In 1993, Ding and Liu announced their result about the multiplicity of solutions to the prescribed Gaussian curvature on compact Riemannian -manifolds of negative Euler characteristic. Their interesting result then published in the journal Trans. Amer. Math. Soc. in 1995, see this link.
Roughly speaking, by starting with the prescribed Gaussian curvature equation, i.e.
when the Euler characteristic is negative, i.e.
they perturbed using
where is a real number and the candidate function is assumed to be
Then they were interested in solving the following PDE
Their main result can be stated as follows:
Theorem (Ding-Liu). There exists a such that
- the PDE has a unique solution for ;
- the PDE has at least two solutions if ; and
- the PDE has at least one solution when .
As always, the first solution was found as local minimizes of the energy functional associated to the PDE, i.e.
To find the second solution, they used the mountain pass lemma. As such, one of the key steps is to check whether or not the functional satisfies the Palais-Smale condition. Loosely speaking, if is any sequence in such that
- for some ; and
- in the dual space of , then
admits a convergent subsequence. To achieve this goal, they proved that the sequence is locally -bounded in the open set
where they denoted . However, there was a sign problem in their paper which is recently pointed out by Michael Struwe and his co-authors in their preprint entitled “Large” conformal metrics of prescribed Gauss curvature on surfaces of higher genus. To be exact, instead of fixing the mistake by Ding and Liu i.e. fixing , they consider a sequence of solution of the PDE when is changing. However, their technique also works for the Ding and Liu situation. And in this note, we show how to fix the error.
To do so, we let and fix a smooth cut-off function supported in and with on . Then let . Using as a test function, it follows from that
Here and in sequel we use to denote various constants depending only on . Note that
and here is the mistake in the Ding and Liu paper because they used a plus sign on the right hand side. Then there exists some small such that
in . Thanks to for any , therefore, we obtain the following estimate
Recalling that and using the Young inequality, we can bound
it is easy to see that the following
holds. Hence and consequently, is -bounded in . The proof is complete.