Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential

TL;DR

This paper proves the radial symmetry of minimizers for a Hartree equation with Coulomb potential by modifying the reflection method and employing Pohozaev identities, addressing challenges posed by the nonlocal term's sign.

Contribution

It introduces a modified reflection method combined with Pohozaev identities to establish symmetry of minimizers in Hartree equations with Coulomb potential, overcoming previous difficulties.

Findings

Minimizers are radially symmetric.

The modified reflection method is effective for nonlocal terms.

Pohozaev identities aid in symmetry proof.

Abstract

In the present article we study the radial symmetry of minimizers of the energy functional, corresponding to the repulsive Hartree equation in external Coulomb potential. To overcome the difficulties, resulting from the "bad" sign of the nonlocal term, we modify the reflection method and then, by using Pohozaev integral identities we get the symmetry result.