Research Article | Open Access
Existence of Conformal Metrics with Prescribed Q-Curvature
We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere . Under the assumption that the order of flatness at critical points of prescribed Q-curvature function is , we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.
1. Introduction and Main Result
The Paneitz operator on -Riemannian manifold is a fourth-order differential operator which arises in conformal geometry and satisfies a certain covariance property (see ). For it is defined by where Such a is a fourth-order invariant called Q-curvature or Paneitz curvature. See [2, 3] for details about the properties of .
If is a conformal metric to , where is a smooth positive function, then the conformal covariance property of the Paneitz operator reads as follows: If one prescribes the Q-curvature for the metric by a function , this leads to the following equation: The literature for the existence of solutions of the prescribed Q-curvature problem on compact manifolds is considerably bigger. In [4–7], existence results for the constant Q-curvature problem in 4-dimensional manifolds are given. On manifolds of dimension greater than 4, existence results were given for Einstein manifolds in . On the sphere , we refer to results of [8–14] and the references therein.
In this paper we continue to study the problem of prescribing Q-curvature on the standard sphere , . The Paneitz operator in this case is coercive on the Sobolev space and has the following expression: where According to problem (4), the problem can be formulated as follows. Given a smooth function on , we look for solutions of The special case where the manifold is a sphere endowed with its standard metric deserves particular attention. Indeed due to Kazdan-Warner type obstructions, see , conditions have to be imposed on the function .
To state our result we set up the following conditions and notation.
Assume that , be a positive function such that for any critical point of , there exists some real number such that in some geodesic normal coordinate system centered at (through the exponential map), we have where , , and as tends to . Here denotes all possible derivatives of order and is the integer part of . For each critical point of , we denote Let, For each -tuple, of distinct points such that for all , , we define a symmetric matrix by where and is the Green function for the operator on . Here is the first component of in some geodesic normal coordinate system.
Let be the least eigenvalue of . It was first pointed out by Bahri  that when the interaction between the different bubbles is of the same order as the self interaction, the least eigenvalue of some matrices like (11) plays a fundamental role in the existence of solutions to problems like (7). Regarding problem (7), such kind of phenomenon appears under condition in ; see .
Assume that for each distinct point .
Now, we introduce the following set: The main result of this paper is the following.
Our argument uses a careful analysis of the lack of compactness of the Euler Lagrange functional associated with problem (7). Namely, we study the noncompact orbits of the gradient flow of the so-called critical points at infinity following the terminology of Bahri . These critical points at infinity can be treated as usual critical points once a Morse lemma at infinity is performed from which we can derive just as in the classical Morse theory the difference of topology induced by these noncompact orbits and compute their Morse index. Such a Morse lemma at infinity is obtained through the construction of suitable pseudogradient for which the Palais-Smale condition is satisfied along the decreasing flow lines, as long as these flow lines do not enter the neighborhood of a finite number of critical points of such that .
The rest of this paper is organized as follows. In Section 2, we set up the variational problem and we recall the expansion of the gradient of the associated Euler-Lagrange functional near infinity. In Section 3, we characterize the critical points at infinity of the associated variational problem. Section 4 is devoted to the proof of the main result Theorem 1, while we give in Section 5 a more general statement than Theorem 1.
2. General Framework and Some Known Facts
2.1. Variational Structure and Lack of Compactness
However, it is delicate from a variational viewpoint because the functional does not satisfy the Palais-Smale condition. This means that there exist sequences along which is bounded, its gradient goes to zero and which not converge. The analysis of sequences failing Palais-Smale condition can be analyzed along the ideas introduced in [16, pages 325 and 334]. In order to describe such a characterization in our case, we need to introduce some notations. For and , let where is the geodesic distance on and is chosen so that is the family of solutions of the problem (see ).
We define now the set of potential critical points at infinity associated with the function . For and , let us define Here, .
For a solution to (7) we also define as If is a function in , one can find an optimal representation, following the ideas introduced in Proposition 5.2 of  and [16, pages 348–350]. Namely, we have the following.
Proposition 2. For any , there is such that if and , then the following minimization problem has a unique solution , up to a permutation.
In particular, we can write as follows: where belongs to and it satisfies , and are the tangent spaces at of the unstable and stable manifolds of for a decreasing pseudogradient of , and is the following: Here, and denotes the scalar product defined on by Notice that Proposition 2 is also true if we take and, therefore, and in .
We are ready now to state the characterization of the sequences failing the Palais-Smale condition. For technical reasons, we introduce the following subsets of . Let and let for positive very small, where . Using the idea introduced in [16, 18], see also , we have the following proposition.
Proposition 3. Let be a sequence in such that is bounded and goes to zero. Then there exists an integer , a sequence , tends to zero, and an extracted subsequence of 's, again denoted , such that where is zero or a solution to (7).
Now arguing as in  (pages 326, 327, and 334), we have the following Morse lemma which completely gets rid of the -contributions and shows that it can be neglected with respect to the concentration phenomenon.
Proposition 4. There is a -map which to each such that belongs to associates such that is unique and satisfies Moreover, there exists a change of variables such that
The following proposition gives precise estimates of .
At the end of this subsection, we give the following definition.
Definition 6 (see [16, 18]). A critical point at infinity of on is a limit of a flow line of the equation such that remains in for . Here is either zero or a solution to (7) and is some positive function tending to zero when . Using Proposition 2, can be written as Denoting , we denote by such a critical point at infinity. If , it is called of -type or mixed type.
2.2. Expansion of the Gradient of the Functional
In this subsection, we recall the expansion of the gradient of the functional in , .
Proposition 7 (see ). For any in , the following expansion holds
(i) where .
(ii) If , and is a positive constant small enough, one has
(iii) Furthermore if , for very small, one then has where .
Proposition 8 (see ). Letting , one has
(i) where .
(ii) If , , one has where and is the component of in some geodesic normal coordinates system.
3. Characterization of the Critical Points at Infinity
This section is devoted to the characterization of the critical points at infinity in , under -flatness condition with . This characterization is obtained through the construction of a suitable pseudogradient at infinity for which the Palais-Smale condition is satisfied along the decreasing flow-lines as long as these flow lines do not enter in the neighborhood of finite number of critical points of such that .
Now we introduce the following main result.
Theorem 9. Assume that satisfies and , .
Let . For , there exists a pseudogradient in so that the following holds.
There exists a constant independent of such that Furthermore is bounded and the only case where the maximum of the 's is not bounded is when with , , .
Proposition 10 (see , Proposition 3.7). For there exists a pseudogradient in such that for all , one has where is a positive constant independent of . Furthermore, one has which is bounded and the only case where the maximum of ’s is not bounded is when , for all , with .
Proposition 11. For , there exists a pseudogradient in so that the following holds.
There exists independent of such that Furthermore is bounded and the only case where the maximum of the 's is not bounded is when with , for all , and for all .
Observe that in the interaction of two bubbles is negligible with respect to the self-interaction. Similar phenomena occur for the scalar curvature problem, see , so the proof of Proposition 11 is similar to the corresponding statement in .
Let be a fixed positive constant large enough and let such that , , for all . For any index , , we define the following vector fields: We claim that is bounded. Indeed, the claim is trivial if . If , for any , , such that , we have the following estimate: Hence our claim follows. Next, we will say that and we will denote by the index satisfying It easy to see that if , then .
Now, we introduce the following two lemmas.
Lemma 12. Let , such that , , for all . One then has where is defined in (45).
Lemma 13. For , such that , , for all , one has where is defined in (45).
Proof of Theorem 9. In order to complete the construction of the pseudogradient suggested in Theorem 9, it only remains (using Propositions 10 and 11) to focus attention on the two following subsets of .
Subset 1. We consider here the case of such that Without loss of generality, we can assume that We distinguish three cases.
Case 1. = , , for all and for all .
Let be the pseudogradient on defined by , where is the vector filed defined by Proposition 11 in . Note that if , then the pseudo-gradient does not increase the maximum of the 's, . Using Proposition 11, we have An easy calculation yields
Fix , we denote by Using (54) and (55), we find that From another part, by Lemma 12 we have Observe that using a direct calculation, we have Since for , we have and for and we have , we obtain . These estimates yield It is easy to see that for any index , we have where is defined in (45) and is large enough. Thus, we derive that
Let be small enough; using Lemma 13, (63), and (55), we get and by (57) we obtain We need to add the remainder indices . Note that . Thus using Proposition 10, we apply the associated vector field which we will denote by . We then have the following estimate: since for and .
Let in this case .
From (65) and (66) we obtain
Case 2. and , , , for all and .
Since , by Proposition 10, we can apply the associated vector field which we will denote by . We get Observe that does not increase the maximum of the 's, , since . Fix and let Using (68) and (55), we get We need to add the indices , . Letting , since , we can apply the associated vector field given by Proposition 10. Let be this vector field. By Proposition 11 we have Observe that and we are in the case where for all , we have . Thus by (55), we get and hence Let in this case