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,

Hence,

This gives

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.

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