The electrostatic Born-Infeld equations with integrable charge densities

We study the minimizer of the electrostatic Born–Infeld energy \begin{equation*} \int_{$\mathbb{R}$^n}1-\sqrt{1-|D v|^2}
dx-\int_{$\mathbb{R}$^n}\rho v\ dx, \end{equation*} which vanishes at infinity. We show that the minimizer $u$ is strictly spacelike and it is a weak solution to \begin{equation*}-\operatorname{div}\Big(\frac{D u}{\sqrt{1-|D u|^2}}\Big)=\rho,\end{equation*} provided that $\rho$ is in the dual space of the solution space and $\rho \in L^p({\mathbb{R}}^n)$, for some $p>n\ge 3$. Moreover, we have $u\in C^{1,\alpha }({\mathbb{R}}^n)$ for some $\alpha \in (0,1)$.