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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair

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Volume 19, Issue 3


Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations

Rainer MandelORCID iD: https://orcid.org/0000-0003-3864-6360
Published Online: 2019-06-13 | DOI: https://doi.org/10.1515/ans-2019-2050


We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions are mildly localized as they lie in the essential spectrum of the corresponding linear operator. As a new auxiliary tool a limiting absorption principle for the curl-curl operator is proved.

Keywords: Nonlinear Helmholtz Equations; Curl-Curl Equations; Limiting Absorption Principles; Herglotz Waves

MSC 2010: 35Q60; 35Q61; 35J91


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About the article

Received: 2018-11-20

Revised: 2019-05-07

Accepted: 2019-05-08

Published Online: 2019-06-13

Published in Print: 2019-08-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: CRC 1173

The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 “Wave phenomena: analysis and numerics”.

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 3, Pages 569–593, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2019-2050.

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