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It is shown that for each finite number of Dirac measures supported at points$s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, thereexists a unique real-valued Lipschitz function $u$, vanishing at infinity,which distributionally solves the quasi-linear elliptic partial differentialequation of divergence form$-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2})=4\pi\sum_{n=1}^N a_n\delta_{s_n}$. Moreover, $u$ is real analytic away from the $s_n$. The resultcan be interpreted in at least two ways: (a) for any number of point charges ofarbitrary magnitude and sign at prescribed locations $s_n$ in three-dimensionalEuclidean space there exists a unique electrostatic field which satisfies theMaxwell-Born-Infeld field equations smoothly away from the point charges andvanishes as $|s|\to\infty$; (b) for any number of integral mean curvaturesassigned to locations $s_n$ there exists a unique asymptotically flat, almosteverywhere space-like maximal slice with point defects of Minkowski spacetime,having lightcone singularities over the $s_n$ but being smooth otherwise, andwhose height function vanishes as $|s|\to\infty$. No struts between the pointsingularities ever occur.

On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4\pi\sum_k a_k \delta_{s_k}$ in Physics and Geometry