Theorem Let , satisfies
If there exists such that
If as , then in .
Proof: Without loss of generality assume on , or we can add a large constant on to make it positive on boundary. , there exists such that
Since , we have
Also on since is non-negative on boundary. Then maximum principle tells in , where . Then we can first take and then take to get in .
Remark The condition is essential, otherwise one can take plus a large constant as a counterexample.