  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.
