I am going to talk about uniform estimates and blow-up phenomena for solutions of

in two dimensions done by Brezis and Merle around 1991 published in *Comm. Partial Differential Equations* [here]. As a first step, I am going to derive some inequality that we need later.

Assume is bounded domain and let be a solution of

together with Dirichlet boundary condition. Here function is assumed to be of class .

**Theorem **(Brezis-Merle). For every we have

where denotes the -norm and a solution to our PDE.

*Proof*. Let

so that for some ball of radius . Extend to be zero outside and set, for ,

so that

on . Note that for since . Besides,

.

It follows from the maximum principle that on and thus

.

We now use the Jensen inequality

with

in order to estimate the RHS. We obtain

But for we have

.

And thus the proof follows.

A simple consequence of the theorem is

**Corollary**. Let be a solution of PDE with . Then for every constant

.

*Proof*. Let

.

We may split as with and . Write where are the solutions of

with Dirichlet boundary condition. Choosing, for example, in the theorem we find

and thus

.

The conclusion follows since and .

This kind of result for biharmonic operator had also been done by C.S.L [here]. It is worth noticing that this result had also been extended to the -Laplacian by Aguilar-Peral [here].

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Hi!

I think that the modulus in the left side of the 11th math line is unnecessary because in Br the used function is non negative.

Just for saving ink!

Comment by Fab — December 10, 2010 @ 21:12

Thank Fab. You are right, thus it reads as the following

Comment by Ngô Quốc Anh — December 10, 2010 @ 21:38