In a previous note, we showed how to prove an boundedness of a sequence of solutions
to the following PDE
in the negative case, i.e. for some
as
. In this note, we do not change
, instead, we are going to change
. More precisely, we are interested in some
boundedness of a sequence of solutions
to the following PDE
for some with
as
. This note is adapted from a recent preprint by Michael Struwe et al., see this.
As always, we assume that is constant and
is a closed, connected Riemannian surface with smooth background metric
. We further assume for the sake of simplicity that
.
Step 1. We claim for sufficiently large that
for some constant depending only on
and on
.
Proof of Step 1. To prove this, we observe that
which implies, by , for large
that