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

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.