On the electrostatic Born-Infeld equation with extended charges

In this paper, we deal with the electrostatic Born-Infeld equation where is an assigned extended charge density. We are interested in the existence and uniqueness of the potential and finiteness of the energy of the electrostatic field . 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 is radially distributed, we recover the weak formulation of and the regularity of the solution of the Poisson equation (under the same smoothness 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 is smooth. Then we analyze the case where the density is a superposition of point charges and discuss the results in (Kiessling, Commun Math Phys 314:509-523, 2012). Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.