Deane Yang, thank you for your answer. I have one more question. The local version of this gradient estimate imply a Harnack inequality: for $x,y \in B(R)$ $u(x) \leq C_1 u(y) e^{C_2 R \sqrt{K+\lambda}}.$ Is it possible to obtain a explicit Harnack inequality like this for subsolutions: $\Delta u + \lambda u \geq 0?$