Abstract
We investigate the existence of solutions \({E:\mathbb{R}^3 \to \mathbb{R}^3}\) of the time-harmonic semilinear Maxwell equation
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.
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Communicated by P. Rabinowitz
The study was supported by research Grant NCN 2013/09/B/ST1/01963.
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Mederski, J. Ground States of Time-Harmonic Semilinear Maxwell Equations in \({\mathbb{R}^3}\) with Vanishing Permittivity. Arch Rational Mech Anal 218, 825–861 (2015). https://doi.org/10.1007/s00205-015-0870-1
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DOI: https://doi.org/10.1007/s00205-015-0870-1