Ground States of Time-Harmonic Semilinear Maxwell Equations in (Formula presented.) with Vanishing Permittivity

Abstract

We investigate the existence of solutions $${E:\mathbb{R}^3 \to \mathbb{R}^3}$$E:R3→R3 of the time-harmonic semilinear Maxwell equation$$abla \times (abla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$∇×(∇×E)+V(x)E=∂EF(x,E)inR3where $${V:\mathbb{R}^3 \to \mathbb{R}}$$V:R3→R, $${V(x) \leqq 0}$$V(x)≦0 almost everywhere on $${\mathbb{R}^3}$$R3, $${abla \times}$$∇× denotes the curl operator in $${\mathbb{R}^3}$$R3 and $${F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}$$F:R3×R3→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 $${L^{3/2}}$$L3/2 -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and $${V(x) = -\mu\omega^2 \varepsilon(x)}$$V(x)=-μω2ε(x) where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field $${\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}$$R{E(x)eiωt} and $${\varepsilon}$$ε is the linear part of the permittivity in an inhomogeneous medium.