Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and

$\Delta u + \lambda u = 0,$

where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's book (Harmonic functions and applications to complete manifolds): Suppose

$Ric \geq -K.$

Then

$ | \nabla \log u |^2 \leq \frac{(n-1)K}{2} - \lambda + \sqrt{\frac{(n-1)^2K^2}{4} - (n-1) \lambda K}.$

My question is the following: if we assume

$\Delta u + \lambda u \geq 0$

do we still have a similar gradient estimate?

Note that by de Giorgi-Nash-Moser theory we still have (non-sharp) a Harnack inequality.

Thanks!