It easy to prove that no non-constant positive concave function exists (for example by integrating: $ u'' \leq 0 \to u' \leq c \to u \leq cx+c_2 $ and since $u>0$ , we obviously get a contradiction.
Can this result be generalized to $ \mathbb{R}^2 $ and the laplacian? Is there an easy way to see this? (i.e.- no non-constant positive real-valued function with non-positive laplacian exists)
Thanks!