The aim of this entry is to derive the -boundedness for a single solution of the following PDE
over a domain . This elegant result had been done by Brezis and Merle around 1991 published in Comm. Partial Differential Equations [here].
There are two possible cases.
The case of bounded domain. Let us assume a solution of the following PDE
where is a bounded domain and is a given function on .
Theorem. If and for some then .
Proof. It first follows from the Brezis-Meler inequality that
which by the Holder inequality gives
Thus, a standard -estimate argument from the elliptic theory implies that is bounded.
The conclusion of our theorem still holds for a solution of
with and for some .
Remark. It is worth noticing that a local version of the corollary mentioned in this entry still holds. Taking into account this result still gives a local version of the theorem above.
Theorem (local version). If and for some then .
The case of the whole space. Let us now turn to the case of . Our PDE is simply
In this circumstance, we get
Theorem. Assume . If and for some then .
Proof. Fix and split as with
Let be the unit ball of radius centered at . We denote by various constants independent of . Let be the solution of
By the Brezis-Meler inequality,
and in particular
We also have
Let so that on . The mean value theorem for harmonic functions implies that
On the other hand we have
we see that
Thus we find that
Finally we write
for some . Using once more the mean value theorem and standard elliptic estimates we deduce that
Since is independent of , we conclude that .
Remark. The mean value theorem for harmonic functions says that if is harmonic then
for any point . Thus
Taking sup both sides we arrive at