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

We consider the electrostatic Born-Infeld energy egin{equation*} int_{R^N}left(1-{sqrt{1-| abla u|^2}} ight), dx -int_{R^N} ho u, dx, end{equation*} where $ ho in L^{m}(R^N)$ is an assigned charge density, $m in [1,2_*]$, $2_*:=rac{2N}{N+2}$, $Ngeq 3$. We prove that if $ ho 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 egin{equation}label{eq:BI-abs} ag{$mathcal{BI}$} -operatorname{div}left(displaystylerac{ abla u}{sqrt{1-| abla u|^2}} ight)= ho quadhbox{in }mathbb{R}^N, end{equation} with the boundary condition $lim_{|x| oinfty}u(x)=0$ and it is of class $C^{1,alpha}_{loc}(RN)$, for some $alpha in (0,1)$.