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

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.