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
We now multiply our PDE by and integrate by parts to find
By the Young inequality, we can bound
Thus, the claim follows.
Step 2. There holds
Proof of Step 2. A proof for this claim is quite obvious since for some . Else, we can use the Jensen and Holder inequalities as follows
Step 3. There exists a uniform constant such that
Proof of Step 3. This is somehow a Poincare type inequality with a modified weight rather than . However, since is again a volume form, this conclusion follows.
Step 4. The sequence is bounded in the following sense
for some where is the energy given as follows
Proof of Step 4. Clearly,
Furthermore, we know from that
for some constant . Hence we can select a sufficiently large constant in such a way that
Using Step 3 and Cauchy-Schwartz inequality, we can verify that
and our claim follows.
Using Step 1, we can conclude that the sequence of solution is somehow bounded in if the energies are also bounded.