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
.
Therefore, if
while if
.
Thus, a standard
-estimate argument from the elliptic theory implies that
is bounded.