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 satisfy
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