I just read this method in a book due to W.X. Chen and C.M. Li published by American Institute of Mathematical Sciences this year 2010.
Let
be a topological vector space. Suppose there are two extended norms (i.e. the norm of an element in
might be infinity) defined on 
.
Let

and
.
Theorem (Regularity Lifting I). Let
be a contraction map from
into itself and from
into itself. Assume that
, and that there exits a function
such that
in
. Then
also belongs to
.
Proof. Firstly, let

be a norm on
. We first show that
is a contraction. Since
is a contraction on
, there exists a constant
,
such that
.
Similarly, we can find a constant
,
such that
.
Let
. Then, for any 
.
Since
is a contraction, given
, we can find a solution
such that
. Notice that
is also a contraction and
, the solution of the equation
must be unique in
. Because both
and
are solutions of the same equation
in
, we deduce that
.
Remark. In practice, we usually choose
to be the space of distributions, and
and
to be function spaces, for instance,
and
. We start from a function
in a lower regularity space
, if we can show that
is a contraction from
to itself and from
to itself, then we can lift the regularity of
to be in
.
Applications to PDEs. Now, we explain how the Regularity Lifting Theorem proved in the previous subsection can be used to boost the regularity of week solutions involving critical exponent
.
Still assume that
is a smooth bounded domain in
with
. Let
be a weak solution of the above PDE. Then by Sobolev embedding
.
We can split the right hand side of the PDE in two parts
.
Then obviously
. Hence, more generally, we consider the regularity of the weak solution of the following equation
.
Theorem. Assume that
. Let
be any weak solution of the foregoing PDE. Then
for any
.
The proof of this theorem can be found in the book mentioned above.
Let us go back to the first PDE. Assume that
is a
weak solution. From the above theorem, we first conclude that
is in
for any
. Then by a standard regularity result known as the
-regularity for a second order uniformly elliptic operator in divergence form,
is in
. This implies that
via Sobolev embedding. Finally, by repeated applications of the Schauder estimates, we derive that
.