On the regularity of the minimizer of the electrostatic Born-Infeld energy

We consider the electrostatic Born-Infeld energy $$int_{R^N}left(1-{sqrt{1-|nabla u|^2}}right), dx -int_{R^N} ho u, dx,$$ where $rho in L^{m}(R^N)$ is an assigned charge density, $m in [1,2_*]$, $2_*:={2N}/{N+2}$, $Ngeq 3$. We prove that if $rho in L^q(R^N) $ for $q>2N$, the unique minimizer $u_ ho$ is of class $W_{loc}^{2,2}(R^N)$. Moreover, if the norm of $ ho$ is sufficiently small, the minimizer is a weak solution of the associated PDE $$-operatorname{div}left( abla u}{sqrt{1-| abla u|^2}} ight)= ho in R^N,$$ with the boundary condition $lim_{|x| o +infty}u(x)=0$ and it is of class $C^{1,alpha}_{loc}(RN)$, for some $alpha in (0,1)$.