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.
so that for some ball of radius . Extend to be zero outside and set, for ,
on . Note that for since . Besides,
It follows from the maximum principle that on and thus
We now use the Jensen inequality
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
We may split as with and . Write where are the solutions of
with Dirichlet boundary condition. Choosing, for example, in the theorem we find
The conclusion follows since and .