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Session 38. Variational Methods in Nonlinear Analysis
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Ground states of time-harmonic semilinear Maxwell equations |
Jarosław Mederski, Nicolaus Copernicus University, Poland
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We investigate the existence and the nonexistence of solutions \(E:\mathbb{R}^3\to\mathbb{R}^3\) of the time-harmonic semilinear Maxwell equation
\[\nabla\times(\nabla\times E) + V(x) E = \partial_E F(x,E) \hbox{ in }\mathbb{R}^3\]
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 in homogeneous
medium.
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References- T. Bartsch, J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain , to appear in Arch. Ration. Mech. Anal. (2014).
- J. Mederski Ground states of time-harmonic semilinear Maxwell equations in \(\mathbb{R ^3}\) with vanishing permittivity, arXiv:1406.4535.
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