## Duke Mathematical Journal

### The scalar-curvature problem on higher-dimensional spheres

#### Article information

Source
Duke Math. J. Volume 93, Number 2 (1998), 379-424.

Dates
First available: 19 February 2004

http://projecteuclid.org/euclid.dmj/1077230885

Mathematical Reviews number (MathSciNet)
MR1625991

Zentralblatt MATH identifier
0977.53035

Digital Object Identifier
doi:10.1215/S0012-7094-98-09313-9

#### Citation

Ben Ayed, Mohamed; Chtioui, Hichem; Hammami, Mokhles. The scalar-curvature problem on higher-dimensional spheres. Duke Mathematical Journal 93 (1998), no. 2, 379--424. doi:10.1215/S0012-7094-98-09313-9. http://projecteuclid.org/euclid.dmj/1077230885.

#### References

• [1] A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman Sci. Tech., New York, 1989.
• [2] A. Bahri, Another proof of the Yamabe conjecture for locally conformally flat manifolds, Nonlinear Anal. 20 (1993), no. 10, 1261–1278.
• [3] A. Bahri, An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension, Duke Math. J. 81 (1996), no. 2, 323–466.
• [4] A. Bahri, On the scalar-curvature equation in dimension $n\geqslant5$, preprint.
• [5] A. Bahri, The scalar-curvature problem on sphere of dimension $n\geqslant7$, preprint.
• [6] A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294.
• [7] A. Bahri and J.-M. Coron, The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal. 95 (1991), no. 1, 106–172.
• [8] A. Bahri and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems of $3$-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 6, 561–649.
• [9] M. Ben Ayed, Y. Chen, H. Chtioui, and M. Hammami, On the prescribed scalar curvature problem on $4$-manifolds, Duke Math. J. 84 (1996), no. 3, 633–677.
• [10] H. Brezis and J.-M. Coron, Convergence of solutions of $H$-systems or how to blow bubbles, Arch. Rational Mech. Anal. 89 (1985), no. 1, 21–56.
• [11] J. Escobar and R. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math. 86 (1986), no. 2, 243–254.
• [12] J. L. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317–331.
• [13] Y. Y. Li, On prescribing scalar curvature problem on $S^3$ and $S^4$, C. R. Acad. Sci. Paris Ser. I Math. 314 (1992), no. 1, 55–59.
• [14] Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems, I, J. Differential Equations 120 (1995), no. 2, 319–410.
• [15] Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems, II: Existence and compactness, Comm. Pure Appl. Math. 49 (1996), no. 6, 541–597.
• [16] Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^ n$, Duke Math. J. 57 (1988), no. 3, 895–924.
• [17]1 P.-L. Lions, The concentration-compactness principle in the calculus of variations, I: The limit case, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201.
• [17]2 P.-L. Lions, The concentration-compactness principle in the calculus of variations, II: The limit case, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121.
• [18] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495.
• [19] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), Lecture Notes in Math., vol. 1365, Springer-Verlag, Berlin, 1989, pp. 120–154.
• [20] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71.
• [21] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511–517.