We consider the minimal surface equation $$ (1+|\nabla u|^2)\Delta u=\sum_{i,j=1}^n\partial_iu\partial_ju\partial_{ij}u\quad\hbox{in $B_1\subset\mathbb R^n.$} $$$$ (1+|\nabla u|^2) \, \Delta u=\sum_{i,j=1}^n\partial_iu \, \partial_ju \, \partial_{ij}u\quad\hbox{in $B_1\subset\mathbb R^n.$} $$ If $u\in C^2(B_1)$ is a positive solution of above equation, does the Harnack inequality hold in $B_{1/2}$? That is, is there a constant $C>0$, which is only dependent on $n$, such that $$\sup_{B_{1/2}}u\leq C\inf_{B_{1/2}}u?$$
If above equation is uniformly elliptic, then Harnack inequality for the uniformly elliptic equation concludes that the constant $C$ may be dependent on the elliptic constants. However, can we derive the Harnack inequality for the minimal surface equation without the uniformly elliptic condition?
