I found the following simple result in a paper due to F. Lin and J.C. Wei published in *Commun. Pure Applied Math. *this year [here].

Lemma 4.2. Let satisfywhere satisfies

Then

.

*Proof*.

By the Poisson formula,

Because of the grown of , it is known that as . We construct suitable supersolutions on . Then the result will follow from the maximum principle.

In fact, let

where and the parameters are chosen so that

.

Then simple computations show that