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

Abstract

We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_{\hat{I}}\pm (\ddot{I}):=\frac{1}{q}{\int }_{{\mathbb{R}}^d}|\ddot{I}(x){|}^{q}dx+\frac{1}{2}{\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{\ddot{I}(x)\ddot{I}(y)}{|x-y{|}^{d-\hat{I}\pm }}dxdy-{\int }_{{\mathbb{R}}^d}V(x)\ddot{I}(x)dx,$$ where $d\ge 2$, $\hat{I}\pm \in (0,d)$ 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 $\hat{I}\pm$ and $q$. If $\hat{I}\pm \in (0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $\hat{I}\pm =2$ 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.