Coulomb law in the nonuniform Euler-Heisenberg theory

We consider the nonlinear classical field theory which results from adding to the Maxwell's Lagrangian the contributions from the weak-field Euler-Heisenberg Lagrangian and a nonuniform part which involves derivatives of the electric and magnetic fields. We focus on the electrostatic case where the magnetic field is set to zero, and we derive the modified Gauss law, resulting in a higher-order differential equation. This equation gives the electric field produced by stationary charges in the higher-order nonlinear electrodynamics. Specializing for the case of a point charge, we investigate the solutions of the modified Gauss law and calculate the correction to the Coulomb law.