Suppose we are in the following loosely described setting: we have a non-negative supersolution $h$ of the following elliptic equation: \begin{equation} \Delta h + \|\nabla h\|^2 + f(x) \geq 0 \end{equation} Having all the needed regularity, is it possible to control \begin{equation} R^{n-1} \int_{B_{R}} \|\nabla h\| \end{equation} from above in terms of $\inf_{B_R}h $ and, perhaps, a correction term coming from $f$ and some operator data? In other words - a question for a weak Harnack-type inequality for the gradient.

Any comments, suggestions and references are welcome. Many thanks!

Suppose we are in the following loosely described setting: we have a non-negative supersolution $h$ of the following elliptic equation: \begin{equation} \Delta h + \|\nabla h\|^2 + f(x) \geq 0 \end{equation} Having all the needed regularity, is it possible to control \begin{equation} R^{1-n} \int_{B_{R}} \|\nabla h\| \end{equation} from above in terms of $\inf_{B_R}h $ and, perhaps, a correction term coming from $f$ and some operator data? In other words - a question for a weak Harnack-type inequality for the gradient.

Any comments, suggestions and references are welcome. Many thanks!