Let us consider the well known Born-Infeld equation $$-{\rm div}\left(\frac{\nabla u}{\sqrt{1-\frac{1}{b^2}|\nabla u|^2}}\right) =g(u).$$ It appears quite naturally in several fields such as electomagnetism, relativity and so on.
My question is: if we consider the previous equation involving a coefficient which depend on the solution itself $A(u)$, I mean $$-{\rm div}\left(\frac{A(u)\nabla u}{\sqrt{1-\frac{1}{b^2}|\nabla u|^2}}\right) =g(u),$$ what kinds of phenomena does it describe? Could someone please tell me something or give some references? (you can also consider, for simplicity, $b=1$).
Thank you in advance.