Dual formulation for constraint solutions of the multi-state Choquard equation

TL;DR

This paper introduces a dual variational formulation for the multi-state Choquard equation, enabling new insights into the existence of steady states in critical and sub-critical cases, relevant to physical phenomena.

Contribution

It presents a novel dual formulation for the variational functional of the multi-state Choquard equation, advancing the analysis of its steady states.

Findings

Established existence results for steady states in critical cases.

Extended analysis to sub-critical cases.

Provided a new mathematical framework for the Choquard equation.

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 Schr\"odinger 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 indiced 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.