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 $E_{\alpha }\left(\rho \right)≔\frac{1}{q}{\int }_{R^d}{\left|\rho \left(x\right)\right|}^{q}dx+\frac{1}{2}{\iint }_{R^d\times R^d}\frac{\rho \left(x\right)\rho \left(y\right)}{{|x-y|}^{d-\alpha }}dxdy-{\int }_{R^d}V\left(x\right)\rho \left(x\right)dx,$where $d\in [2,\infty )$, $\alpha \in (0,d)$, $q\in \left(\frac{2d}{d+\alpha },\infty \right)$ 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 to the cone of non-negative functions. Our study is motivated by recent models of charge screening in graphene, where sign-changing minimizers appear in a natural way.