On the Born-Infeld equation for electrostatic fields with a superposition of point charges

Abstract

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_k\delta_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ where $N\ge3$, $a_k\in\mathbb R$ for all $k=1,\dots,n$, $x_k\in\mathbb R^N$ are the positions of the point charges, possibly non symmetrically distributed, and $\delta_{x_k}$ is the Dirac delta distribution centered at $x_k$. For this problem, we give explicit quantitative sufficient conditions on $a_k$ and $x_k$ to guarantee that the minimizer of the energy functional associated to the problem solves the associated Euler-Lagrange equation. Furthermore, we provide a more rigorous proof of some previous results on the nature of the singularities of the minimizer at the points $x_k$'s depending on the sign of charges $a_k$'s. For every $m\in\mathbb N$, we also consider the approximated problem $$ -\sum_{h=1}^m\alpha_h\Delta_{2h}u=\sum_{k=1}^n a_k\delta_{x_k}\quad\mbox{in }\mathbb R^N, \qquad\lim_{|x|\to\infty}u(x)=0 $$ where the differential operator is replaced by its Taylor expansion of order $2m$, see (2.1). It is known that each of these problems has a unique solution. We study the regularity of the approximating solution, the nature of its singularities, and the asymptotic behavior of the solution and of its gradient near the singularities.