Ground States of Time-Harmonic Semilinear Maxwell Equations in $${\mathbb{R}^3}$$ 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 $$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$ where $${V:\mathbb{R}^3 \to \mathbb{R}}$$ , $${V(x) \leqq 0}$$ almost everywhere 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 the $${L^{3/2}}$$ -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 μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field $${\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}$$ and $${\varepsilon}$$ is the linear part of the permittivity in an inhomogeneous medium.