Dual variational methods for a nonlinear Helmholtz system

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Abstract

This paper considers a pair of coupled nonlinear Helmholtz equations
$$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right) |u|^{\frac{p}{2} - 2}u,\\ -\,\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right) |v|^{\frac{p}{2} - 2}v \end{array}\right. } \end{aligned}$$
on \(\mathbb {R}^N\) where \(\frac{2(N+1)}{N-1}< p < 2^*\). The existence of nontrivial strong solutions in \(W^{2, p}(\mathbb {R}^N)\) is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.

Keywords

Nonlinear Helmholtz sytem Dual variational methods 

Mathematics Subject Classification

Primary 35J50 Secondary 35J05 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyInstitute for AnalysisKarlsruheGermany

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