caa a22 4500 605516022 CHVBK 20210128100712.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00205-015-0870-1 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0870-1 Mederski Jarosław Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland aut Ground States of Time-Harmonic Semilinear Maxwell Equations in $${\mathbb{R}^3}$$ R 3 with Vanishing Permittivity [Elektronische Daten] [Jarosław Mederski] We investigate the existence of solutions $${E:\mathbb{R}^3 \to \mathbb{R}^3}$$ E : R 3 → R 3 of the time-harmonic semilinear Maxwell equation $$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$ ∇ × ( ∇ × E ) + V ( x ) E = ∂ E F ( x , E ) in R 3 where $${V:\mathbb{R}^3 \to \mathbb{R}}$$ V : R 3 → R , $${V(x) \leqq 0}$$ V ( x ) ≦ 0 almost everywhere on $${\mathbb{R}^3}$$ R 3 , $${\nabla \times}$$ ∇ × denotes the curl operator in $${\mathbb{R}^3}$$ R 3 and $${F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}$$ F : R 3 × R 3 → 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 $${L^{3/2}}$$ L 3 / 2 -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and $${V(x) = -\mu\omega^2 \varepsilon(x)}$$ V ( x ) = - μ ω 2 ε ( x ) whereμ>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 ) e i ω t } and $${\varepsilon}$$ ε is the linear part of the permittivity in an inhomogeneous medium. The Author(s), 2015 Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 825-861 0003-9527 218:2<825 2015 218 205 https://doi.org/10.1007/s00205-015-0870-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0870-1 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Mederski Jarosław Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 825-861 0003-9527 218:2<825 2015 218 205