I came across the following quasilinear PDE: $$ \nabla \cdot \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) = 0. $$ This is almost the minimal surface equation, except that there is a minus sign in the denominator instead of a plus sign. I believe this is the equation with the ansatz of static conservative field in the Maxwell-Born-Infeld theory in electromagnetism.
Imposing circular symmetry in 2D (if I am not mistaken), I get the following ODE for $E := \partial_r u$: $$ \partial_r E = - \frac E r (1-E^2), $$ from which I get the solution $\frac 1 {\sqrt{c r^2 +1}}$, with $c > 0$.
My question is, given that solutions to the circular symmetric problem are nice, do we have an existence / regularity theory for large data, say, for the Dirichlet problem with boundary data on a circle in 2D? (I guess for small data this should be obtained by standard methods).
Any help would be much appreciated.