# Optimal decay and regularity for a Thomas--Fermi type variational problem

## TL;DR

This paper investigates the existence, uniqueness, and qualitative properties of minimizers for a nonlocal Thomas--Fermi type energy functional, revealing how decay, positivity, and sign-changing behavior depend on parameters and potential decay.

## Contribution

It provides new results on existence, regularity, and decay of minimizers for a nonlocal energy functional, including conditions for positivity and sign-changing behavior.

## Key findings

- Minimizers exist and are unique under broad conditions.
- Decay at infinity depends on parameters α and q.
- Sign-changing minimizers can occur with fast-decaying potentials.

## Abstract

We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_\alpha(\rho):=\frac{1}{q}\int_{\mathbb{R}^d}|\rho(x)|^q dx+\frac{1}{2}\iint_{\mathbb{R}^d\times\mathbb{R}^d}\frac{\rho(x)\rho(y)}{|x-y|^{d-\alpha}}dx dy-\int_{\mathbb{R}^d}V(x)\rho(x)dx,$$ where $d\ge 2$, $\alpha\in (0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non--trivial way on the choice of $\alpha$ and $q$. If $\alpha\in (0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $\alpha=2$ where the global minimizer has typically compact support. We also show that if $V$ decays sufficiently fast the global minimizer is sign--changing even if $V$ is non--negative. In such regimes we establish a relation between the positive part of the global minimizer and the support of the minimizer of the energy, constrained on the non--negative functions. Our study is motivated by recent models of charge screening in graphene, where sign--changing minimizers appear in a natural way.