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Dual formulation of constrained solutions of the multi-state Choquard equation

Part of: General mathematical topics and methods in quantum theory Elliptic equations and systems

Published online by Cambridge University Press:  13 September 2024

Gershon Wolansky
Gershon Wolansky*
Affiliation:
Department of Mathematics Technion, Israel Institute of Technology, Haifa, Israel (gwolan1@gmail.com)
*
*Corresponding author
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Abstract

The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schrödinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinear optics.

In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wave functions.

In this article, we introduce a dual formulation for the variational functional in terms of the potential induced by the wave function, and use it to explore the existence of steady states of a multi-state version the Choquard equation in critical and sub-critical cases.

Keywords

Shrodinger-PoissonChoquard equationvariational methods

MSC classification

Primary: 35J10: Schrödinger operator
Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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