Definition 6 see [16,18]. Here w is either zero or a solution to 7 and e s is some.
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. Now we introduce the following main result.
~ Abstracts ~
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. Proposition 10 see , Proposition 3. Proposition Observe that in V1 p, e 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 .
We claim that Xi is bounded. Hence our claim follows. Lemma One then has. Proof of Theorem 9. In order to complete the construction of the pseudogradient W suggested in Theorem 9, it only. Notethatifw1 g V1 71;e ,then the pseudo-gradient W1 u1 does not increase the maximum of the A;'s, i e I1. Using Proposition 11, we have. Using 54 and 55 , we find that. These estimates yield. We need to add the remainder indices i e J2. Thus using Proposition 10, we apply. Since u2 e V2 tyI2, e , by Proposition 10, we can apply the associated vector field which we will denote by V1.
We get. Fix i0 e I2 and let. Let V2 be this vector field. By Proposition 11 we have. Thus by 55 , we get. Let V1 resp. Subset 2. In this region, the construction of the pseudo-gradient W proceeds exactly as the proof of Theorem 3. Now, arguing as in Appendix 2 of , see also Appendix B of , claim ii of Theorem 9 follows from i and Proposition 5.
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This completes the proof of Theorem 9. We prove the existence result by contradiction.
It follows from Corollary These manifolds can be easily described once a finite-dimensional reduction like the one we performed in Section 3 is established. By using a deformation lemma see [22, proposition 7. Now, arguing as in the proof of theorem of [18, pages ], 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 K'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 . Lemma 15 see [11, Section 3. Let w be a solution to 7.
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Once the existence of mixed critical points at infinity is ruled out, it follows from the previous arguments that. 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 d given in Theorem 1 is equal to zero; that is, the total sum in d 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.
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Theorem If there exists k e N such that. Let I j be the maximal index over all elements of CTO. Let U 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 U.
By transversality arguments, we can assume that such a deformation avoids all critical points at infinity of index.
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It follows then from a theorem of Bahri and Rabinowitz  that. Regarding the multiplicity result, we observe that for generic K's the functional J admits only nondegenerate critical points. Hence by Lemma 15, the set U will be deformed into. Symmetry, Integrability and Geometry. Methods and Applications, vol. Branson, "Differential operators canonically associated to a conformal structure," Mathematica Scandinavica, vol.
Djadli, E. Hebey, and M. Ledoux, "Paneitz-type operators and applications," Duke Mathematical Journal, vol. Chang and P. Yang, "Extremal metrics of zeta function determinants on S"-manifolds," Annals of Mathematics, vol. Djadli and A.
Malchiodi, "Existence of conformal metrics with constant Q-curvature," Annals of Mathematics, vol. Gursky, "The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE," Communications in Mathematical Physics, vol. Wei and X. Abdelhedi and H. Chtioui, "On the prescribed Paneitz curvature problem on the standard spheres," Advanced Nonlinear Studies, vol.
Bensouf and H.
Andrea Malchiodi: list of publications
Chtioui and A. Paris, vol. Djadli, A. Malchiodi, and M. Ahmedou, "Prescribing a fourth order conformal invariant on the standard sphere. A perturbation result," Communications in Contemporary Mathematics, vol. Felli, "Existence of conformal metrics on S" with prescribed fourth-order invariant," Advances in Differential Equations, vol. Bahri, "An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension," Duke Mathematical Journal, vol. Lin, "A classification of solutions of a conformally invariant fourth order equation in R"," Commentarii Mathematici Helvetici, vol.
Ben Ayed, Y. Chen, H. Chtioui, and M.