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 $$E_{\alpha }(\rho ):=\frac{1}{q}{\int }_{{\mathbb{R}}^d}\mathrm{\mid }\rho (x){\mathrm{\mid }}^{q}dx+\frac{1}{2}{\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{\rho (x)\rho (y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d-\alpha }}dxdy-{\int }_{{\mathbb{R}}^d}V(x)\rho (x)dx,$$ 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.