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 thatwhich by the Holder inequality gives

.

Therefore, if

while if

.

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