We consider the following PDE
via the potential theory, we has already proved that
As such, the analysis of turns out to be the core of the studying of solutions to our PDE. As in this entry, we showed that the following limit
exists for certain function . Not just the behavior at the infinity, as a question proposed also in that entry, we can control the decay rate of
i.e. we need the fact
for some positive constant where is a particular solution to
I do think this result is correct since it has been used once in a paper by X.X. Chen published in Calc. Var. Partial Differential Equations [here] but some idea is involved. I leave here as my own open question needed to be addressed in the future.