In the article , Nagasaki and Suzuki considered the Liouville-type problem
where is a smooth bounded domain, , and is a smooth function such that
Equations of the form (1.1) are of actual interest in several contexts, including turbulent Euler flows, chemotaxis, and the Nirenberg problem in geometry; see, e.g.,  and the references therein. A recent example is given by the mean field equation
which was derived in  for turbulent flows with variable intensities, where is a probability measure related to the vortex intensity distribution. In this case, setting
it is readily seen that if , then along a blow-up sequence, (1.3) is of the form (1.1). See [10, 11, 13, 12] for more details, where the existence of solutions by variational arguments and blow-up analysis are also considered. Blow-up solution sequences for (1.3) have also been recently constructed in  following the approach introduced in .
In , Nagasaki and Suzuki derived a concentration-compactness principle for (1.1), mass quantization, and the location of blow-up points, under some additional technical assumptions for f. More precisely, they assumed that
By a complex analysis approach, they established the following result.
for some , , where denotes the Green’s function for the Dirichlet problem on Ω. Furthermore, at each blow-up point , , there holds that
The original estimates in  are involved and require the technical assumption . It should be mentioned that this assumption was later weakened to the natural assumption in , by taking a different viewpoint on the line of .
Here, we are interested in revisiting the complex analysis framework introduced in . In particular, we study the effect of the lower-order terms which naturally appear when the equation is considered on a compact Riemannian 2-manifold. We observe that, although the very elaborate key -estimate obtained in , namely, Proposition 1.2 below, may be extended in a straightforward manner to the case of manifolds (see Appendix A for the details), the lower-order terms are naturally estimated only in . Therefore, we are led to consider an -framework, which turns out to be significantly simpler and which holds under the weaker assumption . As a byproduct, we obtain a quick proof of mass quantization and blow-up point location for the case .
In order to state our results, for a function , we define the quantity
Then, if u is a solution to (1.1), we have
In particular, in the Liouville case , the function is holomorphic. Therefore, the complex derivative may be viewed as an estimate of the “distance” between the equation in (1.1) and the standard Liouville equation.
We recall that the main technical estimate in  is given by the following proposition.
It is natural to expect that corresponding results should hold on a compact Riemannian 2-manifold without boundary. We show that, in fact, the -convergence as stated in Proposition 1.2 still holds true on M (see Proposition A.1 in Appendix A). However, a modified point of view is needed in order to suitably locally define a function S corresponding to (1.6), such that the lower-order terms may be controlled, as well as to prove its convergence to a holomorphic function in some suitable norm, so that the mass quantization and the location of the blow-up points may be derived. As we shall see, our point of view holds under the weaker assumption and is significantly simpler than the original -framework.
More precisely, on a compact Riemannian 2-manifold without boundary , we consider the problem
where , dx denotes the volume element on M, and denotes the Laplace–Beltrami operator. We assume that satisfies (1.2) and, moreover, that
In the spirit of , we assume that along a blow-up sequence we have
In particular, without loss of generality, we may assume that
We note that (1.9) implies that . We now define the modified quantity corresponding to . Let denote the blow-up set. Let and denote . We consider a local isothermal chart such that , , , , and . For the sake of simplicity, we identify here functions on M with their pullback functions to . We denote by the Green’s function of on B. We set
with defined by
for every ,
for every blow-up point , the function in as , where is holomorphic in B.
Consequently, we derive the following corollary.
Moreover, for all , we have the relation
We provide the proofs of Theorem 1.3 and Corollary 1.4 in Section 2. For the sake of completeness and in order to readily allow a comparison with the -framework employed in , in Appendix A we extend Proposition 1.2 to the case of Riemannian 2-manifolds without boundary.
Throughout this note, we denote by a constant whose actual value may vary from line to line.
2 Proof of Theorem 1.3
We begin by establishing the following result.
Let u be a solution to (1.7). For every , we have
We multiply the equation by . Integrating, we have
since . Using the assumptions on φ, there exists such that for , so that
and the claim follows using again the fact that . ∎
Let u be a solution to (1.7). Then, for every and for every , we have
In view of (1.8), we have
Moreover, (1.10) implies that
Then, for every , using Hölder’s inequality we have
Let . By Lemma 2.1, for
using Hölder’s inequality again, we have
Let . We denote by an isothermal chart satisfying
where denotes a coordinate system on . We consider sufficiently small so that and let . The Laplace–Beltrami operator is then mapped to the operator on , where . By we denote the Green’s function of on B, namely,
We recall from (1.12) that
with the constant defined by (1.13), namely,
where . Then, and
Let be a blow-up solution sequence for (1.7). As , in , for , and in , we have in , so that
We consider the following functions defined in B:
The complex function defined in (2.6) is holomorphic in B and in
By (2.6) we have
Then, using we compute
Using (2) we derive that
At this point, we set and and we observe that by the Cauchy integral formula we may write
and is holomorphic in B. On the other hand, we have
To prove (2.12), it is sufficient to observe that for every , we have and then
and hence, up to subsequences,
so that by (2.9),
This completes our proof. ∎
Finally, we use the following result from .
For , , the conditions , , and in imply that is harmonic in B, where is some constant and
Now, we are ready to prove Corollary 1.4. By we denote the Green’s function on the manifold M, defined by
Proof of Corollary 1.4.
where H is harmonic in B and . Then, using the fact that
By Proposition 2.3, we know that is holomorphic. Hence, we can conclude that and . Since
is harmonic in B, we have
for some harmonic function . Arguing similarly for each , we conclude that
In particular, we obtain
Now, Corollary 1.4 is completely established. ∎
A The -estimate on M
In this appendix, for the sake of completeness and in order to outline the original arguments in , so that the simplification of our -approach may be seen, we check that Proposition 1.2 may be actually extended to (1.7) on a compact Riemannian 2-manifold without boundary with minor modifications. We consider a solution sequence for (1.7). We assume that f satisfies (1.2), (1.4), and (1.5). Moreover, we assume (1.10), so that as . We show the following proposition.
Let u be a solution to (1.7). Then,
The proof relies on the following relation, due to Obata.
Let be a solution to
where F is a -function. Then, there holds the identity
where, in local coordinates,
Let u be a solution to (1.7). Then, for every , there holds
On the other hand, we have
In view of Obata’s identity (A.2), we conclude that
In particular, since
by the last inequality we obtain
and then, since , using (2.1) again we have
For , we define
Then, (A.5) may be written in the form
The proof can be easily obtained as in Lemma 2.1. Let us observe that in our assumption, for every , we have
On the other hand, since , we have
Reducing (A.6) to
and using (A.7) and the Sobolev embedding, we obtain
Moreover, we have
for (). We choose such that
Arguing as in , for every , we obtain
and, for ,
Now, we conclude the proof of Proposition A.1.
Proof of Proposition A.1.
Moreover, by (1.7) we have
Hence, for , we have
Now, observing that , by (A.11) we obtain
for every with
Hence, as , we have
Moreover, if , then
where . By (A.12), for every and since , we have
Using again (A.11), for every , we have
Then, for every , we have
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About the article
Published Online: 2015-12-08
Published in Print: 2016-02-01
The first author acknowledges the support of FP7-MC-2009-IRSES-247486 “MaNEqui”.