Multiplicity results for the prescribed scalar curvature on low spheres

Ben Ayed, Mohamed; Ould Ahmedou, Mohameden

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 609-634.

In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres ${𝕊}^3,{𝕊}^4$. Under generic conditions we establish some Morse Inequalities at Infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments we derive a new existence result on ${𝕊}^4$ through an Euler-Hopf type formula.

[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equatins satisfying general boundary value conditions, I, Comm. Pure Appl. Math. 12 (1959), 623-727. | MR | Zbl

[2] A. Ambrosetti, J. Garcia Azorero and A. Peral, Perturbation of $-\Delta u+u^{\frac{(N+2)}{(N-2)}}=0$, the Scalar Curvature Problem in ${ℝ}^N$ and related topics, J. Funct. Anal. 165 (1999), 117-149. | MR | Zbl

[3] T. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296. | MR | Zbl

[4] T. Aubin, Meilleures constantes de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 32 (1979), 148-174. | MR | Zbl

[5] T. Aubin, “Some nonlinear problems in Riemannian geometry”, Springer Monographs Math., Springer Verlag, Berlin, 1998. | MR | Zbl

[6] T. Aubin and A. Bahri, Méthodes de topologie algébrique pour le problème de la courbure scalaire prescrite. (French) [Methods of algebraic topology for the problem of prescribed scalar curvature], J. Math. Pures Appl. 76 (1997), 525-849. | MR | Zbl

[7] T. Aubin and A. Bahri, Une hypothèse topologique pour le problème de la courbure scalaire prescrite. (French) [A topological hypothesis for the problem of prescribed scalar curvature], J. Math. Pures Appl. 76 (1997), 843-850. | MR | Zbl

[8] T. Aubin and E. Hebey, Courbure scalaire prescrite, Bull. Sci. Math. 115 (1991), 125-132. | MR | Zbl

[9] A. Bahri, “Critical points at infinity in some variational problems”, Pitman Res. Notes Math. Ser. Longman Sci. Tech. Harlow, Vol. 182, 1989. | MR | Zbl

[10] A. Bahri, An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimension, A celebration of J. F. Nash Jr., Duke Math. J. 81 (1996), 323-466. | MR | Zbl

[11] A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of topology of the domain, Comm. Pure Appl. Math. 41 (1988), 255-294. | MR | Zbl

[12] A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal. 95 (1991), 106-172. | MR | Zbl

[13] A. Bahri and P. H. Rabinowitz, Periodic solutions of $3$-body problems, Ann. Inst. H. Poincaré Anal. Non linéaire. 8 (1991), 561-649. | Numdam | MR | Zbl

[14] M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J. 84 (1996), 633-677. | MR | Zbl

[15] Ben Ayed, H. Chtioui and M. Hammami, The scalar curvature problem on higher dimensional spheres, Duke Math. J. 93 (1998), 379-424. | MR | Zbl

[16] H. Brezis and J. M. Coron, Convergence of solutions of $H$-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 (1985), 21-56. | MR | Zbl

[17] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297. | MR | Zbl

[18] S. A. Chang, M. J. Gursky and P. Yang, The scalar curvature equation on 2 and 3 spheres, Calc. Var. Partial Differential Equations 1 (1993), 205-229. | MR | Zbl

[19] K. C. Chang and J. Q. Liu, On Nirenberg's problem, Internat. J. Math. (1993), 53-58. | MR | Zbl

[20] S. A. Chang and P. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math. 159 (1987), 215-259. | MR | Zbl

[21] S. A. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J. 64 (1991), 27-69. | MR | Zbl

[22] C. C. Chen and C. S. Lin, Estimates of the scalar curvature via the method of moving planes I, Comm. Pure Appl. Math. 50 (1997), 971-1017. | MR | Zbl

[23] C. C. Chen and C. S. Lin, Estimates of the scalar curvature via the method of moving planes II, J. Differential Geom. 49 (1998), 115-178. | MR | Zbl

[24] C. C. Chen and C. S. Lin, Prescribing the scalar curvature on $S^n$, I. A priori estimates, J. Differential Geom. 57 (2001), 67-171. | MR | Zbl

[25] H. Chtioui and M. Ould Ahmedou, Conformal metrics of prescribed scalar curvature on $4-$manifolds: The degree zero case, Preprint 2008.

[26] A. Dold, “Lectures on algebraic topology”, Springer Verlag, Berlin, 1995. | MR | Zbl

[27] J. Escobar and R. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math. 86 (1986), 243-254. | MR | Zbl

[28] E. Hebey, Changements de metriques conformes sur la sphere, le problème de Nirenberg, Bull. Sci. Math. 114 (1990), 215-242. | MR | Zbl

[29] E. Hebey, The isometry concentration method in the case of a nonlinear problem with Sobolev critical exponent on compact manifolds with boundary, Bull. Sci. Math. 116 (1992), 35-51. | MR | Zbl

[30] Y. Y. Li, Prescribing scalar curvature on $S^n$ and related topics, Part I, J. Differential Equations, 120 (1995), 319-410. | MR | Zbl

[31] Y.Y. Li, Prescribing scalar curvature on $S^n$ and related topics, Part II : existence and compactness, Comm. Pure Appl. Math. 49 (1996), 437-477. | MR | Zbl

[32] C. S. Lin, Estimates of the scalar curvature via the method of moving planes III, Comm. Pure Appl. Math. 53 (2000), 611-646. | MR | Zbl

[33] P. L. Lions, The concentration compactness principle in the calculus of variations. The limt case, Rev. Mat. Iberoamericana 1 (1985), I:165-201, II: 45-121. | MR | Zbl

[34] J. Milnor, “Lectures on $h$-cobordism”, Princeton University Press, Princeton, N.J., 1965. | MR | Zbl

[35] O. Rey, The role of Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52. | MR | Zbl

[36] M. Schneider, Prescribing scalar curvature on $S^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 563-587. | Numdam | MR | Zbl

[37] R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere, Calc. Var. Partial Differential Equations 4 (1996), 1-25. | MR | Zbl

[38] M. Struwe, A flow approach to Nirenberg problem, Duke Math. J. 128 (2005), 19-64. | MR | Zbl

[39] M. Struwe, “Variational methods: Applications to nonlinear PDE $\&$ Hamilton systems”, Springer-Verlag, Berlin, 1990. | Zbl

[40] M. Struwe, A global compactness result for elliptic boundary value problems involving nonlinearities, Math. Z. 187 (1984), 511-517. | MR | Zbl