Solutions to $\Delta u\ge u^2$

Let $ (M,g)$ be a complete Riemannian manifold. Suppose that $ u$ is a nonnegative solution to $ \Delta_gu\ge u^2$ . Does it follow that $ u$ must be identically 0?

I know that the answer to above question is yes if one assumes that $ Ric(g)$ has a lower bound, which allows for a maximum principle argument, using the distance function to cut-off.

I wonder if this is true in general, with no additional assumptions?