Abstract and Applied Analysis

Volume 2013 (2013), Article ID 568245, 11 pages

http://dx.doi.org/10.1155/2013/568245

Research Article

Existence of Conformal Metrics with Prescribed Q-Curvature

Mohammed Ali Al-Ghamdi,¹ Hichem Chtioui,¹ and Afef Rigane²

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80230, Jeddah, Saudi Arabia

²Department of Mathematics, Faculty of Sciences of Sfax, Road Soukra, Sfax, Tunisia

Received 13 October 2012; Revised 5 February 2013; Accepted 8 February 2013

Academic Editor: Svatoslav Staněk

Copyright © 2013 Mohammed Ali Al-Ghamdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 [1]). 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 [3]. 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 [3], 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 [15] 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 [11].

Assume that for each distinct point .

Now, we introduce the following set:

The main result of this paper is the following.

Theorem 1. Assume that satisfies and , with

If

Then (7) has at least one solution.

Moreover if , for generic one has

where denotes the set of solutions to (7).

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 [15]. 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 .

Similar Morse lemma at infinity has been established for the problem (7) on the sphere , , under the hypothesis that the order of flatness at critical points of is ; see [11].

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

Our problem (7) enjoys a variational structure. Indeed, solutions to (7) correspond to positive critical points of the functional

defined on

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 [17]).

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 [15] 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 [19], 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 [16] (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 .

Proposition 5 (see [11, Lemma 3.1]). Let and let be defined in Proposition 4. One has the following estimates: there exists independent of such that the following holds:

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.

Notice that remains invariant under the flow generated by (see Lemma 5.1 of [20]; see also Lemma 4.1 of [18]).

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 [11]). 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 [11]). 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 , , .

We will prove Theorem 9 at the end of the section. Now we state two results which deal with two specific cases of Theorem 9. Let

We then have the following.

Proposition 10 (see [11], 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 [21], so the proof of Proposition 11 is similar to the corresponding statement in [21].

Before giving the proof of Theorem 9, we state the following notations extracted from [11].

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).

Proof. Using Proposition 7, we have

Observe that for , if , we have

taking large enough. If not, we have

Using the fact that defined in (45) satisfies , if , Lemma 12 follows.

Lemma 13. For , such that , , for all , one has

where is defined in (45).

Proof. Using Proposition 8, we have

Using (43) and the fact that , if , Lemma 13 follows.

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 , small enough.

Using the previous estimate and Lemma 13, we find that

Case  3. and .

Let (resp., ) be the pseudo-gradient in defined by (resp., ) where (resp., ) is the vector field defined by Proposition 11 (resp., Proposition 10) in (resp., ) and let in this case

Using Proposition 10, Proposition 11, and (55) we get

Notice that in the first and second cases, the maximum of the 's, , is a bounded function and hence the Palais-Smale condition is satisfied along the flow lines of . However in the third case all the 's, , will increase and go to along the flow lines generated by .

Subset 2. We consider the case of , such that there exist satisfying .

In this region, the construction of the pseudo-gradient proceeds exactly as the proof of Theorem 3.2, of subset 2, of [11].

Finally, observe that our pseudogradient in satisfies claim (i) of Theorem 9 and it is bounded, since and are bounded. From the definition of , the 's, , decrease along the flow lines of as long as these flow lines do not enter in the neighborhood of finite number of critical points , , of such that .

Now, arguing as in Appendix 2 of [16], see also Appendix of [18], claim (ii) of Theorem 9 follows from (i) and Proposition 5. This completes the proof of Theorem 9.

Corollary 14. Let . The critical points at infinity of in correspond to

where . Moreover, such a critical point at infinity has an index equal to .

4. Proof of Theorem 1

We prove the existence result by contradiction. Assume that has no critical point in . It follows from Corollary 14 that the critical points at infinity of the associated variational problem are in one to one correspondence with the elements of defined in (13).

Notice that, just like for usual critical points, it is associated with each critical point at infinity of stable and unstable manifolds and (see [16, pages 356-357]). These manifolds can be easily described once a finite-dimensional reduction like the one we performed in Section 3 is established.

For any , let denote the associated critical value. Here we choose to consider a simplified situation, where for any , , and thus order the 's, as

By using a deformation lemma (see [22, proposition 7.24 and Theorem 8.2]), we know that if , then

where and denotes retracts by deformation.

We apply the Euler-Poincaré characteristic of both sides of (79); we find that

where denotes the index of the critical point at infinity . Let

Since we have assumed that (4) has no solution, is a retard by deformation of . Therefore , since is a contractible set. Now using (80), we derive after recalling that

Hence if (82) is violated, has a critical point in .

Now, arguing as in the proof of theorem of [18, pages 659-660], we prove that such a critical point is positive.

To prove the multiplicity part of the statement, we observe that it follows from Sard-Smale theorem that for generic 's, the solutions to (7) are all nondegenerate, in the sense that the associated linearized operator does not admit zero as eigenvalue. We need to introduce the following lemma extracted from [11].

Lemma 15 (see [11, Section 3.2]). Let be a solution to (7). Assume that the function satisfies condition , with ; then for each , there are neither critical points nor critical points at infinity in .

Once the existence of mixed critical points at infinity is ruled out, it follows from the previous arguments that

Now using the Euler-Poincaré theorem, we derive that

Hence our theorem follows.

5. A General Existence Result

In this last section of this paper, we give a generalization of Theorem 1. Namely, in view of the result of Theorem 1, one may think about the situation where the degree given in Theorem 1 is equal to zero; that is, the total sum in is equal to 1, but a partial one is not equal to 1. A natural question arises: is it possible in this case to use such information to derive an existence result? In the following theorem we give a partial answer to this question.

Theorem 16. Let be a function satisfying and , with . If there exists such that

then the problem (7) has at least one solution.

Moreover for generic 's, if , then the number of solutions is lower bounded by

Let be the maximal index over all elements of . Please observe that the integer satisfies (1) of Theorem 16; it follows that Theorem 1 is a corollary of Theorem 16.

Proof of Theorem 16. We set

where is the closure of the unstable manifolds of . Observe that is a stratified set of top dimension , which is contractible set in , since is a contractible set. Let denote the image of such a contraction. To prove the first part of Theorem 16, arguing by contradiction, we assume that (7) has no solution. Using the pseudo-gradient constructed in Theorem 9, we can deform . By transversality arguments, we can assume that such a deformation avoids all critical points at infinity of index greater or equal to . Note that, from assumption (1) of Theorem 16, there is no critical point at infinity with index . It follows then from a theorem of Bahri and Rabinowitz [22] that

Hence from the Euler-Poincaré theorem, we get

which is a contradiction with assumption (2) of Theorem 16.

Regarding the multiplicity result, we observe that for generic 's the functional admits only nondegenerate critical points. Hence by Lemma 15, the set will be deformed into

where denotes the set of the critical points of of Morse index less than or equal to , which are dominated by . Finally by using the Euler-Poincaré theorem, the proof of Theorem 16 follows.

References

S. M. Paneitz, “A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds,” SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, vol. 4, article 036, 2008.

T. P. Branson, “Differential operators canonically associated to a conformal structure,” Mathematica Scandinavica, vol. 57, no. 2, pp. 293–345, 1985.

Z. Djadli, E. Hebey, and M. Ledoux, “Paneitz-type operators and applications,” Duke Mathematical Journal, vol. 104, no. 1, pp. 129–169, 2000.

Sun-Y.A. Chang and P. C. Yang, “Extremal metrics of zeta function determinants on -manifolds,” Annals of Mathematics, vol. 142, no. 1, pp. 171–212, 1995.

Z. Djadli and A. Malchiodi, “Existence of conformal metrics with constant -curvature,” Annals of Mathematics, vol. 168, no. 3, pp. 813–858, 2008.

M. J. Gursky, “The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE,” Communications in Mathematical Physics, vol. 207, no. 1, pp. 131–143, 1999.

J. Wei and X. Xu, “On conformal deformations of metrics on ,” Journal of Functional Analysis, vol. 157, no. 1, pp. 292–325, 1998.

W. Abdelhedi and H. Chtioui, “On the prescribed Paneitz curvature problem on the standard spheres,” Advanced Nonlinear Studies, vol. 6, no. 4, pp. 511–528, 2006.

A. Bensouf and H. Chtioui, “Conformal metrics with prescribed -curvature on ,” Calculus of Variations and Partial Differential Equations, vol. 41, no. 3-4, pp. 455–481, 2011.

H. Chtioui and A. Rigane, “On the prescribed -curvature problem on ,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 348, no. 11-12, pp. 635–638, 2010.

H. Chtioui and A. Rigane, “On the prescribed Q-curvature problem on ,” Journal of Functional Analysis, vol. 261, no. 10, pp. 2999–3043, 2011.

Z. Djadli, A. Malchiodi, and M. O. Ahmedou, “Prescribing a fourth order conformal invariant on the standard sphere. I. A perturbation result,” Communications in Contemporary Mathematics, vol. 4, no. 3, pp. 375–408, 2002.

Z. Djadli, A. Malchiodi, and M. O. Ahmedou, “Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications,” Annali della Scuola Normale Superiore di Pisa, vol. 1, no. 2, pp. 387–434, 2002.

V. Felli, “Existence of conformal metrics on with prescribed fourth-order invariant,” Advances in Differential Equations, vol. 7, no. 1, pp. 47–76, 2002.

A. Bahri, Critical Points at Infinity in Some Variational Problems, vol. 182 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1989.

A. Bahri, “An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension,” Duke Mathematical Journal, vol. 81, no. 2, pp. 323–466, 1996.

C.-S. Lin, “A classification of solutions of a conformally invariant fourth order equation in ,” Commentarii Mathematici Helvetici, vol. 73, no. 2, pp. 206–231, 1998.

M. Ben Ayed, Y. Chen, H. Chtioui, and M. Hammami, “On the prescribed scalar curvature problem on -manifolds,” Duke Mathematical Journal, vol. 84, no. 3, pp. 633–677, 1996.

E. Hebey and F. Robert, “Asymptotic analysis for fourth order Paneitz equations with critical growth,” Advances in Calculus of Variations, vol. 4, no. 3, pp. 229–275, 2011.

M. Ben Ayed and K. El Mehdi, “The Paneitz curvature problem on lower-dimensional spheres,” Annals of Global Analysis and Geometry, vol. 31, no. 1, pp. 1–36, 2007.

R. Ben Mahmoud and H. Chtioui, “Prescribing the scalar curvature problem on higher-dimensional manifolds,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 5, pp. 1857–1879, 2012.

A. Bahri and P. H. Rabinowitz, “Periodic solutions of Hamiltonian systems of three body type,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 8, no. 6, pp. 561–649, 1991.