  In this paper, we deal with the electrostatic Born-Infeld equation
\begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} \left\{
\begin{array}{ll} -\operatorname{div}\left(\displaystyle\frac{\nabla
\phi}{\sqrt{1-|\nabla \phi|^2}}\right)= \rho, & \hbox{in } \mathbb{R}^N, \\
\displaystyle\lim_{|x|\to \infty}\phi(x)= 0, \end{array} \right. \end{equation}
where $\rho$ is an assigned extended charge density. We are interested in the
existence and uniqueness of the potential $\phi$ and finiteness of the energy
of the electrostatic field $-\nabla \phi$. We first relax the problem and treat
it with the direct method of the Calculus of Variations for a broad class of
charge densities. Assuming $\rho$ is radially distributed, we recover the weak
formulation of \eqref{eq:BI-abs} and the regularity of the solution of the
Poisson equation (under the same smootheness assumptions). In the case of a
locally bounded charge, we also recover the weak formulation without assuming
any symmetry. The solution is even classical if $\rho$ is smooth. Then we
analyze the case where the density $\rho$ is a superposition of point charges
and discuss the results in [Kiessling, Comm. Math. Phys. 314 (2012), 509--523].
Other models are discussed, as for instance a system arising from the coupling
of the nonlinear Klein-Gordon equation with the Born-Infeld theory.
