On the electrostatic Born-Infeld equation with extended charges

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.