation $$\nabla\times(\nabla\times E) + V(x) E = \partial_E F(x,E) \quad \text{in}\mathbb{R}^3,$$ where $V:\mathbb{R}^3\to\mathbb{R}$, $V(x)\leq 0$ 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 $V(x)=-\mu\omega^2\varepsilon(x)$ 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 $\varepsilon$ is the linear part of the permittivity in an inhomogeneous medium.