Uniqueness of Ground State Solutions for a Defocusing Hartree Equation via Inverse Optimal Problems

TL;DR

This paper proves the existence and uniqueness of ground state solutions for a defocusing Hartree equation using an inverse optimal problem approach, providing new insights into nonlocal Schrödinger operators.

Contribution

It introduces an inverse optimal problem method to establish uniqueness and existence of solutions for a nonlocal Hartree equation, advancing analytical techniques in this area.

Findings

Existence of ground state solutions confirmed.

Uniqueness of ground state solutions established.

Continuous dependence on parameters demonstrated.

Abstract

We study a generalized defocusing Hartree equation with nonlocal exchange potential and repulsive Hartree--Fock interaction. Using an inverse optimal problem (IOP) approach, we prove the existence and uniqueness of ground state solutions. Additionally, we establish the existence of principal solutions, their continuous dependence on parameters, and a dual variational formulation. The IOP method provides a systematic framework for addressing inverse problems in nonlocal Schr\"{o}dinger operators and offers new insights into the structure of solutions for defocusing Hartree-type equations.