## 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?